Computational Diffusion MRI: International MICCAI Workshop, Granada, Spain, September 2018 3030058301, 9783030058302

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Mathematics and Visualization

Elisenda Bonet-Carne · Francesco Grussu · Lipeng Ning · Farshid Sepehrband · Chantal M. W. Tax Editors

Computational Diffusion MRI International MICCAI Workshop, Granada, Spain, September 2018

Mathematics and Visualization Series Editors Hans-Christian Hege, Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB), Berlin, Germany David Hoffman, Department of Mathematics, Stanford University, Stanford, CA, USA Christopher R. Johnson, Scientific Computing and Imaging Institute, Salt Lake City, UT, USA Konrad Polthier, AG Mathematical Geometry Processing, Freie Universität Berlin, Berlin, Germany Martin Rumpf, Bonn, Germany

The series Mathematics and Visualization is intended to further the fruitful relationship between mathematics and visualization. It covers applications of visualization techniques in mathematics, as well as mathematical theory and methods that are used for visualization. In particular, it emphasizes visualization in geometry, topology, and dynamical systems; geometric algorithms; visualization algorithms; visualization environments; computer aided geometric design; computational geometry; image processing; information visualization; and scientific visualization. Three types of books will appear in the series: research monographs, graduate textbooks, and conference proceedings.

More information about this series at http://www.springer.com/series/4562

Elisenda Bonet-Carne Francesco Grussu Lipeng Ning Farshid Sepehrband Chantal M. W. Tax •

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Editors

Computational Diffusion MRI International MICCAI Workshop, Granada, Spain, September 2018

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Editors Elisenda Bonet-Carne Centre for Medical Image Computing University College London London, UK Lipeng Ning Psychiatry Neuroimaging Laboratory, Brigham and Women’s Hospital Harvard Medical School Boston, MA, USA Chantal M. W. Tax Cardiff University Brain Research Imaging Centre (CUBRIC) Cardiff University Cardiff, UK

Francesco Grussu Queen Square Institute of Neurology and Centre for Medical Image Computing University College London London, UK Farshid Sepehrband Laboratory of Neuro Imaging (LONI) Stevens Neuroimaging and Informatics Institute, Keck School of Medicine University of Southern California Los Angeles, CA, USA

ISSN 1612-3786 ISSN 2197-666X (electronic) Mathematics and Visualization ISBN 978-3-030-05830-2 ISBN 978-3-030-05831-9 (eBook) https://doi.org/10.1007/978-3-030-05831-9 Library of Congress Control Number: 2018964925 Mathematics Subject Classification (2010): 00B25, 00A66, 00A72, 42B35, 60J60, 60J65, 62P10, 65CXX, 65DXX, 65Z05, 68R10, 68T99, 92BXX © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover illustration provided by Francesco Grussu. Images produced by merging the Alhambra azulejo pattern with a 3D T1-weighted MPRAGE MRI scan of a healthy control at 3 Tesla. Images produced with the style transfer technique as implemented in the neural-style python package based on TensorFlow (https://github.com/anishathalye/neural-style). This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Programme Committee

Nagesh Adluru Manisha Aggarwal Dogu Baran Aydogan Dafnis Batalle Sylvain Bouix Ryan Cabeen Emmanuel Caruyer Mara Cercignani Suheyla Cetin Jian Cheng Daan Christiaens Alessandro Daducci Tom Dela Haije Matt Hall Enrico Kaden Jan Klein Christophe Lenglet Damien McHugh Dorit Merhof Marco Palombo Torben Schneider Thomas Schultz Filip Szczepankiewicz Pew-Thian Yap Fan Zhang

University of Wisconsin-Madison, USA Johns Hopkins University, USA University of Southern California, USA King's College London, UK Harvard Medical School, USA University of Southern California, USA IRISA, France University of Sussex, UK Karayumak Harvard Medical School, USA National Institutes of Health, USA King's College London, UK University of Verona, Italy University of Copenhagen, Denmark University College London, UK; National Physical Laboratory, Teddington, UK University College London, UK Fraunhofer MEVIS, Germany University of Minnesota, USA University of Manchester, UK RWTH Aachen University, Germany University College London, UK Philips Healthcare, UK University of Bonn, Germany Harvard Medical School, Brigham and Women´s Hospital, USA University of North Carolina at Chapel Hill, USA Harvard Medical School, USA

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Preface

It is our great pleasure to present the proceedings of the 2018 International Workshop on Computational Diffusion MRI (CDMRI’18) and the main results of the Multi-shell Diffusion MRI Harmonization and Enhancement Challenge (MUSHAC). Both were held under the auspices of the International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI), which took place in Granada, Spain, on 20 September 2018. CDMRI’18 and MUSHAC were sponsored by MICCAI, Siemens Healthineers, NVIDIA and Restaurante Oliver Granada and endorsed by the International Society for Magnetic Resonance in Medicine (ISMRM). This volume presents the latest developments in the highly active and rapidly growing field of diffusion MRI. The reader will find numerous contributions covering a broad range of topics, from the mathematical foundations of the diffusion process and signal generation, to new computational methods and estimation techniques for the in vivo recovery of microstructural and connectivity features, as well as harmonization and frontline applications in research and clinical practice. This edition includes invited chapters from high-profile researchers with the specific focus on four new important topics that are gaining momentum within the diffusion MRI community: (i) diffusion MRI signal acquisition and processing strategies; (ii) machine learning for diffusion MRI; (iii) diffusion MRI signal harmonization; and (iv) diffusion MRI outside the brain and clinical applications. Additionally, the volume also includes contributions in the field of tractography and connectivity mapping, which continue being relevant and popular topics across all editions of CDMRI. This volume offers the opportunity to share new perspectives on the most recent research challenges for those currently working in the field, but also offering a valuable starting point for anyone interested in learning computational techniques in diffusion MRI. The book includes rigorous mathematical derivations, a large number of rich, full-colour visualisations and clinically relevant results. As such, it will be of interest to researchers and practitioners in the fields of computer science, MRI physics and applied mathematics.

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Each contribution in this volume has been peer-reviewed by multiple members of the International Program Committee. We would like to express our gratitude to all CDMRI’18 authors and reviewers for ensuring the quality of the presented work and to all the teams who participated in MUSHAC. We are grateful to the MICCAI 2018 chairs for providing a platform to present and discuss the work collected in this volume, to our sponsors and to ISMRM for the endorsement. We also would like to thank the editors of the Springer book series Mathematics and Visualization as well as Martin Peters and Ruth Allewelt (Springer, Heidelberg) for their support to publish this collection as part of their series. Finally, we express our sincere congratulations to the winners of the prizes that were awarded during CDMRI’18 for the first time this year. The prizes were awarded following careful evaluation by a panel of judges, made by the CDMRI’18 and MUSHAC organisers and by the CDMRI’18 and MUSHAC keynote speakers. • Prize for the best paper: Simon Koppers et al. with “Spherical harmonic residual network for diffusion signal harmonization”. • Prize for the best MUSHAC method: Jaume Coll-Font et al. with the “DIAMOND2” method. • Prize for the best oral presentation: Daniel Moyer et al. with “Measures of tractography convergence”. • Prize for the best poster presentation: Santiago Aja-Fernandez et al. with “Return-to-plane probability calculation from single-shell acquisitions”. • Prize for the best MUSHAC team presentation: Suheyla Cetin-Karayuma et al. with the “LinearRISH” method. Granada, Spain September 2018

Elisenda Bonet-Carne, Ph.D. Research Associate Francesco Grussu, Ph.D. Research Associate Lipeng Ning, Ph.D. Instructor Farshid Sepehrband, Ph.D. Assistant Professor Chantal M. W. Tax, Ph.D. Research Fellow

Contents

Part I

Diffusion MRI Signal Acquisition and Processing Strategies

Towards Optimal Sampling in Diffusion MRI . . . . . . . . . . . . . . . . . . . . Hans Knutsson Joint Image Reconstruction and Phase Corruption Maps Estimation in Multi-shot Echo Planar Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . Iñaki Rabanillo, Santiago Sanz-Estébanez, Santiago Aja-Fernández, Joseph Hajnal, Carlos Alberola-López and Lucilio Cordero-Grande Return-to-Axis Probability Calculation from Single-Shell Acquisitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Santiago Aja-Fernández, Antonio Tristán-Vega, Malwina Molendowska, Tomasz Pieciak and Rodrigo de Luis-García A Novel Spatial-Angular Domain Regularisation Approach for Restoration of Diffusion MRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alessandro Mella, Alessandro Daducci, Giandomenico Orlandi, Jean-Philippe Thiran, Maria Deprez and Merixtell Bach Cuadra Dmipy, A Diffusion Microstructure Imaging Toolbox in Python to Improve Research Reproducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . Abib Alimi, Rutger Fick, Demian Wassermann and Rachid Deriche Tissue Segmentation Using Sparse Non-negative Matrix Factorization of Spherical Mean Diffusion MRI Data . . . . . . . . . . . . . . . . . . . . . . . . . Peng Sun, Ye Wu, Geng Chen, Jun Wu, Dinggang Shen and Pew-Thian Yap A Closed-Form Solution of Rotation Invariant Spherical Harmonic Features in Diffusion MRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mauro Zucchelli, Samuel Deslauriers-Gauthier and Rachid Deriche

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Orientation-Dispersed Apparent Axon Diameter via Multi-Stage Spherical Mean Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marco Pizzolato, Demian Wassermann, Rachid Deriche, Jean-Philippe Thiran and Rutger Fick Part II

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Machine Learning for Diffusion MRI

Current Applications and Future Promises of Machine Learning in Diffusion MRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Daniele Ravi, Nooshin Ghavami, Daniel C. Alexander and Andrada Ianus q-Space Learning with Synthesized Training Data . . . . . . . . . . . . . . . . . 123 Chuyang Ye, Yue Cui and Xiuli Li Graph-Based Deep Learning for Prediction of Longitudinal Infant Diffusion MRI Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Jaeil Kim, Yoonmi Hong, Geng Chen, Weili Lin, Pew-Thian Yap and Dinggang Shen Supervised Classification of White Matter Fibers Based on Neighborhood Fiber Orientation Distributions Using an Ensemble of Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . 143 Devran Ugurlu, Zeynep Firat, Ugur Ture and Gozde Unal Part III

Diffusion MRI Signal Harmonisation

Challenges and Opportunities in dMRI Data Harmonization . . . . . . . . . 157 Alyssa H. Zhu, Daniel C. Moyer, Talia M. Nir, Paul M. Thompson and Neda Jahanshad Spherical Harmonic Residual Network for Diffusion Signal Harmonization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Simon Koppers, Luke Bloy, Jeffrey I. Berman, Chantal M. W. Tax, J. Christopher Edgar and Dorit Merhof Longitudinal Harmonization for Improving Tractography in Baby Diffusion MRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Khoi Minh Huynh, Jaeil Kim, Geng Chen, Ye Wu, Dinggang Shen and Pew-Thian Yap Inter-Scanner Harmonization of High Angular Resolution DW-MRI Using Null Space Deep Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Vishwesh Nath, Prasanna Parvathaneni, Colin B. Hansen, Allison E. Hainline, Camilo Bermudez, Samuel Remedios, Justin A. Blaber, Kurt G. Schilling, Ilwoo Lyu, Vaibhav Janve, Yurui Gao, Iwona Stepniewska, Baxter P. Rogers, Allen T. Newton, L. Taylor Davis, Jeff Luci, Adam W. Anderson and Bennett A. Landman

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Effects of Diffusion MRI Model and Harmonization on the Consistency of Findings in an International Multi-cohort HIV Neuroimaging Study . . . . . . . . . . . . . . . . . . . . . . . . . 203 Talia M. Nir, Hei Y. Lam, Jintanat Ananworanich, Jasmina Boban, Bruce J. Brew, Lucette Cysique, J. P. Fouche, Taylor Kuhn, Eric S. Porges, Meng Law, Robert H. Paul, April Thames, Adam J. Woods, Victor G. Valcour, Paul M. Thompson, Ronald A. Cohen, Dan J. Stein and Neda Jahanshad Muti-shell Diffusion MRI Harmonisation and Enhancement Challenge (MUSHAC): Progress and Results . . . . . . . . . . . . . . . . . . . . . 217 Lipeng Ning, Elisenda Bonet-Carne, Francesco Grussu, Farshid Sepehrband, Enrico Kaden, Jelle Veraart, Stefano B. Blumberg, Can Son Khoo, Marco Palombo, Jaume Coll-Font, Benoit Scherrer, Simon K. Warfield, Suheyla Cetin Karayumak, Yogesh Rathi, Simon Koppers, Leon Weninger, Julia Ebert, Dorit Merhof, Daniel Moyer, Maximilian Pietsch, Daan Christiaens, Rui Teixeira, Jacques-Donald Tournier, Andrey Zhylka, Josien Pluim, Greg Parker, Umesh Rudrapatna, John Evans, Cyril Charron, Derek K. Jones and Chantal W. M. Tax Part IV

Diffusion MRI Outside the Brain and Clinical Applications

Diffusion MRI Outside the Brain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Rita G. Nunes, Luísa Nogueira, Andreia S. Gaspar, Nuno Adubeiro and Sofia Brandão A Framework for Calculating Time-Efficient Diffusion MRI Protocols for Anisotropic IVIM and An Application in the Placenta . . . . . . . . . . . 251 Paddy J. Slator, Jana Hutter, Andrada Ianus, Eleftheria Panagiotaki, Mary A. Rutherford, Joseph V. Hajnal and Daniel C. Alexander Spatial Characterisation of Fibre Response Functions for Spherical Deconvolution in Multiple Sclerosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Carmen Tur, Francesco Grussu, Ferran Prados, Sara Collorone, Claudia A. M. Gandini Wheeler-Kingshott and Olga Ciccarelli Edge Weights and Network Properties in Multiple Sclerosis . . . . . . . . . 281 Elizabeth Powell, Ferran Prados, Declan Chard, Ahmed Toosy, Jonathan D. Clayden and Claudia Gandini A. M. Wheeler-Kingshott Part V

Tractography and Connectivity Mapping

Measures of Tractography Convergence . . . . . . . . . . . . . . . . . . . . . . . . 295 Daniel C. Moyer, Paul Thompson and Greg Ver Steeg

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Brain Connectivity Measures via Direct Sub-Finslerian Front Propagation on the 5D Sphere Bundle of Positions and Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 Jorg Portegies, Stephan Meesters, Pauly Ossenblok, Andrea Fuster, Luc Florack and Remco Duits Inference of an Extended Short Fiber Bundle Atlas Using Sulcus-Based Constraints for a Diffeomorphic Inter-subject Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Nicole Labra Avila, Jessica Lebenberg, Denis Rivière, Guillaume Auzias, Clara Fischer, Fabrice Poupon, Pamela Guevara, Cyril Poupon and Jean-François Mangin Resolving the Crossing/Kissing Fiber Ambiguity Using Functionally Informed COMMIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 Matteo Frigo, Isa Costantini, Rachid Deriche and Samuel Deslauriers-Gauthier Voxel-Wise Clustering of Tractography Data for Building Atlases of Local Fiber Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 Irene Brusini, Daniel Jörgens, Örjan Smedby and Rodrigo Moreno Obtaining Representative Core Streamlines for White Matter Tractometry of the Human Brain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Maxime Chamberland, Samuel St-Jean, Chantal M. W. Tax and Derek K. Jones Improving Graph-Based Tractography Plausibility Using Microstructure Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 Matteo Battocchio, Gabriel Girard, Muhamed Barakovic, Mario Ocampo, Jean-Philippe Thiran, Simona Schiavi and Alessandro Daducci Deterministic Group Tractography with Local Uncertainty Quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Andreas Nugaard Holm, Aasa Feragen, Tom Dela Haije and Sune Darkner Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

Part I

Diffusion MRI Signal Acquisition and Processing Strategies

Towards Optimal Sampling in Diffusion MRI Hans Knutsson

Abstract The methodology outlined in this chapter is intended to provide a tool for the generation of sets of MRI diffusion encoding waveforms that are optimal for tissue micro-structure estimation. The methodology presented has five distinct components: 1. Defining the class of waveforms allowed, i.e. defining the measurement space. 2. Specifying the expected distribution of microstructure features present in the targeted tissue. 3. Learning the metric in the chosen measurement space. 4. Designing a continuous parametric functional suitable for approximation of the estimated metric. 5. Finding a distribution of a chosen number of waveforms that is optimal given the continuous metric. The tissue is modeled as a collection of simple elliptical compartments with varying size and shape. Two waveform classes are tested: The classical Stejskal-Tanner waveform and an idealized Laun long-short waveform. The estimation of the metric is based on correlations between measurements obtained at given points in the measurement space using an information theoretical approach. Optimal sets of waveforms are found using a simulated annealing inspired energy minimizing approach. The superior performance of the methodology is demonstrated for a number of different cases by means of simulations. Keywords Learning · Sample space metric · Optimal waveform sets

1 Introduction The discussion concerning strategies for finding optimal sets of diffusion encoding waveforms has been lively from the very start of diffusion imaging and is continuing to be a major topic of research [1–21]. Existing sampling schemes are based on experience combined with more or less ad hoc approaches of which many display interesting features. The presented sampling schemes are almost exclusively based on the concept of q-space, a notion that is only strictly valid for a very limited class H. Knutsson (B) Linköping University, Linköping, Sweden e-mail: [email protected]

© Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_1

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of waveforms [22]. In addition, there is no consensus regarding the optimal choice of a q-sample distribution for a given situation. A number of more advanced approaches using extended waveform classes have also been put forward, e.g. double PFG [5], oscillating gradients [11], quadruple PFG [14], isotropic waveforms [16], rotating gradients [18], Q-space trajectory imaging [19, 20] and quadrature gradients [21]. For most of these approaches the number of parameters, i.e. the measurement space dimension, is far higher than in the traditional q-space approaches. This is probably the reason for the apparent lack of suggestions for assessing the optimality of sets of such waveforms. It is also for this reason that the cases presented in the following sections mainly concerns traditional StejskalTanner waveforms. It deserves to be stressed, however, that the presented approach is generally valid. The following five steps summarizes the approach: 1. Defining the class of waveforms allowed, i.e. defining the measurement space. This definition can in theory be highly complex but will in practice need to be relatively low-dimensional. The class of waveforms used will determine the microstructure feature space that is at all possible to study. In this chapter two waveform classes are tested: The classical Stejskal-Tanner waveform and an idealized Laun long-short waveform [23]. 2. Specifying the expected distribution of microstructure features present in the targeted tissue. In the examples below the tissue is modeled as a collection of simple elliptical compartments with varying size and shape [13]. 3. Learning the metric of the measurement space. The estimation of the metric is based on correlations between measurements obtained at given points in the measurement space using an information theoretical approach [13]. The estimation principle allows for any dimension of the measurement space but the number of examples needed to map out the metric will increase rapidly with dimension. 4. Designing a continuous parametric functional suitable for approximation of the estimated metric. This part is necessarily ad hoc and will be based on experience and trial and error. We previously presented a five parameter metric for this purpose [12]. In this chapter an extended eight parameter functional that can capture the main features of the estimated metric in cases studied is presented and used. 5. Finding a distribution of a chosen number of waveforms that is optimal given the continuous metric. Optimal sets of waveforms are found using a simulated annealing inspired energy minimizing approach [12]. Steps 2 and 3 are treated in Sect. 2. Steps 4 and 5 are treated in Sect. 3. In Sect. 4 the performance of eight different optimized sets of waveforms are presented. The results are obtained through simulation of the diffusion MRI physics [24]. The simulator used, [24], will not be further discussed in this chapter. A cylindrical compartment with a diameter of 15 µm and a length of 40 µm was used as the test compartment. The results clearly demonstrate the importance of carefully choosing the waveform set.

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2 Q-Space Metric Estimation This section presents a method to determine a local q-space metric that is optimal from an information theoretic perspective with respect to the expected signal statistics Sect. 2. It should be noted that the approach differs significantly from the classical estimation theory approach, e.g. one based on Cramer-Rao bounds [25]. The latter requires a pre-defined mathematical representation, the estimator. In contrast, our approach aims at obtaining the maximum amount of information without enforcing a particular feature representation. The estimated metric will be dependent on the q-space location and will indicate the information gain when adding a sample in a second q-space location. The intention is to use the obtained metric as a guide for the generation of specific efficient q-space distributions for the targeted tissue. Note that the approach is general in that it can be used for any set of waveforms.

2.1 Mutual Information The mutual information (originally termed rate of transmission) between to signals relates directly to the entropies involved and can be estimated from the joint signal statistics [26]. Using a Gaussian signal+noise source model, which is a quite reasonable starting point in the present context, the estimate is directly related to the canonical correlation between two signals, a and b, and is given in bits by: 1 Iab = − log2 2



[Caa ||Cbb | |C|



 where

C=

Caa Cab Cab Cbb

 (1)

and Caa , Cab , Cbb are covariance matrices. For the one-dimensional case this reduces 2 ), where ρab is the correlation between the two variables. to: Iab = − 21 log2 (1 − ρab This expression can also be used to estimate the information from a single signal by measuring the correlation between the signal with and added noise realization and the same signal without noise. In order to obtain an estimate of a local information based q-space metric we can compute the information gain, IΔ , from measuring in a second q-space location, qb , given that we already have a measure at a first location, qa . IΔ is obtained as the information due to the second measurement alone minus the mutual information between the two measured signals, i.e. the information that is already present due to the first measurement: IΔ = Ib0 b − Iab

1 = log2 2



2 1 − ρab

1 − ρb20 b

 (2)

In this expression b0 is the true, noise free, source signal at the second location. It can  SNR +1 if the second measurement be noted that Eq. (2) will give IΔ = 21 log2 2SNR +1

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is taken at the same location as the first. For reasonably high SNR (signal to noise √ ratio) this corresponds 0.5 bits or, equivalently, improving measurement SNR by 2 (3 dB). It should also be noted that the Gaussian and additive assumptions are not crucial since mutual information between two variables is a monotonically increasing function of the correlation even in highly non-Gaussian and non-additive cases [27].

2.2 Learning the Metric To obtain the statistics of the q-space signals we generate a large number of q-space response examples. Using these examples correlation estimates between any two q-space locations, as well as correlations between different instances of the same location, can be estimated. From these correlations the added information from measuring in a second q-space location, given a first measurement in any other location, can be found. Each voxel in will contain a huge number of different propagators determining the q-space signals and a substantial intra voxel variation in propagator size and shape can be expected. It is certainly an over-simplification to use a Gaussian as an approximation of the q-space response magnitude. We have, however, used a Gaussian approximation as it suffices to demonstrate the features and the value of our approach. The example generator was set to produce 3D Gaussian q-space responses having one long axis and two equal short axes. All generated distributions had 300 different long axes orientations evenly distributed to cover all 3D orientations. The size of the average propagators was also varied. The total number of the propagator examples of a given ’tissue volume’ was set to vary as the inverse of the volume, i.e. the total volume of the smaller propagators was equal to the total volume of the larger propagators. The average size of the propagators was set to vary logarithmically in the specified range. The ratio between the long and short axes was also set to vary logarithmically in the specified range while keeping the propagator volume constant. The intention is to study a few archetypal situations that can be easily understood, not to mimic real tissue. A multitude of approaches for modelling realistic tissue have been put forward and such models can readily be incorporated in the present framework, doing so is, however, beyond the scope of this chapter. Listed below and shown in Fig. 1 are the three different distributions generated to study the effects different average diffusion propagator distributions will have on the q-space sampling metric. All three generated distributions are rotation invariant. Also note that the individual spatial propagator positions are not important for our analysis. 1. Allsorts—Long/short axis ratios from 1 to 10 and volumes from 0.5 to 2.0. 2. Round—A long/short axis ratio of 1.15 and volumes from 0.22 to 4.5. 3. Stick—A long/short axis ratio of 10 and volumes from 0.5 to 2.0

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Fig. 1 The upper row illustrates three archetypal average propagator distributions, as iso-surfaces of the used Gaussians, randomly distributed in a piece of tissue. From left to right is shown: Allsorts— Varying in orientation, shape and size (Left). Round—Almost spherical propagators of varying in size (Center). Stick—Highly anisotropic only varying in orientation (Right). (See also [13])

2.3 Metric Estimation Results The results of the analysis carried out is visualized in Fig. 2. Below follows a short discussion of the q-space metric features that can be observed. The Allsorts distribution: The upper left plot in Fig. 2 shows the result of the estimated q-space metric for the Allsorts distribution. The lilac colored iso-surfaces show the 3D q-space locations where the information gain from a second sample, given a first sample in the center (yellow), reaches ΔI = 2 bits. Results for five different radii (0, 0.25, 0.5, 0.75 and 1.0) of the first sample are shown. The radii were chosen to highlight the typical information gain behavior that will be present in different parts of q-space. Since the setup is rotationally invariant the results will be the same along any axis through the origin. The upper plot shows iso-surfaces along four different directions in one octant of q-space and is intended to visualize that the metric is rotation invariant. The upper right part of Fig. 2 shows the results obtained for the initial point located at five different positions on the x-axis. The Round and Stick distributions: By studying distribution that are individually more uniform and different from the Allsorts distribution in two different ways further insights can be gained. The bottom left plot of Fig. 2 shows the results when using the Round distribution and bottom right plot shows the results from using the Stick distribution is plotted. A feature that is globally present is that the average information gain from a second sample is much lower for the Round distribution than for the Stick distribution. This is, most likely, a consequence of that the former distribution has a lower overall variability, i.e. lower entropy.

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Fig. 2 Estimated q-space metrics for different average propagator distributions. The top plots are for the Allsorts distribution. The bottom left plot is for the Round distribution and the bottom right plot is for the Stick distribution. The lilac colored iso-surfaces show where the information gain from a second sample, given a first sample in the center (yellow), reaches ΔI = 2 bits. The figure shows the results for five different radii of the first sample. The upper left plot shows one octant of q-space and is intended to demonstrate that the metric is rotation invariant. The upper right and lower plots show only the cases where the first sample is on the x-axes. The multi colored surfaces show the information gain when moving the second sample away from the first, the gain in bits is indicated by the numbers 1–4 on the upper part of the y-axis. The iso-contours on the x-y plane below are drawn for every 0.5 bits. (See also [13])

3 Q-Space Sample Distribution In the effort of extracting meaningful micro-structural properties from diffusion weighted MRI (dMRI) it becomes clear that a large number of acquisitions are required to get the full extent of the information. A multitude of suggestions concerning optimal q-space sampling strategies has been presented. Among the most

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well known approaches is the electrostatic repulsion algorithm suggested by Jones et al. [1]. Jones’ algorithm finds a ’uniform’ single-shell distribution of Q-space sample points by finding the lowest electrostatic energy of a system consisting of N antipodal charge pairs on the surface of a sphere. However, when aiming for a full 3-dimensional reconstruction of the diffusion propagator the sample points should be evenly distributed in the targeted 3-dimensional Q-space. Such a solution can not be attained using the traditional electrostatic repulsion approach since alleviating the single-shell constraint will make the sample distribution expand indefinitely. Further, it is doubtful if the Euclidean vector difference metric traditionally used is the best possible choice. The Q-space samples actually represent orientation, rather than direction. For this reason it would seem more appropriate to consider charges moving in, for example, an outer product tensor space [28]. In this section we present a novel and general framework allowing the generation of full 3-dimensional Q-space sample distributions. Our framework extends the electrostatic charge distribution model in two fundamental ways: 1. Enabling a user specified definition of the 3-dimensional space to be sampled. 2. Enabling a user specified definition of the distance metric to be used.

3.1 Charged Containers for 3D Sampling Jones’ algorithm attempts to find a ‘uniform’ distribution of q-space sample points by searching for the lowest electrostatic energy of a system consisting of N antipodal equal charge pairs on the surface of a sphere. The system energy takes the form: E=

  m

(ˆxm − xˆ n −1 + ˆxm + xˆ n −1 )

(3)

n

where n, m ∈ [1 : N ]. See Fig. 3 left for our simulation result using 500 sample points. An extension beyond the single-shell is not immediately obvious since a full Q-space sampling requires samples that are evenly distributed in 3 dimensions. To extend the sample distribution generation to cover a full 3d Q-space we add a continuously charged volume with a specified shape and charge density. We term this volume the charged container. The total charge of the container is equal to the negative of the sum of the moving point charges. This volume will then attract the point charges and keep them inside the container. To add the corresponding energy term we need to find the added potential field. The sphere is the simplest possible container and one of the very few cases for which a simple closed form interior potential function is known: Vc = −N (3 − r 2 ), where r is the distance from the container center. Note, however, that it is possible to specify any shape of the negatively charged volume as long as a good approximation of the spatial gradient of the potential field

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H. Knutsson

can be found. Alleviating the sphere surface requirement and adding the spherical container contribution to the traditional energy function yields: E=



(xm − xn −1 + xm + xn −1 ) − 2N

m n=m

 (3 − xm 2 )

(4)

m

Here the ’self energy’ contribution from the individual antipodal pairs have been excluded from the summation (n = m). This is consistent with the view that the sole purpose of the pair construction is to implement an appropriate metric, i.e. the antipodal pair points do not repel each other. Minimizing this energy function for 500 charge pairs we get the solution shown in Fig. 3 right.

3.2 Tensor Metrics for Distribution of Q-Space Samples While the energy definition of Eq. (3) works well in the intended single-shell context a different definition is natural for a fully 3-dimensional Q-space sampling system. The Q-space samples represent orientation, rather than direction. For this reason it would seem more appropriate to consider charges moving in an outer product tensor space. To enable the use of different metrics we present a means to employ a generalized metric in the optimization. Thus, each charge act as if positioned in a higher dimensional space that naturally represents the concept of orientation [28]. A five parameter metric class—As demonstrated in Sect. 2.3 the q-space measurement metric is typically non-Euclidean. For this reason we need to find a class of metrics general enough to be able to fit the q-space metric of most common situations. Towards this end we previously introduced the five parameter tensor related metric class given by Eq. (5) [12]. has a number of useful features. The metric is rotation invariant in the sense that it can be expressed in terms of the radii of the two position vectors and the angle between them.

  2 T (5) D2 (xm , xn ) ≡ wr rmα − rnα + wϕ 2 (rm rn )β 1 − (ˆxm xˆ n )γ 

 Dr2

Dϕ2

where r = x, xˆ = xr There are five parameters defining the metric: wr and wϕ α β γ

Weights of the radial and angular parts respectively. Exponents for radial dependence of the radial part of D. Exponent for radial dependence of the angular part of D. Exponent for angular dependence of the angular part of D.

Towards Optimal Sampling in Diffusion MRI

11

Notable special cases of the metric—A number of useful metrics are instances of this general metric. The following examples demonstrate the generality of the proposed metric. A. Euclidian metric [ wr = wϕ = α = β = γ = 1 ] D2 = (rm − rn )2 + 2 rm rn [ 1 − cos(ϕ) ] = xm − xn 2

(6)

which shows that this parameter setting corresponds to the standard Euclidean metric. B. Outer product metric [ wr = wϕ = 1, α = β = γ = 2] D2 = xm xmT − xn xnT 2

(7)

C. Traceless outer product metric [ wr = 1, wϕ = α = β = γ = 2 ] D2 = rm2 + rn2 − 2rm rn cos(2ϕ)

(8)

This metric is sometimes referred to as the double angle metric, it corresponds to a traceless outer product tensor metric. D. N:th order outer product metric [ wr = wϕ = 1, α = β = γ = N ] D2 = xm⊗N − xn⊗N 2

(9)

E. Log-Euclidean metric [ α wr = 1, α = β 0.2. The FA values of each volume are clustered in 5 different classes, calculated with a fuzzy c-means algorithm. The resulting centroids are C L = {0.24, 0.36, 0.51, 0.66, 0.86}. The different points of the white matter are assigned to one cluster, using the FA values and the minimum fuzzy distance. This way, each volume is segmented in 5 different regions with similar diffusion values. We calculate ARTAP using one single-shell. In order to compare results, the median of the values inside each cluster for each b-value is considered. Results are depicted in Fig. 3. A variation with b can be detected in most of the implementations, with the exception of OPDT and pOPDT, that remains stable for the whole range of b-value variation. This variation is related to the fact that the diffusion at different b-values can be in fact measuring different underlying effects. However, since the differences among the different clusters remain, ARTAP can still be used for research studies, with the counterpart that measures acquired at different b-values will not be comparable. But that variability with the acquisition parameters is also detected in common measures, like the FA.

5.5 5 4.5 4 3.5

10000

3

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6000

0.3

8

RTAP(DOT)

RTAP(pOPDT)

10000

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0

4000

0.25 0.2 0.15 0.1 4000

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10000

10

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8000

10000

b value

b value

b value

0.05

8000

10

4

7 6 5 4 3

4000

6000

8000

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Fig. 3 Variation of ARTAP with the b-value, using the real data set. The volume has been clustered in 5 different sets and the median of each set is shown. Centroids of the data C L = {0.24, 0.36, 0.51, 0.66, 0.86}

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4.2 Comparison with EAP Estimators For the next experiment, ARTAP is compared to RTAP calculated with state-of-theart methods for EAP reconstruction: the directional radial basis functions (RBFs, constrained 2 regularization with standard parameters suggested by the authors) [8], the Mean Apparent Propagator (MAP-MRI, an anisotropic basis with radial order set to 6) [10] and the Laplacian-regularized MAP-MRI [6] (MAPL, an anisotropic basis with radial order set to 8 and regularization weighting λ = 0.2). Since we will be using data from the HCP in which only 4 shells are acquired, we have discarded methods that needed a dense sampling of the q-space, as well as those methods based on Cartesian sampling. In the experiment, we use 5 different volumes from the Human Connectome Project: MGH 1007, MGH 1010, MGH 1016, MGH 1018 and MGH 1019. The acquisition presented 40 different baselines that were averaged to reduce the noise.

Table 1 Correlation coefficient between the different methods to estimate RTAP, using 5 different volumes from HCP. For ARTAP, pQBalls and OPDT have been used. Highlighted the higher correlation values pQBalls OPDT RBF MAPL MAP-MRI RBF MAPL MAP-MRI 4 shells

3 shells

2 shells

ARTAP (b = 3000) ARTAP (b = 5000) ARTAP (b = 10000) RBF MAPL ARTAP (b = 3000) ARTAP (b = 5000) ARTAP (b = 10000) RBF MAPL ARTAP (b = 3000) ARTAP (b = 5000) ARTAP (b = 10000) RBF MAPL

0.58

0.75

0.84

0.50

0.75

0.89

0.67

0.84

0.86

0.46

0.75

0.88

0.65

0.87

0.79

0.45

0.77

0.88

– – 0.43

0.58 – 0.82

0.53 0.88 0.84

0.42

0.83

0.85

0.45

0.88

0.86

0.45

0.85

0.87

0.36

0.81

0.75

0.47

0.83

0.86

– – 0.68

0.41 – 0.84

0.39 0.92 0.87

0.83

0.82

0.86

0.67

0.85

0.82

0.82

0.80

0.84

0.56

0.77

0.71

0.83

0.79

0.83

– –

0.79 –

0.81 0.86

Return-to-Axis Probability Calculation from Single-Shell Acquisitions

37

Fig. 4 Visual comparison of RTAP using state-of-the-art methods. Slice 42 of the MGH-1016 volume from HCP. ARTAP is calculated with one single shell. MAPL, MAP-MRI and RBF are calculated using 2, 3 and 4 shells (the value of b in the image indicates the highest shell used)

To reduce the amount of data in the experiments (and, consequently the estimation time), only three different slices are selected from each volume.2 RTAP is calculated for RBF, MAP-MRI and MAPL using the 4 available shells (b = {1000, 3000, 5000, 10000}), 3 shells (b = {1000, 3000, 5000}) and 2 shells (b = {1000, 3000}). ARTAP is calculated using one single shell for three different bvalues: b = 3000, b = 5000 and b = 10000. For ARTAP, only 2 different functions were now considered: pQballs and OPDT. A white matter mask is used and we have only selected those points where FA > 0.2. Then, the correlation between RTAP computed with the different methods are estimated. Results of the correlation between methods are gathered in Table 1. In addition, a visual comparison of RTAP is shown in Fig. 4, where RTAP have been scaled for display purposes. Results show a very high correlation between ARTAP with only one shell and the calculation given by the other techniques, particularly those based on MAP. In many cases, that correlation goes up to 87%, which basically means that those methods are measuring the same underlying features. Surprisingly, the correlation with RBF is lower, with its best value at 68% for pQballs but 83% for OPDT. However, the correlation ARTAP-RBF is similar (even larger) to the one found between RBF-MAP. The variation in the calculation of RTAP among the methods already proposed in the literature is larger than the variation found between those methods and ARTAP. If we focus on the 4-shell approach, it is interesting to point out that RTAP calculated over 2 MGH

1007: 42, 52, 65; MGH 1010: 46, 54, 60; MGH 1016: 42, 55, 68; MGH 1018: 31, 41, 51; MGH 1019: 40, 50, 64.

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one shell, with only 64 points (b = 3000) or 128 (b = 5000) can have a high correlation with methods calculated using 4 shells and 512 points. All in all, ARTAP have shown a greater fidelity to RTAP calculated using MAP methods while correlations with RBF are lower.

4.3 Dependence of EAP Estimators on the Number of Shells One of the drawbacks initially declared of ARTAP is that results can depend on the b-value, making it dependent on the acquisition parameters. The state-of-the-art methods do not depend on one single shell, but on the whole q-space and are therefore supposed to be more stable. To test this assumption we have carried out an experiment similar to that in Fig. 3 for RBF, MAPL and MAP-MRI. The 5 volumes and 3 slices from HCP used in the previous experiment are used again. The different slices are divided in 5 different regions according to their diffusion features. To do that, the FA of all data is first calculated and a mask is applied to the values of FA> 0.2. The FA values of each volume are clustered in 5 different classes, calculated with a fuzzy c-means algorithm with resulting centroids C L = {0.24, 0.33, 0.45, 0.58, 0.76}. The different points of the white matter are assigned to one cluster, using the FA values and the minimum fuzzy distance. We calculate RTAP using 2, 3 and 4 shells and the median of the values inside each cluster is calculated. Results are shown in Fig. 5. The first thing to notice is that there is really a dependence between RTAP and the number of shells used for the calculation of the EAP. This dependence is even more severe than the one detected for ARTAP in Fig. 3. RBF exhibits a crossing of values when a different number of shells are considered. Both MAP methods show a dependence with the number of shells, but they succeed in keeping the different areas separated. To sum up, RTAP calculated with the state-of-the-art methods also present a dependence with the shells considered.

10

1.6

4

1.4 1.2 1 0.8

10 4

6 5

4

4

3

3

2

2 2

2.5

3

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4

1

10 4

5

1 2

2.5

3

3.5

Number of shells

4

0

2

2.5

3

3.5

4

Number of shells

Fig. 5 Variation of RTAP with the number of shells used for calculation, using three slices from 5 volumes. The data has been clustered in 5 different sets and the median of each set is shown. Centroids of the data C L = {0.24, 0.33, 0.45, 0.58, 0.76}

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4.4 Execution Times One of the advantages of ARTAP is that it avoids the calculation of the whole EAP in order to compute the RTAP. As a consequence, it saves a great amount of computation time. In this final section, we compare the execution times of the proposed methodology to the state-of-the-art methods used. To that end, a whole diffusion volume from HCP will be used, specifically HCP MGH 1016. The estimated execution times for the methods are measured on quad-core Intel(R) Core(TM) i7-6700K 4.00GHz processor under Debian GNU/Linux 8.6 operating system. The external code for RBFs3 was run under MATLAB 2013b (The MathWorks, Inc., Natick, MA) and DIPY 0.13.04 library under Python 3.6.4 (scipy 1.0.0) was used for MAP-MRI and MAPL. ARTAP is implemented in MATLAB. The proposed method is calculated only over one shell individually (second, third and fourth). The ARTAP is calculated using pQBalls. Results are included in Table 2. The execution times are just relative and they can be accelerated using optimized code and acceleration techniques, such as GPU and/or parallel implementation. However, the times give a reasonable idea of the complexity of the different methods. Note that the execution times of most of the methods make the unfeasible to be used on a real study. In the case of RBF, the execution times goes from 5 to 24 days per volume, something that goes beyond the capability of clinical groups. Even in the best of the cases, MAPL goes 17 times slower than the proposal. In all the cases, most of the time is spent in calculating the EAP. In MAPL, for instance, only 0.6% of the calculation time corresponds to RTAP, while the other 99.4% is spent in estimating the EAP.

Table 2 Estimated execution times for the calculation of RTAP for the different methods for a single volume (HCP MGH 1016) Method Two shells Three shells Four shells RBFs MAP-MRI MAPL ARTAP

118 h 13 h 2h 3.3 min

332 h 13 h 2h 3.5 min

3 https://github.com/LipengNing/RBF-Propagator. 4 http://nipy.org/dipy.

577 h 16 h 2h 4 min

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5 Conclusions We have presented a new methodology that allows the calculation of RTAP using a single shells acquisition. The main advantage of the proposal is that we do not longer need a dense sampling of the q-space and the subsequent estimation of the EAP to compute the RTAP. As a consequence, we can drastically reduce the amount of acquired points and the computation time. ARTAP is compatible with some standard diffusion acquisitions for DKI, CHARMED and HARDI. This allows the calculation of RTAP to a wider audience, since it limits the requirements of the acquisition process drastically reducing acquisition and processing time. In addition, the use of different ODF to estimate the ARTAP allows introducing a degree of design on the measure. Different ODFs may be biased to different features of the tissue. On the other hand, one of the weaknesses of the method is that the output value depends on the selected b-value. However, this variability can also be found in other state-of-the-art methods, whose results depend on the number of shells used for the estimation of the EAP.

References 1. Boscolo Galazzo, I., Brusini, L., Obertino, S., Zucchelli, M., Granziera, C., Menegaz, G.: On the viability of diffusion MRI-based microstructural biomarkers in ischemic stroke. Front. Neurosci. 12, 92 (2018) 2. Brusini, L., Obertino, S., Zucchelli, M., Galazzo, I.B., Krueger, G., Granziera, C., Menegaz, G.: Assessment of mean apparent propagator-based indices as biomarkers of axonal remodeling after stroke. In: Medical Image Computing and Computer-Assisted Intervention—MICCAI 2015, pp. 199–206. Springer International Publishing, Cham (2015) 3. Canales-Rodríguez, E.J., Melie-García, L., Iturria-Medina, Y.: Mathematical description of q-space in spherical coordinates: exact q-ball imaging. Magn. Reson. Med. 61(6), 1350–1367 (2009) 4. Descoteaux, M., Angelino, E., Fitzgibbons, S., Deriche, R.: Apparent diffusion profile estimation from high angular resolution diffusion images: estimation and applications. Magn. Reson. Med. 56(2), 395–410 (2006) 5. Descoteaux, M., Deriche, R., Le Bihan, D., Mangin, J.F., Poupon, C.: Diffusion propagator imaging: using Laplace’s equation and multiple shell acquisitions to reconstruct the diffusion propagator. In: International Conference on Information Processing in Medical Imaging, pp. 1–13. Springer, Heidelberg (2009) 6. Fick, R.H., Wassermann, D., Caruyer, E., Deriche, R.: MAPL: tissue microstructure estimation using Laplacian-regularized MAP-MRI and its application to HCP data. NeuroImage 134, 365– 385 (2016) 7. Hosseinbor, A.P., Chung, M.K., Wu, Y.C., Alexander, A.L.: Bessel Fourier orientation reconstruction (BFOR): an analytical diffusion propagator reconstruction for hybrid diffusion imaging and computation of q-space indices. NeuroImage 64, 650–670 (2013) 8. Ning, L., Westin, C.F., Rathi, Y.: Estimating diffusion propagator and its moments using directional radial basis functions. IEEE Trans. Med. Imag. 34(10), 2058–2078 (2015) 9. Özarslan, E., Koay, C., Basser, P.: Simple harmonic oscillator based estimation and reconstruction for one-dimensional q-space MR. Proc. Int. Soc. Mag. Reson. Med. 16, 35 (2008)

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10. Özarslan, E., Koay, C.G., Shepherd, T.M., Komlosh, M.E., ˙Irfano˘glu, M.O., Pierpaoli, C., Basser, P.J.: Mean apparent propagator (MAP) MRI: a novel diffusion imaging method for mapping tissue microstructure. NeuroImage 78, 16–32 (2013) 11. Özarslan, E., Sepherd, T.M., Vemuri, B.C., Blackband, S.J., Mareci, T.H.: Resolution of complex tissue microarchitecture using the diffusion orientation transform (DOT). NeuroImage 31, 1086–1103 (2006) 12. Tristan-Vega, A., Aja-Fernández, S., Westin, C.F.: Deblurring of probabilistic ODFs in quantitative diffusion MRI. In: 2012 9th IEEE International Symposium on Biomedical Imaging (ISBI), pp. 932–935. IEEE (2012) 13. Tristán-Vega, A., Westin, C.F., Aja-Fernández, S.: Estimation of fiber orientation probability density functions in high angular resolution diffusion imaging. NeuroImage 47(2), 638–650 (2009) 14. Tristan-Vega, A., Westin, C.F., Aja-Fernandez, S.: A new methodology for the estimation of fiber populations in the white matter of the brain with the Funk-Radon transform. Neuroimage 49(2), 1301–1315 (2010) 15. Tuch, D.S., Reese, T.G., Wiegell, M.R., Wedeen, V.J.: Diffusion MRI of complex neural architecture. Neuron 40, 885–895 (2003) 16. Wu, Y.C., Field, A.S., Alexander, A.L.: Computation of diffusion function measures in q-space using magnetic resonance hybrid diffusion imaging. IEEE Trans. Med. Imag. 27(6), 858–865 (2008)

A Novel Spatial-Angular Domain Regularisation Approach for Restoration of Diffusion MRI Alessandro Mella, Alessandro Daducci, Giandomenico Orlandi, Jean-Philippe Thiran, Maria Deprez and Merixtell Bach Cuadra

Abstract In this paper we tackle the problem of regularisation for inverse problems in single shell diffusion weighted image restoration. Our aim is to recover a highresolution and denoised DWI signal, prior to any model fitting. The main contribution of our method is the combination of two regularization terms, one using the information arising from the spatial domain, hence analysing the single image, while the other uses information coming from the angular domain, thus using the relationships between the values along different directions within a single voxel. We show that our novel regularization method outperforms widely used and recent DWI denoising algorithms. Additionally we demonstrate that the proposed regularisation technique can be successfully applied to the super-resolution reconstruction of high-resolution volume from thick-slice data. Both scenarios are tested on simulated phantom and real DWI data.

1 Introduction Diffusion-weighted magnetic resonance imaging (DWI) allows in-vivo characterization of brain white matter microstructure. However, due to its intrinsic signalto-noise (SNR) characteristics and sensitivity to subject’s motion, DWI suffers of limited to relatively low spatial resolution (of around 2 × 2 × 2 mm3 ) as compared A. Mella (B) University of Bologna, Mathematics, Bologna, Italy e-mail: [email protected] A. Daducci · G. Orlandi Computer Science Department, University of Verona, Verona, Italy J.-P. Thiran · M. Bach Cuadra Radiology Department, Lausanne University Hospital; Center for Biomedical Imaging (CIBM), Lausanne University,Lausanne, Switzerland J.-P. Thiran · M. Bach Cuadra EPFL, Signal Processing Laboratory 5 (LTS5), Lausanne, Switzerland M. Deprez Department of Biomedical Engineering, King’s College London, London, UK © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_4

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to other structural imaging. There is a need for developing new techniques that can achieve enhanced DWI (SNR and spatial resolution) so that tissue microstructure can be evaluated with an improved signal and less partial volume effects. On one side, enhancement of diffusion MR image resolution can be addressed from a hardware/acquisition point of view, tailoring novel accelerated sequences, using ultrahigh magnetic fields, etc. On another side, image post-processing techniques have also been developed to address different problems such as motion, noise, blurring and, dealing with anisotropic volumes. To do so, an inverse-problem formulation is often used, where a functional to be minimized is defined with a composition of a fidelity term, which measures the distance from the original image, and a regularisation term. The latter has the double role of solving the possible ill-posedness of the inverse problem and its minimization solves also some other issue like the presence of noise. In DWI, regularisation can be applied either in the spatial or the angular domain, or as a combination of both. On one side, spatial domain regularisation has been applied for denoising [11] and super-resolution problems [5, 14, 15]. The former aims to denoise each DWI volume independently, using adaptive Total Variation (TV). The latter aims at reconstructing a set of isotropic high-resolution (HR) DWIs from multiple sets of anisotropic low-resolution (LR) acquisitions, prior to any diffusion model fitting. In [14, 15] super-resolution reconstruction (SRR) is solved in each DWI volume separately using 3D Laplacian, while in [5] they obtain a SRR volume using a nonlocal patch-based strategy and a B0 image to constrain the reconstruction, thus not exploiting the relation between nearby angular directions within DWI volumes discussed in [16]. On the other side, the usage of angular domain, i.e. the relation between directions, has been widely exploited in the direct reconstruction of DWI models. Introduced in [17] for the spherical deconvolution used for the estimation of fibre orientation, it became soon part of the regularization terms used for example in the Orientation Distribution Function (ODF) [6]. The idea of combining spatial and angular regularization terms was successfully adopted for the ODF and fiber reconstruction [1, 3, 13]. Though combining spatial and angular regularisation has been proved to be effective in model fitting reconstruction, to the best of our knowledge, it has not been applied for SRR of the diffusion signal, nor for denoising. Here, we present a novel single shell DWI restoration framework that integrates the spatial domain with different angular directions for regularization. Our method is formulated in a variational framework, with a spatial term based on TV regularisation of a single DWI volume and an angular regularisation based on Laplace-Beltrami that encodes the information from neighboring angular directions. As [15], our framework can deal with both denoising and super-resolution problems. We will present quantitative results on phantom data [8] and qualitative results on real data from the Human Connectome Project (HCP) [19].

A Novel Spatial-Angular Domain Regularisation Approach …

45

2 Methodology 2.1 Acquisition Model Let X be the HR image we want to recover and X lk be the l-th observation of the k-th degraded image. The most widely accepted acquisition model avoiding motion effect is represented by equation X lk = Dk G X + 

(1)

where Dk is the downsampling operator, G is the blurring operator and  is Rician noise. The blurring G usually is represented by a 3D Gaussian function with full width half a maximum equal to the slice thickness along the direction of acquisition and 1.2× voxel size in plane [9, 10]; as done in [18], we can decompose the blurring operator in two operators, one acting on the in plane and the other on the slice acquisition direction. Since we want to consider strongly anisotropic volumes, the blurring will be stronger along the direction of acquisition, thus to simplify the model and to reduce computation time we will consider the blurring acting only in this direction. From this functional we can easily derive the one for the denoising problem by setting the blurring operator G to be the identity.

2.2 Spherical Harmonics Framework The values taken a voxel along different directions can be seen as the sampling along a certain direction of a function on the sphere; since the diffusion along a certain angle is expected to be the same, this function will also be antipodally symmetric. A natural choice for a basis in this setting is the Spherical Harmonics (SH) basis, as suggested in [6]. They proposed a particular choice of a basis {Yi }0≤i≤R which is antipodally symmetric, orthonormal (for the complete construction of such basis see [6]). Given the family of SH and given the set of sampling directions (φ j , θ j ) we can build the following matrix ⎤ Y1 (θ1 , φ1 ) Y2 (θ1 , φ1 ) . . . Y R (θ1 , φ1 ) ⎥ ⎢ .. .. .. .. B=⎣ ⎦ . . . . Y1 (θ N , φ N ) Y2 (θ N , φ N ) . . . Y R (θ N , φ N ) ⎡

(2)

and write the set of equations as an overdetermined linear system X = Bc, where c is the vector containing the SH coefficient. Thus we can pass from the angular representation to the spatial representation of the signal simply multiplying by B. Since Laplacian (Δ) is one of the methods allowing to measure the smoothness of a function, we can use it as a regularization term. As shown in [6], the advantage

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of using the SH representation is that the elements of SH basis are the eigenfunction of the Laplace-Beltrami operator. Thus the action of ΔYi = λi Yi , where λi is the corresponding eigenvalue. From [6] we know that λi = −l(l  + 1) where l is the order of the i−th SH. Thus if we can write a function f = ci Yi we have that the following energy functional E( f ) =

S2

(Δf )2 ds =

2  Δ ds = c Lc ci Yi

(3)

S2

where c is the vector of coefficients and L is the matrix with diagonal λi2 .

2.3 Spatial-Angular Super-Resolution Functional Variational methods for SR are based on the minimization of a convex functional composed by a fidelity term and a regularization term. Here we choose a least-square fidelity term of the form 1 k X 0l − Dk G X lk 2 (4) 2 k,l k is the original image, while X lk is our estimate. where X 0l For the regularization term we want to use both the information arising from the single volume itself (spatial domain) and also the angular correlation with the other volumes (angular domain). For the spatial domain many terms have been studied in the time, usually related to a certain norm of the gradient, each with different properties. The Total Variation we use here is defined as the L1-norm of the gradient ∇ X 1 and has the property of being sparsity promoting, but it is non-smooth. For the angular domain we choose the Laplace-Beltrami operator defined in Eq. 3. Then, the complete restoration functional

J ( Xˆ ) = ∇ Xˆ 1 + λ

1 k,l

2

k X 0l − DG B Xˆ lk 2 + μ



c Lc

(5)

vox

where λ and μ are two regularization terms balancing the effect of the various components, and in the last term we use the SH representation for Xˆ . As in [18], due to the non-smoothness of the TV term, the minimization of the functional was performed using the primal-dual algorithm presented in [2].

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3 Experiments We present results on both phantom and real DWI acquisitions of adult brain in two different scenarios: denoising and super-resolution reconstruction. Simulated data comes from the HARDI reconstruction challenge held in ISBI 2013 [8]. For denoising, we use the dMRI with S N R = 20, simulated along 32 directions and b-value b = 1200 s/mm2 . For the SR experiment, we degradate the Ground Truth (GT) of the previous phantom image (same directions and b-value) of the denoising experiment with a blurring kernel along the acquisition direction, adding Rician Noise and simulating a downsampling giving a volume with voxel of size 1 × 1 × 3 mm3 . It has been proven empirically that the blurring Gaussian function has the full width at half maximum equal to the slice thickness in the slice select direction [10]. Real data comes from the Human Connectome Project [19]. Acquisitions were performed with a Siemens 3T Connectom scanner, which is a modified 3T Skyra system (MAGNETOM Skyra Siemens Healthcare), equipped with 64-channel, tight-fitting brain array coil. 1.5 × 1.5 × 1.5 mm3 ; 140×140×96 voxels; TR/TE = 8800/57 ms; δ/Δ (ms) 12.9/21.8; In our experiments we used only the image corresponding to the 64 diffusion direction acquired with b-value = 1000 s/mm2 . We consider the HCP data of high quality and free of artefacts, thus we add Rician noise to the image as input for the denoising evaluation (see Fig. 1, panels a and b). As previously, we blur along the acquisition direction and add Rician noise to create the input for the SRR reconstruction. For the phantom data parameters were chosen by the maximization of the Peak Signal to Noise Ratio (PSNR), while for the in vivo data an empirical tuning was performed. For the phantom data the performances were evaluated with the PSNR and with the mean square error (MSE) of the generalized fractional anisotropy (gFA) against the ground truth signal and gFA values. Generalized FA is computed using Dipy library. Since human DWI was good quality, but no ground truth was available, the computation of PSNR is avoided, and the performances are evaluated with qualitative analysis. Qualitative analysis will be based on Figs. 1, 3, 4 and 5, in which are compared the effect of the different algorithms on a slice of the diffusion signal at certain direction.

3.1 Denoising We compare our results to two state-of-the-art methods (an optimised block-wise non-local means (NLM) 3-d filter [4] and a Principal Component Analysis (PCA) [20, 21]) that are freely available in Dipy [7] and MRtrix [12] software libraries. We tested different parameters for both algorithms. In all the cases the default parameters were the best performing, except in the in vivo experiments where for Dipy we set

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(a) Simulated data

(b) NLM-Dipy

(d) Noisy data

(e) PCA-MRtrix

(c) TV, λ = 8

(f) TV-Lap, λ = 10, μ = 0.05

Fig. 1 Qualitative results for simulated DWI denoising methods

the patch_radius = 2 and block_radius = 3. In order to evaluate the contribution of the angular domain, we will also compare our TV-Laplace-Beltrami regularisation (TV-Lap) to a spatial regularisation only (TV). Qualitative results on simulated and real data are shown in Figs. 1 and 3, respectively. Additionally, gFA scalars after denoising simulated data are shown in Fig. 2. In both scenarios, the NLM approach has the strongest regularizing effect, leading to the disappearance of some fibres and structures in the simulated data and a blurring effect on real data. On phantom data, PCA and TV based algorithms perform similarly, though PCA seems to better preserve fibre boundaries than TV. Our approach based on TV and Laplacian regularisation results in a stronger noise removal and a better preservation of boundaries. These results are supported by both PSNR and MSE of the gFA values, where we can see that on the phantom data the angular domain is overcoming the other methods with PSNR = 27.34 dB and MSE of gFA = 7.9384e-05, while the second one is TV with PSNR = 22.01 dB and MSE of gFA = 8.5779e-05 (see Table 1). Differences between PCA, TV and TV-Lap are less obvious on real data. However, deep sulci seem better recovered by TV and TV-Lap approaches (red and blue box). Moreover, cortical ribbon seems better denoised by TV-Lap.

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Table 1 Quantitative results on denoising phantom data: PSNR (top row) and MSE of gFA (bottom row). Comparison of our method with exising techniques TV PCA-MRtrix NLM-Dipy TV-Lap PSNR MSE of gFA

22.0139 8.5779e-05

(a) Simulated data

16.0391 1.0669e-04

19.5586 1.5539e-04

(b) NLM-Dipy

(d) PCA-MRtrix

27.3355 7.9384e-05

(c) TV

(e) TV-Lap

Fig. 2 Generalized FA maps from the denoised simulated data. MSE with respect ground truth gFA values is shown in Table 1

3.2 Super-Resolution Reconstruction In this section we reconstruct a SR volume from anisotropic DWI. Our proposed spatial-angular (TV-Lap) framework will be compared to state-of-the-art TV regularization. Results on simulated and real data are shown in Fig. 4. We can see that our method preserves boundaries of the structures better than TV and that fibres are better denoised. This is supported also by quantitative measures (PSNR), which shows how combining both spatial and angular domains can improve the results (20.48 db over 18.61 dB when using TV only). Results on real data (Fig. 5) are more difficult to

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(a) HCP data

(b) NLM-Dipy

(c) TV, λ = 150

(d) Noisy data

(e) PCA-MRtrix

(f) TV-Lap λ = 170, μ = 0.1

Fig. 3 Qualitative comparison of denoising real DWI data

evaluate. The sulcus highlighted by the red square almost disappeared on LR images due to the blurring and noise. We can see that the TV fails to recover the sulcus, while adding the information arising from angular directions in the regularisation term allow this sulcus better.

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Fig. 4 Qualitative and quantitative results for simulated DWI SRR methods. Parameters setting and PSNR: c λ = 25, 18.61 dB, d λ = 30, μ = 0.2, 20.48 dB

(a) HCP data

(d) LR+Noise data

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Fig. 5 Quantitative results for real DWI super-resolution reconstruction methods. Parameters setting: b λ = 110, c λ = 110, μ = 0.1, e λ = 170, f λ = 170, μ = 0.1

4 Conclusion In this paper we presented a novel spatial-angular regularization for diffusion MRI data. Effectively, we have combined a TV with a Laplace-Beltrami regularisation for spatial and angular domains respectively. Our framework has been evaluated for denoising and super-resolution of simulated and real data. Our preliminary results

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prove that integrating information from both of these domains improves the restoration in quantitative (PSNR) as well as visual terms, with better preservation of structure and edges in denoising and SRR problems. This idea is not limited to the particular regularisation and fidelity functionals we have presented here, but in the future more complex terms could be included, such as multi shell approachs, or fidelity terms better fitted to the noise distribution. Moreover, extensive testing on different noise-level phantoms and also with DWI at different b-values should be performed. Acknowledgements This work was supported by the CIBM of the Unil, the Swiss Federal Institute of Technology Lausanne, the University of Geneva, the CHUV, the Hôpitaux Universitaires de Genève, the Leenaards and Jeantet Foundations. This work was also supported by the Swiss National Science Foundation grant SNSF-IZK0Z2_170894.

References 1. Aura Rasclosa, A.: Sparse inverse problems for Fourier imaging applications to Optical Interferometry and Diffusion Magnetic Resonance Imaging. Ph.D. thesis, Lausanne (2017) 2. Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imag. Vis. 40(1), 120–145 (2011) 3. Chen, Y., Dai, Y.H., Han, D.: Fiber orientation distribution estimation using a PeacemanRachford splitting method. SIAM J. Imag. Sci. 9(2), 573–604 (2016) 4. Coupé, P., Yger, P., Prima, S., Hellier, P., Kervrann, C., Barillot, C.: An optimized blockwise nonlocal means denoising filter for 3-d magnetic resonance images. IEEE Trans. Med. Imag. 27(4), 425–441 (2008) 5. Coupé, P., Manjón, J.V., Chamberland, M., Descoteaux, M., Hiba, B.: Collaborative patchbased super-resolution for diffusion-weighted images. NeuroImage 83, 245–261 (2013) 6. Descoteaux, M., Angelino, E., Fitzgibbons, S., Deriche, R.: Regularized, fast, and robust analytical q-ball imaging. Mag. Res. Med. 58(3), 497–510 (2007) 7. Diffusion Imaging in PYthon (DIPY). http://nipy.org/dipy/ 8. HARDI Reconstruction Challenge, ISBI 2013. http://hardi.epfl.ch/static/events/2013_ISBI/ 9. Jiang, S., Xue, H., Glover, A., Rutherford, M., Rueckert, D., Hajnal, J.V.: MRI of moving subjects using multislice snapshot images with volume reconstruction (SVR): application to fetal, neonatal, and adult brain studies. IEEE Trans. Med. Imag. 26(7), 967–980 (2007) 10. Kuklisova-Murgasova, M., Quaghebeur, G., Rutherford, M.A., Hajnal, J.V., Schnabel, J.A.: Reconstruction of fetal brain MRI with intensity matching and complete outlier removal. Med. Image Anal. 16(8), 1550–1564 (2012) 11. Liu, R.W., Shi, L., Huang, W., Xu, J., Yu, S.C.H., Wang, D.: Generalized total variation-based MRI Rician denoising model with spatially adaptive regularization parameters. Magn. Reson. Imag. 32(6), 702–720 (2014) 12. MRtrix. http://www.mrtrix.org/ 13. Ouyang, Y., Chen, Y., Wu, Y., Zhou, H.: Total variation and wavelet regularization of orientation distribution functions in diffusion MRI. Inv. Prob. Imag. 7, 565–583 (2013) 14. Scherrer, B., Gholipour, A., Warfield, S.: Super-resolution in diffusion-weighted imaging. In: MICCAI 2011, pp. 124–132 (2011) 15. Scherrer, B., Gholipour, A., Warfield, S.K.: Super-resolution reconstruction to increase the spatial resolution of diffusion weighted images from orthogonal anisotropic acquisitions. Med. Image Anal. 16(7), 1465–1476 (2012) 16. Tobisch, A., Neher, P.F., Rowe, M.C., Maier-Hein, K.H., Zhang, H.: Model-based superresolution of diffusion MRI. In: Computational Diffusion MRI and Brain Connectivity, pp. 25–34. Springer, Cham (2014)

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17. Tournier, J.D., Calamante, F., Gadian, D.G., Connelly, A.: Direct estimation of the fiber orientation density function from diffusion-weighted MRI data using spherical deconvolution. NeuroImage 23(3), 1176–1185 (2004) 18. Tourbier, S., Bresson, X., Hagmann, P., Thiran, J.P., Meuli, R., Cuadra, M.B.: An efficient total variation algorithm for super-resolution in fetal brain MRI with adaptive regularization. NeuroImage 118, 584–597 (2015) 19. Van Essen, D.C., Smith, S.M., Barch, D.M., Behrens, T.E., Yacoub, E., Ugurbil, K., Consortium, W.M.H., et al.: The WU-Minn human connectome project: an overview. Neuroimage 80, 62–79 (2013) 20. Veraart, J., Fieremans, E., Novikov, D.S.: Diffusion MRI noise mapping using random matrix theory. Magn. Reson. Med. (2016) 21. Veraart, J., Novikov, D.S., Christiaens, D., Ades-Aron, B., Sijbers, J., Fieremans, E.: Denoising of diffusion MRI using random matrix theory. NeuroImage 142, 394–406 (2016)

Dmipy, A Diffusion Microstructure Imaging Toolbox in Python to Improve Research Reproducibility Abib Alimi, Rutger Fick, Demian Wassermann and Rachid Deriche

Abstract Non-invasive estimation of brain white matter microstructure features using diffusion MRI—otherwise known as Microstructure Imaging—has become an increasingly diverse and complicated field over the last decade. Multi-compartmentbased models have been a popular approach to estimate these features. In this work, we present Diffusion Microstructure Imaging in Python (Dmipy), a diffusion MRI toolbox which allows accessing any multi-compartment-based model and robustly estimates these important features from single-shell, multi-shell, and multidiffusion time, and multi-TE data. Dmipy follows a building block-based philosophy to microstructure imaging, meaning a multi-compartment model can be constructed and fitted to dMRI data using any combination of underlying tissue models, axon dispersion-or diameter distributions, and optimization algorithms using less than 10 lines of code, thus helps improve research reproducibility. In describing the toolbox, we show how Dmipy enables to easily design microstructure models and offers to the users the freedom to choose among different optimization strategies. We finally present three advanced examples of highly complex modeling approaches which are made easy using Dmipy. Keywords Diffusion MRI · Microstructure imaging · dMRI · Multi-compartment models · Python · Free open source software

Abib Alimi and Rutger Fick share first authorship for this work. A. Alimi (B) · R. Deriche Athena, Inria, Université Côte d’Azur, Sophia Antipolis, France e-mail: [email protected] R. Fick TheraPanacea, Paris, France D. Wassermann Parietal, Inria, CEA, Université Paris-Saclay, Paris, France © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_5

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1 Introduction Diffusion MRI (dMRI) is sensitive to the micrometer-scale displacement of water molecules and has, therefore, become an invaluable clinical diagnostic tool, particularly in neuroimaging. In principle, dMRI can provide information about the tissue microstructure on a much smaller scale than that of the actual image resolution: micrometers versus millimeters. The practice of estimating microstructural tissue features using dMRI is referred to as Microstructure Imaging. In Microstructure Imaging, the observed diffusion signal is related to tissue structure using biophysical models (see a partial taxonomy in [1]). A combination of biophysical models constitutes a microstructure model. In particular, multi-compartment (MC)-based models have been a popular approach to estimate these features. Different toolboxes have been proposed so far. The Microstructure Imaging Sequence Simulation Toolbox (MISST) [2, 3] applies a generalized gradient waveform to microstructure models and eventually estimates their parameters. However, the main goal of MISST is to accurately simulate the diffusion signal of microstructure models generally hard-coded in the toolbox. The recently introduced CUDA diffusion modeling toolbox (cuDIMOT) [4] proposes to use Graphical Processing Units (GPUs) in order to accelerate the computationally expensive biophysical modeling and microstructure mapping to diffusion measurements as well as tractography and connectivity estimation. In [5] the authors go further in term of reducing computation time in their Microstructure Diffusion Toolbox (MDT) in which they use parallelized GPUs and multi-core CPU in diffusion modeling in large group studies. To provide access to any MC-based model and improve reproducibility in MCbased research, we propose Diffusion Microstructure Imaging in Python (Dmipy). Dmipy is a diffusion MRI toolbox which allows accessing any MC-based model. It uses multi-core CPUs and modularity is a key feature that simplifies both its implementation e.g. future extension and its usage. Furthermore, unlike the abovementioned toolboxes, not only does Dimpy propose to use ‘standard’ MC-modeling strategy but also offers the spherical mean (SM) of any microstructure model, as well as a generalization of the Constrained Spherical Deconvolution (CSD) [6] technique and adds new tissue models to CSD and SM. Dmipy robustly estimates these models’ important features from single-shell, multi-shell, and multi-diffusion time diffusion data, using less than 10 lines of code and therefore helps improve research reproducibility. It is an open source package currently available for download on https://github.com/AthenaEPI/dmipy/. This paper extends our previous work [7, 8] and gives a further explanation on the design of microstructure models and the optimization algorithms available in Dmipy to estimates microstruction features as illustrated in Fig. 1.

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Fig. 1 Dmipy workflow: Modular microstructure model setup and parameter estimation. Different biophysical tissue models (see Fig. 2) are dispersed and/or distributed and combined together in a multi-compartment model, which is then fit to diffusion data using a chosen optimization algorithm to estimate tissue feature parameters, reconstruct FODs and quantify the quality of the fitting

2 Dmipy Signal Representation and Multi-compartment Modeling 2.1 Diffusion Contrast In this work, we focus on probing the tissue microstructure using the standard Pulsed Gradient Spin-Echo sequence (PGSE) [9] to obtain diffusion-weighted images (DWIs). The measurement of the diffusion signal is directly related to the concept of attenuation. In the presence of diffusion, the signal intensity S(b) is lower than the non-diffusion-weighted signal S(0). The signal attenuation is then expressed as E(b) = S(b)/S(0). Without restricting boundaries in the diffusion process, the signal attenuation is then given as a Gaussian as E(b) = exp(−bD) with b the b-value and D the water diffusivity [9].

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It is possible, but impractical, to densely sample all of E(q, τ ) and non-parametrically recover the displacement probability P(r, τ ) [10], from which some information about the tissue structure can be inferred. A more efficient approach has been to study the geometry of different tissue types to come up with mathematical representations which approximate the diffusion in those tissues. For example, diffusion inside axons with some radius is typically modeled as diffusion inside cylinders with the same radius, which has an analytical expression (e.g. [11]). Combinations of these representations can then be fitted to more sparsely measured DWIs, after which the recovered representation parameters (such as the diameter of the cylinder) are used to interpret the structure of the tissue. Such an approach is called multicompartment modeling or Microstructure Imaging and is the focus of the Dmipy software project.

2.2 Biophysical Modeling Biophysical models are mathematical representations of different tissue types, based on assumptions found in cited references. Here we show an extensive overview of these models as illustrated in Fig. 2. Models of Intra-cellular Diffusion (I1 through I4). Diffusion in the intracellular space is represented using axially symmetric and restricted geometric (cylinders, spheres, planes) models, with the orientation parallel to the axis. The three-dimensional diffusion signal in these models is given as the separable product of parallel and restricted perpendicular diffusions [12]. For each geometry, Dmipy offers different approximations i-e Solderman, Gallaghan and Gaussian Phase approximations. More details can be found in references [11, 13–17]. Axonal Diameter Distribution (DD1). Histology studies [18] showed that the diameters of the axons are not constant throughout the brain, but they follow a distribution. In Microstructure Imaging, the true axon diameter distribution is usually modeled by a Gamma distribution [19], which is implemented in Dmipy for the aforementioned restricted geometries. Models of Extra-cellular Diffusion (G1 through G3). Hindered extra-cellular diffusion is often modeled as an anisotropic, axially symmetric gaussian called “Zeppelin” (G2) in [1]. However, recent works argue that hindered diffusion is actually slowerthan-exponential over diffusion time τ due to how the external axon boundaries still hinder diffusing particles [20]. Later on, [21] proposed a modification of the Zeppelin taking into account the time-dependence of the diffusion, which Dmipy dubbed the temporal zeppelin (G3). Parameter links. Note that our toolbox enables to relate the volume fractions of the different compartments by modeling a first-order tortuosity (T1) approach to modulate their diffusivities as in [22].

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Fig. 2 A schematic of most biophysical models used in PGSE-based Microstructure Imaging. Using different combinations of these “components”, any microstructure model can be assembled using Dmipy

Models of Axonal Orientation Distributions (SD1 through SD3). Histology studies show that the orientation of axons within one white matter bundle are dispersed around the central bundle direction [23]. Therefore, the diffusion signal can be modeled as the spherical convolution of a spherical distribution and a kernel which describes the signal of a single axon micro-environment. Dmipy models this spherical distribution using, as most generally, the spherical harmonics (SD3) functional basis [24, 25] to give the so-called fiber Orientation Distribution Function (fODF) or Fiber Distribution Function (FOD). The toolbox also provides Bingham (SD2) and Watson (DS1) distributions [26–28], more straightforward to represent anisotropic and isotropic dispersions, respectively.

2.3 Multi-compartment Modeling Using Dmipy, any selection of biophysical models can be combined into a multicompartment model, of which the framework provides three different types:

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• multi-compartment (MC) or ‘standard’ MC, • multi-compartment spherical mean (MC-SM) and • multi-compartment constrained spherical deconvolution (MC-CSD). The general formulations of the MC and MC-SM models are as follows: MC : E(g) ˜ =

N 

 Ci (g; ˜ x),

MC-SM :

i=1

S2

E(g)dn ˜ =

N   i=1

S2

Ci (g; ˜ x)dn (1)

where C(g; ˜ x) represents any (dispersed and/or distributed) compartment model shown in Fig. 2, (g) ˜ is shorthand for all PGSE acquisition parameters and x for all biophysical model parameters. In MC-CSD, the biophysical models are not directly summed to approximate the signal, but constitute the convolution kernel that convolved with a (model-wise) Fiber Orientation Distribution (FOD): MC-CSD : E(g) ˜ =

N 

Ci (g; ˜ x) ∗S2 F O Di (n).

(2)

i=1

More details about the MC, MC-SM and MC-CSD models are given in Sect. 3. The associated generated models are then fit to data using different optimizers.

2.4 Modular Optimization Algorithms Based on Dmipy’s modular philosophy, each modeling framework also has a modular selection of optimization algorithms. Because of their structural similarity, MC and MC-SM models share the same optimizer choices (brute2fine and mix), while MCCSD has algorithms specialized for estimating FODs. brute2fine The “brute2fine” optimizer uses of scipy’s brute force and L-BFGS-B algorithms for constrained optimization of the MC parameters without needing an initial guess. mix Dmipy also includes a generalized implementation of the recent Microstructure Imaging in Crossing (MIX) optimizer [29], which is useful when fitting highly complex models. csd For MC-CSD models, Dmipy provides a completely generalized multi-shell constrained spherical deconvolution approach similar to Jeurissen et al. (2014) [30]. For any kernel composition, it allows to enforce unity and positivity constraints on the volume fractions and FOD respectively, as well as smoothness.

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3 Reproducible Research with Dmipy The main purpose of Dmipy is to help increase research reproducibility in diffusion Microstructure Imaging. Dmipy is agnostic about model design, giving the user the freedom to easily create any model which is applicable to their research question, and fit it to the PGSE data they have with the optimizer they want. Here is presented a basic overview of how to design microstructure models with the combination of biophysical models presented in the previous section. To set up a Microstructure model in Dmipy, a ‘MultiCompartmentModel’ object is created by combining one or more biophysical models to represent tissue features such as water diffusivity, axon diameter distributions, axon dispersion, axon bundle crossings and extra-cellular diffusion. A constructed model object can also include custom or known parameter constraints such as axon tortuosity [22]. The following code snippet illustrates the generation of Ball and Stick [14] model in all three estimation frameworks presented in Sect. 2.3.

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Snippet 1: MC-models from dmipy.signal_models import cylinder_models, gaussian_models from dmipy.core.modeling_framework import *

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stick = cylinder_models.C1Stick() ball = gaussian_models.G1Ball()

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BAS = MultiCompartmentModel(models=[ball, stick]) BAS_SM = MultiCompartmentSphericalMeanModel(models=[ball, stick]) BAS_CSD = MultiCompartmentSphericalHarmonicsModel(models=[ball, stick])

The three different MC-models generated above each use different representations of biophysical models: ‘standard’ for MC, ‘spherical mean’ for MC-SM and ‘rotational harmonics’ for MC-CSD. Every model has these representations, meaning nearly any combination of cylinders, spheres, Gaussians and distributed models (see Sect. 4.2) can be combined in any MC representation. Note that the MC representation is independent of any data which is to be fitted. Fitting a Dmipy model requires only a description of the acquisition parameters (gradient strength, directions, pulse length/duration, TE) and an ND-array of dMRI data. In this way, Dmipy’s fitting interface is designed very similar to dipy [31] and other successful packages. After fitting any fitted parameters, fitting errors, FODs and others can be easily recovered from the fitted model object:

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Snippet 2: MC-CSD BAS_fit = BAS.fit(scheme, data, solver='brute2fine') BAS_fit.fitted_parameters # also mse, fod, R2, etc.

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The modular design of Dmipy provides the users with the means to explore their diffusion data using any model and global optimization approaches. This modularity also enables them to easily adapt/extend and also share their model implementations in their work, allowing for the addition of new models or optimizers, thus improving the reproducibility of Microstructure Imaging research.

4 Dmipy Advanced Examples Dmipy can be used for a large range of applications, from modular design of multicompartment models to signal simulation and tissue feature estimation, using usually less than 10 lines of code. In this section, we show three advanced examples of highly complex modeling approaches which are made easy using Dmipy.

4.1 MC-SM to MC-CSD with Voxel-Varying Kernel During the 2017 ISMRM: TraCED robust tractography challenge1 the winning results were obtained using CSD with a voxel-varying convolution kernel, estimated using MC-MDI [32] (an MC-SM technique). Such a method using a chain of multiple estimation frameworks is tricky to implement, yet using Dmipy it can be done with little effort, see Fig. 3:

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Snippet 3: MC-SM to MC-CSD stick = cylinder_models.C1Stick() zeppelin = gaussian_models.G2Zeppelin() mcsm = MultiCompartmentSphericalMeanModel(models=[stick, zeppelin])

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mcsm.set_tortuous_parameter('G2Zeppelin_1_lambda_perp', 'C1Stick_1_lambda_par', 'partial_volume_0', 'partial_volume_1') mcsm.set_equal_parameter('G2Zeppelin_1_lambda_par', 'C1Stick_1_lambda_par')

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vox_varying_csd = mcsm_fit.return_spherical_harmonics_fod_model() vox_varying_csd_fit = vox_varying_csd.fit(scheme_data, data) fods = vox_varying_csd_fit.fod(sphere)

1 https://my.vanderbilt.edu/ismrmtraced2017/.

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Fig. 3 Estimated FODs and intra-axonal water fraction through Multi Compartment Spherical Mean Model

4.2 A “Complete” White Matter Representation Recent works consider axon-dispersion and the axon-diameter-distribution to be relevant phenomena for characterizing the white matter [19, 33, 34], as well as the temporal dependence of the extra-axonal space [21]. Modeling all these effects together in one representation, with all its parameters, is quite complicated. Using Dmipy, even creating such a complicated model is very intuitive by just applying distribution effects one after another:

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Snippet 4: Dispersed and distributed MC from dmipy.distributions import distribute_models cylinder = cylinder_models.C4CylinderGaussianPhaseApproximation() temporal_zeppelin = gaussian_models.G3TemporalZeppelin()

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bingham_bundle = distribute_models.SD2BinghamDistributed( models=[cylinder, temporal_zeppelin]) gamma_bingham_bundle = distribute_models.DD1GammaDistributed( models=[bingham_bundle], target_parameter='C4CylinderGaussianPhaseApproximation_1_diameter') white_matter_mc_model = MultiCompartmentModel(models=[gamma_bingham_bundle]) white_matter_mc_model.visualize_model_setup()

Note that despite its complexity, this model can be used to fit and simulate data like any other,2 and can even be used as a CSD convolution kernel. Line 11 can be used to visualize any model as a graph such as Fig. 4.

2 Although

finding a unique minimum will be difficult given its many parameters.

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G3TemporalZeppelin

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Fig. 4 Graph representation of the constructed complete white matter model

4.3 Multi-tissue CSD and Multi-tissue SMT Recent FOD estimation methods such as Multi-Tissue CSD (MT-CSD) obtain more robust tractograms using non-parameters tissue response kernels - data-estimated kernels representing white/grey matter and CSF [30]. Dmipy can estimate these kernels directly from the dMRI dataset using the method of [35] and represents them as standard biophysical models, meaning they can be used together with parametric models by all three estimation frameworks. In the following code snippet, we show the Dmipy code to estimate tissue response models and construct both MT-CSD and multi-tissue spherical mean representation. We show the tissue volume fractions of both in Fig. 5.

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Snippet 5: MT-CSD and MT-SMT from dmipy.tissue_response.three_tissue_response import ( three_tissue_response_dhollander16) wm, gm, csf = three_tissue_response_dhollander16(scheme_data, data) mtcsd = MultiCompartmentSphericalHarmonicsModel([wm, gm, csf]) mtcsd_fit = mtcsd.fit(scheme_data, data)

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mtsmt = MultiCompartmentSphericalMeanModel([wm, gm, csf]) mtsmt_fit = mtsmt.fit(scheme_data, data)

5 Discussion This work introduces Dmipy, a software toolbox that allows to design multicompartment diffusion models and to estimate their parameters. It is an open source project already available online. Therefore, users can easily tailor the software to investigate their own research and also to reproduce the models presented in the Microstructure Imaging literature.

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(a) Multi-Tissue CSD

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(b) Multi-Tissue SMT

Fig. 5 White Matter, Grey Matter and CSF volume fractions, estimated using a MT-CSD and b MT-SMT

Future work will target the implementation of a generalized gradient waveform like in MISST [2, 3] to account for sequences such as STEAM or OGSE best suited for achieving, for instance, long and short diffusion times [20, 21, 36]. We will also compare Dmipy’s performance in terms of computation time and fitting-quality with cuDIMOT [4] and MDT [5] diffusion toolboxes. Dmipy provides many advantages. Indeed, it is an easy way to set up microstructure models (few lines of codes) with a high-level interaction. It also proposes different global optimizers including MIX [29]. Dmipy is, to our knowledge, the only software solution that allows different and complementary microstructure estimation strategies (Spherical Mean, and generalized CSD, etc.) for all tissue models while taking into account or not their dispersion and/or distributions. It is different from dipy [31] because Dmipy is a “model generator”. That is, it does not hard-code various models. However, one can generate all existing models on the fly. Furthermore, thanks to its modular design, it is very easy to add new optimization algorithms or tissue models to be used in the framework, thus helps improve reproducibility in Microstructure Imaging research. Acknowledgements This work was partly supported by ANR “MOSIFAH” under ANR-13MONU-0009-01, the ERC under the European Union’s Horizon 2020 research and innovation program (ERC Advanced Grant agreement No 694665:CoBCoM).

References 1. Panagiotaki, E., et al.: Compartment models of the diffusion MR signal in brain white matter: a taxonomy and comparison. NeuroImage 59(3), 2241–2254 (2012) 2. Ianu¸s, A., Alexander, D.C., Drobnjak, I.: Microstructure imaging sequence simulation toolbox. In: International Workshop on Simulation and Synthesis in Medical Imaging, pp. 34–44. Springer, Cham. (2016) 3. Drobnjak, I., et al.: The matrix formalism for generalised gradients with time-varying orientation in diffusion NMR. J. Magn. Reson. 210(1), 151–157 (2011)

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4. Hernandez-Fernandez, M., Reguly, I., Jbabdi, S., Giles, M., Smith, S., Sotiropoulos, S.N.: Using GPUs to accelerate computational diffusion MRI: from microstructure estimation to tractography and connectomes (2018) 5. Harms, R.L., et al.: Robust and fast nonlinear optimization of diffusion MRI microstructure models. Neuroimage 155, 82–96 (2017) 6. Tournier, J.D., Calamante, F., Connelly, A.: Robust determination of the fibre orientation distribution in diusion mri: non-negativity constrained super-resolved spherical deconvolution. NeuroImage 35, 14591472 (2007) 7. Fick, R., Wassermann, D., Deriche, R.: Mipy: an open-source framework to improve reproducibility in brain microstructure imaging. In: OHBM 2018 - Human Brain Mapping, pp. 1–4, June 2018, Singapore, Singapore (2018). (hal-01722146) 8. Fick, R., Deriche, R., Wassermann, D.: Dmipy: An Open-source Framework for Reproducible dMRI-Based Microstructure Research (Version 0.1), Zenodo. (2018). https://doi.org/10.5281/ zenodo.1188268 9. Stejskal, E., Tanner, J.: Spin diffusion measurements: spin echoes in the presence of a timedependent field gradient. J. Chem. Phys. 42, 288292 (1965) 10. Wedeen, V.J., Hagmann, P., Tseng, W.Y.I., Reese, T.G., Weisskoff, R.M.: Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging. Magn. Reson. Med. 54, 13771386 (2005) 11. Callaghan, P.T.: Pulsed-gradient spin-echo NMR for planar, cylindrical, and spherical pores under conditions of wall relaxation. J. Magn. Reson. Ser. A 113(1), 53–59 (1995) 12. Assaf, Y., Freidlin, R.Z., Rohde, G.K., Basser, P.J.: New modeling and experimental framework to characterize hindered and restricted water diffusion in brain white matter. Magn. Reson. Med. 52, 965978 (2004) 13. Kroenke, C.D., Ackerman, J.J.H., Yablonskiy, D.A.: On the nature of the NAA diffusion attenuated MR signal in the central nervous system. Magn. Reson. Med. 52, 1052–1059 (2004) 14. Behrens, T., Woolrich, M., Jenkinson, M., Johansen-Berg, H., Nunes, R., Clare, S., Matthews, P., Brady, J., Smith, S.: Characterization and propagation of uncertainty in diffusion-weighted mr imaging. MRM 50, 10771088 (2003) 15. Söderman, O., Jönsson, B.: Restricted diffusion in cylindrical geometry. J. Magn. Reson. Ser. A 117, 9497 (1995) 16. Vangelderen, P., DesPres, D., Vanzijl, P., Moonen, : C.: Evaluation of restricted diffusion in cylinders. Phosphocreatine in rabbit leg muscle. J. Magn. Reson., Ser. B 103, 255–260 (1994) 17. Neuman, C.: Spin echo of spins diffusing in a bounded medium. J. Chem. Phys. 60, 45084511 (1974) 18. Aboitiz, F., Scheibel, A.B., Fisher, R.S., Zaidel, E.: Fiber composition of the human corpus callosum. Brain Res. 598, 143153 (1992) 19. Assaf, Y., Blumenfeld-Katzir, T., Yovel, Y., Basser, P.J.: Axcaliber: a method for measuring axon diameter distribution from diffusion mri. MRM 59, 13471354 (2008) 20. Novikov, D.S., Jensen, J.H., Helpern, J.A., Fieremans, E.: Revealing mesoscopic structural universality with diffusion. Proc. Nat. Acad. Sci. 201316944 (2014) 21. Burcaw, L.M., Fieremans, E., Novikov, D.S.: Mesoscopic structure of neuronal tracts from time-dependent diffusion. NeuroImage 114, 1837 (2015) 22. Szafer, A., Zhong, J., Gore, J.C.: Theoretical model for water diffusion in tissues. Magn. Reson. Med. 33, 697712 (1995) 23. Leergaard, T.B., et al.: Quantitative histological validation of diffusion MRI fiber orientation distributions in the rat brain. PloS one 5(1), e8595 (2010) 24. Tournier, J.D., Calamante, F., Gadian, D.G., Connelly, A.: Direct estimation of the fiber orientation density function from diffusion-weighted mri data using spherical deconvolution. NeuroImage 23, 11761185 (2004) 25. Descoteaux, M., Deriche, R., Anw, A., Bio, T., Odyssée, P.: Deterministic and probabilistic qball tractography: from diffusion to sharp fiber distributions. Technical report 6273, INRIA Sophia Antipolis (2007b)

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26. Tariq, M., Schneider, T., Alexander, D.C., Wheeler-Kingshott, C.A.G., Zhang, H.: Binghamnoddi: Mapping anisotropic orientation dispersion of neurites using diffusion mri. NeuroImage 133, 207223 (2016) 27. Bingham, C.: An antipodally symmetric distribution on the sphere. Ann. Stat. 1201–1225 (1974) 28. Kaden, E., Knsche, T.R., Anwander, A.: Parametric spherical deconvolution: inferring anatomical connectivity using diffusion MR imaging. NeuroImage 37(2), 474–488 (2007) 29. Farooq, H., Xu, J., Nam, J.W., Keefe, D.F., Yacoub, E., Georgiou, T., Lenglet, C.: Microstructure imaging of crossing (MIX) white matter fibers from diffusion MRI. Sci. Rep. 6, 38927 (2016) 30. Jeurissen, B., Tournier, J.D., Dhollander, T., Connelly, A., Sijbers, J.: Multi-tissue constrained spherical deconvolution for improved analysis of multi-shell diffusion MRI data. NeuroImage 103, 411–426 (2014) 31. Garyfallidis, E., Brett, M., Amirbekian, B., Rokem, A., Van Der Walt, S., Descoteaux, M., Nimmo-Smith, I., Contributors, D.: Dipy, a library for the analysis of diffusion MRI data. Front. Neuroinformatics 8 (2014) 32. Kaden, E., Kelm, N.D., Carson, R.P., Does, M.D., Alexander, D.C.: Multi-compartment microscopic diffusion imaging. NeuroImage (2016) 33. Zhang, H., Schneider, T., Wheeler-Kingshott, C.A., Alexander, D.C.: Noddi: practical in vivo neurite orientation dispersion and density imaging of the human brain. NeuroImage 61, 10001016 (2012) 34. Pizzolato, M., Wassermann, D., Deriche, R., Fick, R.: Orientation-dispersed apparent axon diameter via multi-stage spherical mean optimization. In: CDMRI (2018) 35. Dhollander, T., et al.: Unsupervised 3-tissue response function estimation from single-shell or multi-shell diffusion MR data without a co-registered T1 image. In: ISMRM Workshop on Breaking the Barriers of Diffusion MRI, vol. 5 (2016) 36. De Santis, S., Jones, D.K., Roebroeck, A.: Including diffusion time dependence in the extraaxonal space improves in vivo estimates of axonal diameter and density in human white matter. NeuroImage 130, 91103 (2016)

Tissue Segmentation Using Sparse Non-negative Matrix Factorization of Spherical Mean Diffusion MRI Data Peng Sun, Ye Wu, Geng Chen, Jun Wu, Dinggang Shen and Pew-Thian Yap

Abstract In this paper, we present a method based on sparse non-negative matrix factorization (NMF) for brain tissue segmentation using diffusion MRI (DMRI) data. Unlike existing NMF-based approaches, in our method NMF is applied to the spherical mean data, computed on a per-shell basis, instead of the original diffusionweighted images. This is motivated by the fact that the spherical mean is independent of the fiber orientation distribution and is only dependent on tissue microstructure. Applying NMF to the spherical mean data will hence allow tissue signal separation based solely on the microstructural properties, unconfounded by factors such as fiber dispersion and crossing. We show results explaining why applying NMF directly on the diffusion-weighted images fails and why our method is able to yield the expected outcome, producing tissue segmentation with greater accuracy. Keywords Tissue segmentation · Sparse NMF · Spherical mean · Diffusion MRI

This work was supported in part by NIH grants (NS093842, EB022880, EB006733, EB009634, AG041721, MH100217, and AA012388) and NSFC grants (11671022 and 61502392). P. Sun · Y. Wu · G. Chen · D. Shen · P.-T. Yap (B) Department of Radiology and BRIC, University of North Carolina, Chapel Hill, NC, USA e-mail: [email protected] P. Sun School of Computer Science, Northwestern Polytechnical University, Xi An, China Y. Wu Institute of Information Processing and Automation, Zhejiang University of Technology, Hangzhou, China J. Wu School of Electronics and Information, Northwestern Polytechnical University, Xi An, China © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_6

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1 Introduction In diffusion MRI (DMRI), tissue segmentation is a key step to tease apart signal contributions of white matter (WM), gray matter (GM), and cerebrospinal fluid (CSF) for fine-grained analysis [1]. For example, it has been demonstrated in [2] that the overestimation of apparent WM density in voxels containing GM and CSF can be mitigated by explicitly considering the response functions associated with the individual tissue types. Unlike segmentation of anatomical (e.g., T1-weighted) images, which is typically intensity-based, DMRI segmentation is based on diffusion-attenuated signals stemming from the motion of water molecules. Recently, nonnegative matrix factorization (NMF) has been proposed for tissue segmentation in DMRI [3]. NMF is a data-driven approach for blind source separation and can be viewed as a clustering algorithm [4–9]. The goal of NMF is to approximate a nonnegative data matrix by the product of two unknown matrices: A basis matrix and a weight matrix. The rank of the factorization is generally smaller than that of the data matrix and the product can hence be seen as a compressed representation of the data. The solution to the plain vanilla NMF [4] is not unique. However, this nonuniqueness problem can be mitigated by imposing sparsity or convexity constraints [3, 9, 10]. Jeurissen et al. [3] applied NMF directly to the diffusion-weighted (DW) images for tissue-type segmentation. The data matrix is formed with each row corresponding to a different DW image and each column corresponding to a different voxel. The factorization rank is set to 3 to recover the three main tissue types in the brain. Each column of the basis matrix returned by NMF is a basis vector that best represents a certain tissue type and the corresponding row of the weight matrix contains the weights of the voxels in relation to the tissue type. We note that the method described above, while seemingly effective, suffers from a major limitation. Although it is straightforward to understand that CSF and GM can each be represented by a basis vector in the basis matrix, it is unclear how WM, which can come in a large variety of orientations and configurations, can be sufficiently captured using a single basis vector. In fact, we will show in this paper that this is an unreasonable representation and causes significant errors in segmentation. We will address the above problem by applying NMF to spherical mean DMRI data. The spherical mean is computed for each shell associated with a diffusion weighting (i.e., b-value). The spherical mean is independent of the fiber orientation distribution and is only dependent on tissue microstructure [11]. Applying NMF on the spherical mean data will hence allow tissue classification based solely on the microstructural properties, unconfounded by factors such as fiber dispersion and crossing. We will show experimental results demonstrating why applying NMF directly on the DW images is not ideal and why our method is able to overcome the problem, producing tissue segmentation results with greater accuracy.

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2 Approach Our approach to tissue segmentation is inspired by the work of Jeurissen et al. [3]. However, instead of applying NMF directly to the DW images, we apply NMF on the spherical mean, which is computed on a per-shell basis.

2.1 The Spherical Mean The spherical mean is not dependent on the fiber orientation distribution and is a function of per-axon diffusion characteristics [11]. This is based on the observation that for a fixed b-value the spherical mean of the diffusion-attenuated signal over the gradient directions, i.e.,  1 Sb (g)dg (1) S¯b = 4π S2 does not depend on the fiber orientation distribution. Assuming that the signal can be represented  as the spherical convolution of a fiber orientation distribution p(ω) ( p(ω) ≥ 0, S2 p(ω)dω = 1, p(ω) = p(−ω), ω ∈ S2 ) with an axial and antipodal symmetric kernel h b (g, ω) ≡ h b (|g, ω|), i.e.,  Sb (g) = S0 it can be shown that [11]

S2

h b (g, ω) p(ω)dω,

S¯b = S0 h¯ b .

(2)

(3)

This shows that the spherical mean is independent of the orientation distribution.

2.2 Tissue Segmentation via Sparse NMF Brain tissue segmentation can be cast as a blind source separation problem, where the sources and the mixing weights are determined concurrently. NMF is an ideal tool for this problem owing to its ability to represent the data as an additive nonnegative combination of a set of nonnegative basis vectors, resulting in a partsbased decomposition. NMF factorizes a nonnegative data matrix V ∈ R M×N into two unknown nonnegative matrices: The basis matrix W ∈ R M×K and the weight matrix H ∈ R K ×N : V ≈ W × H. (4) In Jeurissen et al.’s [3] approach, each row of V corresponds to a different DW image and each column corresponds to a different voxel inside the brain. Instead

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of taking this approach, we show that a better approach is to perform segmentation based on the spherical mean. That is, each row of V should correspond to a shell associated with a b-value and each column to the voxels. Upon performing NMF, each column of W is a basis vector that best represents a certain tissue type and the corresponding row of the H contains the weights (also known as volume fractions when normalized to have a sum of 1) of the voxels in relation to the tissue type. The parameter K of the factorization should be usually chosen to be smaller than M and N . Naturally, for brain tissue segmentation into WM, GM, and CSF, it should be set to 3. The value can be increased beyond 3 to account for pathology such as lesions. In general, NMF is undetermined and the factorization is obtained by solving the following problem min V − W H 2F , wi, j ≥ 0, h i, j ≥ 0. W,H

(5)

The non-uniqueness of NMF can be controlled by sparsity penalization [9]: min V − W H 2F + λ1



W,H

h j 1 + λ2



j

h j 22 , wi, j ≥ 0, h i, j ≥ 0,

(6)

j

where h j is j-th column vector of H . This form of penalization is reasonable since partial volume occurs mostly at tissue boundaries and hence most parts of the brain should be comprised of single tissue types [3]. The solution to (6) can be obtained by alternating between solving two subproblems, i.e., a non-negative least-squares problem: min V − W H 2F , wi, j ≥ 0, W

(7)

when H is fixed, and an elastic net problem min V − W H 2F + λ1 H

 j

h j 1 + λ2



h j 22 , h i, j ≥ 0,

(8)

j

when W is fixed. We use SPAMS [12–14] in our implementation. Using the spherical mean instead of the original signal in V allows us to solve the problem inherent in Jeurissen et al.’s method [3]. This is because the spherical mean captures the per-axon diffusion characteristics independent of the orientation distribution. For example, voxels with very different fiber configuration, but with the same per-axon microstructural parameters, can be represented using exactly the same spherical mean vector. Our formulation of the problem also means that a smaller matrix V needs to be decomposed since it has a significantly smaller number of rows. This implies that tissue segmentation can be performed much faster using our method.

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3 Experiment We evaluated our method with Jeurissen et al.’s [3] method as the comparison baseline. Both qualitative and quantitative results are reported.

3.1 Synthetic Data Three different sets of synthetic datasets were generated for evaluation. The parameters used for generating the synthetic data were consistent with the real data described in the next section: b = 500, 1000, 1500, 2000, 2500, 3000 s/mm2 , with respectively 9, 12, 17, 24, 34, and 48 non-collinear gradient directions, 20 × 20 voxels. Rician noise with level comparable to the real data was added to the synthetic data. The three sets of synthetic datasets model three different scenarios: (1) Single: WM with single fiber orientations + GM + CSF; (2) Crossing: WM with two crossing fiber directions + GM + CSF; and (3) Partial Volume: WM with two crossing fiber directions with partial volume of varying degree with GM + CSF. The voxel number ratio is 9:9:2 for WM:GM:CSF, giving 45% WM voxels, 45% GM voxels, and 10% CSF voxels. The ground truth synthetic data with the fiber orientations (FOs) and color-coded tissue volume fractions are shown on the far left of Fig. 1. Comparing the estimation results obtained using the baseline and the proposed method, we can see that the baseline method is easily confused by the different WM orientations, resulting in incorrect volume fraction estimates. Note that the GM and CSF volume fraction estimates are also affected by the WM errors. On the other hand, the proposed method performs consistently for the different datasets, confirming the fact that it is not sensitive to variation in WM configurations. We compute the mean absolute difference (MAD) between the estimated and the ground truth volume fractions. Table 1 shows the MAD values for the datasets, indicating that the proposed method yields estimates that are significantly closer to the ground truth. The baseline method, on the other, deviates significantly from the ground truth. Note that since the volume fraction is a normalized quantity, the estimation errors of one tissue type will affect others.

3.2 Real Data The DW images of an adult were acquired using a Siemens 3T Prisma MR scanner with the following protocol: 140 × 140 imaging matrix, 1.5 mm × 1.5 mm × 1.5 mm resolution, TE = 88 ms, TR = 2365 ms, 32-channel receiver coil, and multi-band factor 9, 12, 17, 24, 34, and 48 non-collinear directions respectively for b = 500, 1000, 1500, 2000, 2500, 3000 s/mm2 . A non-DW image with b = 0 s/mm2 was collected for every 24 images, giving a total of 6.

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Crossing

Single

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FOs

Ground Truth

Baseline

Proposed

Fig. 1 Synthetic Datasets. Synthetic datasets with different configurations and with/without partial volume effects. Different fiber orientations are simulated. Volume fractions are colored coded with blue for WM, green for GM, and red for CSF. The ground truth volume fractions and those estimated by the baseline and the proposed method are shown Table 1 MAD between estimated volume fraction and ground truth Method Dataset WM GM Baseline

Proposed

Single Crossing Mixture Single Crossing Mixture

0.6206 0.4411 0.4775 0.0050 0.0020 0.0430

0.5584 0.6512 0.2728 0.0043 0.0028 0.0403

CSF 0.0167 0.0237 0.0219 0 0 0

Figure 2 shows the volume fraction maps of the baseline and the proposed method. One can appreciate that with the proposed method a clearer separation of the different types can be obtained. This is apparent from the better contrast of the volume fraction maps given by the proposed method. The WM map given by the proposed method is also more homogeneous in terms of intensity. This is an indicator that the proposed method is less affected by the WM orientation distribution.

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GM

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Fig. 2 Volume Fractions. Volume fraction maps given by the baseline and the proposed method Proposed

Baseline

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1.2 1.0 Signal Value

Signal Value

1.0 0.8 0.6 0.4

0.8 0.6 0.4 0.2

0.2

0

0 0

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2000

3000

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2000

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Fig. 3 Basis. Plots of basis (normalized to 1 at b = 0 s/mm2 )

Figure 3 shows the plots for the basis vectors determined using the two methods. For the baseline method, the values (in W ) are averaged over each shell to obtain the curves. The curves are expected to decay with diffusion weighting due to the increase in signal attenuation. It can be observed from the figure that the proposed method yields results that are closer to this expectation.

4 Conclusion In this paper, we have shown that improved brain tissue segmentation can be improved by applying NMF on the spherical mean DMRI data. Specifically, we have demonstrated that, in contrast to Jeurissen et al.’s method, the proposed method is uncon-

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founded by the differences in fiber orientation distributions. The proposed method will hence allow tissue segmentation to be performed more robustly using DMRI data.

References 1. Yap, P.T., Zhang, Y., Shen, D.: Multi-tissue decomposition of diffusion MRI signals via 0 sparse-group estimation. IEEE Trans. Image Process. 25(9), 4340–4353 (2016) 2. Jeurissen, B., Tournier, J.D., Dhollander, T., Connelly, A., Sijbers, J.: Multi-tissue constrained spherical deconvolution for improved analysis of multi-shell diffusion MRI data. NeuroImage 103, 411–426 (2014) 3. Jeurissen, B., Tournier, J.D., Sijbers, J.: Tissue-type segmentation using non-negative matrix factorization of multi-shell diffusion-weighted MRI images. In: The Annual Meeting of the Internal Society for Magnetic Resonance in Medicine, vol. 23, p. 0349 (2015) 4. Lee, D.D., Seung, H.S.: Learning the parts of objects by non-negative matrix factorization. Nature 401(6755), 788–791 (1999) 5. Zdunek, R., Cichocki, A.: Blind image separation using nonnegative matrix factorization with gibbs smoothing. In: International Conference on Neural Information Processing, Springer, pp. 519–528 (2007) 6. Wang, J., Lai, S., Li, M., Lies, D.: Medical image retrieval based on an improved non-negative matrix factorization algorithm. J. Digit. Inf. Manag. 13(6) (2015) 7. Matloff, N.: Quick introduction to nonnegative matrix factorization (2016) 8. Rajapakse, M., Wyse, L.: Face recognition with non-negative matrix factorization. In: Visual Communications and Image Processing, International Society for Optics and Photonics, vol. 5150, pp. 1838–1848 (2003) 9. Hoyer, P.O.: Non-negative matrix factorization with sparseness constraints. J. Mach. Learn. Res. 5(Nov), 1457–1469 (2004) 10. Christiaens, D., Maes, F., Sunaert, S., Suetens, P.: Convex non-negative spherical factorization of multi-shell diffusion-weighted images. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, Springer, pp. 166–173 (2015) 11. Kaden, E., Kruggel, F., Alexander, D.C.: Quantitative mapping of the per-axon diffusion coefficients in brain white matter. Magn. Reson. Med. 75(4), 1752–1763 (2016) 12. Mairal, J., Bach, F., Ponce, J., Sapiro, G., Jenatton, R., Obozinski, G.: SPAMS: A SPArse Modeling Software, v2.3 (2012) 13. Mairal, J., Bach, F., Ponce, J., Sapiro, : G.: Online dictionary learning for sparse coding. In: International Conference on Machine Learning, ACM, pp. 689–696 (2009) 14. Mairal, J., Bach, F., Ponce, J., Sapiro, G.: Online learning for matrix factorization and sparse coding. J. Mach. Learn. Res. 11(Jan), 19–60 (2010)

A Closed-Form Solution of Rotation Invariant Spherical Harmonic Features in Diffusion MRI Mauro Zucchelli, Samuel Deslauriers-Gauthier and Rachid Deriche

Abstract Rotation invariant features are an indispensable tool for characterizing diffusion Magnetic Resonance Imaging (MRI) and in particular for brain tissue microstructure estimation. In this work, we propose a new mathematical framework for efficiently calculating a complete set of such invariants from any spherical function. Specifically, our method is based on the spherical harmonics series expansion of a given function of any order and can be applied directly to the resulting coefficients by performing a simple integral operation analytically. This enable us to derive a general closed-form equation for the invariants. We test our invariants on the diffusion MRI fiber orientation distribution function obtained from the diffusion signal both in-vivo and in synthetic data. Results show how it is possible to use these invariants for characterizing the white matter using a small but complete set of features. Keywords Diffusion MRI · Rotation invariant · Gaunt coefficients · Spherical harmonics

1 Introduction Brain tissue microstructure influences the diffusion Magnetic Resonance Imaging (MRI) signal in-vivo in multiple ways. Tissue microstructure in gray and white matter can be subdivided into two main categories: tissue composition and tissue orientation. Tissue composition corresponds to the heterogeneity of the neurites in the voxel, namely the presence and relative fraction of a certain neurite population in the voxel. Axonal diameter and intra-axonal signal fraction are two of the most studied tissue composition microstructural features in diffusion MRI. Tissue orientation, on the other hand, corresponds to the geometrical arrangement of the neurites in a given voxel. White matter axonal bundles (also known as “fiber bundles”), because of their high anisotropy, represent the key factor which influences the diffusion MRI signal M. Zucchelli (B) · S. Deslauriers-Gauthier · R. Deriche Athena Project-Team, Inria Sophia Antipolis - Méditerranée, Biot, France e-mail: [email protected] © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_7

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via tissue orientation. It is common practice in diffusion MRI to model the tissue orientation information in each voxel using the so-called fiber Orientation Distribution Function (fODF) [18]. The fODF represents the probability distribution function of the fibers, namely the probability of having a fiber oriented in a given direction. Such a distribution is commonly obtained from the diffusion signal by deconvolving it with a single fiber response which, in theory, should correspond to the tissue composition in the same location [3, 18, 21]. In diffusion MRI, fODF is modeled using its real symmetric Spherical Harmonics (SH) series expansion, and therefore, each fODF is completely characterized by its SH coefficients. Being a well defined numerical quantity, the idea of using the SH coefficients directly as a biomarker for tissue orientation is appealing but it suffers from the fact that even a small rotation of the same fODF will produce a different set of coefficients, which makes their use unsuitable for classification and clustering tasks. An intuitive solution to this problem is to use the SH coefficients to calculate Rotation Invariant Features (RIF) which, as the name implies, have values which are independent of the direction of the fODF. The first and most well-known fODF RIF which has been developed for diffusion MRI is the Generalized Fractional Anisotropy (GFA) [19]. GFA represents the fODF evolution of the diffusion tensor-derived Fractional Anisotropy (FA) with the advantage of being able to better represent crossing fibers. However, reducing the rich directional information present in the fODF into a single numerical value severely limits its descriptive power. Bloy and Verma [1] proposed to use the power spectrum of the SH coefficients to register diffusion MR images. The power spectrum provides a richer description with respect to GFA but is still incomplete with respect to the theoretical number of independent invariants obtainable. Ghosh et al. [6] and Papadopoulo et al. [13] proposed a method for recovering all 12 invariants obtainable from a 4th order tensor while Caruyer and Verma [2] derived a framework for obtaining all the algebraically independent invariants for any given order of SH. Such a framework initially derives all the polynomials invariant to a rotation of 1◦ with respect to the x and z axes, which are proven to be invariant to any additional rotation. After collecting all the possible invariants, a pruning algorithm is applied to the polynomials to reduce them to the minimal set of algebraically independent invariants. The main limitations of this technique is that the invariants cannot be expressed in closed-form and that it produces a huge number of polynomials to be tested for independence (more than 30,000 for a SH order of 6). The goal of this work is to propose a new framework for generating a set of invariants in closed-form (Sect. 2) with an elegant mathematical framework. Our solution is able to produce fewer polynomials to be tested for independence compared to previous published related work. In Sect. 3 we will present the synthetic and invivo fODF data which we will use to test our RIF. Results show the potential for these invariants to disambiguate complex fiber geometries using only a small set of invariants (Sect. 4).

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2 Theory 2.1 A New Set of Invariants for SH Given a spherical function f (u) defined on the three dimensional unit vector u we can represent it as a linear combination of SH as f (u) =

l ∞  

clm Ylm (u)

(1)

l=0 m=−l

where Ylm are the real SH functions [4, 7], and clm ∈ R are the SH coefficients. The coefficients clm are calculated as the inner product f and Ylm  clm =

S2

f (u)Ylm (u)du.

(2)

A rotation R applied to the SH functions can be written as RYlm (u) = Ylm (R −1 u) =

l 



Dml  ,m (R)Ylm (u)

(3)

m  =−l

with Dml  ,m (R) defined as 

Dml  ,m (R) = e−im α dml  ,m (β)e−imγ

(4)

where α, β, and γ are the Euler angles and dm  ,m is the Wigner d function. Therefore, we can define h(u) = R f (u) the function f rotated by R and whose SH expansion is given by l ∞   clm Ylm (R −1 u) h(u) = R f (u) = l=0 m=−l

=

l ∞   l=0

(5)



glm  Ylm (u).

m  =−l

The rotated SH coefficients for a given l are given by g

lm 

=

l 

clm Dml  ,m (R).

(6)

m=−l

A RIF of a Spherical Function f can be expressed as a function I for which I [ f (u)] = I [R f (u)], ∀R

(7)

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If f is expressed in terms of its SH expansion, we have that I [c] = I [g]

(8)

where c and g are the SH coefficients of f (u) and h(u), respectively. It is straightforward to see that the integral of f (u) represents a RIF, in fact: 

 I[ f ] =

f (u)du =

S2

S2

f (R −1 u)du

(9)

this equivalence follows the natural law that the area of a function does not change √  while rotating the function. In particular in this case I [ f ] = S 2 f (u)du = c00 4π, √ and therefore, normalizing by the constant factor 4π, the polynomial P(c) = c00 represents a first RIF for f . Similarly, we can prove that if we raise the function f to the power d, the integral of such a function is still a RIF. In fact  Id[ f ] =

 S2

[ f (u)]d du =

 S2

f (R −1 u)

d

du

(10)

Although it is an interesting RIF, I d [ f ] represents only a partial representation of f . In order to overcome this limitation we can define a new set of RIF from I d [ f ]. In particular, we observe that we can write I d [ f ] as Id[ f ] =

∞ 

···

l1 =0

= = =

∞ 

∞   d  S2

ld =0

···

⎡ ⎣

i=1

∞  

li 

l1 

l1 =0

ld =0

∞ 

∞ 

l1 

l1 =0

ld =0 m 1 =−l1

∞ 

∞ 

l1 =0

···

cli m i Ylmi i (u)⎦ du

m i =−li

S 2 m =−l 1 1

···

⎤

···

ld 

···

cl1 m 1 · · · cld m d Ylm1 1 (u) · · · Ylmd d (u)du

m d =−ld ld 

cl1 m 1 · · · cld m d

m d =−ld

Ild1 ...ld [ f ] =

ld =0



(11)

 S2

Ylm1 1 (u) · · · Ylmd d (u)du

Ild [ f ]

l

with l = [l1 . . . ld ]. Ild [ f ] can be evaluated as Ild1 ...ld [ f ] =

l1  m 1 =−l1

···

ld 

cl1 m 1 · · · cld m d G(l1 , m 1 | · · · |ld , m d )

(12)

m d =−ld

where G(l1 , m 1 | · · · |ld , m d ) represents the generalized Gaunt coefficient calculated as the integral of the product of d SH of the same argument [7]. We can also view Ild as a polynomial of degree d of the SH coefficients clm .

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Section 6 provides a proof that Eq. (12) generates rotation invariant features of the spherical function f (u) given the set of SH indices l. The function Ild shares some of the properties of the Gaunt coefficients, in particular it is zero for most combination of l and it is invariant to the permutations of the indices. In this work, we will refer to the set of all the non-zero invariants of maximal SH order lmax and degree d as Ildmax .

2.2 Example 1: Degree 2 Invariant-Polynomials In the case of polynomials of degree d = 2 and considering a maximal SH order lmax = 2, Ild corresponds to Il21 ,l2 [

f] =

l1 

l2 

cl1 m 1 cl2 m 2 G(l1 , m 1 |l2 , m 2 )

(13)

m 1 =−l1 m 2 =−l2

with l = [(0, 0), (0, 1), · · · (2, 2)] considering only the sorted couple in ascending order of (l1 , l2 ) because of the symmetry property of the Gaunt coefficients. ,m 2 . This G(l1 , m 1 |l2 , m 2 ) represents the integral of two SH which is equal to δll12,m 1 means that the invariant will be non-zero only when l1 = l2 and m 1 = m 2 . We can thus rewrite the degree two invariants as Il2 [ f ] =

l  

cl,m

2

(14)

m=−l

which corresponds to the power spectrum of the SH basis. The power spectrum is one of the most widely used invariants in pattern recognition application and has been recently introduced in diffusion MRI for tissue microstructure estimation [12].

2.3 Example 2: Degree 3 Invariant-Polynomials The invariants of degree 3 can be calculated as Il31 ,l2 ,l3 [ f ] =

l1 

l2 

l3 

cl1 m 1 cl2 m 2 cl3 m 3 G(l1 , m 1 |l2 , m 2 |l3 , m 3 )

(15)

m 1 =−l1 m 2 =−l2 m 3 =−l3

It is possible to show that Il31 ,l2 ,l3 [ f ] are actually equivalent to the bispectrum invariants proposed in [9–11] (results not shown due to space limitation). The main difference with respect to the bispectrum is that in this case we have the Gaunt coefficients obtained from the integral of three SH instead of the Clebsch-Gordan coefficients.

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It is possible to show that both Il21 ,l2 and Il31 ,l2 are not only invariants but they also form a set of algebraically independent invariants. The next subsection will provide the formal definition of algebraically independence and how to apply it in order to verify the independence of any set of Ildmax .

2.4 Algebraically Independence Definition 1 A set of m polynomials P1 , . . . , Pm ∈ R[X 1 , . . . , X n ] are called algebraically independent over the field R if ∀Q ∈ R[Y1 , . . . , Ym ] Q(P1 , . . . , Pm ) = 0 ⇐⇒ Q = 0 Example Given P1 = 2x 2 − 4x y + 2y 2 and P2 = x − y, the set [P1 , P2 ] is not algebraically independent. In fact, considering Q = 2P22 − P1 it leads to: 2(x − y)2 − (2x 2 − 4x y + 2y 2 ) = 0. An efficient way for verifying whether the invariants Ildmax are algebraically independent is the Jacobian theorem proposed by [5]: Theorem 1 The polynomials P1 , . . . , Pm ∈ R[X 1 , . . . , X n ] with m ≤ n are algebraically independent iff the Jacobian matrix (∂ Pi /∂ X j )1≤i≤m,1≤ j≤n has full rank. In our case, the Jacobian matrix corresponds to the partial derivatives of our invariants Ild with respect to the SH coefficients clm . The testing for full-rankness is performed taking advantage of the Schwartz-Zippel polynomial identity testing lemma [16, 20] which states that given a random vector of coefficients c, the Jacobian rank corresponds to the rank of (∂ Pi /∂ X j ) X =c with a probability that is proportional to the probability of extracting c randomly. In this work, we consider 100 random instances of clm in order to assess the independence of Ild with sufficient accuracy. This approach has already been used for the same purpose by [2], where the authors also theorized that the maximal number of independent coefficients is equal to the maximal number of SH coefficients used, n c , minus three, the degrees of freedom of the rotation. Table 1 show the number of invariants calculated for the symmetric SH basis [4]. In bold we highlight the number of algebraically independent invariants and we also add an asterisk if this number corresponds to n c − 3. For the complete SH basis we observed that using d = 4 we are able to obtain all the invariants up to lmax = 6, while for the symmetric SH basis our technique require d = 5 in order to obtain the 12th invariant for lmax = 4. Caruyer and Verma [2] were able to obtain all 12 invariants for lmax = 4 at d = 4 using a different technique, and we are also able to obtain the complete set of invariants at d = 4 if we include also the odd terms of the basis (see Table 2). It is possible that with our technique the 12th invariant can be expressed as a combination of two mixed even-odd invariants and appears as a function of purely even coefficients only at d = 5. Note that with our technique all the invariants at a certain degree include also all the 2 4 represents the same invariant as I0,0,2,2 ). Therefore, invariants of lower degree (i.e. I2,2

A Closed-Form Solution of Rotation Invariant Spherical Harmonic …

83

Table 1 Number of RIF for symmetric (even) SH basis given the maximum value of l and the degree. In bold we report the number of algebraically independent invariants, with an asterisk if the number corresponds to the maximum theoretical number of invariants d=1 d=2 d=3 d=4 d=5 lmax = 2 lmax = 4 lmax = 6

1 (1) 1 (1) 1 (1)

2 (2) 5 (3) 7 (4)

3 (3*) 7 (7) 13 (13)

4 (3*) 12 (11) 28 (25*)

5 (3*) 18 (12*) 49 (25*)

Table 2 Number of RIF given the maximum value of l and the degree. In bold we report the number of algebraically independent invariants, with an asterisk if the number corresponds to the maximum theoretical number of invariants d=1 d=2 d=3 d=4 lmax lmax lmax lmax lmax

=2 =3 =4 =5 =6

1 (1) 1 (1) 1 (1) 1 (1) 1 (1)

3 (3) 4 (4) 5 (5) 6 (6) 7 (7)

5 (5) 8 (8) 14 (14) 20 (20) 30 (30)

8 (6*) 17 (13*) 33 (22*) 57 (33*) 94 (46*)

in order to calculate all the algebraically independent invariants it is sufficient to consider only the degree d that provides the maximal number of invariants, without having to compute all the invariants at lower degree. For example, taking Table 1 first row, we can observe that we have two invariants at d = 2 of which one is a new invariant and the other is the same invariant that appears at d = 1. In practice, in order to obtain the number of new invariants appearing only at degree d we have to subtract the numbers of invariants of degree d − 1.

3 Materials and Methods Synthetic fODF dataset: We created a synthetic fODF dataset consisting of crossings of two delta functions on the sphere. We can obtain the SH coefficients of two delta functions with equal volume fraction centered around v1 and v2 , respectively, as cr ossing

cl,m

=

1 m 1 Y (v1 ) + Ylm (v2 ) 2 l 2

(16)

In this dataset, we will use the cosine of the crossing angle, v1 · v2 , as the reference value for the invariants. Since the maximum absolute value of our RIF is obtained when the considered fODF is a delta function (O D = 0 or v1 · v2 = ±1) we can use this value as a normalization factor for our invariants. In this work we will call the normalized fODF invariants Iˆld [ρ] to distinguish them from the unnormalized version.

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Human Connectome Project (HCP) dataset: In order to test our RIF in-vivo we considered one subject of the HCP diffusion MRI dataset [17]. The HCP acquisition scheme presents 18 b-value 0 s/mm2 volumes, and 90 volumes at b-value 1000, 2000, and 3000 s/mm2 , respectively. All the three shell were acquired using an unique gradient direction. HCP pulse separation time was set at  = 43.1ms and pulse width at δ = 10.6 ms. Each MRI volume consists of 145 × 174 × 145 voxels with a resolution of 1.25 × 1.25 × 1.25 mm3 .

4 Results Figure 1 shows the proposed algebraically independent RIF given a maximum SH order lmax = 4 for the synthetic fODF dataset, before (left) and after (right) the normalization by the maximum theoretical value. This dataset was generated by taking 1000 fODF directions v1 and v2 randomly, and the relative invariants was ordered according to the value of v1 · v2 . Because of the randomness of the orientation of the crossings, the smoothness of the curves itself can be view as an empirical proof of the rotation invariance property of our RIF. In fact, two adjacent points in the graph had been generated by crossings oriented, up to a rotation, in two completely different directions. If any of the Ild was not invariant its curve would have appear as a broken line in the graph. We also test the robustness of the invariants with respect to noise. We simulate diffusion signal arising from crossing fibers at 0, 30, 60, and 90◦ . We add Rician noise with a signal to noise ratio equal to 30, generating 100 noise realization per crossing angle. The fODF have been calculated using constrained spherical deconvolution algorithm with SH order equal to 4. Table 3 reports the mean and standard deviation of the twelve algebraically independent invariants. In general, our invariants result to be stable with values of standard deviation fifteen times lower than the mean in the worst case. Invariants including higher harmonics terms seem to present higher standard deviation (e.g l = (2, 2) is more stable than l = (4, 4)). This is probably due to the fact that higher harmonics tend to model signal noise more with respect to the lower ones that are too smooth to capture rapid signal changes. In order to test the invariants in-vivo we consider one subject of the HCP dataset. The main advantage of this dataset is the high spatial resolution which enables us to observe the fine changes of our RIF in the white matter. From the diffusion signal, a voxel-dependent response function was obtained using the multi-compartment model presented in [8]. Each response function has been used for calculating the fODF with constrained spherical deconvolution (lmax = 4) as in [21]. Figure 2 shows the 12 algebraically independent invariants after normalization for a coronal slice of the HCP subject. As for the synthetic data, the √ first invariant l = (0) is constant because the first SH coefficient c00 is always 1/ 4π in order to have the integral of the fODF equal to 1 [12]. The average values of the invariants generally decrease with the increase of each li and with the degree of the polynomials. For example, l = (2, 2) has an average of 0.12 while l = (2, 2, 2, 2, 4) has an average of only 0.0009 in white matter. Conversely, the contrast between white matter and gray

A Closed-Form Solution of Rotation Invariant Spherical Harmonic …

Iˆld

Ild 1.0

1.0 l=(0)

0.8 0.6

l=(0)

l=(2,2) l=(4,4) l=(2,2,2) l=(2,2,4) l=(2,4,4)

0.4 0.2

0.8

l=(2,2)

0.6

l=(2,2,2)

0.4

l=(4,4,4) l=(2,2,2,4)

0.2

l=(2,2,4,4) l=(2,4,4,4)

l=(4,4) l=(2,2,4) l=(2,4,4) l=(4,4,4) l=(2,2,2,4) l=(2,2,4,4) l=(2,4,4,4)

0.0

l=(4,4,4,4)

0.0

85

l=(2,2,2,2,4)

l=(4,4,4,4) l=(2,2,2,2,4)

−1.00 −0.75 −0.50 −0.25

0.00

v1 · v2

0.25

0.50

0.75

1.00

−1.00 −0.75 −0.50 −0.25

0.00

v1 · v2

0.25

0.50

0.75

1.00

Fig. 1 Unnormalized (left) and normalized (right) visual representation of the 12 algebraically independent invariants for lmax = 4 with respect to different crossing angles Table 3 Mean and standard deviation of the 12 algebraically independent invariants for lmax = 4 with respect to different crossing angles 0◦ 30◦ 60◦ 90◦ l = (0) l = (2, 2) l = (4, 4) l = (2, 2, 2) l = (2, 2, 4) l = (2, 4, 4) l = (4, 4, 4) l = (2, 2, 2, 4) l = (2, 2, 4, 4) l = (2, 4, 4, 4) l = (4, 4, 4, 4) l = (2, 2, 2, 2, 4)

1.00/0.000 0.89/0.012 0.67/0.021 0.85/0.017 0.73/0.017 0.63/0.021 0.54/0.026 0.69/0.019 0.60/0.022 0.51/0.025 0.45/0.028 0.66/0.022

1.00/0.000 0.75/0.012 0.38/0.014 0.64/0.016 0.45/0.012 0.30/0.012 0.19/0.012 0.38/0.013 0.28/0.012 0.17/0.010 0.15/0.011 0.34/0.014

1.00/0.000 0.39/0.009 0.23/0.012 0.13/0.008 0.09/0.006 0.04/0.004 0.02/0.009 -0.02/0.003 0.05/0.004 -0.00/0.002 0.05/0.005 0.03/0.003

1.00/0.000 0.23/0.007 0.46/0.016 -0.11/0.005 0.07/0.003 0.21/0.010 0.28/0.018 -0.03/0.002 0.06/0.003 0.06/0.003 0.21/0.014 0.02/0.001

matter generally increases with the degree. A similar trend can be observed in the contrast between single bundles and crossings in white matter. Some of our RIF present negative values in certain areas of the brain, in particular in fiber crossing regions. In our simulations we observed that negative values for certain invariants are associated with a characteristic crossing angle (e.g 90◦ crossing for l = (2, 2, 2)). However, our synthetic data is formed only of 2-fiber crossings, whereas crossings of three or even more fibers are common in the human brain. Therefore, a direct mapping of the value of the RIF to the fiber geometry is not trivial and will require more advanced mathematical tools.

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0.0

0.2

0.5

l = (2, 2)

0.8

1.0

0.00

0.09

l = (2, 2, 4)

-0.001

0.027

0.055

0.083

0.0106

0.0212

0.0318

0.27

0.36

0.00

0.04

l = (2, 4, 4)

0.111

l = (2, 2, 4, 4)

0.0000

0.18

l = (4, 4)

-0.010

0.010

0.029

0.049

-0.0025

0.0047

0.0118

0.0190

0.13

0.17

l = (4, 4, 4)

0.069

l = (2, 4, 4, 4)

0.0424

0.09

l = (2, 2, 2)

-0.002

0.012

0.027

0.042

0.0000

0.0072

0.0145

0.0217

0.000

0.053

0.105

0.158

l = (2, 2, 2, 4)

0.056

l = (4, 4, 4, 4)

0.0261

-0.052

-0.0087

0.0060

0.0207

0.0353

0.0500

l = (2, 2, 2, 2, 4)

0.0290

-0.00004 0.00819 0.01642 0.02466 0.03289

Fig. 2 All the 12 algebraically independent invariants calculated on the fODF of a coronal slice of one of the HCP subjects

5 Discussion and Conclusion In this work, we proposed a new framework for generating algebraically independent rotation invariant features for the diffusion MRI signal. One of the main results of our technique is that, conversely with respect to other works, we were able to provide a general closed-form for our invariants. The core mathematical tool of our formulation are the Gaunt coefficients originating from the integral of multiple SH. Gaunt coefficients also appear in the invariants for diffusion MRI proposed in [15], although using a completely different approach and without testing the resulting invariants for completeness or independence. Rotation invariant features are not only a subject of interest in diffusion MRI but also in several other fields including computer vision and pattern recognition [9, 10]. We believe that our framework will be advantageous for non-diffusion applications given the simplicity of the mathematical formulation of our RIF. Our mathematical formulation makes them very fast to compute even for large SH order compared to other techniques. Recent works [12, 14] show the importance of rotation invariant features such as the power spectrum for estimating tissue microstructural properties. We will focus

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our future works in trying to establish whether including the complete set of invariants will result in a more precise estimation of brain microstructure. Acknowledgements This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (ERC Advanced Grant agreement No 694665 : CoBCoM—Computational Brain Connectivity Mapping). Data were provided by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University.

6 Relation Between Signal SH Coefficients and fODF SH Coefficients Ild [ f ] is rotation invariant iff Ild [ f ] = Ild [R f ] = Ild [h]. Given the rotated SH expansion h(u) we can calculate Ild [h] as Ild1 ...ld [h] =

=

=

=

  d S 2 i=1

  d S 2 i=1

  d S 2 i=1

  d

=

li 

⎣ ⎡

⎡

li 

li 

⎣

=

···

cli m i Dmli  ,m i (R)Yli (u)⎦ du i

li  m i =−li

⎤ m i

Dmli  ,m i (R)Yli (u)⎦ du i

⎤

cli m i Ylm (R −1 u)⎦ du ld 

···

···

cl1 m 1 · · · cld m d Ylm1 1 (R −1 u) · · · Ylmd d (R −1 u)du

m d =−ld ld  m d =−ld

m 1 =−l1

m i

m i =−li

m 1 =−l1 l1 

cli m i

m i =−li

m 1 =−l1

l1 

⎤

li 

m i =−li m i =−li

l1  S2

gli m i Yli (u)⎦ du

li 

⎣ ⎡

⎤ m i

m i =−li

⎣

S 2 i=1

 =

⎡

ld 

cl1 m 1 · · · cld m d

 S2

Ylm1 1 (R −1 u) · · · Ylmd d (R −1 u)du

cl1 m 1 · · · cld m d G(l1 , m 1 | · · · |ld , m d )

m d =−ld

= Ild1 ...ld [ f ] which proves the invariance property Ild [ f ] = Ild [R f ].

(17)

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Orientation-Dispersed Apparent Axon Diameter via Multi-Stage Spherical Mean Optimization Marco Pizzolato, Demian Wassermann, Rachid Deriche, Jean-Philippe Thiran and Rutger Fick

Abstract The estimation of the apparent axon diameter (AAD) via diffusion MRI is affected by the incoherent alignment of single axons around its axon bundle direction, also known as orientational dispersion. The simultaneous estimation of AAD and dispersion is challenging and requires the optimization of many parameters at the same time. We propose to reduce the complexity of the estimation with an multi-stage approach, inspired to alternate convex search, that separates the estimation problem into simpler ones, thus avoiding the estimation of all the relevant model parameters at once. The method is composed of three optimization stages that are iterated, where we separately estimate the volume fractions, diffusivities, dispersion, and mean AAD, using a Cylinder and Zeppelin model. First, we use multi-shell data to estimate the undispersed axon micro-environment’s signal fractions and diffusivities using the spherical mean technique; then, to account for dispersion, we use the obtained micro-environment parameters to estimate a Watson axon orientation distribution; finally, we use data acquired perpendicularly to the axon bundle direction to estimate the mean AAD and updated signal fractions, while fixing the previously estimated diffusivity and dispersion parameters. We use the estimated mean AAD to initiate the following iteration. We show that our approach converges to good estimates while being more efficient than optimizing all model parameters at once. We apply M. Pizzolato (B) · J.-P. Thiran Signal Processing Lab (LTS5), École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland e-mail: [email protected] D. Wassermann Parietal, Inria, CEA, Université Paris-Saclay, Palaiseau, France R. Deriche Athena, Inria, Université Côte d’Azur, Sophia Antipolis, France J.-P. Thiran University Hospital Center (CHUV) and University of Lausanne (UNIL), Lausanne, Switzerland R. Fick TheraPanacea, Paris, France © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_8

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our method to ex-vivo spinal cord data, showing that including dispersion effects results in mean apparent axon diameter estimates that are closer to their measured histological values. Keywords Diffusion MRI · Axon diameter · Dispersion · Spherical mean

1 Introduction Axon diameter estimation in white matter (WM) tissue is one of the most challenging tasks in Diffusion MRI (dMRI), due to demanding acquisition requirements and to the presence of confounding factors, such as axon orientational dispersion [1]. These estimates can be improved by using complex models that account for dispersion [1]. However, the optimization of these models can be computationally expensive, since the presence of local minima often requires using global optimization strategies over a huge parameter space. Moreover, the degeneracy of the solution space further complicates the problem [2]. In this work, we propose a method inspired by alternate convex search [3] that decomposes the estimation problem into three simpler subproblems, thus reducing the complexity while converging to desired results. The knowledge of axon diameters in the WM tissue is important for a variety of applications, spanning from anatomical to functional imaging. In dMRI, however, based on modeling and physical considerations one could often compute estimates of an apparent axon diameter (AAD). The AAD can be estimated by acquiring data using strong diffusion sensitizing gradients directed perpendicularly to the axons, in combination with multiple diffusion times [4]. In some cases, a 3D acquisition is added to compensate for eventual misalignment with the principal axons bundle direction [5]. Typically, multi-compartment models composed of intra-axonal Cylinder and extra-axonal Gaussian models are used to estimate the AAD from this data [5–9]. The parameter estimation is confounded by the presence of axon dispersion around the principal diffusion direction [1]. However, the simultaneous estimation of both AAD and dispersion involves fitting many model parameters in a space that has several local minima. Classical approaches to find the global minimum in these situations use methods like brute force or differential evolution based optimization [10]. However, the computational cost of fitting models such as the one considered in this work, with many non-linear parameters, quickly becomes prohibitively expensive. The method we propose, allows for the estimation of orientation-dispersed axon diameters with reduced computational complexity. In our proposed multi-stage spherical mean (MSSM) approach, we consider a two compartments model composed of an impermeable Cylinder [11] and a Gaussian Zeppelin, representing the intra- and extra-axonal diffusion micro-environments. This model is then dispersed over the sphere with a Watson distribution to represent orientational dispersion [12, 13]. The complete model has many free, mostly nonlinear parameters: the mean apparent axon diameter, intra/extra signal fractions, parallel and perpendicular diffusivities, angular orientation, and dispersion. We propose

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to estimate these parameters in three main stages. In the first stage, we use multi-shell (MS) data to estimate the micro-environment model parameters with the spherical mean of each shell, in analogy with Multi-Compartment Microstructure Diffusion Imaging (MC-MDI) [14]. The signal fractions and the diffusivities estimated in this first stage are used to estimate the parameters of the Watson distribution from the MS data samples [12, 14], i.e. the orientation and the dispersion. Finally, the previous estimates are used to fix some of the corresponding parameters of the complete dispersed model which is fitted using the perpendicularly acquired data in order to estimate the mean AAD, an updated version of the perpendicular diffusivity, and the intra/extra-axonal compartment signal fractions. The newly estimated AAD and perpendicular diffusivity are fixed as parameters of the micro-environment model of the first stage, and the process is iterated until convergence. The method is implemented using the open-source Dmipy framework for reproducible microstructure research1 [15, 16], which facilitates the concatenation and construction of multi-compartment models in a parallelized and simple manner. We validate the approach with simulations and on a spinal cord diffusion MRI dataset with histology [17, 18]. Results show that the method accurately captures dispersion improving AAD estimates over a conventional non-dispersed estimation, showing an improved spatial correlation with histology.

2 Theory and Methods Orientation dispersion can be taken into account by defining a microscopic axonal model kernel, which is then convolved with a Watson distribution to produce the orientation-dispersed axon model [13]. This is iteratively fitted to MS and perpendicular data with a three-stage procedure, where at each stage the parameter space is constrained by earlier estimates, and where estimates are updated at each iteration. We first present the model, then describe the proposed fitting procedure, and provide a description of the adopted datasets.

2.1 Orientation-Dispersed Axon Model The non-dispersed kernel model describes the microscopic axonal environment, and is composed of a restricted and a hindered diffusion compartments [4, 6]. The restricted compartment is described by a Gaussian Phase Approximation (GPA) cylinder [11] with diameter a, whereas the hindered compartment is described by a zeppelin [19]. Note that the choice of a zeppelin implies orientational dependence of

1 https://github.com/AthenaEPI/dmipy.

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the hindered compartment, corresponding to anisotropic diffusion profiles. The nondispersed kernel is then convolved with a Watson distribution to give the following formulation for the signal attenuation ker nel

 ⎤          E(·) = W (κ, µ) ∗S2 ⎣ f h E h (·|λ⊥ , λ ) + fr Er (·|λ , a)⎦       Watson

 ⎡



Zeppelin

Hindered Extra-Axonal

GPA cylinder

(1)

Intra-Axonal

where “·” stands for the set of experimental variables g, δ,  including the gradient vector, pulse gradient duration, and separation of a PGSE sequence [20], µ is the main orientation in spherical coordinates, κ is the concentration parameter of the Watson distribution W (·), λ ≥ λ⊥ are the undispersed parallel and perpendicular diffusivities of the microscopic axonal environment, and f h , fr are the corresponding hindered and restricted signal fractions, with fr = 1 − f h . Note that both the hindered E h (·) and restricted Er (·) signal attenuations consider the full 3D signal, where the direction is determined by the Watson distribution.

2.2 Multi-Stage Spherical Mean (MSSM) Optimization We propose a multi-stage method with three optimization stages for estimating the parameters a, κ, µ, λ , λ⊥ , and fr of the model presented in Eq. 1. At each stage, we estimate a different subset of parameters while fixing the others to initial values or to estimates obtained with a previous estimation. In this way, we obtain a cascade of simpler fittings that are computationally more efficient, and possibly less degenerate, than the global problem. Stage 1: estimating diffusivities using shells spherical mean. The first step of the procedure uses the spherical mean of each shell of the MS dataset to estimate the parameters of the micro-environment without the influence of dispersion [14]. The kernel in Eq. 1 is fitted by accounting for the spherical means of three or more shells i, s¯ (bi ), which allows estimating the signal fraction fr (and f h ), λ , and λ⊥ , regardless of the presence of dispersion fˆr , λˆ  , λˆ ⊥ = argmin ¯s (b) − (1 − fr )e¯h (b, λ , λ⊥ ) − fr e¯r (b, λ , a)22 fr ,λ ,λ⊥

(2)

where e¯h (·) and e¯r (·) are the hindered and the restricted spherical means of the corresponding hindered and restricted compartments, and where b is the b-value. Note that e¯r is the dispersion-invariant signal spherical mean of a GPA cylinder, which we compute by approximation with spherical harmonic bases. The cylinder diameter a can initially be fixed to a large value, e.g. 6μm, or to zero which would correspond to calculating the spherical mean signal of a stick.

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Stage 2: estimating orientation and dispersion. In the second stage, we consider the full model in Eq. 1, where we fix ( fr , λ , λ⊥ ) to those estimated in the previous stage with Eq. 2. This time, we optimize for the parameters of the Watson distribution, W (κ, µ) [14], minimizing the error norm of the model over the multi-shell diffusionweighted images (DWIs) κ, ˆ µ ˆ = argmin S(bn)/S0 − E(κ, µ|bn, fˆr , λˆ  , λˆ ⊥ )22 κ,µ

(3)

where S(bn)/S0 is the measured signal attenuation with S0 being the non attenuated ˆ µ ˆ is used to constrain the fitting signal. The estimated set of parameters λˆ  , λˆ ⊥ , κ, of the model in Eq. 1 for the next optimization stage, whereas fˆr may be used as initial guess when it applies. Stage 3: estimating the diameter using a perpendicular acquisition. The third stage optimization involves minimizing the error norm with respect to the signal attenuation measured along the perpendicular direction n⊥ . For this reason, this is the most suitable acquisition for estimating the diameter, a, and updating the value of λˆ ⊥ , and of fˆr a, ˆ λˆ ⊥ , fˆr = argmin S(bn⊥ )/S0 − E(a, λ⊥ , fr |bn⊥ , κ, ˆ µ, ˆ λˆ  )22 a,λ⊥ , fr

(4)

where now all of the parameters of the model in Eq. 1 have been estimated. In the following iteration, the newly estimated values aˆ and λˆ ⊥ are now fixed in Eq. 2. The three stages are repeated until convergence.

2.3 Implementation, MRI and Histology Data The dataset consists of a four shell DWIs collection (796 samples) and two radial acquisitions (1791 samples) with gradient direction perpendicular to the axon bundle axis of an ex-vivo cat spinal cord [17], which is available at the White Matter Microscopy Database [18]. Maximum gradient strength for the MS and perpendicular acquisitions were G ≈ 0.6 T/m and G ≈ 0.85 T/m respectively, with diffusion time τ = 29 ms for the MS and τ ∈ {6,12,17,22,27,32,37,39} ms for the perpendicular acquisitions. Note that the data was acquired with a different echo-time for each diffusion time, which requires making some assumptions as discussed in Sect. 4. The synthetic datasets were generated using the same acquisition scheme for consistency. Mean axon diameter histology is also present, already in MRI space. In all optimization stages we used the global MIX optimizer [10], made exception for stage 2, where differential evolution [21] was employed. All implementations were done and can be reproduced in the Dmipy software framework for reproducible microstructure research [15, 16].

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3 Experiments and Results To validate the approach, we performed experiments on synthetic and acquired data. In all experiments, we initialized the MSSM method with a “large” mean diameter of 6 µm and let it converge in a fixed number of iterations. Results show that the proposed algorithm reaches convergence in few iterations. Moreover, obtained AAD estimates in the presence of various levels of dispersion are more precise compared to when dispersion is not modeled, as expected. The non-dispersed technique, consisting on fitting only the kernel of the model in Eq. 1 with a ball instead of a zeppelin [4, 5], was informed of the angular axon bundle main direction, µ, estimated with stages 1 and 2 of the proposed method. For this reason, differences in the comparison only account for dispersion. Moreover, we apply our method to the cat spinal cord dataset, where we report mean apparent axon diameter estimates closer to histological values and with comparable or better spatial correlation with respect to the non-dispersed case.

3.1 Synthetic Results We generated the synthetic signal attenuation for a grid of orientation dispersion index (ODI) values (related to κ [1]), ODI ∈ {0.10, 0.15, 0.20, 0.25}, and cylinder diameters, a ∈ {1, 2, 3, 4, 5, 6} µm. For the experiment, we set µ = (0, 0), λ = 1.1e − 9 m2 /s, λ⊥ = 0.8λ , and fr = 0.6. Figure 1 reports the AAD estimation absolute error obtained with the proposed method with respect to the ground truth, for some iterations of the algorithm. We find that for smaller diameters the error is larger. This is expected since dMRI is less sensitive to small diameters at these

Fig. 1 Convergence of the proposed approach for a grid of parameters (diameter,ODI) over iterations (first three images). Intensities represent under (blue) and over-estimation (red). On the right, the error of the non-dispersed technique

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Fig. 2 Convergence of diameter and ODI estimates for proposed method over iterations. On the right, means and standard deviations of axon diameter estimates as function of the SNR, for the zeroth iteration, and at convergence

gradient strengths. For larger diameters and ODIs, the method converges to the correct estimates, often in just two iterations. As expected, when not accounting for dispersion, diameters are overestimated up to 3 µm (Fig. 1, right). We more clearly show the convergence of the method in the left and central plots of Fig. 2. We also perform an evaluation of the robustness to noise for different signal-to-noise ratios, SNR = 1/σ, with σ being the standard deviation of the real and imaginary Gaussian distributions generating the added Rician noise. This evaluation is illustrated in the right plot of Fig. 2, which reports the mean and standard deviation of the absolute error of the estimated diameter for the case of a 3 µm apparent axon diameter with dispersion index ODI = 0.15. The values were obtained after repeating the MSSM method with 100 different noise realizations for each SNR, and were reported for the zeroth iteration (yellow), and at convergence (blue). The plot shows that the estimation of axon diameters converges to the right values fot SNR > 30. We believe that the suboptimal convergence at SNR = 20, 30 is due to the Rician bias which is not taken into account in the optimization. With respect to computational efficiency we find that one iteration of our MSSM approach takes up to 10s, whereas a naive global implementation using MIX takes about 2 min for SNR = 20 on a Intel® Core™ i7-3840QM with 32 Gb of RAM.

3.2 Histological Validation and Comparison Figure 3 reports the histological mean AAD maps, and the corresponding maps calculated without accounting for dispersion and for the proposed method. In the latter case, we also report the estimated dispersion index map, i.e. the ODI. The AAD map computed with the proposed method reports values clearly closer to the histology compared to the case in which dispersion is neglected. This confirms findings in the synthetic experiments. In fact, the ODI map shows a medium level of dispersion almost everywhere that the proposed method is able to capture, thus rendering apparent axon diameter estimates that are more coherent with histology. We also computed the spatial correlation between the axon diameter map obtained with the proposed method, or with the non-dispersed one, and the histology. Results that

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Fig. 3 Mean apparent axon diameter maps reported from histology, and calculated with the nondispersed and with the proposed method. Right, the estimated orientation dispersion index for the proposed method

consider axons with histological diameter in range 2–5 µm are shown in Fig. 4. We note an improved spatial correlation and a higher dynamic range of diameter estimates obtained with the proposed method. In the same figure, we report the difference between histological and computed AAD separately for group of axons with histological diameter in range [1,2], [2–3], [3–4], and [4–5] µm, which confirms that accounting for dispersion renders estimates closer to histology.

4 Discussion, Limitations, and Future Directions This work presents a new method for estimating apparent axon diameters while accounting for orientational dispersion by means of a spherical mean-based multistage search approach. Our method reduces the complexity of the estimation compared to the naive global optimization while showing convergence. Results on ex-vivo data showed that the proposed approach was able to capture dispersion and render AAD estimates better aligned with histological values. The presented method however makes some assumptions that lead to some limitations. From the point of view of modeling, we have neglected the extra-axonal diffusion time dependence [22–24]. Although assuming only intra-axonal time dependence might be a good approximation at very high gradient strengths, depending on the combination of (δ, ), at lower diffusion sensitization regimes it could possibly lead to a wrong description of the signal. In the future, we would like to account for this phenomenon by accordingly changing the extra-axonal model [22]. As a procedural limitation, we mention that the estimation of the signal fraction and diffusivities (parallel and perpendicular) obtained using shells spherical mean accounts only for the diffusion time (more generally for δ and ) of the MS acquisition. As a consequence, the parameters estimated at stage 1 might not be coherent with those of stage 3 where the model parameters should actually explain time dependence with respect to δ and . This issue is partially mitigated for the signal fraction and the perpendicular diffusivity, since they are re-estimated at stage 3, but it is present for the parallel diffusivity and for the orientation parameters propagated through stage

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Fig. 4 The absolute AAD error estimated w.r.t. histology for the proposed (light blue) and the non-dispersed (red) methods, for increasing ranges of histological axon diameters. On the right, the correlation of MRI-based axon diameter estimates with histological values of axons with diameter in range 2–5 µm

2. In our synthetic results, however, we could not directly observe the effects of this phenomenon even if a more accurate investigation should be performed. This limitation could be prevented with a dataset containing MS data at different diffusion times, since the spherical mean of the cylinder, in the case of the proposed model, is dependent on diffusion time. Alternatively, in acquisitions like the one used for this work, we speculate that it could be possible to use the parameters estimated at stage 1 and 2 as initial guess for performing a gradient-descent non-linear fitting at stage 3 where all the parameters could be estimated at once. Also, the proposed processing assumes that the transverse relaxation times of the intra- and extra-axonal compartments are the same. Where this assumption did not hold, then relaxation should be modeled. This would be particularly relevant in datasets like the one used in this work, which includes different echo-times. In general, we point out that several choices have been made in this work that could be changed and/or improved upon. For instance, we decided to estimate λ⊥ , although initial tortuosity constraints could be imposed, as for the MC-MDI model [14], thus reducing the number of parameters. Moreover, the initialization value of the diameter could be fixed with a previous non-dispersed fitting, and a step-size could be introduced to improve the MSSM convergence rate. At the same time, the use of global optimizers might not be necessary at each step/iteration. In addition to proposals made above, future work includes performing a more thorough comparison with global optimization results and investigating more deeply on the convergence properties of the proposed approach.

5 Conclusion In this work, we propose the use of spherical mean data to reduce the computational complexity of apparent axon diameter estimates from a compartmental model. Despite the above mentioned limitations, the adopted strategy effectively splits the

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estimation of a complex compartmental model into several stages leading to simpler and faster estimation steps. Further investigation is required to improve the method along this line. In fact a similar approach, eventually applied to time-dependent multishell data, might reveal helpful for obtaining reliable estimates in human datasets, especially where the richness of the data used for this work may not be achievable. Acknowledgements This work is supported by the Swiss National Science Foundation under grant number CRSII5_170873 (Sinergia project) and by the ERC Advanced Grant agreement No. 694665 (CoBCoM).

References 1. Zhang, H., et al.: Axon diameter mapping in the presence of orientation dispersion with diffusion MRI. NeuroImage 56(3), 1301–1315 (2011) 2. Jelescu, I.O., et al.: Degeneracy in model parameter estimation for multicompartmental diffusion in neuronal tissue. NMR Biomed. 29(1), 33–47 (2016) 3. Gorski, J., et al.: Biconvex sets and optimization with biconvex functions: a survey and extensions. Math. Method Oper. Res. 66(3), 373–407 (2007) 4. Alexander, D.C.: A general framework for experiment design in diffusion MRI and its application in measuring direct tissue-microstructure features. MRM (2008) 5. Alexander, D.C., et al.: Orientationally invariant indices of axon diameter and density from diffusion MRI. NeuroImage 52(4), 1374–1389 (2010) 6. Assaf, Y., et al.: AxCaliber: a method for measuring axon diameter distribution from diffusion MRI. MRM, 1347–1354 (2008) 7. Huang, S.Y., et al.: The impact of gradient strength on in vivo diffusion MRI estimates of axon diameter. NeuroImage 106, 464–472 (2015) 8. De Santis, S., et al.: Including diffusion time dependence in the extra-axonal space improves in vivo estimates of axonal diameter and density in human white matter. NeuroImage 130, 91–103 (2016) 9. Daducci, A., et al.: Accelerated microstructure imaging via convex optimization (AMICO) from diffusion MRI data. NeuroImage 105, 32–44 (2015) 10. Farooq, H., et al.: Microstructure imaging of crossing (MIX) white matter fibers from diffusion MRI. Nature Sci. Rep. 6, 38927 (2016) 11. Vangelderen, P., et al.: Evaluation of restricted diffusion in cylinders: Phosphocreatine in rabbit leg muscle. J. Magn. Reson. Ser. B 103(3), 255–260 (1994) 12. Kaden, E., et al.: Parametric spherical deconvolution: inferring anatomical connectivity using diffusion MR imaging. NeuroImage 37(2), 474–488 (2007) 13. Zhang, H., et al.: NODDI: practical in vivo neurite orientation dispersion and density imaging of the human brain. NeuroImage 61(4), 1000–1016 (2012) 14. Kaden, E., et al.: Multi-compartment microscopic diffusion imaging. NeuroImage 139, 346– 359 (2016) 15. Fick, R., et al.: Dmipy: an open-source framework to improve reproducibility in Brain Microstructure Imaging. In: HBM (2018). https://github.com/AthenaEPI/dmipy 16. Fick, R., et al.: Dmipy: an open-source framework for reproducible dMRI-based microstructure research (Version 0.1). Zenodo (2018). https://doi.org/10.5281/zenodo.1188268 17. Duval, T., et al.: Validation of quantitative MRI metrics using full slice histology with automatic axon segmentation. In: ISMRM, p. 928 (2016) 18. Cohen-Adad, J., et al.: White matter microscopy database (2017). https://doi.org/10.17605/ OSF.IO/YP4QG

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19. Panagiotaki, E., et al.: Compartment models of the diffusion MR signal in brain white matter: a taxonomy and comparison. NeuroImage 59(3), 2241–2254 (2012) 20. Tanner, J.E., Stejskal, E.O.: Restricted self-diffusion of protons in colloidal systems by the pulsed-gradient, spin-echo method. JCP 49(4), 1768–1777 (1968) 21. Storn, R., Price, K.: Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optim. 11, 341–359 (1997) 22. Burcaw, L.M., et al.: Mesoscopic structure of neuronal tracts from time-dependent diffusion. NeuroImage 114, 18–37 (2015) 23. Fieremans, E., et al.: In vivo observation and biophysical interpretation of time-dependent diffusion in human white matter. NeuroImage 129, 414–427 (2016) 24. Lee, H.H., et al.: What dominates the time dependence of diffusion transverse to axons: intra-or extra-axonal water? NeuroImage (2017)

Part II

Machine Learning for Diffusion MRI

Current Applications and Future Promises of Machine Learning in Diffusion MRI Daniele Ravi , Nooshin Ghavami , Daniel C. Alexander and Andrada Ianus

Abstract Diffusion-Weighted Magnetic Resonance Imaging (DW-MRI) explores the random motion of diffusing water molecules in biological tissue and can provide information on the tissue structure at a microscopic scale. DW-MRI in used in many applications both in the brain and other parts of the body such as the breast and prostate, and novel computational methods are at the core of advancements in DWMRI, both in terms of research and its clinical translation. This article reviews the ways in which machine learning and deep learning is currently applied in DW-MRI. We will also discuss the more traditional methods used for processing diffusion MRI and the potential of deep learning in augmenting these existing methods in the future. Keywords Diffusion-weighted · MRI · Machine learning · Deep learning

1 Introduction to Diffusion MRI DW-MRI probes the random motion of diffusing water molecules in biological tissue [1]. Since the diffusion pattern is influenced by the presence of cellular membranes and their configuration, DW-MRI is an indirect probe of tissue structure at the microscopic scale [2]. Standard DW-MRI measurements are made by applying a pair of magnetic field gradients with opposite polarities, usually referred to as diffusion gradients, which encode displacement along their direction [1]. In DW-MRI contrast, the larger the molecular displacement, the more the signal is attenuated. Thus, tissues with highly packed cells, such as some tumours, appear bright, and areas where water diffuses freely, such as cerebral spinal fluid, D. Ravi (B) · N. Ghavami · D. C. Alexander · A. Ianus University College London, London, UK e-mail: [email protected]

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appear dark [3, 4]. DW measurements are often described in terms of their gradient orientation and b-value, which reflects the diffusion weighting power and depends on the gradient strength, duration and diffusion time [2]. The simplest DW-MRI technique is mapping the Apparent Diffusion Coefficient (ADC), which requires at least two measurements, with and without diffusion gradients. The contrast in an ADC map is inverted from the DW-MRI contrast. Thus areas of low ADC values correspond to tissue with tightly packed cells, while high ADC values correspond to areas where molecules can diffuse more. The simple ADC model assumes isotropic diffusion, thus it is often used in stroke and cancer imaging [3, 4]. In order to probe neural tissue, which is highly anisotropic in white matter, DWMRI techniques need to account for anisotropic diffusion. Thus, a methodology commonly applied for brain imaging is Diffusion Tensor Imaging (DTI) [5], which is an extension of the diffusion coefficient in three dimensions, and requires at least 6 DW measurements with non-colinear gradient directions to be estimated. DTI can report on the main fibre orientation and provides scalar metrics which reflect diffusion properties of the tissue, for instance, axial and radial diffusivity or fractional anisotropy. DTI is employed in a wide range of applications involving brain [6] and central nervous system [7] imaging. Such applications include mapping white matter tracts [8], studying development [9] and ageing [10] as well as investigating various diseases, such as, Multiple Sclerosis (MS), Alzheimer’s disease, diffuse axonal injury, etc [11, 12]. Although DTI is simple and robust, it cannot describe multiple fibre populations and/or diffusion in a more complex environment with restricting barriers [13]. Over the last decades, a significant effort has been made to improve the measurement protocols and data analysis tools for DW-MRI in order to have better sensitivity and specificity to various tissue characteristics, while keeping the acquisition time clinically feasible [14–16]. Novel computational methods, alongside improvements in scanner capabilities, are at the core of advancements in DW-MRI, both in terms of research and its clinical translation. The rest of the paper will look at the different methods currently used for acquiring, processing and applying DW-MRI, with a focus on machine learning which is becoming an important tool used to efficiently solve computational problems in this wide range of applications.

2 Mathematical Versus Computational Modelling Current research on DW-MRI includes the development of methods for the entire imaging pipeline, for instance designing better acquisitions [17], image reconstruction [18], microstructure parameter mapping [19, 20], image quality transfer [21], tractography [22], parcellation [23], detection of lesions [24], monitoring treatment response [25], etc. State-of-the-art methods on DW-MRI can be divided into two main categories: those based on mathematical modelling and the others based on

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Fig. 1 Main models used in the processing and analysis of DW-MRI

(a)

(b)

Fig. 2 a Distribution of published papers in the field of DW-MRI. b Percentage of increased publications with respect to 2010. The results are obtained from Google Scholar; the search phrase is defined using the key word DW-MRI plus the name of popular models in the correlated category e.g., “DW-MRI” “Deep learning” OR CNN OR Autoencoder

machine learning, as shown in Fig. 1. Usually, mathematical modelling exploits a set of assumptions or theoretical foundations [26, 27] to model some physical property of the DW-MRI. Instead, the methods in the latter category are data-driven computational methods that model the output from its input while making the smallest number of assumptions possible.

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Table 1 Table containing references and applications that use machine learning and deep learning for DW-MRI Class Target Method Task References Machine learning

Deep learning

Breast

Linear regression

Lesions classification

[24]

Rectum Brain

Linear regression Linear regression

[30] [31]

Brain Brain Brain Brain Brain Brain Brain Brain

RF RF RF RF RF RF Bayesian model Bayesian model

Brain Brain

SVM SVM

Brain Prostate Prostate Brain

SVM CRF RVM ANN

Malignant tumour classification Age-related microstructural differences Permeability map Assess biomarkers correlation Investigate brain changes Tracrography Lesion classification Image quality transfer Estimate microstructural properties Investigate uncertainty in tractography Survival time prediction Classification of high angular resolution Decoding gender dimorphism Cancer localization Cancer localization Predicting ischemic infarction

Brain Brain Brain Brain Brain Brain Rectum Brain Prostate Brain Kidney Brain Brain Prostate Brain Kidney

ANN DNN DNN DNN DNN CNN CNN CNN CNN 3D-CNN U-net Bayesian CNN CNN Deep autoencoder Deep autoencoder Deep autoencoder

[49] [50, 51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65, 66]

Brain Brain Prostate

VAE GAN GAN

Parameter mapping Image acquisition Permeability estimation Survival time prediction Cancer detection Estimation of fiber orientations Cancer detection White matter tract segmentation Cancer segmentation Survival time prediction Image registration Image quality transfer Image quality transfer Cancer grade groups Spatio-temporal denoising Classification of renal rejection types Detection of q-space abnormalities Spatial super-resolution Semantic segmentation

[20, 32] [34] [35] [36] [37] [21, 38] [19] [42] [43] [44] [45] [46] [47] [48]

[67] [68] [69]

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In general, mathematical models are used when the process that needs to be described is partly or completely known (e.g. in the case of image reconstruction or mapping parameters in DTI). However, sometimes accurate mathematical models that express relationships to the DW-MRI signal are intractable. In such situations, as well as when a more complex and largely unknown phenomenon is under examination, computational methods and especially machine learning methods are more adequate. In particular, in the last few years, a sub-category of machine learning called deep learning has become very popular [28]. The theoretical principles behind deep learning are based on the Artificial Neural Network (ANN) concepts. The main difference between the basic ANN and advanced deep learning is that the latter exploits more hidden neurons, more layers and new training paradigms to process a larger amount of data. This results in an effective high-level abstraction enabling automatic feature extraction, which otherwise would have to be done by explicitly deriving a set of hand-crafted features before applying the model. Rapid improvements in computational power, fast data storage, and parallelisation have contributed to making deep learning very popular. This also applies to the field of DW-MRI and is confirmed from our analysis shown in Fig. 2. In Fig. 2a we can see that the number of papers published today exploiting machine learning or deep learning methods is comparable to the number of papers based on mathematical modelling, and, as explicitly illustrated in Fig. 2b, since 2010 their numbers increased much faster compared to the papers based on mathematical approaches which instead have remained approximately unaltered. For this reason, in this chapter, we focus our analysis mainly on methods based on machine and deep learning, (i.e. the orange section of Fig. 1) providing a comprehensive up-to-date overview of the current state-of-the-art in DW-MRI. Both the machine learning and deep learning parts will be subdivided into different methods, within which a variety of applications will be described, as summarised in Table 1.

3 Standard Machine Learning Models In this section, we focus our analysis on the main machine learning methods used for processing DW-MRI. We will divide the section up into the different methods and describe the applications that each is used for.

3.1 Linear Regression Linear regression is the simplest machine learning approach, which linearly models the response of a dependent variable to one or more independent variables. It can be subdivided into simple linear regression (when there is only one independent

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variable), or multiple linear regression (which as the name suggests involves more than one independent variable). In the following paragraphs, we will describe some of the applications in DW-MRI. Linear regression is mainly used to fit many of the standard models to DW-MR data such as ADC, DTI, etc. [29] and it has shown to be particularly useful to study changes in diffusion metrics, for instance with disease or ageing. One application of this method is for distinguishing benign and malignant tumours in the breast as described in [24]. Linear regression was used in this last paper to compare mean ADC values and cellularity values between the different types of tumours and was able to show statistically significant differences between the malignant and benign tumours for both measures. Another application of this method was demonstrated in [30] for rectal cancer. The authors use the fact that ADC values of water in a tumour can show how much necrosis there is, to find a correlation between these ADC values and responses of a tumour to chemotherapy treatment. Using the regression model, the authors report a negative correlation between ADC values before and after chemotherapy. Finally, the authors in [31] focus on the age-related microstructural differences through DTI. By computing diffusion parameter maps and using a mass-univariate linear regression model on these maps as part of the analysis, fractional anisotropy in white matter area was shown to decrease with age in a linear fashion. These class of models do come with a few limitations. Firstly, as the name suggests, they are limited to only modelling linear relationships, and this isn’t necessarily the case with many data. In addition to this, the method is also limited by the fact that it can be very sensitive to outliers, and these outliers can significantly affect the regression results.

3.2 Random Forest Random Forests (RFs) are an ensemble learning method and a type of supervised learning algorithm which can be used for both classification and regression tasks. It works by constructing a number of decision trees (forest) during training, one to each bootstrap sample drawn from the full set of training data. Once trained, the model typically outputs the ‘majority’ class in classification tasks or the ‘mean’ class for regression over the full set of trees in the forest. RF has been used in a wide range of applications, such as super-resolution, correlation of biomarkers with disease/ageing, classification of lesions, microstructure parameter mapping etc. Initially, RF has been employed to tackle the intractability for some of the mathematical models of DW-MRI, such as in the microstructural parameter mapping task. For example, Nedjati-Gilani first in [32] and later in [20], proposed an RF approach to map permeability in the brain. In particular, they highlight that traditional mathematical models of diffusion through semi-permeable membranes assume strong mixing, i.e. the average diffusing particle spends time in multiple environments during the experiment. However, this assumption breaks down with typical brain-

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compartment sizes and diffusion times in DWI. The computational model encoded in the RF is able to learn an implicit model from Monte-Carlo simulations thereby providing a mechanism to solve the inverse problem of estimating tissue parameters from measurements. Computation times prevent usage of the simulation directly for this purpose. A similar approach is presented in [33] to map axon diameter from DW-MRI with a RF trained on matching histological data. Other DW-MRI related applications of RF are for studying biomarker changes. For example, [34] assesses the correlation between cognitive impairment and structural damage in the central nervous system for MS. RF has been shown to be useful not only for diseased states but also in healthy ageing as can be seen in [35] which use a Conditional Inference RF on DTI to investigate reductions in white matter integrity with age. In [36] a RF classifier has been combined with a voting procedure to directly processing raw diffusion-weighted signal intensities and obtain fibre tractography. Quantitative and qualitative evaluation show the potential of this technique in comparison with the tractography pipelines that rely on mathematical modelling. RF has been also used to solve classification problems, for example to segment lesion volumes on multiparametric-MRI (mp-MRI), including ADC maps, in patients following ischemic stroke, as detailed in Mitra et al. [37]. RF classifiers showed high accuracy for the detection of these lesion areas. An emerging application enabled by the use of RF is the Image Quality Transfer (IQT) [21]. In this paper, the authors proposed a method to learn and transfer the fine structural detail of images in high-quality data sets, which have longer acquisition times, to enhance lower quality data sets, which have more standard acquisition times. Later in [38] the same authors used the proposed method on the Human Connectome Project (HCP) showing, in more details the benefits of this quality transfer for both brain connectivity mapping and microstructure imaging. However, the aforementioned approaches do not give any indication of confidence in the output, which can be a significant barrier to adopting IQT in clinical practice. For this reason, Tanno et al. [39], presented a Bayesian formulation of RF which enables efficient and accurate quantification of uncertainty in terms of enhanced super-resolution performance. Even with the many applications of RF described, this machine learning model does have its shortcomings. The main limitation of RF is its poor generalisability outside the training set, due to overfitting, particularly when presented with noisy data.

3.3 Bayesian Models Bayesian modelling, also known as Bayesian inference, relies on Bayes’ theorem for updating the probability of a hypothesis as more evidence is obtained. Bayesian modelling is especially useful when data is limited, allows the problem of overfitting to be avoided and is also useful as an approach for modelling uncertainty [40, 41]. For

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example, Reisert et al. [19] have proposed a supervised machine learning approach based on a Bayesian estimator to disentangle the microscopic cell properties of the human brain from the effects of the mesoscopic structure. Bayesian approaches also have an application in the field of tractography. The model presented in [42] is used to investigate the uncertainty associated with estimated white matter fibre paths, in addition to computing the probability of connections between different brain areas. The model is both simple to implement and has the ability to handle noise. The main limitation when it comes to Bayesian approaches is the requirement of a prior, which can sometimes be very difficult to formulate. In terms of the computation power, this approach also requires a high computational cost, especially for more complex models with a large number of parameters.

3.4 Support Vector Machines One of the most popular machine learning approaches is the class of Support Vector Machines (SVMs). SVMs are supervised learning models used for both classification and regression, based on the concept of finding the best set of hyperplanes in a high dimensional feature space that best separates the different classes. Intuitively, a good separation is achieved by the hyperplanes with the largest distance to the nearest training data point of any class. One of the main advantages of SVMs over the other methods described so far is its ability to generalise well and avoid overfitting. This is due to the fact that once the SVM has found the optimum hyperplane, any small changes to the data won’t be able to greatly affect the hyperplane. SVMs have been widely used in brain and cancer imaging for many applications, such as to estimate the survival rates in patients with grade II–IV gliomas from diffusion and perfusion-weighted MRI [43] or to classify High Angular Resolution Diffusion Imaging (HARDI) in vivo data [44]. In this last approach the SVM was trained to discriminate between six different classes: parallel neuronal fibre bundles in white matter, crossing neuronal fibre bundles in white matter, grey matter, partial volume between white and grey matter, background noise and cerebrospinal fluid. The features used were rotation invariant and obtained by exploiting a spherical harmonic decomposition of the HARDI signal. Another interesting approach based on SVM was developed in [45], in order to perform multimodal classification and exploit the relative strengths of each modality. The paper is based on the idea that female brains contain a larger proportion of grey matter tissue, whereas the male brain consists of more white matter, and this can be used to study dimorphism of the human brain. However, the authors show that a single imaging modality is not enough and instead mp-MRI, through the use of T1, T2, and DW-MRI, is required to obtain good performance. The results from their study show that combining these different modalities yields a significantly higher balanced classification accuracy than using a single one. On the same wave is the approach developed in [46] for localizing cancer within the prostate, also through the use of mp-MRI that includes Dynamic Contrast Enhanced (DCE) MRI, DW-MRI

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and T2-weighted MRI. Here, the authors developed a new segmentation method that combines a Conditional Random Field (CRF) with a cost-sensitive framework. The CRF is a class of statistical modelling methods used in machine learning to include structural information in the prediction. For the similar task of prostate cancer localization, Ozer et al. [47] propose, instead, to combine the pharmacokinetic parameters derived from DCE MRI with T2 MRI and DW-MRI. To process these images, they use a Bayesian formulation of the SVM called Relevance Vector Machines (RVM). The accuracy obtained by the RVM has been compared with the SVM and results showed that it can produce more accurate and more efficient segmentation. Although SVMs have been used in many applications, they suffer from an interpretability problem, with both the final model and the variable weights being hard to understand. SVMs, as well as the other ‘classical’ machine learning methods described so far, also suffer from a few general limitations. Firstly, an important issue is scalability. Even when these classical methods work well on small amounts of data, they fail to give the same accuracy on larger datasets and would require significantly larger computational power to analyse. Another big limitation is the need for complex feature engineering, a challenging task which requires a significant amount of time and effort for finding the best features. Moreover, these features are often limited in terms of adaptability or transferability to other domains/applications and extensive study within each different area may be required to transfer their usability. All of these major limitations can be overcome by the introduction of deep learning, which is the focus of the rest of the paper.

4 Deep Learning In the last few years, requests of processing large amount and complexity of data, have driven the evolution of machine learning to the so-called deep learning paradigm. Although deep learning is a relatively new field, many different models have already been proposed. The design of a deep learning model mainly depends on: (i) the amount of data available during the training and the related data-dimensionality (e.g. scalar data, images, 3D images), (ii) the training procedure (supervised or unsupervised), (iii) the desired goal (e.g. classification, regression, segmentation), and (iv) last but not least, the computation resources (cloud computing, Graphics Processing Units (GPU), memory, Central Processing Units (CPU), etc.).

4.1 Artificial Neural Network Before starting to describe the advanced methods of deep learning we will briefly overview the foundation behind it which comes from the ANN theory. An ANN is an ensemble of perceptrons, where a perceptron is defined by a non-linear transfer function f and by two set of parameters: W (the weights) and b (the bias). The output

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of each neuron is the linear combination of the input x with the W added to the bias b, followed by the application of the transfer function (i.e. sigmoid or hyperbolic tangent function). One of the most popular ANNs is the Multi-Layered Perceptrons (MLP) that organizes the neurons in many different layers. ANNs are usually trained through different steps, where at each step a new input sample or batch of samples are presented to the network. In this process, the weights are adjusted using a delta rule and a back-propagation function. Initially, random values are usually assigned to the network parameters and through this iterative training process, the parameters are updated to minimize the difference between the network predictions and the desired outputs. In DW-MRI, ANN has been successfully used, for example, to determinate a voxel-by-voxel forecast of the chronic T2-Weighted images with the purpose of predicting ischemic infarction [48] or for a voxelwise parameter estimation of the combined intravoxel incoherent motion and kurtosis model in [49].

4.2 Deep Neural Network To abstract more complex concepts or extract more sophisticated features, many hidden layers can be added to an ANN, defining a deep architecture referred to as a Deep Neural Network (DNN). Training a DNN is not always trivial, mainly due to possible numerical instabilities that could make the updated weights negligible (vanishing gradient problem), and adequate training countermeasures must be considered to avoid these issues. DNN in DW-MRI was used in many applications from improving acquisition to parameter mapping and studying patient outcome. For example, [50] combined a DNN with spherical harmonics to learn a non-linear mapping that makes it possible to augment existing shells exploiting shells already acquired. This can be particularly useful to reduce the acquisition time of HARDI data. On a similar task, Golkov et al. [51] proposed a method called q-space deep learning which allows mapping scalar parameters, such as diffusion kurtosis or orientation dispersion from significantly reduced acquisitions and detecting abnormalities without the intermediate steps of diffusion models. The authors in [54] proposed a Deep Learning Diffusion Fingerprinting (DLDF) method based on DNN and used to classify DW-MRI voxels. The authors showed that this model can learn even with limited training samples, and the method is capable to segment brain tumours, distinguish between young and older tumours and detect whether or not a tumour has been treated with chemotherapy. The paper concluded also that DLDF can localize changes in the underlying tumour microstructure, which are not evident using conventional measurements of the ADC. Mapping tissue microstructure features, especially permeability, is also investigated in [52], where the authors show that the results provided by the DNN are better than those obtained from a RF regressor model. DNN has also been used to post-process DWMRI images and predict, for example, survival time in amyotrophic lateral sclerosis using as input data the structural connectivity and brain morphology [53].

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4.3 Convolutional Neural Network The main limitation of DNNs is that they do not scale well with multidimensional inputs, such as an image. The reason for this is that the number of nodes and the number of parameters required by a DNN in these cases could become very large, and therefore, their use is not practical. Convolutional Neural Networks (CNNs) have been proposed exactly to overcome this issue and today they are the most successful models for image analysis in deep learning. The design of a CNN is largely inspired by the neurobiological model of the visual cortex and can be summarized in three main steps: (i) The input image is convolved using several small kernels, (ii) the output of the previous step is subsampled and (iii) the first 2 steps are repeated until highlevel features are extracted. Differently from DNNs, CNNs use shared weights -the kernels- to perform the convolutions. This enables the network to process efficiently multidimensional data such as medical images. In the field of DW-MRI, CNNs are mainly used in the post-analysis step, for example, to perform tractography, for tissue segmentation or for predicting survival time. In [55] a CNN is used to reconstruct fibre orientation, which is useful for studying brain connectivity. Rather than fitting a diffusion model, the authors proposed to solve a voxel classification problem in order to estimate the fibre orientation. In [56], instead, a CNN was trained for segmentation of tumours in patients with rectal cancer on mp-MRI, classifying each voxel into a tumour or non-tumour class. On a similar task, Clark et al. [58] proposed a CNN to delineate the prostate gland. The proposed algorithm first detects the slices that contain a portion of the prostate gland within the three-dimensional DW-MRI volume and then uses this information to segment the prostate gland and the related transition zone. TractSeg proposed by Wasserthal et al. in [57] is also a method based on CNNs for fast and accurate white matter tract segmentation. This novel framework directly segments tracts in the field of fibre orientation distribution function peaks without using tractography, image registration or parcellation. This allows to speed up the computational time with respect to traditional methods while providing unprecedented accuracy. Many extensions of the CNN have been proposed in the literature. In [59], the authors use a 3D-CNN that processes multi-modal preoperative brain images and learns a set of supervised features processed by a SVM to predict survival time in patients with high-grade glioma. Another interesting framework based on CNN is the U-net [70], which is developed mainly for biomedical image segmentation and is able to work with few training images or to provide better accuracy. In [60] the U-net is used to segment the kidney and improve the registration accuracy while avoiding artefacts during the free-breathing multi-b-value DW-MRI acquisition. Similarly to the previous work based on RF, Tanno et al. in [61] show that IQT is a highly ill-posed problem, and training a standard CNN could create inevitable ambiguity in the learning procedure. For this reason, they propose to model uncertainty through a per-patch heteroscedastic noise and Bayesian inferences that extend the standard CNN concept. They show that this leads to obtaining convincing superresolved DW-MRI and produces tangible benefits in downstream tractography.

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Finally, a particular formulation of the CNN developed to control memory usage during network training was proposed in [62]. This framework, evaluated in the paradigm of IQT, is able to improve the prediction results by exploiting a deeper network which, however, requires only a slight memory increase.

4.4 Deep Autoencoder Most of the approaches described so far use a supervised learning strategy to train the networks. This means that each sample has a label used as ground truth during the training. However, in some cases, labels are missing or only partially available and the models need to be trained in an unsupervised fashion. An autoencoder is a special ANN designed exactly for this purpose. In this network, the number of input and output nodes are the same, and the training procedure doesn’t require any labels. An autoencoder is based on recreating the input data rather than assigning a class label to it. This procedure enables to encode the data in a lower dimensional space and extract the most discriminative features. With the recent advances in deep learning, many autoencoders can be stacked together to create a deep autoencoder architecture. An example of this framework was proposed in [64] for a spatio-temporal denoising of contrast-enhanced MRI sequences, nevertheless, it could be applied to DWMRI data as well. Here, each autoencoder on the stack is trained on a specific subset of the input space to accommodate different noise characteristics and curve prototypes. Spatial dependencies of the pharmacokinetic dynamics are captured by incorporating the curves of neighbouring voxels in the entire process. A stack autoencoder was also used in [65, 66] which propose a new non-invasive approach for early classification of renal rejection types after transplant. The developed framework mainly consists of three steps: (1) data co-alignment using a 3D B-spline-based approach and segmentation of kidney tissue with an evolving geometric deformable model guided by a voxel-wise stochastic speed function, (2) construction of a cumulative empirical distribution of ADC at low and high b-values of the segmented kidney taking into account blood perfusion and water diffusion, and (3) classification of acute renal transplant rejection types based on a deep autoencoder with non-negative constraints. A computer-aided classification method of prostate cancer grade groups using a stacked sparse autoencoder for mp-MRI was also proposed in [63]. This special deep framework based on the extraction of high-level features from hand-crafted texture features achieved the first place in a related challenge, winning over 43 methods submitted by 21 groups. In [67] instead, a Variational Autoencoder (VAE), that is a particular formulation of the standard autoencoder where strong assumptions are made on the distribution of latent variables, was proposed to detect diffusion-space abnormalities in DW-MRI scans of MS patients. Here, only samples from the normal class are available during the training, while test samples are classified as normal or abnormal through the assignment of a novelty score efficiently computed by VAE.

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4.5 Generative Adversarial Network A Generative Adversarial Network (GAN) is a deep framework that is particularly useful when the training set is only partially labelled. GAN is a class of ANN which train two separate models that challenge each other in a zero-sum game: the first model is a generative model G, and the second is a discriminative model D. The first model is trained to learn how to generate data that looks like the real available data, and the second is trained to discriminate the sample generated by G between the real one. Since the aim of G is to understand how to fool the D network this leads to a better generation of realistic samples. In medical imaging, GAN has shown its success in generating synthetic underrepresented data or for the super-resolution task applied to different image modalities, such as for endomicroscopy images [71], or DW-MRI [68]. Specifically, in this last study, GAN was used to obtain high-resolution images without requiring larger magnetic fields or long scan times. Apart from these examples, GAN has been also used for generating realistic segmentation such as in [69] where an adversarial network which discriminates between expert and generated annotations is used to detect aggressive prostate cancer. This framework, which combines an adversarial training with the U-Net model, obtains good results on segmenting the prostates regions along with the targeted cancer nodules.

5 Discussion and Future Perspectives In this review, we have described the current trends and main methods used in machine learning and deep learning for DW-MRI. We have shown that many machine learning methods have already been successfully applied to solve a large variety of problems in DW-MRI, for instance designing better acquisitions and reconstruction tools, mapping microstructure parameters, classifying biomarkers or predicting disease outcomes. We have organized this paper by grouping the methods according to the main methodological classes, explaining their applications and stating where and why that class can be particularly useful. Specifically, in the last few years, a sub-field of machine learning called deep learning has shown a large interest in research and has been employed in many applications, including in the field of DW-MRI. The most likely reason for its success in this area is that DW-MRI has high dimensionality and/or is often coupled with other MRI contrasts, leading to a very large amount of data to be processed, that standard machine learning methods are not able to handle. Moreover, handcrafted features can be difficult to extract in these scenarios since they may be suitable for only one single modality but not adequate for the others. Deep learning -a technique based on the ANN principles- seems particularly suitable to process a large amount of data allowing the model itself to automatically extract the best features. This characteristic seems particularly important to improve the final results of a model.

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In conclusion, in the most recent methods described in this review, we report the use of deep learning, not as a single black box, but rather integrated with mathematical modelling to improve the training procedure. For this reason, we believe that in the near future we will see more hybrid models where characteristics of deep learning and traditional modelling are combined together for better analysis and results. Acknowledgements DR and DCA are supported by a project that has received funding from the European Unions Horizon 2020 research and innovation programme under grant agreement number 666992. EPSRC grants EP/M020533/1 and EP/N018702/1 support AI and DCAs work on this topic. UCL EPSRC Centre for Doctor Training in Medical Imaging (EP/L016478/1) funds NG.

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q-Space Learning with Synthesized Training Data Chuyang Ye, Yue Cui and Xiuli Li

Abstract q-Space learning has been developed to improve tissue microstructure estimation on diffusion magnetic resonance imaging (dMRI) scans when only a limited number of diffusion gradients are applied. However, the training samples for q-space learning are obtained from high-quality diffusion signals densely sampled in the q-space, which are acquired on the same scanner of the test scans with a large number of diffusion gradients, and they may not be available for existing or ongoing datasets. In this work, we explore q-space learning with synthesized training data so that it can be applied to datasets where training signals are not available. We seek to synthesize diffusion signals densely sampled in the q-space, whose corresponding undersampled signals should match the distribution of observed undersampled diffusion signals. Specifically, by drawing samples from a simple distribution and feeding them into a generator defined by a multiple layer perceptron, we synthesize the continuous SHORE signal representation, from which both densely sampled and undersampled synthesized diffusion signals can be computed. The weights in the generator are learned by minimizing the distribution difference, which is measured by the maximum mean discrepancy, between the synthesized and observed undersampled signals. In addition, regularization terms are added to discourage unrealistic synthetic signals. With the learned generator, densely sampled diffusion signals can be synthesized for q-space learning. The proposed approach was applied to microstructure estimation on dMRI scans acquired with a limited number of diffusion gradients.

C. Ye (B) School of Information and Electronics, Beijing Institute of Technology, Beijing, China e-mail: [email protected] Y. Cui Brainnetome Center, Institute of Automation, Chinese Academy of Sciences, Beijing, China X. Li Deepwise Inc., Beijing, China © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_10

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The results demonstrate the benefit of using synthetic training signals for q-space learning when actual training data are not acquired. Keywords Diffusion MRI · q-space learning · Training data synthesis

1 Introduction Diffusion magnetic resonance imaging (dMRI) provides a unique tool for noninvasively probing neural tissue architecture [20]. However, accurate estimation of tissue microstructure described by complicated biophysical models can be challenging for dMRI scans acquired with a limited number of diffusion gradients, e.g., clinical dMRI scans. To improve the quality of tissue microstructure estimation when the number of diffusion gradients is limited, deep learning based approaches have been developed [6, 19]. In these works, deep networks are designed to learn the mapping from diffusion signals to tissue microstructure. Because the diffusion signal in each diffusion gradient represents a measurement in the q-space, the learning of such a mapping is termed q-space learning [6]. q-Space learning requires training diffusion signals densely sampled in the qspace, which are acquired on the same scanner of the test scans with a large number of diffusion gradients for a few training subjects. From these signals high-quality estimation of tissue microstructure is computed to train the deep networks [6, 19]. However, acquisitions of training diffusion signals with dense q-space sampling may not be available for existing or ongoing dMRI datasets, which limits the use of q-space learning in practice. Motivated by the successful use of synthesized data for training deep networks in computer vision tasks [17], in this work we explore q-space learning with synthesized training data so that it can be applied to datasets where training signals are not available. Because it is desired that training samples accurately represent the distribution of test samples [11], we seek to synthesize diffusion signals that are densely sampled in the q-space and ensure that their undersampled version matches the distribution of observed diffusion signals. Specifically, we synthesize a continuous representation of diffusion signals using the SHORE basis [14], from which both densely sampled and undersampled synthetic diffusion signals can be computed. This synthesis is achieved by drawing samples from a simple distribution and feeding them into a generator defined by a multiple layer perceptron (MLP). The weights in the generator are learned by minimizing the distribution difference, which is measured by the maximum mean discrepancy (MMD) [8], between the synthesized and observed undersampled diffusion signals. In addition, regularization terms are added to discourage unrealistic synthetic signals. With the learned generator, densely sampled diffusion signals can be synthesized for q-space learning. The proposed approach was applied to microstructure estimation on dMRI data acquired with a limited number of diffusion gradients, and the results demonstrate the benefit of using synthetic training signals for q-space learning when actual acquisitions of training data are not available.

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2 Methods 2.1 Background: q-Space Learning In dMRI, a vector of diffusion signals is acquired at each voxel by applying a set of diffusion gradients. For convenience, in this work the diffusion signals refer to the signals that have been normalized by the b0 signal acquired without diffusion weighting. Each entry in the diffusion signal vector then represents a sample in the so-called q-space [6], and the coordinate is determined by its diffusion gradient. By designing models relating tissue organization and diffusion signals, tissue microstructure can be inferred from the observed signals at each voxel by solving regularized least squares problems [3, 14, 20]. To accurately describe tissue microstructure, the signal model can be complicated and consist of a number of parameters. Reliable tissue microstructure estimation could thus require diffusion signals densely sampled in the q-space using a large number of diffusion gradients, and may be challenging for dMRI scans where only undersampled diffusion signals are acquired with a limited number of diffusion gradients. Therefore, q-space learning has been developed to improve tissue microstructure estimation from undersampled diffusion signals [6]. Specifically, for a dMRI dataset with undersampled diffusion signals, densely sampled training diffusion signals are acquired on the same scanner of this dataset for a few training subjects. Then, deep networks are trained by associating the high-quality tissue information computed from these training signals with the undersampled version of these training signals, where the mapping from undersampled diffusion signals to high-quality tissue microstructure estimation is learned. q-Space learning can achieve results that compare favorably with conventional approaches [6, 19]. However, the acquisitions of training signals with dense q-space sampling may not be available for many existing or ongoing dMRI datasets, which limits the application of q-space learning to these data.

2.2 Synthesis of Training Diffusion Signals The goal of this work is to allow q-space learning on dMRI scans acquired with undersampled diffusion signals where densely sampled training signals are not available by synthesizing the training samples. Since it is desirable to have training samples that accurately represent the distribution of test samples to enhance the generalization ability [11], we seek to generate densely sampled training signals and ensure that the distribution of the corresponding undersampled synthetic signals matches that of the observed signals. We denote the set of diffusion gradients associated with the observed real undersampled diffusion signals yr by G and the set of diffusion gradients densely sampling ˜ The synthesized diffusion signals corresponding to G and G˜ are the q-space by G.

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denoted by ys and ˜ys , respectively, where ys is the undersampled version of ˜ys . Because diffusion signals can be continuously represented by the SHORE basis functions with corresponding coefficients [14], it is convenient to synthesize these coefficients, from which both ys and ˜ys can be computed. Specifically, we denote the synthesized coefficients by cs , and the synthesized signals can be represented as ys = G cs and ˜ys = G˜ cs ,

(1)

where G and G˜ are dictionary matrices determined by G and G˜ [14]. To synthesize cs , we draw random samples from a simple distribution and then pass them through a generator that can transform the simple distribution into an arbitrary distribution. We select the MLP as the generator due to its express power [1]. Specifically, the MLP transforms the input random vector z drawn from a multivariate Gaussian distribution N (0, I) into cs . We denote this transformation by cs = g(z; w), where w represents the unknown weights in the MLP. Then, according to Eq. (1) the synthesized diffusion signals are ys = G g(z; w) and ˜ys = G˜ g(z; w).

(2)

The generator should ensure that the distributions of synthesized and observed undersampled diffusion signals are consistent. To measure the difference between these two distributions, the MMD [8] is used, which provides a distance measure between two distributions by comparing their embeddings in the reproducing kernel Hilbert space (RKHS). Briefly, the idea behind the MMD is that if two distributions are identical, all their statistics are the same. Formally, suppose the distributions of yr and ys are r and s, respectively; then the MMD DH (r, s) between r and s is defined as [8] 2 (r, s) = sup ||E yr ∼r [ f ( yr )] − E ys ∼s [ f ( ys )]||2 , DH || f ||H ≤1

(3)

where || f ||H ≤ 1 denotes the set of functions in the unit ball of an RKHS H, and E yr ∼r [·] and E ys ∼s [·] represent the expectations with respect to r and s, respectively. 2 (r, s) = 0 [8]. If H is a universal RKHS, e.g., the Gaussian RKHS, then r = s ⇔ DH 2 An empirical estimate of DH (r, s) is given as [8] 2 (r, s) = Dˆ H

1 Ns2

Ns  Ns  i=1 j=1

k( yis , ysj ) +

1 Nr2

Nr  Nr  i=1 j=1

k( yir , yrj ) −

2 Ns Nr

Ns  Nr  i=1 j=1

k( yis , yrj ). (4)

Here, k(·, ·) is a kernel function; yir and yis are the i-th samples drawn from r and s, respectively; and Nr and Ns are the total numbers of samples drawn from r and s, respectively. For more details about MMD, we refer readers to [8]. In this work, we use a Gaussian kernel, which is a common choice in distribution matching tasks [13],   and set k(x, x  ) = exp −||x − x  ||2 .

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Combining Eqs. (2) and (4), we can learn the generator by minimizing the following objective function to match the distributions of yr and ys : Ns Nr Ns  Nr  1  1  k(G g(z i ; w), G g(z j ; w)) + 2 k( yir , yrj ) LMMD (w) = 2 Ns i=1 j=1 Nr i=1 j=1

−

Ns  Nr 2  k(G g(z i ; w), yrj ), Ns Nr i=1 j=1

(5)

where z i is the i-th input sample drawn from N (0, I). Note that because the diffusion signals normalized by b0 signals should be in the range (0, 1], to discourage unrealistic synthetic diffusion signals, we also use a regularization term Ns  1  h(G g(z i ; w)) + h(1 − G g(z i ; w)) + LR (w) = − Ns i=1  h(G˜ g(z i ; w)) + h(1 − G˜ g(z i ; w)) ,

(6)

where h(x) is a function that averages the negative part of its input x ∈ R Nx (N x is Nx 1 min(0, xl ). LR (w) penalizes synthetic diffusion the length of x): h(x) = Nx l=1 signals that are negative or greater than one. Then, the overall objective function to minimize is L(w) = LMMD (w) + βLR (w),

(7)

where β is a weighting constant. In this work, we empirically set β = 1.0.

2.3 Implementation Details and Evaluation The SHORE representation is implemented using Dipy [4] with the parameters suggested at http://nipy.org/dipy/examples_built/reconst_shore.html, and thus the length of cs is 50. The length of the input vector z is the same as the length of cs . The MLP has three hidden layers, each with 1024 units and a ReLU activation [15]; the output layer has an identity activation. The Adam algorithm [12] is used to minimize the objective function in Eq. (7), where the learning rate is 0.0001, the batch size is 2048, and 50 epochs are used. To demonstrate the application of the proposed method, we use the synthesized diffusion signals to learn the mapping from undersampled diffusion signals to the NODDI microstructure [20], which includes the intra-cellular volume fraction vic , cerebrospinal fluid (CSF) volume fraction viso , and orientation dispersion (OD). The NODDI model is selected because it has shown great potential in providing biomark-

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ers for studies of brain diseases and development [2, 5, 7, 10]. The proposed method could be used for other signal models as well, such as LEMONADE [16] and SMT [9]. For the estimation of NODDI microstructure, the deep network proposed in [19] is considered, which uses a network structure inspired by a sparse reconstruction framework and outperforms the generic MLP regression in [6]. Like in [19], the densely sampled signals ˜ys are used to compute the training microstructure with the AMICO algorithm [3], which can reduce the computation time by orders of magnitude compared with the original NODDI fitting in [20]. Then, the deep network is trained by the synthetic undersampled signals ys (input) and the training microstructure (output), and can thus estimate microstructure from observed undersampled signals yr without actual acquisitions of densely sampled training signals. To evaluate the estimation accuracy, like in [19], for each subject actual diffusion signals densely sampled in the q-space are acquired at each voxel and used to compute the gold standard microstructure with AMICO [3]. Then, the estimation results are compared with the gold standard by measuring the mean absolute difference for each subject.

3 Results We randomly selected 25 subjects from the Human Connectome Project (HCP) dataset [18]. The resolution of the dMRI scans is 1.25 mm isotropic, and 270 diffusion gradients over three shells (b = 1000, 2000, 3000 s/mm2 ) were used. Sixty diffusion gradients were selected as the undersampled diffusion gradients G, and the real signals associated with G represent the observed undersampled signals yr . These 60 diffusion gradients consist of 30 gradient directions on each of the shells b = 1000, 2000 s/mm2 and resemble clinically achievable protocols. We synthe˜ which contains 180 diffusion sized densely sampled signals ˜ys associated with G, gradients, 90 on each of the shells b = 1000, 2000 s/mm2 , and undersampled signals ys associated with G. The full set of real signals was used to compute the gold standard NODDI microstructure for evaluation. Because in practice we may have existing dMRI scans that can be used for training the generator as well as newly scanned data that have not been used in the training, we split the subjects into two groups S1 and S2 to represent these two cases. S1 has five subjects which provided the distribution of observed signals yr for training the generator; S2 has 20 subjects that were not used in training the generator. We trained the generator and synthesized 8 × 106 training samples on an NVIDIA Tesla K80 GPU, which altogether took about 12 h. First, we demonstrate that the synthesized training samples are consistent with the distribution of observed signals. The marginal probability density functions (PDFs) of the synthesized and observed undersampled signals in two representative diffusion gradients (one on each of the shells b = 1000, 2000 s/mm2 ) are compared in Fig. 1a, where the synthesized signals have marginal PDFs similar to those of the observed signals. In addition, the correlations of signals between different diffusion gradients

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Fig. 1 Comparison between the synthesized and observed undersampled diffusion signals: a marginal PDFs in two representative diffusion gradients (one on each of the shells b = 1000, 2000 s/mm2 ); b the correlation of signals between diffusion gradients

are compared between the synthesized and observed undersampled signals in Fig. 1b, where the relationship of the signals in different diffusion gradients is preserved for the synthesized signals. Then, the synthesized signals were used for q-space learning to estimate the NODDI microstructure as described in Sect. 2.3. A cross-sectional view of the estimated microstructure maps of a representative subject in S2 is shown in Fig. 2. (Results on subjects in S1 have similar quality and are thus not shown here.) Compared with the gray matter, the white matter has larger vic and smaller OD, as they should be; in the CSF viso is large, as it should be. Note that when viso is one, the estimation of vic and OD is degenerate, thus these regions will be excluded in the following quantitative evaluation. The means and standard deviations of estimation errors computed using all 25 subjects are shown in Fig. 3. These results are compared with those computed by AMICO [3] using the observed undersampled signals, which represents the conventional approach when training data are not available. The errors of groups S1 and S2 are also shown separately. In all cases, the proposed approach achieves smaller errors, and the difference is highly significant ( p < 0.001) using a paired Student’s t-test.

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Fig. 2 A cross-sectional view of the NODDI microstructure maps estimated by q-space learning with synthesized training signals. Orientations are indicated on the vic map

Fig. 3 Means and standard deviations of the estimation errors. Asterisks (***) indicate that the difference of errors between methods is highly significant ( p < 0.001) using a paired Student’s t-test

4 Conclusion In this work, we have explored q-space learning with synthesized training data. A generator learns the synthesis of densely sampled training signals by matching the distributions of their undersampled version and observed signals. Experiments on dMRI data with undersampled diffusion signals demonstrate the benefit of qspace learning with synthesized training samples when actual acquisitions of training signals are not available.

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Acknowledgements This work is supported by the National Natural Science Foundation of China (NSFC 61601461) and Beijing Institute of Technology Research Fund Program for Young Scholars. Data were provided by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University. Declaration of Conflict or Commercial Interest We declare that we have no conflict or commercial interest that can inappropriately influence our work.

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Graph-Based Deep Learning for Prediction of Longitudinal Infant Diffusion MRI Data Jaeil Kim, Yoonmi Hong, Geng Chen, Weili Lin, Pew-Thian Yap and Dinggang Shen

Abstract Diffusion MRI affords great value for studying brain development, owing to its capability in assessing brain microstructure in association with myelination. With longitudinally acquired pediatric diffusion MRI data, one can chart the temporal evolution of microstructure and white matter connectivity. However, due to subject dropouts and unsuccessful scans, longitudinal datasets are often incomplete. In this work, we introduce a graph-based deep learning approach to predict diffusion MRI data. The relationships between sampling points in spatial domain (x-space) and diffusion wave-vector domain (q-space) are harnessed jointly (x-q space) in the form of a graph. We then implement a residual learning architecture with graph convolution filtering to learn longitudinal changes of diffusion MRI data along time. We evaluate the effectiveness of the spatial and angular components in data prediction. We also investigate the longitudinal trajectories in terms of diffusion scalars computed based on the predicted datasets. Keywords Brain development · Longitudinal prediction · Diffusion MRI · Graph representation · Graph convolution · Residual graph neural network

1 Introduction Diffusion MRI is an attractive imaging modality for longitudinal studies of the developing infant brain, owing to its capability in assessing brain microstructure even in the pre-myelinated brain [1]. To chart temporal brain changes, longitudinal diffusion MRI data need to be collected. However, due to subject dropouts and unsuccessful scans, longitudinal datasets are often incomplete. For T1-weighted and T2-weighted J. Kim School of Computer Science and Engineering, Kyungpook National University, Daegu, South Korea J. Kim · Y. Hong · G. Chen · W. Lin · P.-T. Yap · D. Shen (B) Department of Radiology and BRIC, University of North Carolina at Chapel Hill, Chapel Hill, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_11

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imaging, a number of methods have been developed for longitudinal image prediction, i.e., extrapolation using geodesic regression models [2] and patch-wise sparse representation of image metamorphosis paths [3]. Work on diffusion MRI prediction is however scarce. Longitudinal prediction of diffusion MRI data is more challenging than T1weighted and T2-weighted data, partly due to the high variability in q-space sampling. The sampling domain is not necessarily Cartesian and can vary from shell-based to even random. It is therefore not straightforward to extend advances such as convolutional neural network (CNN) based image generation techniques [4] to diffusion MRI. In this article, we will introduce a graph-based deep learning approach for longitudinal prediction of diffusion MRI data in the first year of life. This work is inspired by [5], where q-space matching with the help of graph representation was harnessed for denoising of diffusion MRI data. In our work, this graph representation is further extended to the x-q space by considering both spatial and angular domains. This allows us to adopt a graph CNN approach for prediction. Convolutions of functions on graphs are implemented as multiplications in the graph spectral domain [6]. In our approach, we introduce a new residual neural network with graph convolutional filtering to predict diffusion MRI data of target time points in a patch-wise manner. We train the residual learning model with a Huber loss for minimizing image differences and L1 regularization on the weights of the convolutional filters to avoid over-fitting.

2 Method In the following sections, we will introduce (1) graph-based representation of the spatio-angular x-q space, (2) convolutional filtering of a function defined on a graph, and (3) a residual neural network with graph spectral filtering for DW image prediction.

2.1 Graph Representation of x-q Space Consider an undirected graph G = (V,E), where V = {vi : i = 0, . . . , n − 1} is a set of vertices, and E is a set of edges connecting the vertices. We denote by wi, j the non-negative weight of an undirected edge ei, j ∈ E between two vertices,  vi and v j . If vi and v j are not connected, wi, j is 0. The adjacency matrix A = wi, j is symmetric.  Given A, the degree matrix D is diag {d0 , d1 , · · · , dn−1 }, where di = j=i wi, j . In this work, we utilized a symmetric normalized form of the graph Laplacian: L = I − D −1/2 AD −1/2 . In diffusion MRI, a vertex vi can be thought as corresponding to a sampling point at spatial location xi of a diffusion-weighted (DW) image with b-value bi and gradient

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direction qi . The geometric structure of the sampling domain is encoded using an adjacency matrix A, with weight wi, j between two points vi and v j defined as wi, j

   √    2 

2   xi − x j 2 bi − b j 1 − qi q j = exp − exp − exp − , (1) 2σx2 2σq2 2σb2

where σx , σb , and σq are parameters for controlling the width of the Gaussian functions. The weight wi, j reflects the spatial distance, the dissimilarity of gradient directions, and the difference in diffusion weightings between the two points.

2.2 Convolution Filtering on Graph Convolutions of functions defined on a graph can be implemented as linear operators in the graph spectral domain [6]. The Fourier basis of the graph representation can be determined as the eigenvectors of graph Laplacian. The Laplacian L is diagonalized by the Fourier basis U = u 0 , u 1 , · · · , u n−1 ∈ Rn×n such that n−1 are orthonormal eigenvectors with nonnegative eigenL = U U  , where {u i }i=0 values  = diag {λ0 , λ1 , · · · , λn−1 } ∈ Rn×n . The Fourier transform of a graph s defined on the graph is U  s ∈ Rn , where the i-th element of s (i.e., si ) corresponds to vertex vi . The convolution of s can be implemented as s  = U f θ ()U  s

(2)

where f θ is a diagonal matrix parameterized by θ, representing the Fourier coefficients of the filter. The filter can be localized in space by representing f θ in the form of a polynomial [7]: K

θk k (3) f θ () = k=0

where the parameters θ ∈ R K +1 are now in the form of polynomial coefficients. The spectral filter with polynomials of degree K is K -localized in the graph. For fast filtering, Chebyshev polynomials are used, allowing the convolution to be realized ¯ with a recursive formula without explicit directly using a rescaled version of L, i.e., L, eigendecomposition of L [6]. That is, based on (3), the filter can be implemented as 

f θ L¯ without the Fourier transform (U  ) and the inverse Fourier transform (U ) in (2). A feature map can be obtained by combining the outcomes of spectral filtering sout =

Fin

l=1



f θl L¯ sl

(4)

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Fig. 1 Residual neural network with graph convolution blocks. GCB: graph convolution block, RGCB: residual graph convolution block; RT-ReLU: randomly-translated leaky ReLU; Fout : the number of output feature maps; and K : the kernel size of the graph convolutional filter

where {sl } are the input feature maps, sout ∈ Rn is an output feature map, and Fin is the number of the input feature maps. Consequently, the number of trainable parameters of the convolutional layer is Fin × Fout × (K + 1).

2.3 Residual Neural Network for DW Image Prediction We implemented a deep residual network with graph convolutions to capture brain longitudinal changes for prediction of missing diffusion MRI data in a patch-wise manner. Figure 1 shows the architecture of the proposed residual neural network. A basic building block is the graph convolutional block (GCB) with graph convolution followed by a randomly-translated leaky rectified linear unit (RT-ReLU) [8] and a group normalization layer [9]. The RT-ReLU is more robust to image noise and jitter by adding small Gaussian noise before the non-linear activation function. Due to the large size of the Laplacian matrix, only small batch sizes are used for training. To prevent the degradation of learning accuracy, we normalize the feature maps within each group. Note that RT-ReLU and the group normalization layers are not used at the first and last GCB layers. In the architecture, we build a residual graph convolution block (RGCB) that consists of two GCBs with element-wise addition. Five RGCBs are connected consecutively. The goal of our model is to learn changes between two time points in order to predict DW volumes at a target time point. We train the model using Huber loss [10]:  h loss =

star − sout 22 , for star − sout 1 < δ 1 2 δ star − sout 1 − 2 δ , otherwise 1 2

(5)

where sout is the output diffusion signals of the model, star is the target diffusion signals, and δ is the parameter to determine a point where the loss function changes from quadratic to linear. The L1 loss is usually employed in image-to-image trans-

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lation, however it is more difficult to converge at small errors than L2 loss. Huber loss combines the advantages of L1 and L2 losses—the sensitivity of L2 loss and the robustness of L1 loss. We also employ L1 regularization on filter coefficients to avoid overfitting.

3 Experiments 3.1 Materials We evaluated the effectiveness of the proposed method by means of the longitudinal prediction of diffusion MRI for 3 and 6-month-old using neonate data set. DW data collected from 28 infants, all born at full term, were used. For each subject, we acquired 42 DW volumes using a spin-echo echo planar imaging sequence with TR/TE = 7680/82 ms, resolution 2 × 2 × 2 mm3 , and b = 1000 s/mm2 using a 3T Siemens Allegra scanner. Seven non-DW (b = 0 s/mm2 ) reference scans were also acquired. The image dimension is 128 × 96 × 60. On average, each subject was scanned 1.2 times.

3.2 Implementation Details For training, we used the diffusion MRI datasets of 20 subjects, covering neonates, 3-month-olds, and 6-month-olds. We used the DW images of 1 randomly selected subject for validation. For model evaluation, we utilized the DW images of 6 subjects, acquired consecutively at three time-points: birth and 3 and 6 months of age. All DW images were processed using the FSL software package [11], involving correction of eddy-current distortion and brain extraction. We aligned all source and target DW images to a longitudinal template [12]. We reoriented the diffusion signals using the spatial warping method described in [13]. After alignment, we extracted patches of size 5 × 5 × 5 × 43 (42 DW volumes + 1 non-DW volume) for training and testing. In our network architecture, the kernel size K of RGCBs was 3 and the number of the output feature maps was 128 for each RGCB and GCB, except the last GCB (see Fig. 1). σx , σb , and σq in (1) for the spatio-angular distance were determined via a grid search in range of 0.0 to 2.0. The point parameter (δ) of Huber loss was 0.5 and the weight parameter for the L1 regularization was 0.00001. The number of the feature groups for the group normalization was 4. The network was trained up to 500 epochs using the ADAM optimizer with initial learning rate = 0.0001 and decay rate = 0.95. Training was terminated based on a validation loss threshold.

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Table 1 Comparison between graph convolutional neural networks: spatio-angular versus spatialonly model MAE (DW volumes) MAE (FA maps) PSNR (FA maps) 3-month-old from neonates Spatial only 47.4±17.9 Spatio-angular 44.4±17.5 6-month-old from 3-month-olds Spatial only 39.5±10.6 Spatio-angular 40.1±10.6

0.075±0.008 0.066±0.008

35.6±1.3 36.5±1.3

0.080±0.004 0.064±0.003

35.1±0.5 38.4±0.4

MAE: mean absolute error; PSNR: peak signal-to-noise ratio; DW: diffusion-weighted; FA: fractional anisotropy; Bold: Significant improvement of spatio-angular over spatial-only model

3.3 Quantitative Analysis The accuracy of the trained model is assessed in terms of mean absolute error (MAE) and peak signal-to-noise ratio (PSNR) of the predicted images with respect to the target images. We compare the proposed model with spatial-only (SO) convolution. This is done by keeping the network architecture but limiting convolution to the spatial domain. We also compare the longitudinal trajectory of diffusion scalars, i.e., fractional anisotropy (FA) and mean diffusivity (MD), which are computed from the predicted DW images of neonates, 3-month-olds, and 6-month-olds.

3.3.1

Comparison with Spatial-Only Convolution

Table 1 shows the accuracy of the proposed model in comparison with the SO model. The proposed model shows significantly improved performance in 3-month-old DW image prediction from neonate images across all test subjects. The FA maps of the DW images predicted using the proposed model were more consistent with the target FA maps than those of the DW images generated by the SO model. In 6-month-old DW image prediction, there was no significant difference between the accuracy of the two models in the DW image prediction. However, the FA maps of the proposed model were more similar to the target FA maps than the SO model. Figure 2 shows the FA maps of the two models and those of the target DW images. As can be observed in Fig. 2, the FA maps of the proposed model show very similar patterns of FA values in white matter regions with the targets.

3.3.2

Longitudinal Trajectory of Diffusion Scalars

Figure 3 shows the FA and MD values of the corpus callosum, measured at corresponding regions of interest in the predicted and target DW images of test subjects

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Fig. 2 FA images of the predicted DW images

at each time point (neonates to 6-month-olds). The predicted values have a similar trajectory as the target values with temporally increasing FA and decreasing MD. However, the predicted FA was slightly lower and the predicted MD was higher than the target values. This may be due to the smoothing effects associated with the averaging operation in patch-wise image generation.

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Fig. 3 Longitudinal trajectories of FA and MD of the corpus callosum

4 Conclusion We have introduced a graph-based residual neural network for longitudinal prediction of diffusion MRI data, which takes into account signal measurements in the joint x-q space. This allows missing data to be imputed so that longitudinal analysis can be performed to study brain development. Acknowledgements This work was supported in part by NIH grants (NS093842, EB022880, EB006733, EB009634, AG041721, MH100217, and AA012388), an NSFC grant (11671022, China), and Institute for Information & communications Technology Promotion (IITP) grant (MSIT, 2018-2-00861, Intelligent SW Technology Development for Medical Data Analysis, South Korea).

References 1. Qiu, A., Mori, S., Miller, M.I.: Diffusion tensor imaging for understanding brain development in early life. Ann. Rev. Psychol. 66(1), 853–876 (2015) 2. Niethammer, M., Huang, Y., Vialard, F.X.: Geodesic regression for image time-series. In: Medical Image Computing and Computer-Assisted Intervention–MICCAI 2011, pp. 655–662 (2011) 3. Rekik, I., Li, G., Wu, G., Lin, W., Shen, D.: Prediction of infant MRI appearance and anatomical structure evolution using sparse patch-based metamorphosis learning framework. In: PatchBased Techniques in Medical Imaging, pp. 197–204, October 2015 4. Isola, P., Zhu, J.Y., Zhou, T., Efros, A.A.: Image-to-image translation with conditional adversarial networks (2016) 5. Chen, G., Dong, B., Zhang, Y., Shen, D., Yap, P.T.: Neighborhood matching for curved domains with application to denoising in diffusion MRI. In: Medical Image Computing and Computer Assisted Intervention – MIC-CAI, pp. 629–637. Springer, Cham (2017) 6. Defferrard, M., Bresson, X., Vandergheynst, P.: Convolutional neural networks on graphs with fast localized spectral filtering, pp. 3844–3852 (2016) 7. Bronstein, M.M., Bruna, J., LeCun, Y., Szlam, A., Vandergheynst, P.: Geometric deep learning: going beyond euclidean data. IEEE Signal Process. Mag. 34(4), 18–42 (2017)

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8. Cao, J., Pang, Y., Li, X., Liang, J.: Randomly translational activation inspired by the input distributions of ReLU. Neurocomputing 275, 859–868 (2018) 9. Wu, Y., He, K.: Group Normalization (2018) 10. Huber, P.J.: Robust estimation of a location parameter. Ann. Math. Stat. 35(1), 73–101 (1964) 11. Jenkinson, M., Beckmann, C.F., Behrens, T.E.J., Woolrich, M.W., Smith, S.M.: FSL. NeuroImage 62(2), 782–790 (2012) 12. Kim, J., Chen, G., Lin, W., Yap, P.T., Shen, D.: Graph-constrained sparse construction of longitudinal diffusion-weighted infant atlases. Medical Image Computing and Computer-Assisted Intervention, pp. 49–56. Springer, Cham (2017) 13. Chen, G., Zhang, P., Li, K., Wee, C.Y., Wu, Y., Shen, D., Yap, P.T.: Improving estimation of fiber orientations in diffusion MRI using inter-subject information sharing. Sci. Rep. 6, 37847 (2016)

Supervised Classification of White Matter Fibers Based on Neighborhood Fiber Orientation Distributions Using an Ensemble of Neural Networks Devran Ugurlu, Zeynep Firat, Ugur Ture and Gozde Unal

Abstract White matter fibers constitute the main information transfer network of the brain and their accurate digital representation and classification is an important goal of neuroscience image computing. In current clinical practice, the reconstruction of desired fibers generally involves manual selection of regions of interest by an expert, which is time-consuming and subject to user bias, expertise and fatigue. Hence, automation of the process is desired. To that end, we propose a supervised classification approach that utilizes an ensemble of neural networks. Each streamline is represented by the fiber orientation distributions in its neighborhood, while the resolved fiber orientations are obtained by generalized q-sampling imaging (GQI) and a subsequent diffusion decomposition method. In order to make the supervised fiber classification succeed in a real scenario where a substantial portion of reconstructed fiber tracts contain spurious fibers, we present a way to create an “invalid” class label through a dedicated training set creation scheme with an ensemble of networks. The performance of the proposed classification method is demonstrated on major fiber pathways in the brainstem. 30 subjects from Human Connectome Project (HCP)’s publicly available “WU-Minn 500 Subjects + MEG2 dataset” are used as the dataset. Keywords Ensemble neural networks · Supervised white matter fiber classification · Brain fiber pathways

1 Introduction White matter tissue in the brain consists of highly myelinated axons that quickly transmit signals over long distances and act as the information transfer network among gray matter regions in the brain, in addition to carrying information from/to D. Ugurlu Sabanci University, Istanbul, Turkey Z. Firat · U. Ture Yeditepe University Hospital, Istanbul, Turkey G. Unal (B) Sabanci University, Istanbul Technical University, Istanbul, Turkey e-mail: [email protected] © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_12

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the brain to/from the body. Study of white matter is hence of tremendous importance in understanding human central nervous system functionality and neurological disorders. As diffusion MRI (DMRI) allows in vivo imaging of white matter, DMRI images are routinely used in clinical practice and neuroscientific studies to reconstruct digital representations of white matter fiber trajectories as a set of “streamlines”. Each streamline is an ordered set of points in 3D space. The process of reconstructing streamlines is called tractography [1, 2]. A full brain tractography, that is, a tractography that uses all voxels in the brain as seeds has several problems that limit its use. Meaningful analysis of white matter fibers requires streamlines that correspond to true anatomical fiber bundles, which are defined through ex vivo fiber dissection [19] or tracking techniques [11]. Even when using sophisticated tractography algorithms, most of the constructed streamlines in a full brain tractogram are invalid, that is, they do not correspond to any true tract [12]. Hence, in order to have an adequate number of streamlines that represent true tracts in a full brain tractogram, millions of streamlines need to be constructed which then need to be classified into classes of known tracts or as invalid. In clinical practice, streamlines that correspond to true bundles are usually reconstructed in two steps that require extensive human interaction. In the first step, in order to reconstruct streamlines that belong to a certain bundle class according to the anatomical definition of the bundle, regions of interest (ROIs) that the bundle is required to pass through are selected by a neuroanatomy expert. In the second step, the neuroanatomy expert removes invalid streamlines by selecting regions of avoidance (ROAs) [9, 18]. Due to the extensive human interaction required, this method can be seen as the manual approach to the fiber classification problem. While providing the most accurate results, the manual approach suffers from significantly long time requirements from experts in addition to operator bias and fatigue. As a consequence, development of automatic fiber classification methods is a vital need and challenge in the area of neurological image computing. Automatic fiber classification methods can be grouped into purely supervised methods and a combination of clustering with some form of supervision. Existing purely supervised methods are usually an attempt to mimic the manual method with automatic ROIs defined on a template space and registering them to the subject space [15, 22, 27]. While producing true bundles in the ideal case, pure automatic ROI methods are reliant on accurate registration and are sensitive to individual variation and pathology. Clustering methods define a fiber-to-fiber dissimilarity or distance measure according to which the fibers are clustered [14, 17]. Clustering normally requires computation of all pairwise distances but since this can be very computation intensive, either a random subset of streamlines or a faster clustering method such as QuickBundles [3] is used in practice. The clusters created by a clustering method may not be valid clusters representing a true bundle, hence clustering is usually combined with some supervision to obtain true bundles [4, 6, 20, 21]. There has been little research that explores the use of a classical supervised classification approach, that is, a method based on learning a mapping from fiber features to class probabilities using a labeled training set of fibers. In [10], the authors used

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heuristic geometric fiber features and a Viola-Jones object detection framework to classify fibers into three classes of bundles. In [7, 8], a volumetric parametrization method is used to allow consistent representation of streamlines without registration and a convolutional neural network (CNN) architecture is used to classify streamlines. Only four training subjects were used for the study however, and it is unclear how invalid streamlines are addressed in the test phase, hence it is not possible to assess the real utility of that method. In a different supervised approach, that can be called a direct volumetric segmentation of fiber bundles, the authors take tractography out of the process and train an encoder-decoder fully connected convolutional neural network (FCNN) to segment binary volume masks for each fiber tract of interest [24]. This method is very fast and volumetrically accurate compared to previous fiber clustering approaches but if streamlines representation of tracts is desired, additional steps are required to construct the fiber tracts. Using a similar network architecture, the authors later created tract orientation maps (TOM) for each bundle [23]. This approach is more suitable for constructing streamlines as the voxels now contain fiber-specific orientation information. In this paper, we propose a streamline-based supervised learning method using an ensemble of neural networks that each map an individual streamline to class probabilities. In our previous work [20], we introduced a fiber representation based on the neighborhood resolved fiber orientation distributions (NRFOD) along the fiber. This representation is translation-invariant and insensitive to small rotations which makes the representation suitable for the supervised learning approach. Each streamline is converted to this NRFOD representation which can be regarded as a heuristic feature vector. An additional contribution of this paper is a novel method to create training sets that include invalid class labels to alleviate the problem of false positives which constitutes a significant problem in any supervised learning approach to fiber classification. The utility of the proposed method is demonstrated on the major brainstem bundles: corticospinal tract (CST), medial lemniscus (ML), middle cerebellar peduncle (MCP), inferior cerebellar peduncle (ICP) and superior cerebellar peduncle (SCP).

2 Materials and Methods 2.1 Dataset, Reconstruction of ODFs and Tractography Diffusion-weighted images (DWI) and corresponding T1 images are from 30 unrelated subjects in the Human Connectome Project (HCP)’s publicly available “WUMinn 500 Subjects + MEG2 dataset”.1 The preprocessed data is used. The ODF

1 http://www.humanconnectome.org/documentation/S500/.

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(Orientation Distribution Function) reconstruction is performed by DSIStudio2 with the generalized q-sampling imaging (GQI) method [25]. At most three orientations are resolved per voxel using an ODF decomposition method [26] and a diffusion anisotropy value (QA) that quantifies the spin population in each resolved orientation is also obtained. For tractography, the method presented in [20] is utilized.

2.2 Manual ROI Selection on Standard Space to Constrain Input Tract Sets As typically, a whole brain tractogram contains millions of streamlines that may not correspond to any true tract, it is generally a good idea to use a rough preprocessing step to reduce the size of the streamline set to be input to a more sophisticated classification algorithm. Five sets of ROIs are defined on the standard MNI152 space [13] to reconstruct a collection of streamlines that will include the following sets of tracts: left projection fibers (left CST and ML), right projection fibers (right CST and ML), SCP, ICP and MCP. Naturally, the ROIs are chosen large enough to account for possible registration errors and individual variations, since this is a preprocessing step meant to reduce computation cost and does not aim to achieve an accurate classification by itself.

2.3 Construction of the Training and Validation Sets Figure 1 depicts an illustration of a crucial procedure in our supervised fiber classification method, which inserts an “invalid” class that represents the class of invalid streamlines that do not belong to any true bundle. We reconstruct the following nine bundles of interest in the brainstem using the manual method for the 30 subjects in the dataset: left CST, right CST, left ML, right ML, MCP, left ICP, right ICP, left SCP and right SCP. ROIs and ROAs are selected by the help of the radiology and neuroanatomy experts. An important issue that needs to be addressed when implementing a streamline-based supervised classification method is how to deal with invalid streamlines. Due to the existence of invalid streamlines, the classification problem is an “open set” recognition problem [16], which means that during the test phase, we expect to encounter streamlines that do not belong to any of the bundles the classifier has been trained on. Our approach in this study is to create a class of invalid streamlines in addition to the true bundles of interest, hence turning the problem into “closed set” recognition. Of course, the problem of how to create a training set for the “invalid” class remains. One way would be to use the streamlines that were deleted by the ROAs during manual classification, as in [7]. However, this would capture only a small subset of invalid streamlines that are likely to be present 2 http://dsi-studio.labsolver.org.

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in an input set of streamlines encountered during the test phase because the ROIs used in manual reconstruction are much stricter and more accurate than the automatic ROIs used to constrain the streamline sets for the automatic classification. Another approach would be to use the automatic ROIs and many ROAs in the manual reconstruction, however this has the downside of requiring an extreme amount of time and adding additional human bias to the training set as well as being specific to the automatic ROIs chosen for this particular study, hence limiting the generalizability of the approach. Instead, we propose the following practical and important component to creating the training set which is also illustrated in Fig. 1: 1. Convert the 9 bundles created with the manual classification method to binary volume masks. 2. Create an automatic streamline set using the automatic ROIs defined in Sect. 2.2. 3. Convert each streamline in the automatic input streamline set to a binary volume mask and calculate the one-way overlap to each bundle mask representing the true bundles and calculate the maximum overlap. 4. If the maximal overlap to true classes is higher than threshold Th , assign to the streamline the class label of the true class for which the maximum overlap is achieved. 5. Otherwise, if the maximal overlap to true classes is lower than threshold Tl , assign to the streamline the label of the invalid class. Notice that streamlines for which the maximal overlap is between Tl and Th will not be assigned any label. This is on purpose, as we do not want to assign a label to streamlines for which it is unclear whether they are invalid or not. These streamlines are removed from the training set, for the purpose of avoiding potential false positives in the classification. The above procedure is repeated for every subject data in the training set. Using this approach, we create six different training and validations using all combinations of threshold parameters Tl = {0.70, 0.75, 0.80}, Th = {0.85, 0.90}. These parameters are selected heuristically. The first 20 subjects of the 30 subjects are used for the training set and the next 5 subjects are used for the validation, leaving the last 5 subjects for the testing phase. A maximum of 200 streamlines per true class per subject and 2000 streamlines per invalid class per subject are used.

2.4 Network Architectures and Training Since the NRFOD representation we previously proposed in [20] is found to be an expressive feature vector, a simple neural network classifier is used with 3 fully connected layers and ReLU activation functions. The network architecture is illustrated in Fig. 2. The NRFOD representation of a fiber is a set of orientation histograms calculated at each fiber point and accumulated over multiple directions on an approximately spherical region centered at the given point. NRFOD parameters were set as K = 50, M = 20, N = 10, h = 6, r = 3 same as in [20]. Since we have 6 different

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Fig. 1 Illustration of the proposed training set creation scheme. For every subject in the training set, Step 1: Manually created streamlines for each fiber tract are converted to a volumetric mask. Depicted tract masks are colored as follows: Red: MCP, Blue: left CST, Green: left ML, Yellow: left ICP, Pink: left SCP. Step 2: A collection of streamlines are constructed by tractography using automatic ROIs. In step 3 and later, the process is illustrated for one streamline. This is repeated for every streamline created in step 2

training sets, as described in Sect. 2.3, we train one classifier using each training set to obtain a total of 6 different classifiers. The networks are implemented in PyTorch,3 stochastic gradient descent is used for optimization with learning rate 0.001, and momentum 0.9. Loss function is chosen as the cross entropy loss. The 6 classifiers are separately trained for 5 epochs with a batch size of 10 on all the training data. After each epoch, accuracy on the validation is calculated and the final model of each classifier in the ensemble is selected as the model that achieves the best accuracy.

2.5 Testing The test set consists of streamlines reconstructed from 5 subjects using the automatic ROIs. During the testing phase, we require unanimity from all 6 classifiers on the

3 http://pytorch.org.

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Fig. 2 The architecture of the classifiers in the ensemble

class assignment of an input streamline for that streamline to be assigned a label of a true fiber bundle. If there is no unanimity, the streamline is considered invalid.

3 Results and Discussion The performance of the proposed method is quantitatively compared to our previous method proposed in [20] which exemplifies a recent fiber-to-fiber distance based clustering method. Two quantitative measures, Bundle-based Minimum Distance (BMD) [5] and Cohen’s Kappa are used to assess the “closeness” of the bundles created by the automatic methods to the manually created bundles. Since manual bundles are regarded to be the closest approximation to the ground truth, which is unknown, they are used as “ground truth” for comparing automatic methods. For calculation of Cohen’s Kappa values, all bundles are converted to binary volume masks using nearest neighbors. The quantitative comparison results are given in Table 1. Visualization of the bundles extracted with the proposed method and the manual method are given in Fig. 3 for qualitative comparison. In addition, separate visualizations of each bundle, the input set of streamlines and streamlines classified as invalid for one subject in the test set are visualized in Fig. 4. Note that the difference between the input set and streamlines classified as invalid is hard to see in the visualization despite the fact that the input set contains 25,000 streamlines and the invalid set contains 12,433 streamlines. There are two main reasons for this. One is that small variations of individual streamlines from the true tract are very difficult to see because of the clutter caused by thousands of streamlines. The second reason is that some streamlines may be falsely classified as invalid because the proposed classifier is quite strict with the unanimity requirement from the ensemble. We made this choice because false negatives are not really a problem as long as enough streamlines are classified as valid to reconstruct the tracts with their true thickness. Even one false positive on the other hand, can significantly change the volume of the bundle and is also easily noticeable on the visualization. Despite the simplicity of the proposed method, it generally outperforms a recent state-of-the-art fiber classification method based on clustering with respect to the quantitative measures. Moreover, visual comparison with manually created bundles

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Fig. 3 Visualization of bundles of interest extracted from the 5 test subjects. A: Manual method for subjects 1–3; B: Proposed method for subjects 1–3; C: Manual method for subjects 4–5; D: Proposed method for subjects 4–5. streamlines are colored according to their bundle label as follows: CST Left: blue; CST Right: purple; ML Left: pink; ML Right: teal; ICP Left: yellow; ICP Right: green; SCP Left: brown; SCP Right: white; MCP: red

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Fig. 4 Visualization of input streamlines and results of the proposed method on one subject in the test set. In top-bottom, left-right order: The input set of streamlines created with the automatic ROIs; invalid streamlines; MCP; left CST; right CST; left ML; left SCP; right SCP; right ML; left ICP; right ICP

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Table 1 Quantitative comparison of the proposed method and our previous clustering-based method [20]. Manually created bundles are used as “ground truth” and the mean ± standard deviation of BMD and Cohen’s Kappa values over the test set between the bundles created by the automatic methods and the manual bundles are given for each true bundle class. The best results for both measures are highlighted in bold. Note that BMD is a dissimilarity measure and Cohen’s Kappa is a similarity measure, i.e. higher values represent better accuracy for Cohen’s Kappa and lower accuracy for BMD CST_Left

Proposed_BMD

UgurluClustering_BMD

Proposed_Kappa

UgurluClustering_Kappa

1.49 ± 0.28

3.12 ± 3.62

0.81 ± 0.03

0.62 ± 0.22

CST_Right

1.26 ± 0.22

1.07 ± 0.23

0.84 ± 0.01

0.80 ± 0.02

ML_Left

1.80 ± 0.58

1.40 ± 0.47

0.77 ± 0.02

0.68 ± 0.11

ML_Right

2.92 ± 1.76

4.62 ± 7.37

0.72 ± 0.13

0.62 ± 0.14

SCP_Left

4.64 ± 2.41

6.40 ± 4.90

0.73 ± 0.10

0.64 ± 0.12

SCP_Right

4.43 ± 1.72

6.67 ± 5.70

0.71 ± 0.08

0.63 ± 0.17

ICP_Left

2.08 ± 1.76

2.11 ± 1.23

0.71 ± 0.08

0.63 ± 0.14

ICP_Right

1.92 ± 0.90

1.46 ± 0.77

0.68 ± 0.09

0.70 ± 0.05

MCP

5.46 ± 2.05

6.19 ± 1.70

0.74 ± 0.05

0.73 ± 0.05

indicate that the proposed method indeed usually produces anatomically meaningful classifications. Since the proposed method does not require pairwise fiber-to-fiber distances unlike most previous clustering approaches, it is faster on the online classification phase in comparison and scales better with increasing number of fibers in the input set. These results have two important implications: (i) Classical supervised classification approach is suitable for the fiber classification task and should be investigated further. (ii) Representing a fiber based on the orientation distributions in its neighborhood is a powerful approach that allows robust comparison of fibers extracted from different subject spaces.

4 Conclusion We presented a novel fiber classification method that applies the classical supervised classification approach, which maps an input vector to class probabilities, to the fiber classification task. The proposed method quantitatively outperformed a recent exemplary fiber clustering method, and was also shown to be visually satisfactory. The proposed training set creation method that includes labeling a wide variety of invalid streamlines, combined with the ensemble of networks that utilizes the created training sets; (i) works in disfavor of false positives; (ii) allows the usage of bundles constructed by a routine manual extraction without requiring any extra manual effort. This makes it easier to extend the training data to include more subjects and more bundle classes in the future. Future work involves comparing the proposed method to a greater variety of other methods in literature and for more classes of tracts.

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Furthermore, one direction our work will be extended is to train deeper networks that directly use the orientation information in the ODF volume, and compare it to the use of the neighborhood orientation information in the NRFOD feature vector. Acknowledgements This work was supported by TÜBITAK (The Scientific and Technological Research Council of Turkey) Grant No. 118E169.

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14. O’Donnell, L.J., Kubicki, M., Shenton, M.E., Dreusicke, M.H., Grimson, W.E.L., Westin, C.F.: A method for clustering white matter fiber tracts. AJNR Am. J. Neuroradiol. 27(5), 1032–1036 (2006) 15. Oishi, K., Zilles, K., Amunts, K., Faria, A., Jiang, H., Li, X., Akhter, K., Hua, K., Woods, R., Toga, A.W., Pike, G.B., Rosa-Neto, P., Evans, A., Zhang, J., Huang, H., Miller, M.I., van Zijl, P.C., Mazziotta, J., Mori, S.: Human brain white matter atlas: Identification and assignment of common anatomical structures in superficial white matter. NeuroImage 43(3), 447–457 (2008) 16. Scheirer, W.J., de Rezende Rocha, A., Sapkota, A., Boult, T.E.: Toward open set recognition. IEEE Trans. Pattern Anal. Mach. Intell. 35(7), 1757–1772 (2013) 17. Siless, V., Chang, K., Fischl, B., Yendiki, A.: AnatomiCuts: hierarchical clustering of tractography streamlines based on anatomical similarity. NeuroImage 166, 32–45 (2018) 18. Stieltjes, B., Kaufmann, W.E., van Zijl, P.C., Fredericksen, K., Pearlson, G.D., Solaiyappan, M., Mori, S.: Diffusion tensor imaging and axonal tracking in the human brainstem. NeuroImage 14(3), 723–735 (2001) 19. Ture, U., Yasargil, M.G., Friedman, A.H., Al-Mefty, O.: Fiber dissection technique: lateral aspect of the brain. Neurosurgery 47(2), 417–426 (2000) 20. Ugurlu, D., Firat, Z., Ture, U., Unal, G.: Neighborhood resolved fiber orientation distributions (NRFOD) in automatic labeling of white matter fiber pathways. Med. Image Anal. (2018) 21. Wassermann, D., Bloy, L., Kanterakis, E., Verma, R., Deriche, R.: Unsupervised white matter fiber clustering and tract probability map generation: applications of a gaussian process framework for white matter fibers. NeuroImage 51(1), 228–241 (2010) 22. Wassermann, D., Makris, N., Rathi, Y., Shenton, M., Kikinis, R., Kubicki, M., Westin, C.F.: The white matter query language: a novel approach for describing human white matter anatomy. Brain Struct. Funct. 221(9), 4705–4721 (2016) 23. Wasserthal, J., Neher, P.F., Maier-Hein, K.H.: Tract orientation mapping for bundle-specific tractography. ArXiv e-prints, June 2018 24. Wasserthal, J., Neher, P.F., Maier-Hein, : K.H.: TractSeg - fast and accurate white matter tract segmentation (2018). CoRR abs/1805.07103. arXiv:1805.07103 25. Yeh, F.C., Wedeen, V.J., Tseng, W.Y.I.: Generalized q-sampling imaging. IEEE Trans. Med. Imaging 29(9), 1626–1635 (2010) 26. Yeh, F.C., Tseng, W.Y.I.: Sparse solution of fiber orientation distribution function by diffusion decomposition. PLOS ONE 8(10), 1–12 (2013) 27. Zhang, Y., Zhang, J., Oishi, K., Faria, A.V., Jiang, H., Li, X., Akhter, K., Rosa-Neto, P., Pike, G.B., Evans, A., Toga, A.W., Woods, R., Mazziotta, J.C., Miller, M.I., van Zijl, P.C., Mori, S.: Atlas-guided tract reconstruction for automated and comprehensive examination of the white matter anatomy. NeuroImage 52(4), 1289–1301 (2010)

Part III

Diffusion MRI Signal Harmonization

Challenges and Opportunities in dMRI Data Harmonization Alyssa H. Zhu, Daniel C. Moyer, Talia M. Nir, Paul M. Thompson and Neda Jahanshad

Abstract Advances in diffusion MRI (dMRI) have led to discoveries of factors that affect brain microstructure and connectivity in health and disease. The small size of many neuroimaging studies led to concerns about poor reproducibility of research findings, and calls for the comparison and pooling of multi-cohort datasets to establish the consistency of reported effects. Across studies diffusion MRI protocols vary in spatial, angular and q-space resolution, b-value, as well as hardware used—all of which affect measured diffusion parameters. Efforts to compare and pool dMRI measures use meta- or mega- analytical techniques to compensate for these sources of variance. Meta-analytical methods gauge the consistency of effects, and megaanalytical methods involve mathematical or statistical transformations of the data. Here, we review some recent advances that allowed the diffusion community to create large scale population studies with greater rigor and generalizability than was previously attainable by individual studies. Keywords Multi-site · Harmonization · diffusion MRI · DTI · DWI

1 Introduction Harmonization, or standardization of scientific data to enable comparisons, is vital to improve the interpretability of research findings. Recent concerns about the poor reproducibility of research findings in neuroscience have stimulated interest in data harmonization—such methods are crucial to enable comparison and pooling of data from diverse sources. This in turn enables researchers to aggregate and compare datasets and pool evidence from different sources, increasing power. Comparisons of datasets are crucial to enable us to test reproducibility of research findings, and

A. H. Zhu · D. C. Moyer · T. M. Nir · P. M. Thompson · N. Jahanshad (B) Imaging Genetics Center, Stevens Neuroimaging and Informatics Institute, Keck School of Medicine of the University of Southern California, Marina del Rey, CA 90292, USA e-mail: [email protected]

© Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_13

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allow us to be more confident in the generalizability (or otherwise) of findings that have been made in single cohorts. As recently as 10 years ago, most brain imaging studies were limited to a single site, with a single scanner, and single acquisition protocol. Many studies aimed to understand population differences between one group and another, yet attempts at replication were not often considered. Over the last several years, researchers in the brain imaging field—and the biological sciences in general—began to realize that they were faced with a reproducibility crisis [1–3]. Null findings were rarely reported, and findings from small cohorts with nominally significant p-values were dominating publications with few attempts at replication. In the search for consistent patterns in the underlying neurobiology of disease, often studies of different cohorts did not agree on which brain regions were significantly different between different diagnostic groups—such as patients with schizophrenia and healthy controls [4]—or any group A and group B. Differences in results across studies can arise due to both methodological differences and biological differences in the cohorts assessed, or due to differences in selection criteria for a given study. However, a complete lack of reproducibility of results may suggest that findings are false positives. In particular, for diffusion MRI (dMRI), where slight variations in acquisition protocols or processing pipelines can affect extracted measures, it is important to know the sources of discrepancies among published findings, what factors contribute to the variations in findings, and whether they can be addressed. Button et al. [5] drew attention to the low statistical power in many areas of neuroscientific research, while others criticized the alarmingly poor reproducibility of scientific research across many fields [6]. In both functional and structural neuroimaging, many studies of disease or genetic effects on the brain had examined imaging data from less than 50 individuals, limited by the cost of data collection and the time needed to recruit and scan a large cohort. Many study designs were insufficiently powered to detect the effects of interest, yet null findings often went unreported. Just as in the behavioral genetics field several years earlier, some initial studies reported effects that were overly optimistic; the failure to replicate many initial findings in both neuroimaging and genetics led some to note that underpowered studies were often subject to the ‘winner’s curse’—initial reports of effects that were weaker or not detectable at all when tested in new and independent data. A separate, but related, concern was that some widely-used methods to adjust for multiple comparisons were failing to control the rate of false positives when tested empirically [7] leading to calls for researchers to either register their hypotheses in advance and replicate findings when a vast number of hypotheses could be tested a priori [8]. Underpowered studies can lead to both false positives and false negatives, which can damage efforts to advance science and misdirect future scientific efforts. Inflated effect sizes also pose a problem, as new studies planned using these effect sizes as benchmarks would vastly underestimate the sample sizes needed to detect true

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differences, leading to cycles of underpowered small studies with false, unreplicated, results. The realization that findings were failing to replicate began to break this cycle, and some funding agencies mandated that researchers plan to assess the rigor and reproducibility of findings.1 Now, the need to harmonize data—to test for reproducibility—has become vital and more universally recognized. Large national biobanks, such as the UK Biobank (UKB), are now being established to overcome the problem of small sample sizes—in UKB, nearly 100,000 individuals over age 45 from the British population are being scanned, with a target completion date of 2022 [9]. This unique sample provides sufficient statistical power and new opportunities to identify factors that affect brain structure or function, such as single genetic variants that typically explain around 0.5–1% of the population variance in a trait, such as volume measures from MRI [10]. While single-scanner biobank studies avoid the need for inter-scanner harmonization, intra-scan reproducibility may still need to be addressed, and scanner drifts monitored. Gathering hundreds or thousands of imaging data samples, from a single scanner, is often not feasible due to cost and time constraints. In particular, for studies of rare disorders, there may only be a relatively few number of patients in a given geographic region, limiting the achievable sample size. Even the UKB is a cohort that is somewhat healthier than the general British population, is limited in the age range assessed, the geographic reach, and in the number of patients with neuropsychiatric conditions in the sample. By pooling data across centers recruiting targeted populations, larger sample sizes can be achieved, and hence greater statistical power to detect reliable case versus control effects. Data harmonization methods are now allowing smaller (non-biobank) studies to become much more valuable in aggregate than on their own. This can allow for findings to be reproduced with greater confidence, and provide the foundation to understand common and consistently identifiable underlying neurobiological mechanisms for neuropsychiatric disease in groups of individuals around the world. Comparison of data across cohorts can also help in identifying cohort specific effects that are unique to a cohort and may not be found in other settings. In short, whether the effects are consistent universally or found only in some settings, the generality and scope of the effects can be better identified by bringing data together from multiple sources, and data harmonization is crucial for doing this. A range of brain imaging data harmonization approaches, applications, along with challenges, caveats and opportunities for advancements, will be discussed in this chapter, with specific attention to diffusion MRI.

1 https://grants.nih.gov/reproducibility/index.htm.

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2 Challenges Underlying Diffusion MRI Data Harmonization for Prospective and Retrospective Data Collection Depending on the study design, and the data to be harmonized, different approaches to harmonization are available. Here, we first introduce some of the more prevalent study designs in neuroimaging, and delve into the challenges and variations in diffusion imaging protocols that lead to the need for harmonization.

2.1 Study Designs Prospective: Multi-site data collection has enabled large-scale imaging initiatives such as the North American Alzheimer’s Disease Neuroimaging Initiative (ADNI [11, 12]) and the Parkinson’s Progression Markers Initiative (PPMI [13]). These initiatives scan participants around North America or around the world, boosting the power to address specific questions about neurodegeneration. We refer to these initiatives as prospective multi-site studies because efforts were made to harmonize some aspects of image collection in advance. For example in ADNI-2 - the second phase of ADNI but the first to include dMRI—the same dMRI acquisition protocol was used at every scanning site, on scanners from the same manufacturer, at the same field strength. The ADNI2 diffusion sequence can be found at: http://adni.loni.usc. edu/wp-content/uploads/2010/05/ADNI2_GE_22_E_DTI.pdf. Prospective studies may use phantoms, either manufactured to have a prescribed geometry and chemical composition, or ‘human phantoms’ to scan at each site in a study, in an effort to quantify inter-site variability [14, 15]. Retrospective: As public data repositories are becoming more common, many studies are being planned using available, already collected, data; some aim to pool data from several cohorts. This presents an added challenge of not being able to collect new data to quantify site specific variations, although site effects can be modeled analytically, so long as they are not confounded with other variables of interest in the study design. These studies have motivated many methodological developments in data harmonization, and much of the focus of our chapter will be on methods to harmonize retrospective data collections.

2.2 No Two Diffusion Images Are the Same Single scanner variability. Before we delve into the challenges of multi-site harmonization, it is important to understand sources of variability in the images produced from a single scanner, even if the same subject or object is scanned repeatedly with

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Fig. 1 Test-retest diffusion scans of a single subject from the Consortium for Reliability and Reproducibility dataset. a, b, d, e Raw b0 images with the subject in different positions. The red line in the sagittal images denotes the axial slices. c, f FA images, each registered to the same atlas, displaying a difference of 0.2 in the right cerebellar peduncle (red circle) between scans

the same protocol for longitudinal studies. For example, a subject’s position in the scanner can affect dMRI results as demonstrated in Fig. 1. Scanner drift is the instability of scanner parameters over time; such variability within a single scanner means that scans taken of the same object with the same acquisition parameters, but at different times are not identical [16, 17]. This signal drift may result from factors such as heating of the gradient system [18], or changes in the calibration of the magnetic field gradients in the scanner. This is particularly apparent in gradient-heavy sequences, including diffusion-weighted imaging, where the echo-planar imaging approach increases distortions in the phase direction and decreases the efficacy of fat saturation pulses [19]. To adjust for a shift within a scan session, Vos and colleagues [17], applied a correction that relies on evenly spaced b0 scans—a scanning protocol requirement specified in advance as part of the data collection paradigm. Variability in acquisition protocols and scanning hardware. Inter-site variability of MRI scans may be driven by differences in acquisition parameters, head coils, imaging gradient non-linearity, software version, image reconstruction algorithm, scanner magnetic field strength, and scanner manufacturer [31–35]. All the parameters and factors affecting standard anatomical MRI also affect diffusion imaging,

4T 3T 4T simulated 1.5 T 4T

3T

Scanner field strength (Siemens and Philips)

Scanner field strength (Siemens)

Voxel size (1.8, 2, 2.5, isotropic mm3 ) b-values (700, 1000 s/mm2 ) b-values (700, 1000, 1300 s/mm2 ) Number of b-values/shells

Angular resolution Angular resolution

Spatial (2.0, 2.5, 2.7 isotropic mm3 ) versus angular resolution Spatial (2.0, 2.5, 2.7 isotropic mm3 ) versus angular resolution

Pagani et al. [22]

Zhan et al. [23]

Papinutto et al. [24] Bisdas et al. [25] Papinutto et al. [24] Correia et al. [26]

Giannelli et al. [27] Zhan et al. [28]

Zhan et al. [29] 3T

1.5 T, 3 T

1.5 T, 3 T

White matter: FA more similar intra-site than inter-site Significant site differences in FA; Field strength had no detectable effect on FA No detectable effect on FA or apparent diffusion coefficient (ADC) Field Strength had no detectable significant effect on FA, increases diffusivity FA measures were similar between the two; some regional differences exist; diffusivity was lower with 7 T; 7 T had higher brain network efficiency Larger voxels: Increased diffusivity; decreased FA Higher b-values: Generally lower FA Higher b-values: Lower diffusivity and FA Increasing the number of shells improves FA and ADC accuracy FA changes significantly; MD does not More gradients provide higher signal-to-noise ratio (SNR); profile of increase in SNR varies for FA, diffusivity and non-tensor metrics Higher spatial resolution, even at the cost of angular resolution, can improve tractography There is a trade-off between spatial and angular resolution; in a given time frame, both should be considered

Findings

Here we list a selection of papers, noting the particular scanning parameter tested, the field strengths of the scanners used, and a brief description of the findings

Jahanshad et al. [30]

3 T, 7 T

Scanner field strength (Siemens)

Brander et al. [21]

3T 1.5 T, 3 T

Scanner (GE) Scanner field strength (Siemens)

Vollmar et al. [14] White et al. [20]

Field strength

Variable/Parameter evaluated

Publication

Table 1 Effects of differences in scan parameters on diffusion metrics have been assessed in several publications

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yet there are several other sources of additional variability that also affect diffusion MRI measures, such as the angular resolution of the data and the protocol b-values (diffusion weightings)—all aspects of the overall q-space sampling scheme used to collect the information on diffusion. Table 1 summarizes a selection of studies that evaluate parameter changes. In the next section, we describe potential options for harmonization and suggest when certain methods may be more appropriate for a given study.

3 Current Harmonization Approaches The methods for combining data across multiple sites can be categorized into two broad categories: meta-analysis, where statistical inferences are pooled across datasets, and mega-analysis, where individual level data are combined before statistical inferences are made. Multiple approaches are available for these categories. Here, we briefly describe meta-analysis, and two broad forms of mega-analysis including: (1) global shifting of site measures in the pooled analysis by modeling site effects at the time of overall group-inference; (2) transforming individual data prior to the multi-site group-level statistics. As we will describe, the latter option does not necessarily preclude the use of statistical adjustments at the multi-site level. Each category of harmonization methods has its benefits and drawbacks, and there is no uniformly “best” solution. The method of choice may depend on the study design and the feasibility of the method given the data available.

3.1 Meta-Analysis In a meta-analysis, a statistical model is fitted to data for a number of different studies, and a weighted mean of the effects across studies is calculated. Individual study effects are weighted by either their sample size or by some measure of the precision of the estimates, as in inverse-variance weighted meta-analysis. Recent efforts in consortia, such as ENIGMA [36, 37], have introduced coordinated meta-analysis to the field of neuroimaging. Diffusion data have been analyzed similarly across multiple sites and data at each site was fit with the same statistical model; this harmonizes the data analysis across each site. Resulting effect sizes at each site are then tested for heterogeneity and pooled [4, 38]. Historically, metaanalysis studies in neuroimaging, and many other fields, pool published results; however, as discussed previously, these may be biased by the tendency to publish significant findings, and not null results.

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3.2 Mega-Analysis We use ‘mega-analysis’ to refer to the case where the individual subject-level data is available for harmonization. Unlike reports of measures derived from structural MRI, where for the same test, mega-analysis has been found to be more powerful than meta-analysis [39], for diffusion imaging, Jahanshad and colleagues have found that meta-analysis consistently results in lower standard errors on the effects, and tighter confidence intervals [40]. While measures from structural MRI, such as regional volumes, represent physical quantities, measures from diffusion imaging are derived from mathematical models of diffusion that depend on the acquisition protocol. Regression level corrections. When the individual data measures, such as regional fractional anisotropy (FA) per subject, are available as opposed to study summary statistics, it is possible to pool these measures, and calculate effect sizes across the entire group; even so, it is necessary to model the site differences to account for the sources of variability mentioned in the previous section. We can either model the mean effect of each site as being simply different from the next (fixed), or consider the site as a random sample from a population of sites (random). A fixed effects model includes site effects as a categorical covariate, equivalent to a series of site indicator variables. These models extend the general linear regression model by adding additional covariates (specifically one less than the number of sites). Fixed effects models place no constraints on the distribution of site coefficients as a whole, but merely adjust for the fact that the groups of variables (individual measures from different sites) have different means and variances; i.e., in the general linear regression model, each group is assumed to have a normal distribution, but with fixed effects modeling, we acknowledge group differences in means and spread. However, two studies may use data from a different set of sites, each with their own distributions and the site difference effects (z-score corresponding to the site covariate) would not translate from one multi-site study to another. In multi-site studies, scanner effects are often confounds rather than variables of interest. If enough sites are being pooled to model the sites as random samples from a population of sites or scanners [41], this could provide a more reproducible solution. Random effects models treat the site-wise effect as random draws from a (parametric) distribution of possible effects. This amounts to finding both site-specific effects as well the distribution parameters for those effects [42], often using maximum likelihood (ML) or restricted maximum likelihood (REML) methods. As detailed in Table 1, intra-subject variability tends to be higher across sites than within sites, even when acquisition protocols are kept identical across scanners, and scanner manufacturer and field strength are consistent. Even studies that collect images with the same protocol to minimize variability may benefit from adjusting for the inter-site variability during the analysis. For such multi-site cohorts, where the diffusion imaging protocol is the same across sites, random effects models may be successful, as in the case of ADNI2, for example [43, 44].

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Data transformation before multi-site statistics. In addition to the modeling of site effects in the statistical analysis, individual-level measures may also be transformed before statistical inferences are made. A rigorous option to validate harmonization methods—which can also guide harmonization itself—is the collection of a ‘traveling substrate’ dataset [15] where either human subjects or physical, manufactured, phantoms are sent to each site and scanned. Here, the assumption is that precisely the same object is being scanned. While this assumption may not always hold true,2 knowing how the image of the same object is different from one scanner to the next, or even from one diffusion acquisition protocol to the next, can guide how to transform images from one dataset to another. Many multi-site studies have used phantoms to guide the post hoc harmonization efforts such as [46]. Venkatraman et al. [47] scanned participants at multiple scanners and time points to create region-specific linear correction factors from cross-sectional and longitudinal training datasets. The cross-sectional linear correction factor was derived from a general linear model with scanner and age as covariates. The longitudinal linear correction factor was derived from a linear mixed effects model with scanner, time interval, and age as fixed effects and included random effects as subject-specific intercepts. Reliability estimates of test datasets showed improvement with linear correction factors from both training sets, though the longitudinal dataset provided greater improvement. In [38], Jahanshad et al. formulated meta-analyses for variance components to estimate heritability (i.e., the proportion of the total variance explained by additive genetic factors) in regional FA measures across two sites. In a follow-up work, Kochunov et al. [48] extended the number of cohorts used to assess FA heritability from two to five. Here, the FA measures for each cohort were adjusted for demographic factors including age and sex, and an inverse Gaussian normalization of the distribution of FA measures was performed for each site. Once all sites had a normal distribution, data were pooled for mega-analysis. This mega-analysis approach was compared to the meta-analysis and found to be comparable. Fortin et al. [49] evaluated several harmonization techniques for diffusion tensor imaging (DTI)-based measures and found that the ComBat method [50] appeared to outperform others in reducing inter-site variability. ComBat was originally developed to correct for batch effects in gene expression assays, extending previous models [51] into an empirical Bayes framework. The ComBat method has informative parametric priors set via an empirical Bayes iterative updating scheme, leveraging conjugacy for efficient computation. The authors claim that such a fitting procedure makes model robust to larger numbers of sites with smaller within-site sample sizes. ComBat has also been used to improve multi-site harmonization of neurite orientation dispersion and density imaging (NODDI) measures [52]. 2 This

assumption is not necessarily true for either type of phantom: for the human phantom, slight brain morphometry changes have been observed in the course of a day [45], and repeat scans on the same scanner will not be identical; for a manufactured phantom, transportation of the phantom may affect its geometry.

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When pooling data across studies it is important to account for population differences between merged cohorts or sites. In [49], the cohorts pooled were matched for age, sex and other demographic variables. Cohort demographics, such as mean age and sex distribution, have also been shown to affect study results [53–57]. Harmonization methods often aim to find a correspondence between matched groups in the different datasets. These demographically matched, often “normal” samples should have similar distributions of non-imaging level covariates, such as age, sex, ethnicity or other factors. This highlights the need for caution, as the desired output is to harmonize with respect to acquisition protocols, and we may not want to attempt to match the data from sites exactly if basic population characteristics are different. The methods described above use summary, or regional measures, already extracted from the individual images (e.g., the mean FA of the corpus callosum, mean diffusivity (MD) of the whole brain, generalized FA, etc.). Harmonizing one of these summary measures will almost certainly require a different transform than harmonizing another; even region by region transformations are often different [47]. Instead of matching the distribution of derived summary measures, we could attempt to create correspondences between the image volumes themselves, or use information from the images as whole to adjust for acquisition variability. Using standard quality assurance practices, Kochunov and colleagues [58], model the noise in the datasets and use this information as empirical shifting parameters in the regression model. While this covariate is applied to the regressions performed on the derived measures, the noise information is obtained on the level of the image volume. In a recent line of work, Mirzaalian and colleagues [59] map site correspondences between raw diffusion images, moving away from derived measures such as FA. They register a voxel-wise spherical harmonic projection of the images to a template to derive local correction parameters. This has only been applied to multi-site data with similar protocols, but extensions of this work are underway.

4 Building Blocks for Diffusion and Tractography Harmonization: Reproducibility and Validation Diffusion MRI simulation methods could provide simple, reproducible baseline data for different harmonization methods. The most recent update to the FSL’s Physics-Oriented Simulated Scanner for Understanding MRI (POSSUM) framework allows for diffusion MR simulations [60, 61] based on specified radiofrequency pulses, direction and magnitude of diffusion weighting, and artifacts. By modifying these parameters, it is possible to simulate data collected across several sites. In [61], Graham and colleagues address the framework’s ability to assess the efficacy of post-processing methods on data acquired with varying scan parameters but do not use it to address single scanner variability.

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Harmonization methods for tractograms (the set of streamlines that result from tractography) and resulting connectomes are still in their infancy. Studies evaluating their validation and reproducibility are currently underway. Many tractography algorithms have been developed, but different methods typically produce different results from the same starting set of dMRIs, leading to different classification accuracies for diseases such as Alzheimer’s disease [64, 65]. Their reproducibility can be evaluated by comparing tract overlap, tract summary measures (e.g., mean FA), and connectivity profiles; reproducibility rates differ across diffusion reconstructions, and can be inconsistent across acquisitions [66]. The lack of ground truth makes it difficult to evaluate the accuracy of the tractograms and connectomes; and this has launched a series of international diffusion challenges over the last several years (see Table 2). The International Society for Magnetic Resonance in Medicine (ISMRM) 2015 Tractography Challenge used Human Connectome Project (HCP) data to simulate a dMRI and invited researchers from all over the world to submit their tractography pipelines for evaluation [62]. They found that most tractography pipelines were

Table 2 Recent diffusion imaging and tractography challenges Conference and Task (goal) Findings challenge title ISMRM 2015— Tractography Challenge

Reconstruct a tractogram simulated from HCP data; validity

ISMRM 2017— Tractographyreproducibility Challenge with Empirical Data (TraCED) CDMRI 2017—Diffusion MRI data harmonisation

Create tractograms on a scan-rescan data set; reproducibility

Harmonize single-shell data within a standard protocol and between high and low image quality; harmonization ISBI 2018—3D Reconstruct tractograms Validation of of one phantom and two Tractography with histologically traced Experimental primate brains; validity MRI (3D VoTEM) CDMRI Harmonize multi-shell 2018—Multi-shell data within a standard Diffusion MRI protocol and between Harmonisation high and low image Challenge 2018 quality; harmonization (MUSHAC)

Most pipelines recovered 90% of the ground truth bundles to some extent, but false-positive bundles were common The datasets remain available online for researchers to test their tractography algorithms

Publication Maier-Hein et al. [62]

In progress

The lack of reverse Journal paper in phase-encoding acquisitions progress for susceptibility distortion correction could be an issue for data harmonization [63] NA NA

At the time of writing, this challenge has not been completed

NA

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able to recover 90% of the ground truth bundles to some extent. Unfortunately, false positive bundles were also common, and some were even consistent between pipelines. Diffusion challenges have also been held on tractography reproducibility. ISMRM 2017 hosted a Tractography-reproducibility Challenge with Empirical Data (TraCED). These challenges and studies establishing the validity and reproducibility of tractography build the foundation necessary for future tractogram harmonization efforts. The current Computational Diffusion MRI (CDMRI) challenge is structured around data from two scanners using two different protocols each. The competition is to map scans from one scanner and one protocol to each of the other scanner/protocol pairs. The success of these estimates is then evaluated in independent, ‘held out’ scans. The ability to map from one scanner to many others is a harmonization task, yet another possibility could be to map many heterogeneous scanner/protocol pairs to a single standard. Current methods often use information from only one source and one target domain; however, we could instead take advantage of patterns in the mappings between scanners, as demonstrated in multi-task learning settings [67]. In the 2017 CDMRI challenge, participants predominately approached the problem using deep learning, which is likely to play an important harmonization role in the near future. In addition to the data available from these challenges, there is a growing number of publicly available dMRI datasets that can be used to test retrospective harmonization procedures. • The Consortium for Reliability and Reproducibility (CoRR; http://fcon_1000. projects.nitrc.org/indi/CoRR/html) collects and shares dMRI and resting state functional MRI data from scanners worldwide to enable evaluations of the testretest reliability and reproducibility of structural and functional connectomics. There are numerous datasets that make up CoRR. • Human Connectome Project (HCP; https://www.humanconnectome.org/) has acquired and released a high-quality dataset of at least 84 individuals with both 3 T and 7 T data, allowing for harmonization across higher field strengths. • ADNI (http://www.adni.loni.usc.edu) updates their imaging protocols to reflect major advancements in the imaging field and there are currently 80+ subjects scanned with two protocols. – ADNI2 acquired dMRI data from approximately one third of enrolled participants in the subset of sites with 3 T GE scanners (using one acquisition protocol). Many subjects were rescanned within three months of the original visit, providing rescan data for older individuals. – ADNI3 incorporated dMRI protocols for Siemens and Philips scanners to maximize the number of participants scanned [11], resulting in data acquired with seven different acquisition protocols, varying in angular resolutions; one protocol includes a multi-shell design.

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5 Moving Forward: Remaining Roadblocks, and Emerging Opportunities and Applications There are several key challenges that remain in pooling data from multiple cohorts scanned with diffusion MRI. At the most fundamental level, different protocols obtain different kinds of information on diffusion, which affects the mean values, variance, and reproducibility of several commonly used diffusion parameters, from microstructural measures to measures derived from tracts and networks. Currently the availability of high quality multishell diffusion datasets allows the direct validation of approaches to recover information from lower quality datasets and compare it to other high quality data. What is more challenging is the development of methods for tractography and network analyses that are invariant to the scanning protocols used. Inevitably, the lack of ground truth and the diverse range of methods available makes it harder to determine the validity of network measures across cohorts, but initial efforts to test multiple methods on the same data go some way towards assessing the sources of variation across datasets that need to be harmonized. A further opportunity in harmonization lies in developing methods that are invariant to the pulse sequence parameters chosen. Even if data cannot be perfectly imputed from one protocol to another, the analyses of invariant measures will allow highly powered assessments of data across a range of older and newer protocols. Acknowledgements The work was supported in part by U54 EB020403. Additional support was provided by R01MH116147, P41 EB015922, RF1 AG051710, RF1 AG041915 and and Michael J. Fox Foundation grant 14848.

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Spherical Harmonic Residual Network for Diffusion Signal Harmonization Simon Koppers, Luke Bloy, Jeffrey I. Berman, Chantal M. W. Tax, J. Christopher Edgar and Dorit Merhof

Abstract Diffusion imaging is an important method in the field of neuroscience, as it is sensitive to changes within the tissue microstructure of the human brain. However, a major challenge when using MRI to derive quantitative measures is that the use of different scanners, as used in multi-site group studies, introduces measurement variability. This can lead to an increased variance in quantitative metrics, even if the same brain is scanned. Contrary to the assumption that these characteristics are comparable and similar, small changes in these values are observed in many clinical studies, hence harmonization of the signals is essential. In this paper, we present a method that does not require additional preprocessing, such as segmentation or registration, and harmonizes the signal based on a deep learningresidual network. For this purpose, a training database is required, which consist of the same subjects, scanned on different scanners. The results show that harmonized signals are significantly more similar to the ground truth signal compared to no harmonization, but also improve in comparison to another deep learning method. The same effect is also demonstrated in commonly used metrics derived from the diffusion MRI signal. Keywords Harmonization · Diffusion imaging · Magnetic resonance imaging · Machine learning · Deep learning · SHResNet

1 Introduction Diffusion imaging, with diffusion tensor imaging (DTI) being the most commonly used strategy to represent the signal, is quickly becoming a critical non-invasive tool in the fields of neuroscience and neuropsychology. Based on magnetic resoS. Koppers (B) · D. Merhof Institute of Imaging & Computer Vision, RWTH Aachen University, Aachen, Germany e-mail: [email protected] L. Bloy · J. I. Berman · J. C. Edgar Department of Radiology, Children’s Hospital of Philadelphia, Philadelphia, USA C. M. W. Tax CURBRIC, Cardiff University, Cardiff, UK © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_14

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nance imaging (MRI), diffusion imaging is able to provide important insight into the structure of the human brain in both healthy- and patient populations. Its ability to provide quantitative measurements sensitive to tissue microstructure, such as fractional anisotropy (FA) or mean diffusivity (MD), has lead to its widespread adoption in imaging research protocols. With the recent increase in multi-site imaging studies, it has become apparent that small differences in acquisition protocols across sites, but also differences of the scanner’s vendor, can lead to significant differences in diffusion measurements even within the same subject. Diffusion measurements between scanners can vary in the range of 5% for white matter and 15% for grey matter, potentially overwhelming the ≤5% effect sizes commonly found in group studies [13] and greatly offsetting the increases in statistical power that multi-site studies and larger sample sizes can provide. Data harmonization is a collection of techniques designed to minimize interscanner and intra-site variance and to improve comparability of data acquired at different sites. Currently, there are two strategies for harmonizing diffusion datasets. The first uses a statistical normalization to create a common feature space [4]. Data from each site is normalized using site-specific means and variances. A drawback is that each derived metric (FA, MD, etc.) must be treated separately. Furthermore, this approach doesn’t allow the definition of a characteristic range of metrics, which would be important for comparing single subjects or other multi-site studies. The second harmonization approach is to learn a mapping between the data of all the sites and a pre-defined standard system [11]. In this case, the measured signals, and therefore all derived diffusion metrics, are harmonized and made comparable. This allows the same analysis pipeline to be subsequently applied to all data records regardless of acquisition site. Currently, there are only few published methods [10, 11], which are model-independent methods that harmonizes diffusion data based on a template-based mapping of rotation invariant spherical harmonic (RISH) features. RISH features are first calculated for every subject and used to create a single template mapping for each RISH order. Afterwards, every subject can be corrected based on the corresponding RISH template mapping, which is a rotation invariant mapping, maintaining the original fiber orientations. However, the disadvantage of this method is that an incorrect registration during the template generation, but also during application, leads to an unreliable correction template. In addition, morphological changes, such as tumor resections or edemas, might lead to additional registration artifacts. While not developed explicitly for harmonization, a number of recently developed deep learning techniques can potentially be adapted for this purpose. In general, deep learning methods [5] can be used to build an internal predictive model of diffusion signals, and have shown good results in interpolating diffusion signals, but also predicting complete shells based on only a few gradient directions [9]. In this work, we propose a novel customized deep learning method based on a residual structure to harmonize diffusion data from different systems, which only has to learn the difference between two signals. In addition, it is independent of registration during application, which is why morphological changes have only minor

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influences on its harmonization performance. In this case the full spherical signal based on spherical harmonics (SH) is predicted, while for multi-shell data other general bases are possible.

2 Method The goal of diffusion signal harmonization is to increase the comparability of diffusion signals measured on two different MRI systems. Our approach is to use a spherical harmonic residual network to predict the data acquired on the second scanner using the data from the first scanner. Once the network is trained on a small number of subjects acquired on both scanners, it can be applied to new datasets acquired on the 1st scanner and outputs diffusion data that matched the properties of the second scanner. In addition, a RISH projection as in [10] is performed on every resulting harmonized signal in order to enforce the same fiber orientations as in the input signal.

2.1 Spherical Harmonic Residual Network The Spherical Harmonic Residual Network (SHResNet) can be described as a method that takes a 3 × 3 × 3 spherical harmonic input patch, acquired on scanner 1, and predicts the spherical harmonic coefficients of the central voxel, within the same patch, in scanner 2. It is based on a residual structure [6], which is generally known to be very robust due to their forward pass and are therefore able to train very deep network structures without losing the ability to be trained based on the vanishing gradient problem [3]. The network itself can be separated into three parts as shown in Fig. 1. The core structure is the residual block (ResBlock), which can be stacked multiple times, keeping spatial and coefficient dimensions. Its main element is called functional unit, which consists of three 3D convolutional layers (kernel size: 3 × 3 × 3 with padding) predicting a single SH order. Depending on the corresponding SH order, the number of output neurons needs to be adjusted (SH order 0: 1 neuron; SH order 2: 5 neurons; SH order 4: 9 neurons). The first two convolutional layers utilize a rectified linear unit (ReLU) as activation function. The resulting coefficients are concatenated and subtracted from the functional unit’s input. The full network uses two 3D convolutional layers (kernel size: 3 × 3 × 3 with padding) as preprocessing and postprocessing layers, with multiple ResBlocks in the middle. After that, an additional residual pathway subtracts the preprocessed input from the postprocessed output of all ResBlocks. At the final stage, 3D convolutional layers, employing spatial neighboring information, are utilized to reduce the input dimensions from 3 × 3 × 3 to the single central voxel.

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Fig. 1 Structure of the complete SHRestNet

After each signal is harmonized in SH space, it is corrected for possible changes in fiber orientation. For this purpose, each signal’s RISH features are projected onto the corresponding input signal based on  Si

= Si · √

R I S Hi R I S Hi

,

(1)

where Si describes the harmonized and Si the non-harmonized spherical harmonic coefficients of order i, while R I S Hi and R I S Hi describe the RISH feature of the harmonized and the input signal, respectively. In the end, the signal space is reconstructed by sampling each gradient direction utilizing the predicted SH coefficients, while every predicted gradient is multiplied by the non-diffusion weighted signal s (b = 0 mm 2 ) of the initial measurement in the input system. Training The network is trained on a sample of subjects scanned on both systems. For each training subject, diffusion data from the first scanner acquisition is spatially registered to the target, and training voxels (grey and white matter) are identified using FAST [14]. Training is performed by looping over each training subject, attempting to predict the second scanner’s data using the diffusion data from the first. The network is trained in 2 stages, the first utilizing the Adam optimizer (learning rate of 0.001, batch size: 256 3D samples), which is replaced, in the 2nd stage after the first five epochs,

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by an SGD optimizer. During the 2nd stage the batch size is reduced to 128 samples and the learning rate decreases by ten percent if the validation performance does not improve over five epochs. Training is completed after validation performance does not improve for more than ten epochs. Overall, the mean squared error (MSE) in signal space is utilized as loss function. Furthermore, RISH projection is not part of the training cycle and was only applied during the application stage.

3 Results We present the results for two experiments, the first (Sect. 3.2) investigates the sensitivity of the method to the number of ResBlocks n required to achieve optimal harmonization. In the second section (Sect. 3.3), we evaluate our methods ability to reduce inter-scanner variance. As our method harmonizes the diffusion signal directly it subsequently harmonizes any DTI marker computed from the diffusion signal. For the purposes of our comparison we investigate differences in FA, MD [7] and the signal itself between our method (SHResNet), an adapted version of the network of Golkov et al. [5], hereinafter referred to as the Golkov method, and non-harmonized values. The Golkov method utilizes a three-layer neural network and is usually used to interpolate diffusion signals, but also to reconstruct specific diffusion quantities. It’s output is adapted i.e. it is able to predict SH coefficients.

3.1 Material For evaluating the proposed method, 10 uncorrelated healthy subjects were selected from the CDMRI Harmonization Challenge [12] database, while grey and white matter [14] are evaluated separately to have a more detail insight into the method, while a segmentation is not utilized during training.. Each subject was scanned on both a 3T GE Excite-HD and a 3T Siemens Prisma scanner utilizing the same protocol. Both scans have a isotropic resolution of 2.4 mm3 , and 30 gradient directions s s at b = 1200 mm 2 , as well as a minimum of 4 baseline measurements with b = 0 mm 2 . All data were corrected for motion and eddy current-distortions [1], while the Prisma data was corrected for susceptibility distortions [2]. Afterwards, the GE dataset was spatially nonlinearly registered to the corresponding Prisma dataset [8]. In order to be applicable to every gradient setting, each dataset was preprocessed by dividing every voxel’s diffusion signal by its non-diffusion measurement (b = s 0 mm 2 ) and further converted utilizing spherical harmonics (SH) of order four, limited due to the amount of gradient directions, and a regularization of λ = 0.006, resulting in 15 SH coefficients.

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Table 1 Resulting MSE Loss (scaled by 100) for different number of ResBlocks during optimization Loss 1 2 3 6 7 9 11 MSE

0.375

0.372

0.375

0.377

0.41

0.41

0.41

3.2 Evaluation of Number of ResBlocks To estimate the optimal number of ResBlocks n, harmonization is evaluated utilizing the mean cross validation loss over 10 subjects. As shown in Table 1, the SHResNet reaches its most stable accurate performance utilizing n = 2 ResBlocks within the network, while higher n result to a high error. In addition, it can be seen that n = 1 ResBlocks already results in a similar MSE compared to n = 2 ResBlocks.

3.3 Evaluation of Harmonization In order to evaluate our method we preformed a 10 fold cross validation. For each fold a SHResNet (n = 2 ResBlocks) was trained on 8 subjects, with 1 validation subject, using all grey and white matter voxels, to predict the Prisma diffusion data using the GE data as input. Afterwards, the trained SHResNet was used to predict the Prisma diffusion signal of the test subject. For comparison the Golkov method was also used to generate a predicted Prisma diffusion subject using the same training strategy. After looping over each fold, for each subject, there are 3 predicted Prisma datasets (SHResNet, Golkov and the unharmonized GE dataset) that were then compared to the ground truth Prisma diffusion dataset for that subject. We utilized the normalized mean squared error (NMSE) (see Eq. 2) of the diffusion signal its resulting FA and MD, to contrast the four methods.

NMSE =

2   ˆ  Y − Y 2

Y22

,

(2)

ˆ is the predicted vector. Furthermore, where Y represents the ground truth vector and Y a pair-wise Wilcoxon signed rank test was performed on the resulting differences to evaluate the statistical improvement. Figures 2, 3 and 4 present the resulting NMSE comparisons for white and grey matter voxels respectively. It can be seen that the SHResNet achieves a significantly lower NMSE in every scenario for white matter and a significant lower error in grey matter for the signal and the MD, while the improvement in FA only results in a non-significant p-value of ≈0.08. Further, it should be noted that the Golkov method

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(b) Grey matter

Fig. 2 Resulting NMSE for the signal in white and grey matter (∗ : p ≤ 0.05 and ∗∗ : p ≤ 0.01)

(a) White matter

(b) Grey matter

Fig. 3 Resulting NMSE for the FA in white and grey matter (∗ : p ≤ 0.05 and ∗∗ : p ≤ 0.01)

improves the harmonization accuracy for the signal, the SH coefficients and the MD, while it results in a decreased performance for the FA in white and grey matter. On average, the signal error decreased by ≈35% in white matter and ≈29% in grey matter utilizing the SHResNet. A similar improvement can also be seen for the FA (white: ≈18%; grey: ≈9%) and the MD (white: ≈40%; grey: ≈28%).

4 Discussion Diffusion acquisitions are increasingly be included in large multi-site studies. While the large sample sizes afforded by these types of studies offer great promise in illucidating population differences, the high levels of inter-site variability, particularly

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(a) White matter

(b) Grey matter

Fig. 4 Resulting NMSE for the MD in white and grey matter (∗ : p ≤ 0.05 and ∗∗ : p ≤ 0.01)

in diffusion but also in other imaging modalities, greatly diminishes the ability to observe small effects. Data harmonization represents a post processing technique designed to minimize inter-site variability. In this paper, we present a novel diffusion data harmonization technique, the SHResNet, based on deep learning, and demonstrate its capability to reduce inter-site and inter-scanner variance. As shown in Table 1, training on deeper networks (increased number of ResBlocks) does not improve harmonization. This effect can be due to several reasons. First, deeper networks are generally more difficult to train, because of the significantly increased number of parameters, while at the same time each voxel also contains a complete diffusion signal. Furthermore, the input size was limited to 3 × 3 × 3 voxels. Deeper networks, however, are primarily intended for larger region-specific neighborhood relationships, which are not desirable in our case, since it could lead to problems in scenarios of anatomical changes. The evaluation of our method (see Figs. 2, 3 and 4) shows that it achieves a significant improvement compared to non-harmonized data, while the Golkov method seem to have difficulties reconstructing the correct signal in white matter, resulting in a higher FA NMSE in comparison to the non-harmonized case. Similar, while not always significant, effects can be seen in the area of grey matter, even though the underlying microstructure is much more complex as in white matter. Moreover, we also participated with a simpler version of this model (SHResNet with n = 1 without RISH projection) in the CDMRI Harmonization Challenge 2017 and consistently showed a very stable and good performance in all given tasks [12]. Despite these promising results, there are also some limitations. The biggest obstacle of this algorithm and harmonization in general is the applicability in a clinical study environment. Since it is only able to reconstruct scenarios represented by the utilized training set, unique features occurring in clinical cases may not be preserved through the harmonization process. Thus for population studies it is important that the training sample, that is scanned on both systems, have similar symptomatology

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and demographic distributions as the entire study sample. Using synthetic data that has similar diffusion characteristics as found in the clinical subjects, the method may be better able to handle these use cases. These types of comparisons are beyond the scope of this work, but represent the next steps needed to use data harmonization in large studies. Furthermore, this method is not free of any registration, since a intersubject registration is required between scanners to generate the training dataset.

5 Conclusion Overall, the SHResNet has been shown to be a very fast, uncomplicated and uniform approach to harmonize diffusion data. It is validated utilizing the NMSE with a significant improvement after harmonization found for the signal, the FA and the MD in white and grey matter. Acknowledgements The authors would like to thank the 2017 computational dMRI challenge organizers (Francesco Grussu, Enrico Kaden, Lipeng Ning and Jelle Veraart) for help with data acquisition and processing, as well as Derek Jones, Umesh Rudrapatna, John Evans, Slawomir Kusmia, Cyril Charron, and David Linden at CUBRIC, Cardiff University, and Fabrizio Fasano at Siemens for their support with data acquisition. This work was supported by a Rubicon grant from the NWO (680-50-1527), a Wellcome Trust Investigator Award (096646/Z/11/Z), and a Wellcome Trust Strategic Award (104943/Z/14/Z). The data were acquired at the UK National Facility for In Vivo MR Imaging of Human Tissue Microstructure funded by the EPSRC (grant EP/M029778/1), and The Wolfson Foundation. This work was supported by the International Research Training Group 2150 of the German Research Foundation (DFG).

References 1. Andersson, J.L., Skare, S., Ashburner, J.: How to correct susceptibility distortions in spinecho echo-planar images: application to diffusion tensor imaging. Neuroimage 20(2), 870–888 (2003) 2. Andersson, J.L., Sotiropoulos, S.N.: An integrated approach to correction for off-resonance effects and subject movement in diffusion MR imaging. Neuroimage 125, 1063–1078 (2016) 3. Bengio, Y., Simard, P., Frasconi, P.: Learning long-term dependencies with gradient descent is difficult. IEEE Trans. Neural Netw. 5(2), 157–166 (1994) 4. Fortin, J.P., Parker, D., Tunç, B., Watanabe, T., Elliott, M.A., Ruparel, K., Roalf, D.R., Satterthwaite, T.D., Gur, R.C., Gur, R.E., Schultz, R.T., Verma, R., Shinohara, R.T.: Harmonization of multi-site diffusion tensor imaging data. NeuroImage 161, 149–170 (2017). http://www. sciencedirect.com/science/article/pii/S1053811917306948 5. Golkov, V., Dosovitskiy, A., Sperl, J.I., Menzel, M.I., Czisch, M., Sämann, P., Brox, T., Cremers, D.: q-space deep learning for twelve-fold shorter and model-free diffusion MRI scans. IEEE Trans. Med. Imaging 35(5), 1344–1351 (2016) 6. He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. In: CVPR, pp. 770–778 (2016)

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7. Jenkinson, M., Beckmann, C.F., Behrens, T.E., Woolrich, M.W., Smith, S.M.: Fsl. NeuroImage 62(2), 782–790 (2012). http://www.sciencedirect.com/science/article/pii/ S1053811911010603, 20 YEARS OF fMRI 8. Klein, S., Staring, M., Murphy, K., Viergever, M.A., Pluim, J.P.: elastix: a toolbox for intensitybased medical image registration. IEEE Trans. Med. Imaging 29(1), 196–205 (2010) 9. Koppers, S., Haarburger, C., Merhof, D.: Diffusion MRI signal augmentation: from single shell to multi shell with deep learning. In: CDMRI, pp. 61–70. Springer (2016) 10. Mirzaalian, H., Ning, L., Savadjiev, P., Pasternak, O., Bouix, S., Michailovich, O., Karmacharya, S., Grant, G., Marx, C.E., Morey, R.A., Flashman, L.A., George, M.S., McAllister, T.W., Andaluz, N., Shutter, L., Coimbra, R., Zafonte, R.D., Coleman, M.J., Kubicki, M., Westin, C.F., Stein, M.B., Shenton, M.E., Rathi, Y.: Multi-site harmonization of diffusion MRI data in a registration framework. Brain Imaging Behav. 12(1), 284–295 (2018). https://doi.org/10. 1007/s11682-016-9670-y 11. Mirzaalian, H., de Pierrefeu, A., Savadjiev, P., Pasternak, O., Bouix, S., Kubicki, M., Westin, C.F., Shenton, M.E., Rathi, Y.: Harmonizing diffusion MRI data across multiple sites and scanners. In: Navab, N., Hornegger, J., Wells, W.M., Frangi, A. (eds.) Medical Image Computing and Computer-Assisted Intervention - MICCAI 2015, pp. 12–19. Springer International Publishing, Cham (2015) 12. Tax, M.W., C., Grussu, F., Kaden, E., Ning, L., Rudrapatna, U., Evans, J., St-Jean, S., Leemans, A., Puch, S., Rowe, M., Rodrigues, P., Prˆckovska, V., Koppers, S., Merhof, D., Ghosh, A., Tanno, R., C Alexander, D., Charron, C., Kusmia, S., EJ Linden, D., K Jones, D., Veraart, J.: Cross-vendor and cross-protocol harmonisation of diffusion tensor imaging data: a comparative study. ISMRM-ESMRMB (2018). https://projects.iq.harvard.edu/cdmri2017 13. Vollmar, C., Identical et al.: But not the same: Intra-site and inter-site reproducibility of fractional anisotropy measures on two 3.0 T scanners. Neuroimage 51(4), 1384–1394 (2010) 14. Zhang, Y., et al.: Segmentation of brain MR images through a hidden Markov random field model and the expectation-maximization algorithm. IEEE Trans. Med. Imaging 20(1), 45–57 (2001)

Longitudinal Harmonization for Improving Tractography in Baby Diffusion MRI Khoi Minh Huynh, Jaeil Kim, Geng Chen, Ye Wu, Dinggang Shen and Pew-Thian Yap

Abstract The human brain develops very rapidly in the first years of life, resulting in significant changes in water diffusion anisotropy. Developmental changes pose significant challenges to longitudinally consistent white matter tractography. In this paper, we will introduce a method to harmonize infant diffusion MRI data longitudinally across time. Specifically, we harmonize diffusion MRI data collected at an earlier time point to data collected at a later time point. This will promote longitudinal consistency and allow sharpening of fiber orientation distribution functions (ODFs) based on information available at the later time point. For this purpose, we will introduce an approach that is based on the method of moments, which allows harmonization to be performed directly on the diffusion-attenuated signal without the need to fit any diffusion models to the data. Given two diffusion MRI datasets, our method harmonizes them voxel-wise using well-behaving mapping functions (i.e., monotonic, diffeomorphic, etc.), parameters of which are determined by matching the spherical moments (i.e., mean, variance, skewness, etc.) of signal measurements on each shell. The mapping functions we use is isotropic and does not introduce new orientations that are not already in the original data. Our analysis indicates that longitudinal harmonization sharpens ODFs and improves tractography in infant diffusion MRI. Keywords Longitudinal harmonization · Diffusion MRI · Method of moments · Tractography

K. M. Huynh · D. Shen · P.-T. Yap Biomedical Engineering Department, University of North Carolina, Chapel Hill, USA K. M. Huynh · J. Kim · G. Chen · Y. Wu · D. Shen · P.-T. Yap (B) Department of Radiology and BRIC, University of North Carolina, Chapel Hill, USA e-mail: [email protected] © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_15

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1 Introduction White matter contains axonal tracts that relay and coordinate communication between different brain regions. Myelination begins in the last trimester with myelin sheaths wrapping around axons and significantly boosting nerve conduction velocity [1]. Rapidly ongoing myelination during infancy leads to dynamic development in verbal, motor, and cognitive functions [2]. As a fatty substance, myelin sheath limits water diffusion in direction perpendicular, but not parallel, to the tracts, causing diffusion anisotropy and giving contrast to diffusion-weighted (DW) images. Tractography is significantly affected by changes in diffusion anisotropy. This stems from the fact that, in general, tractography methods are designed to traverse areas of high anisotropy and avoid areas of low anisotropy. In the developing brain where the anisotropy is typically low and progressively increases with maturation, far fewer voxels are included in the pathway at birth than at successive time points. As a result, even there is a profusion of tracts in an infant brain, incomplete myelination makes reconstruction of the tracts via diffusion tractography challenging. An often used remedy is to lower the anisotropy threshold to allow a tractography algorithm to generate streamlines that traverse some regions with low anisotropy. This approach, however, might result in spurious streamlines owing to directional uncertainty caused by the lack of anisotropy. To tackle the problem, we propose in this paper a longitudinal harmonization method to improve tractography for infant diffusion MRI. Based on the method of moments, our approach involves a model-free harmonization strategy that can be applied directly on longitudinal data without fitting of diffusion models. This is achieved with mapping functions that are designed to be smooth (no abrupt changes), one-to-one (topology preserving), non-decreasingly monotonic (contrast preserving), and invertible (data recoverable). The parameters of the mapping functions are determined by matching the spherical moments (i.e., mean, variance, skewness, etc.) of signal measurements on each shell. These functions are applied isotropically and do not introduce new orientations that are not already in the original data. Our analyses indicate that longitudinal harmonization sharpens fiber orientation distribution functions (ODFs) and improves tractography without introducing false-positive streamlines.

2 Methods In shell-based sampling, the diffusion-attenuated signal Sb is sampled using a set of non-collinear gradient directions with a fixed diffusion weighting (i.e., b-value). The n-th raw spherical moment is defined as  E[Sbn ] =

S2

Sbn (g)dg.

(1)

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The spherical mean (M1,Sb ) and variance (M2,Sb ) can be defined based on the first and second raw spherical moments as M1,Sb = S¯b = E[Sb ],

(2)

M2,Sb = E[(Sb − E[Sb ]) ]. 2

(3)

Taking the dataset at a later time point as the reference R and the dataset at an earlier time point as the target T , we want to determine a mapping function f θ (S) that results in the matching of the two moments: M1,R = E[R] = M1, fθ (T ) = E[ f θ (T )],

(4)

M2,R = E[(R − E[R]) ] = M2, fθ (T ) = E[( f θ (T ) − E[ f θ (T )]) ], 2

2

(5)

where Mn, fθ (T ) is the n-th moment of the target after mapping via f θ (S) whereas Mn,T is the moment before mapping. It is shown in [3] that the spherical mean is independent of the fiber orientation distribution and can be used to characterize peraxon microstructural properties. The spherical variance can be seen as a measure of fiber orientation consistency (opposite to fiber dispersion). To maintain signal contrast, the mapping function must be monotonic. In this work, using the first two moments, we employ a first-order mapping function f θ={α,β} (S) = αS + β and (4) and (5) can therefore be written as M1,R = αM1,T + β and M2,R = α2 M2,T ,

(6)

allowing the parameters to be determined as α=



M2,R /M2,T and β = M1,R − αM1,T .

(7)

To ensure that the contrast is not inverted, we constrain α > 0. The estimation of parameter α is problematic in isotropic regions where the variance is a primarily due to noise, resulting in instability in parameter estimation. For greater robustness, we estimate the parameters using regularized least-squares, as described next. Assuming that the reference and target moment images are spatially aligned, moment matching is performed voxel-wise. For each voxel v, the mapping parameters are determined by minimizing the cost function J (α(v), β(v)) of the unconstrained multivariable optimization problem with Quasi-Newton algorithm:    M1,R (φ−1 (v)) − M1,T ( f θ={α(v),β(v)} (v)) 2  + ψ(α(v), β(v)),  J (α(v), β(v)) = M2,R (φ−1 (v)) − M2,T ( f θ={α(v),β(v)} (v)) 2 (8)

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where φ(v) is a deformation field that spatially aligns the reference to the target. The regularizer is defined as ψ(α, β) = λ



 β2 (α−1)2 + 1 − e− 2 , 1 − e− 2

(9)

where λ is a tunning parameter. The signal Sb (g|v) at spatial location v and for gradient direction g is harmonized via Sb (g|v) = α(v)Sb (g|v) + β(v).

(10)

Note that the mapping does not change with the gradient direction. We found that the first order mapping function is sufficient for our application. Our method allows more sophisticated mapping functions to be used as long as basic properties, such as nondecreasing monotonicity, are met. The parameters, if more than two, can be estimated using higher-order moments. Our method only requires warping of the moment images, but not the DW images, for matching; therefore issues related to signal reorientation [4] can be avoided.

3 Experiments We study the effects of longitudinal harmonization using both synthetic and real in-vivo baby diffusion MRI data.

3.1 Synthetic Data We generated multi-shell synthetic data simulating a complex phantom using Phantomas with 90 DW images in each shell (b = 1000, 2000, and 3000 s/mm2 ) and 18 non-diffusion-weighted B0 images [5]. The reference was created using a 27-bundle configuration with λ1 = 1.7 × 10−3 mm2 /s and λ2 = λ3 = 4.35 × 10−4 mm2 /s to simulate high fractional anisotropy (FA = 0.7). The target was created using only a 8-bundle configuration (subset of the 27 bundles) with λ1 = 1.7 × 10−3 mm2 /s and λ2 = λ3 = 1.2 × 10−3 mm2 /s to simulate low anisotropy (FA = 0.2). Rician noise was added. To illustrate the effects of harmonization, we performed global tractography [6] on both unharmonized and harmonized datasets. Figure 1 shows that after harmonization, the streamlines are cleaner and denser with less false positives. Harmonization also increases the anisotropy of the diffusion signal profiles. Table 1 summarizes the Tractometer [7] statistics, indicating that harmonization increases valid connections and decrease invalid connections. With harmonization, the 8 streamline bundles can be fully recovered without any false positives that may be introduced by the reference.

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Fig. 1 Synthetic Data. Tractography results of the 8-bundle phantom (middle) and exemplar diffusion signal profiles (right) before and after harmonization. Note that even though the reference has 27 bundles, harmonization does not introduce any false-positive bundles Table 1 Tractometer statistics for synthetic data Tracts VC(%) IC (%) Unharmonized Harmonized Reference

4870 8052 10962

2.5 11.5 12.89

7.1 3.2 3.22

NC(%)

VC VC+IC

90.4 85.3 83.89

26.04 78.2 80.01

(%)

VB

IB

8 8 27

12 9 127

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3.2 Real Data We used a diffusion MRI dataset of a 9-month old subject as the reference to harmonize the dataset of the same subject collected at 3 months of age. Each dataset consist of 74 DW images distributed on two shells (b = 1500 and 3000 s/mm2 ) and 5 non-DW B0 images. Eddy-current distortion, motion, and susceptibility artifacts were corrected prior to harmonization [8]. The dataset of the later time point was registered to the earlier time point using multi-channel diffeomorphic registration [9] with fractional anisotropy (FA) and B0 images. Signal normalization was carried out by dividing the DW images with the average of the non-DW images. Diffusion Indices: Fractional anisotropy (FA), mean diffusivity (MD), per-axon FA (μFA), and per-axon MD (μMD) were compared before and after harmonization. Figure 2 illustrates the effects of harmonization on the indices. The anisotropy of the 3-month dataset increases noticeably after harmonization. Fiber ODFs: Multi-tissue constrained spherical deconvolution [10] was performed to estimate the fiber ODFs. From Fig. 3, one can appreciate the increase in sharpness 1

0

0.02

0

1

0

0.03

9-Month Reference

Harmonized

Unharmonized

0

FA

MD

µFA

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Fig. 2 Diffusion Indices. Fractional anisotropy (FA), mean diffusivity (MD), per-axon FA (μFA), and per-axon MD (μMD) of the 3-month dataset before (top) and after (middle) harmonization, and the 9-month reference dataset (bottom)

189

Harmonized

Unharmonized

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Fig. 3 Fiber ODFs. Left to Right: DW image, tissue volume fraction map (Blue: White matter; Green: Gray matter; Red: Cerebrospinal fluid), Close-up view of forceps minor, and an FODF in forceps minor before and after harmonization

of the FODFs after harmonization. The tissue volume fraction map also reveals greater details with further extension into cortical regions. Tract Probability: The tract probability maps shown in Fig. 4 were obtained using tractography with second-order integration over the fiber ODFs [11]. After harmonization, all 20 major tracts described in [12] show increased tract probabilities. Detailed quantitative results are provided in Table 2.

Unharmonized

50

5

Harmonized

50

5

Fig. 4 Tract Probability. Probability (%) of the presence of a tract at a particular location, determined via probabilistic tractography using unharmonized and harmonized data

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Table 2 Probalility (%) of major tracts before and after harmonization Tract

Unharmonized

Harmonized

Anterior thalamic radiation (L)

10.79

20.57

Anterior thalamic radiation (R)

8.45

16.37

Corticospinal tract (L)

16.42

22.21

Corticospinal tract (R)

21.38

33.81

Cingulum (cingulate gyrus) (L)

7.63

13.96

Cingulum (cingulate gyrus) (R)

5.40

11.65

Cingulum (hippocampus) (L)

6.65

11.52

Cingulum (hippocampus) (R)

6.45

10.83

Forceps major

14.90

23.56

Forceps minor

12.86

24.03

Inferior fronto-occipital fasciculus (L)

14.24

25.37

Inferior fronto-occipital fasciculus (R)

13.78

23.77

Inferior longitudinal fasciculus (L)

9.06

16.50

Inferior longitudinal fasciculus (R)

12.27

23.57

Superior longitudinal fasciculus (L)

13.36

25.92

Superior longitudinal fasciculus (R)

13.96

25.44

Uncinate fasciculus (L)

10.48

24.25

Uncinate fasciculus (R)

9.76

16.60

Superior longitudinal fasciculus (L)

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Tractogram: Whole brain tractograms were constructed using global tractography [13]. Figure 5 confirms the effectiveness of harmonization as it significantly improved tractography results.

Fig. 5 Tractograms. The effects of harmonization shown using tractograms

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4 Conclusion We have introduced a model-free approach, based on the method of moments, for longitudinal harmonization. We have demonstrated its effectiveness in sharpening fiber ODFs and in improving tractography in infant diffusion MRI. Acknowledgements This work was supported in part by NIH grants (1U01MH110274, NS093842, EB022880, and MH100217) and the efforts of the UNC/UMN Baby Connectome Project Consortium. Declarations The authors declare that there is no conflict or commercial interest and the work is in compliance with the IRB regulations.

References 1. Hüppi, P.S., Maier, S.E., Peled, S., Zientara, G.P., Barnes, P.D., Jolesz, F.A., Volpe, J.J.: Microstructural development of human newborn cerebral white matter assessed in vivo by diffusion tensor magnetic resonance imaging. Pediat. Res. 44(4), 584 (1998) 2. Dubois, J., Dehaene-Lambertz, G., Kulikova, S., Poupon, C., Hüppi, P.S., Hertz-Pannier, L.: The early development of brain white matter: a review of imaging studies in fetuses, newborns and infants. Neuroscience 276, 48–71 (2014) 3. Kaden, E., Kruggel, F., Alexander, D.C.: Quantitative mapping of the per-axon diffusion coefficients in brain white matter. Magn. Reson. Med. 75(4), 1752–1763 (2016) 4. Yap, P.T., Shen, D.: Spatial transformation of DWI data using non-negative sparse representation. IEEE Trans. Med. Imaging 31(11), 2035–2049 (2012) 5. Caruyer, E., Daducci, A., Descoteaux, M., Houde, J.C., Thiran, J.P., Verma, R.: Phantomas: a flexible software library to simulate diffusion MR phantoms. In: The Annual Meeting of the International Society for Magnetic Resonance in Medicine (2014) 6. Wu, H., Chen, G., Jin, Y., Shen, D., Yap, P.T.: Embarrassingly parallel acceleration of global tractography via dynamic domain partitioning. Front. Neuroinformatics 10(25) (2016) 7. Côté, M.A., Girard, G., Boré, A., Garyfallidis, E., Houde, J.C., Descoteaux, M.: Tractometer: towards validation of tractography pipelines. Med. Image Anal. 17(7), 844–857 (2013) 8. Andersson, J.L., Sotiropoulos, S.N.: An integrated approach to correction for off-resonance effects and subject movement in diffusion MR imaging. NeuroImage 125, 1063–1078 (2016) 9. Avants, B.B., Tustison, N., Song, G.: Advanced normalization tools (ANTS). Insight J. 2, 1–35 (2009) 10. Jeurissen, B., Tournier, J.D., Dhollander, T., Connelly, A., Sijbers, J.: Multi-tissue constrained spherical deconvolution for improved analysis of multi-shell diffusion MRI data. NeuroImage 103, 411–426 (2014) 11. Tournier, J.D., Calamante, F., Connelly, A.: Improved probabilistic streamlines tractography by 2nd order integration over fibre orientation distributions. In: The Annual Meeting of the International Society for Magnetic Resonance in Medicine (2010) 1670 12. Mori, S., Wakana, S., Van Zijl, P.C., Nagae-Poetscher, L.: MRI Atlas of Human White Matter. Elsevier (2005) 13. Christiaens, D., Reisert, M., Dhollander, T., Sunaert, S., Suetens, P., Maes, F.: Global tractography of multi-shell diffusion-weighted imaging data using a multi-tissue model. NeuroImage 123, 89–101 (2015)

Inter-Scanner Harmonization of High Angular Resolution DW-MRI Using Null Space Deep Learning Vishwesh Nath, Prasanna Parvathaneni, Colin B. Hansen, Allison E. Hainline, Camilo Bermudez, Samuel Remedios, Justin A. Blaber, Kurt G. Schilling, Ilwoo Lyu, Vaibhav Janve, Yurui Gao, Iwona Stepniewska, Baxter P. Rogers, Allen T. Newton, L. Taylor Davis, Jeff Luci, Adam W. Anderson and Bennett A. Landman

Abstract Diffusion-weighted magnetic resonance imaging (DW-MRI) allows for non-invasive imaging of the local fiber architecture of the human brain at a millimetric scale. Multiple classical approaches have been proposed to detect both single (e.g., tensors) and multiple (e.g., constrained spherical deconvolution, CSD) fiber population orientations per voxel. However, existing techniques generally exhibit low reproducibility across MRI scanners. Herein, we propose a data-driven technique using a neural network design which exploits two categories of data. First, training data were acquired on three squirrel monkey brains using ex-vivo DW-MRI and histology of the brain. Second, repeated scans of human subjects were acquired on two different scanners to augment the learning of the network proposed. To use these data, we propose a new network architecture, the null space deep network (NSDN), to simultaneously learn on traditional observed/truth pairs (e.g., MRI-histology V. Nath (B) · P. Parvathaneni · C. B. Hansen · J. A. Blaber · I. Lyu · B. A. Landman EECS, Vanderbilt University, Nashville, TN 37203, USA e-mail: [email protected] C. Bermudez · K. G. Schilling · C. B. Hansen · A. W. Anderson · B. A. Landman BME, Vanderbilt University, Nashville, TN 37203, USA A. E. Hainline Biostatistics, Vanderbilt University, Nashville, TN 37203, USA S. Remedios CS, Middle Tennessee State University, Murfressboro, TN 37132, USA I. Stepniewska Psychology, Vanderbilt University, Nashville, TN 37203, USA V. Janve · Y. Gao · B. P. Rogers · A. T. Newton · A. W. Anderson · B. A. Landman VUIIS, Vanderbilt University, Nashville, TN 37232, USA L. T. Davis VUMC, Vanderbilt University, Nashville, TN 37203, USA J. Luci BME, University of Texas at Austin, Austin, TX 78712, USA © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_16

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voxels) along with repeated observations without a known truth (e.g., scan-rescan MRI). The NSDN was tested on twenty percent of the histology voxels that were kept completely blind to the network. NSDN significantly improved absolute performance relative to histology by 3.87% over CSD and 1.42% over a recently proposed deep neural network approach. More-over, it improved reproducibility on the paired data by 21.19% over CSD and 10.09% over a recently proposed deep approach. Finally, NSDN improved generalizability of the model to a third in-vivo human scanner (which was not used in training) by 16.08% over CSD and 10.41% over a recently proposed deep learning approach. This work suggests that data-driven approaches for local fiber reconstruction are more reproducible, informative, precise and offer a novel, practical method for determining these models. Keywords Diffusion · HARDI · DW-MRI · Null space · CSD · Harmonization · Inter-scanner · Deep learning · Histology

1 Introduction Diffusion-weighted MRI (DW-MRI) provides orientation and acquisition-dependent imaging contrasts that are uniquely sensitive to the tissue micro-architecture at a millimeter scale [1]. Substantial effort has gone into modeling the relationship between observed signals and underlying biology, with a tensor model of Gaussian processes being the most commonly used model [2]. Voxel-wise models that characterize higher order spatial dependence than tensors fall under the moniker of higher angular resolution diffusion imaging (HARDI) [3]. Recently, a myriad of techniques has emerged to estimate local structure from these diffusion measures [4–7]. However, broad adoption and clinical translation of specific methods has been hindered by a lack of reproducibility [8, 9], inter-scanner stability [10, 11], and anatomical specificity when compared to a histologically defined true micro-architecture [12]. There are known critical issues of the inter-scanner diffusion harmonization that go beyond noise effects [13–15]. Recently, it has become feasible to apply a data-driven approach to estimate tissue micro-architecture from in-vivo diffusion weighted MRI using deep learning [16]. This approach relied on a histologically defined truth with correspondingly paired voxels with DW-MRI data. Yet, no approaches to date have addressed inter-scanner variation and scan-rescan reproducibility. Moreover, traditional deep learning architecture do not specifically create models that have these necessary characteristics for clinical translation. Here, we propose a new learning architecture, the null space deep network (NSDN), to address the short comings of precision and reproducibility across scanners. Within the NSDN framework, we use inter-scanner paired in vivo human data to stabilize the data driven approach linking pre-clinical DW-MRI with histologicaldata. Using a withheld dataset, the NSDN method is compared against a previously published fully connected network and the leading model-based approach, super resolved constrained spherical deconvolution (CSD) [4] in terms of the

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precision with which the model captures histologically defined truth from DW-MRI data, the reproducibility of the approach on in-vivo human data, and the generalizability of the model to in-vivo data acquired on an additional MRI scanner. The remainder of this manuscript is organized as follows. Section 2 presents the acquisition and processing of all the data that has been used for the study. Section 3 presents the design and the parameters of the proposed network architecture. Section 4 presents the results. Section 5 presents the conclusion.

2 Data Acquisition and Processing Three ex-vivo squirrel monkey brains were imaged on a Varian 9.4T scanner (Fig. 1). A total of 100 gradient volumes were acquired using a diffusion-weighted echo-planar imaging (EPI) sequence at a diffusivity value of 6000 s/mm2 , acquired at an isotropic resolution of 0.3 mm. Once acquired, the tissue was sectioned and stained with fluorescent dil and imaged on a LSM710 Confocal microscope using following procedures outlined in [12]. A similar procedure is outlined by [17]. The histological fiber orientation distribution (HFOD) was extracted using 3D structure tensor analysis. A multi-step registration procedure was used to determine the corresponding diffusion MRI signal. A total of 567 histological voxels were processed. 54 voxels of these were labelled as outliers qualitatively and were rejected from the analysis. A hundred random rotations were applied to the remaining voxels for both the MR signal and the HFOD to augment the data and bringing the total to 51,813 voxels [18]. A withheld set of 72 test voxels was maintained for validation. With rotations, these total to 7,272 voxels.

Fig. 1 Formation of the training dataset where 2D histology was performed on the squirrel monkey brains and FODs were constructed per voxel basis using ensemble structure tensor analysis which correspond to ex-vivo MRI acquisition of the squirrel monkey brains. Comparative analyses were performed between the reconstructed histology FODs and FODs from CSD, DN and NSDN

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The in-vivo acquisitions of the human subject’s data were acquired on three different sites, referred to as A, B and C. Three healthy human subjects were acquired with a scan each at the sites in the following manner. Subject 1: Site A and Site B. Subject 2: site A and site B. Subject 3: site B and site C. Structural T1 MPRAGE were acquired for all subjects at all sites. The diffusion acquisition protocol and scanner information are listed for each of the sites as follows: Site A was equipped with a 3T scanner with a 32-channel head coil. The scan was acquired at a diffusivity value of 2000 s/mm2 (approximating diffusion contrast of fixed ex-vivo scan at a b-value of 6000 s/mm2 ). 96 diffusion weighted gradient volumes were acquired with a b0. Briefly the other parameters are: SENSE = 2.5, partial Fourier = 0.77, FOV = 96 × 96, Slice = 48, isotropic resolution: 2.5 mm. Site B was equipped with a 3T scanner with a 32-channel head coil. All the parameters of the scan acquisition were as of scanner at site A except for the isotropic resolution which was 1.9 mm × 1.9 mm × 2.5 mm and up-sampled to 2.5 mm isotropic. Site C was equipped with a 3T scanner with a 32-channel head coil. The scan acquisition parameters were same as that of site A, except for the number of slices (n = 50) and GRAPPA = 2 (instead of SENSE). All in-vivo acquisitions were pre-processed with standard procedures eddy, topup, b0 normalization, outlier imputation and then registered pairwise per subject [19–22]. T1’s were registered and transformed to the diffusion space. Brain extraction tool was used for skull stripping [21]. FAST white matter (WM) segmentation was performed using the T1 for the in-vivo data [23]. Outliers were removed using. Note that there were three pairs of pre-processed acquisitions in total. The pair of data from Subject 1 along with the histology data set was used for the training of NSDN . The pairs of data from Subject 2 and 3 were used for quantitative and qualitative evaluation of the network. No site C data were used in training.

3 Method: Network Design Our proposed null space architecture is motivated by the linear algebra null spaces in that we need to design/constrain the aspect of the network that has no impact on the outcome. This work is inspired by [24] in which a person re-identification classification problem in computer vision was addressed using a Siamese architecture deep network. The novelty of our approach is that we use paired (but unlabeled) data to train the data-driven network to ignore potential features that would lead it to differentiate between the paired data. The proposed network design takes three inputs of 8th order spherical harmonic (SH) coefficients (Fig. 2). Each input provides an orthonormal representation of the DW-MRI signal and is known to characterize the angular diffusivity signal well [25]. The network outputs a 10th order SH FOD. The base network consists of five fully connected layers; the numbers of neurons per layer are 45, 400, 66, 200 and 66 in the respective order. Activation functions of ReLU have only been used for the first two layers. They have not been used for the remainder of the layers to allow for

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Fig. 2 Network design for the null space architecture. The architecture depicts how pairwise inputs of in-vivo voxels can be incorporated in a deep neural net architecture and can be added in the loss function as a noise enhancement/augmentation technique

negativity in the network because SH coefficients need not be positive. The three outputs obtained by the network are merged with a common loss function which optimizes on the assumption that the pairwise difference should be zero given the subject is the same and there should not be a difference in the FOD being predicted. For implementation simplicity, a modified weighted square loss function was defined as (here in λ = 1): L=

m 1  (ytr uei − y pr edi )2 + λ(Pai − Pbi )2 m i=1

(1)

where m is the total number of samples. Pa and Pb are paired in-vivo voxels. A sample size of 37,648 pairs of paired WM voxels were extracted from subject 1 using the acquisitions from site A and site B. A random selection of 37,648 data points was made from the training data set of the histology voxels. While training the network a K-fold cross-validation was used with K = 5. The cross-validation set size was set to 0.2. RMSProp was used as the optimizer of the network [26]. The number of iterations was determined at 3 using cross-validation. A batch size of 100 has been used. To evaluate the performance, we use Angular Correlation Coefficient (ACC) which describes the correlation between two FODs on a scale of -1 to 1 [6], where 1 is the best outcome.

4 Results The median of the ACC computed from the blind set of 7,272 augmented histology voxels for CSD, DN and NSDN were 0.79, 0.81, and 0.82, respectively. Nonparametric signed rank test for all pairs of distributions were found to be p < 0.01.

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Fig. 3 Representative voxels are shown for the 25th, 50th, and 75th percentiles of CSD ACC along with corresponding DW-MRI, DN and NSDN glyphs and ACCs

Fig. 4 a Histogram peaks of ACC per bin distributed over 100 bins for subject 2 of CSD, DN and NSDN. b Histogram peak of ACC per bin distributed over 100 bins for subject 3 of CSD, DN and NSDN

Qualitatively, we explore the results relative to the truth voxel in Fig. 3. At 25th percentile it can be observed that CSD and DN show a crossing fiber structure when compared to HFOD. NSDN is representative of more similar single fiber structure of histology. At 50th percentile CSD tends to show a crossing fiber structure, while DN and NSDN show a higher ACC and are like the structure of histology. At 75th percentile all three methods closely resemble the histology. For subject 2, the histogram distribution of ACC for NSDN is most skewed (towards higher ACC) compared to DN and CSD (Fig. 4a). The median values for the ACC distributions of CSD, DN and NSDN are: 0.67, 0.74 and 0.82. The gain in performance (calculated by the difference of the medians) is 21.19% for (CSD,

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Fig. 5 The selected ROI shows left side frontal lobe of WM. The image underlay for the ROIs shown is Angular Correlation Coefficient (ACC) to indicate areas of agreement between scan-rescan (vertical pairs). CSD shows high correlation for core white matter where single fiber orientation exists (observer diagonal pattern of high ACC). DN shows increased corre-lation over a broader region which encompasses crossing fibers. NSDN shows a higher correlation across the most extended anatomical area

NSDN) and 10.09% for (DN, NSDN). Non-parametric signed rank test for all pairs of ACC metrics per voxels for subject 2 resulted in p < 0.01. Qualitatively, we explore the spatial diffusion inferred structure in the WM of the frontal lobe of the middle axial brain slice (Fig. 5). CSD (A & D) show low correlation and spurious fibers in the crossing fiber regions. DN (B & E) improves correlation in crossing fiber regions however NSDN (C & F) shows the highest correlation for crossing fiber regions. For single fibers, all three methods show high correlation. In the quantitative results for subject 3, we observe that the skewed distribution towards higher ACC for NSDN is the highest as compared to both the other methods (Fig. 4b). The median for three distributions of CSD, DN, and NSDN are 0.62, 0.67 and 0.74. The performance gain of NSDN over CSD is 16.08% and DN is 10.41%. Non-parametric signed rank test for all pairs of voxels for subject 3 show p < 0.01.

5 Conclusion The NSDN method for reconstructing local fiber architecture is (1) more accurate when compared to histologically defined FODs, (2) more reproducible qualitatively and quantitatively on scan-rescan data, and (3) more reproducible on previously

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unseen scanners. While histological-MRI paired datasets are exceedingly rare, scanrescan data are ubiquitous and often acquired as part of multi-site studies. The NSDN method provides a natural framework for harmonization that can use already acquired scan-rescan data to ensure that analysis methods are as reproducible across all sites. A much wider comparative study with multiple different HARDI methods and using multiple scanners is warranted. It would be interesting to explore the impact of including data from diffusion phantoms to enhance the diversity of signals captured in a data-driven approach. While this work focused on DW-MRI, the NSDN approach can naturally be applied to other deep learning based networks with two relatively simple modifications. First, one needs to construct a multiple channel network graph of the same form as base network, but with shared weights for all channels and without crossconnections between the channels. This will ensure that one input can be placed per channel and all inputs will see the same base network. Second, the loss function needs to be modified so it combines a traditional loss with a reproducibility loss. The traditional loss comes with (without loss of generality) from the first channels output relative to a traditionally provided truth dataset. The reproducibility loss is then computed by a metric of reproducibility between the remaining channels (herein a weighted squared error metric, but Dice, surface distance, etc. could be used as appropriate for the datatype). The potential synergies with data augmentation and neighborhood information have yet to be explored. Acknowledgements This work was supported by R01EB017230 (Landman), RO1NS058639 (Anderson), S10RR17799 (Gore) and T32EB001628. This work was conducted in part using the resources of the Advanced Computing Center for Research and Education at Vanderbilt University, Nashville, TN. This project was supported in part by the National Center for Research Resources, Grant UL1 RR024975-01, and is now at the National Center for Advancing Translational Sciences, Grant 2 UL1 TR000445-06. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIH. This work has been supported by Nvidia with supplement of hardware resources (GPUs). Glyph visualizations were supported using [27].

References 1. Le Bihan, D.: Looking into the functional architecture of the brain with diffusion MRI. Nature Rev. Neurosci. 4(6), 469 (2003) 2. Basser, P.J., Mattiello, J., LeBihan, D.: MR diffusion tensor spectroscopy and imaging. Biophys. J. 66(1), 259–267 (1994) 3. Tournier, J., Calamante, F., Connelly, A.: How many diffusion gradient directions are required for HARDI. In: Proceedings of the International Society Magnetic Resonance in Medicine (2009) 4. Tournier, J.-D., Calamante, F., Connelly, A.: Robust determination of the fibre orientation distribution in diffusion MRI: non-negativity constrained super-resolved spherical deconvolution. Neuroimage 35(4), 1459–1472 (2007) 5. Tuch, D.S.: Qball imaging. Magn. Reson. Med. 52(6), 1358–1372 (2004) 6. Anderson, A.W.: Measurement of fiber orientation distributions using high angular resolution diffusion imaging. Magn. Reson. Med. 54(5), 1194–1206 (2005)

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Effects of Diffusion MRI Model and Harmonization on the Consistency of Findings in an International Multi-cohort HIV Neuroimaging Study Talia M. Nir, Hei Y. Lam, Jintanat Ananworanich, Jasmina Boban, Bruce J. Brew, Lucette Cysique, J. P. Fouche, Taylor Kuhn, Eric S. Porges, Meng Law, Robert H. Paul, April Thames, Adam J. Woods, Victor G. Valcour, Paul M. Thompson, Ronald A. Cohen, Dan J. Stein and Neda Jahanshad

Abstract HIV-related white matter (WM) differences reported across studies are inconsistent. This is due to clinical and demographic heterogeneity of HIV infected populations, and variations in diffusion MRI (dMRI) acquisition, processing, and analysis methods across studies. Therefore, reliable neuroanatomical consequences of infection and therapeutic targets are difficult to identify. Here, we pooled data from six existing HIV studies from around the world as part of the ENIGMA-HIV consortium to evaluate (1) the effects of harmonization of dMRI measures across sites using ComBat, and (2) whether an improved, higher-order tensor dMRI model, the tensor distribution function (TDF), and derived scalar index (FATDF ) conferred higher sensitivity across heterogeneous sites to understand the effect of HIV on WM microstructure. This study suggests that improved dMRI indices and harmonization of these measures across cohorts, may be helpful for detecting consistent effects of Neda Jahanshad for the ENIGMA-HIV Working Group T. M. Nir · H. Y. Lam · P. M. Thompson (B) · N. Jahanshad Imaging Genetics Center, Mark and Mary Stevens Neuroimaging and Informatics Institute, University of Southern California, Marina del Rey, CA, USA e-mail: [email protected] J. Ananworanich HIV-NAT, Thai Red Cross AIDS Research Centre, Bangkok, Thailand J. Boban Faculty of Medicine, Diagnostic Imaging Center, University of Novi Sad, Novi Sad, Serbia B. J. Brew · L. Cysique Department of Neurology, St Vincent’s Hospital, and University of New South Wales, Sydney, Australia J. P. Fouche · D. J. Stein Department of Psychiatry and Mental Health, University of Cape Town, Cape Town, South Africa T. Kuhn · A. Thames Semel Institute for Neuroscience and Human Behavior, University of California, Los Angeles, CA, USA © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_17

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disease on the brain in international multi-site studies, while preserving biological differences. Keywords HIV · Multi-site · Harmonization · TDF · Diffusion MRI · ComBat · DTI

1 Introduction The human immunodeficiency virus type 1 (HIV) enters the brain soon after infection, instigating an inflammatory cascade that typically leads to neural dysfunction, axonal damage, and diffuse myelin pallor [1]. As a result, many brain imaging studies of HIV+ individuals have used measures derived from diffusion tensor imaging (DTI), such as fractional anisotropy (FADTI ), to study white matter (WM) microstructural abnormalities, and frequently identify lower FADTI with infection. However, inconsistencies in the effect sizes, regional distribution, and even direction of FADTI changes reported across studies have limited the generalizability of the conclusions drawn to date [2]. Understanding common neurobiological substrates of HIV could help lead to improved therapeutic targets, and surrogate markers to evaluate treatment effects, but heterogeneity in study findings makes it challenging to identify neuropathogenic pathways that generalize across international populations. Sources of heterogeneity between findings of single cohort studies may be due to methodological as well as population differences, including age, sex, and environmental, socioeconomic or lifestyle attributes of the cohort. While this is true for studies across multiple diagnostic conditions, HIV is further complicated by different viral load status, comorbidities and co-infections, drug use, age at infection, duration of infection, treatment regimen, and degree of neurocognitive impairment, which can all differ drastically E. S. Porges · A. J. Woods · R. A. Cohen Institute on Aging, Department of Aging and Geriatric Research, School of Medicine, University of Florida, Gainesville, FL, USA M. Law Department of Radiology, Keck School of Medicine, University of Southern California, Los Angeles, CA, USA R. H. Paul Missouri Institute of Mental Health, University of Missouri in Saint Louis, Saint Louis, MO, USA A. Thames Department of Psychology, University of Southern California, Los Angeles, CA, USA V. G. Valcour Memory and Aging Center, Department of Neurology, University of California, San Francisco, San Francisco, CA, USA R. A. Cohen Department of Psychiatry and Human Behavior, Warren Alpert Medical School, Brown University, Providence, RI, USA

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across study populations. Other sources of heterogeneity may come from variations in study sample sizes and analysis techniques that can also affect reported findings. It is important to account for technical and methodological differences across studies without compromising or washing out true biological differences. To address methodological differences, and boost statistical power, the ENIGMAHIV Working Group was established to harmonize data analysis from neuroimaging studies around the world. However, beyond coordinated diffusion MRI (dMRI) processing and analysis at each site, differences in scanners and acquisition protocol variables, including b-values, angular and spatial resolution have been shown to affect FADTI and other standard diffusion indices [3–5]. Meta-analysis of effects across sites alleviates the need to directly pool data, and is convenient for case/control studies; however, studies of rare modulators of effects (only a few participants per cohort) may also be possible, retrospectively, if data could be reliably pooled across sites. Several harmonization methods have been proposed, and a recent comparison of such techniques [6] suggested that ComBat [7] is one of the most effective harmonization techniques for DTI measures. The effect of harmonization techniques may not be independent of the measures being harmonized. It is important to select dMRI models and derived scalar indices that can detect disease effects and their modulators with maximal sensitivity and power. The DTI or single diffusion tensor model [8] is the most frequently used dMRI model in clinical studies, yet the model is limited as it can only model a single fiber population per voxel, with a single dominant orientation. At the typical resolution of dMRI, estimates of the proportion of WM voxels containing crossing fibers are as high as 90% [9, 10]. Among other possible models, the tensor distribution function (TDF) as proposed by [11], is a multi-tensor model that models crossing fibers as a probabilistic ensemble of Gaussian tensors. FA derived from the TDF (FATDF ) has been shown to be more sensitive to neurodegenerative disease-related differences than FADTI , regardless of angular resolution, and q-ball imaging (QBI) generalized FA (GFA) [12, 13]. This offers a potentially more powerful and suitable option for our ENIGMA-HIV study, where the dMRI data are from clinical studies, limiting reliable reconstruction of many higher-order diffusion models that require high angular resolution (HARDI) or multi-shell protocols (e.g. DSI [14], DKI [15], NODDI [16], and others). Prior studies have shown the TDF and resulting FATDF have less error in the fit and improved intra-study test-retest reliability across scans [12, 17], relative to FADTI , but multi-site reliability and harmonization across studies have not yet been evaluated. Here, we pooled and analyzed dMRI data from six HIV neuroimaging studies from around the world as part of the ENIGMA-HIV consortium, to evaluate (1) the effects of harmonization of dMRI measures across sites using ComBat, and (2) whether FATDF confers higher sensitivity across heterogeneous sites to detect the effect of HIV on WM microstructure. While future studies are needed to further quantify effects, we aimed to perform an initial investigation into the harmonization of diffusion measures, to better detect consistent and cohort-specific effects across heterogeneous studies of HIV infection.

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Table 1 Demographic characteristics of participants and DWI protocols, by study Study

Total N

Age, years (Mean ± SD)

Sex (% Male)

MRI scanner

Voxel size (mm3 )

DWI protocol (s/mm2 )

UCSF, USA [24]

52 HIV+ 28 HIV−

64.6 ± 3.5

92.5%

3T Siemens

2×2×2

1 b0 /64 DWI b = 2000

UNSW, Australia [25]

80 HIV+ 40 HIV−

55.0 ± 6.6

100%

3T Philips

2.5 × 2.5 × 2.5

2 b0 /32 DWI b = 1000

UCLA, USA [26]

61 HIV+ 33 HIV−

50.4 ± 13.2

73.4%

3T Siemens

1.98 × 1.98 × 2

7 b0 /64 DWI b = 1000

Brown, USA [27] ARCHa , USA

51 HIV+ 27 HIV−

42.8 ± 10.6

62.8%

3T Siemens

1.77 × 1.77 × 1.8

11 b0 /64 DWI b = 1000

26 HIV+ 40 HIV−

39.3 ± 10.6

45.5%

3T Siemens

2×2×2

[28]

11 b0 /64 DWI b = 1000

Serbia [29]

72 HIV+ 75 HIV−

40.0 ± 11.4

81.0%

3T Siemens

2.03 × 2 × 2

3 b0 /64 DWI b = 1500

a

Alcohol Research Center in HIV (ARCH)

2 Methods 2.1 Study Samples We analyzed data from six independent HIV studies from the U.S., Australia, and Serbia as part of the ENIGMA-HIV DTI consortium. T1-weighted and diffusionweighted MRI (DWI) were collected from 342 HIV-infected individuals (mean age: 50.6 ± 11.9 yrs; 301 M/46 F) and 243 seronegative controls (mean age: 45.3 ± 14.3 yrs; 163 M/80 F). Study specific demographic and clinical data, scanner and acquisition parameters are detailed in Tables 1 and 2. As part of an effort to understand effects of scanner hardware and software upgrades during the course of the Thai Resilience study of adolescents with perinatally acquired HIV [18], four individuals were scanned twice, on average 2.8 months apart (±2.0 mo) using two different scanners and protocols. These eight scans were used to better understand acquisition variability, as opposed to individual or cohort level differences. DWI from the first protocol were acquired on a 1.5 T GE scanner with 1 b0 , 25 b = 1000 s/mm2 volumes, and 3.5 × 3.5 × 3.5 mm isotropic voxels acquired in duplicate, while the second protocol DWI were acquired on a 3 T Philips scanner with 2 b0 and 32 b = 1000 s/mm2 volumes with 2 × 2 × 2 mm isotropic voxels. Each site obtained approval from their local ethics committee and/or institutional review board; participants signed an informed consent form at each participating site.

2.2 Image Preprocessing DWI images were denoised using the LPCA filter [19] and corrected for motion and eddy current distortions. T1-weighted images were denoised using the nonlocal means filter [20] and underwent N3 intensity inhomogeneity normalization [21],

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Table 2 Clinical characteristics of HIV+ participants, by study Study

% Detectable viral load

Mean current CD4+ (Mean ± SD cells/mm3 )

Mean Nadir CD4+ (Mean ± SD cells/mm3 )

% On cARTa

UCSF

32.6

517.04 (220.48)

191.42 (146.81)

98.0

UNSW

1.3

556.49 (278.25)

177.09 (125.69)

100.0

UCLA

35.7

603.11 (297.77 )

254.71 (224.57)

100.0

Brown

32.0

529.24 (271.10)

189.70 (171.43)

80.4

ARCH

34.6

499.92 (222.59)

182.08 (154.91)

92.0

Serbia

26.4

576.04 (330.56)

302.04 (187.32)

69.0

a Combination

antiretroviral therapy (cART)

and brain extraction [22]. These images were linearly aligned to diffusion images, and diffusion images were then non-linearly warped to their respective T1-weighted scans to correct for echo-planar imaging (EPI) induced susceptibility artifacts [23]; sites had not collected field maps and all protocols used single phase acquisition. Diffusion gradient directions were rotated to accommodate linear registrations.

2.3 Anisotropy Indices from DTI and TDF Models A single diffusion tensor (DTI) was modeled at each voxel in the brain from the corrected DWI scans, and fractional anisotropy (FADTI ) scalar maps were calculated [8]. In contrast to the single tensor model, the tensor distribution function (TDF) represents the diffusion profile as a probabilistic mixture of tensors that optimally explain the observed DWI data, allowing for the reconstruction of multiple underlying fibers per voxel, together with a distribution of weights [11]. As in [12], the TDF corrected form of FA (FATDF ) was calculated from the resulting TDF.

2.4 ENIGMA-DTI Processing Protocol Using publically available harmonized ENIGMA-DTI protocols (http://enigma.usc. edu/protocols/dti-protocols/) [30], individual subject FADTI maps were warped to the ENIGMA-DTI FA template with ANTs [23] and the transformations applied to respective FATDF maps. FA indices were projected onto the ENIGMA-DTI template skeleton with TBSS [31]. Using the Johns Hopkins University (JHU) WM atlas [32], mean skeletonized FA indices were extracted from 25 regions of interest (ROIs). Here, we focus on three ROIs in particular: the corpus callosum (CC), a region with highly coherent fiber organization, the corona radiata (CR), a region with large numbers of crossing fibers, and the full WM skeleton (Fig. 1a).

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Fig. 1 a The ENIGMA skeletonized FADTI template is displayed along with the three ROIs assessed in the study: the corona radiata (CR), corpus callosum (CC), and full WM skeleton. b b0 volumes from control participants in each study visually highlight differences between dMRI acquisition protocols

2.5 ComBat DWI Harmonization ComBat, originally developed to model and remove batch effects from genomic microarray data [7], has recently been applied to multi-site cortical thickness [33], functional MRI connectivity [34], and DTI measures [6]. ComBat uses an empirical Bayes framework to remove variability from multi-site data introduced by differences in acquisition protocols, while aiming to preserve the biological variability of site effect estimates. An empirical test of this method on data with diagnostic heterogeneity as complex as global HIV infection has yet to be performed. We used the ComBat algorithm to harmonize data across sites for each FA measure. For FATDF and FADTI , diagnosis, age, sex, and information from all 25 ROIs were used to inform the statistical properties of the site effects.

2.6 Statistical Analyses For each of the three ROIs, we fit a linear regression, covarying for age, sex, and age*sex interaction, to test for differences in FA between HIV+ individuals and seronegative controls at each site. A fixed-effects, inverse variance weighted meta-

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Fig. 2 For each FA measure, a meta- and mega-analysis was conducted on both the original and harmonized data

analysis of Cohen’s d effect sizes was conducted to combine effects across sites. Heterogeneity scores (I 2 ) for each test were also computed, indicating the percentage of the total variance in effect size explained by heterogeneity between sites. We also pooled the data and performed an analogous random effects mega-analysis across sites, grouping by site. After harmonization of data across sites, meta- and megaanalyses were repeated for comparison with unharmonized results (Fig. 2). The false discovery rate (FDR) procedure was used to correct for multiple comparisons across ROIs (q = 0.05) [35]. For each site, a two-tailed, one-sample T -test was used to identify significant changes (%) in full WM FA values after harmonization ((FAharmonized -FAoriginal )/ FAoriginal ); a two-sample paired T -test was used to assess significant differences between DTI and TDF percent change in each site. To identify significant differences between sites before and after harmonization, an ANCOVA was used controlling for diagnosis, age, sex, age*sex, and site as a random variable, followed by post-hoc pairwise tests between sites.

3 Results 3.1 DWI Harmonization b0 volumes from control participants in each study allow us to visualize differences across dMRI protocols from each site (Fig. 1b). An ANCOVA testing for significant site differences in FA measures confirmed significant differences in both full WM FADTI (P = 1.6*10−76 ; F = 196.8) and FATDF (P = 1.1*10−75 ; F = 99.9) between sites. In pairwise tests between sites, significant differences in 12 of 15 pairwise FADTI tests were detected (FDR q < 0.05; ARCH, UNSW, and Serbia were not different from each other, despite differences in scanner manufacturer, b-value, angular and

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spatial resolution). FATDF was significantly different in 11 of 15 tests (Brown, Serbia, and UCLA were not significantly different from each other, nor were Serbia and UCSF). After harmonization with ComBat, an ANCOVA detected significant differences in both full WM FADTI (P = 2.2*10−15 ; F = 16.7) and FATDF (P = 1.2*10−20 ; F = 22.5) between sites, but with reduced effect sizes (F-value) compared to the original, unharmonized effect sizes. Pairwise tests revealed no significant differences in FADTI between sites after multiple comparisons correction, yet a significant difference between FATDF from Brown and all sites but ARCH remained. Density plots of the pre-ComBat full WM TDF and DTI FA from HIV+ and seronegative control participants from each site revealed the distribution of one site, Brown, deviated the most from the other sites for FADTI but not FATDF (Fig. 3a). This was rectified after harmonization with ComBat (Fig. 3b). All sites showed significant changes in FA values after harmonization (Fig. 3c). While FADTI values did not consistently show a larger percent change in the full WM after harmonization compared to FATDF across sites, the largest percent change in FA was detected for FADTI in Brown (24%), followed by FADTI in UCSF (10%). FATDF changed at most 5% (UNSW). Similarly, box plots of TDF and DTI FA in the full WM before and after ComBat, in controls only, show the greatest changes in Brown and UCSF (Fig. 3d).

3.2 HIV Diagnosis Effect Sizes Significantly lower FATDF was detected in HIV+ individuals compared to seronegative controls in the full WM, CC, and CR in both mega- and meta-analyses (FDR q < 0.05); the largest effects were detected in the CR (Cohen’s d = −0.32; Fig. 4a). Significantly lower FADTI was only detected in the CC, a region with highly coherent WM organization, where effect sizes were comparable to those for FATDF . There were no major differences between meta- or mega-analysis effect sizes. Effect sizes were marginally lower after harmonization of FA measures across sites, but still within the standard error bounds; ComBat did not alter the statistical significance for any ROI. Heterogeneity scores (I 2 ) revealed approximately two times the variance explained between sites for FATDF effect size estimates compared to FADTI across ROIs, regardless of harmonization (Fig. 4b). There was effectively no variance across sites (I 2 = 0) in effects detected by FADTI in the CR. A comparison of effect sizes from each site in the CR—the region with crossing fibers and the largest FATDF effect size—showed (1) no major differences between original and harmonized analyses except for Brown, where FATDF showed improved effect sizes, and (2) improvements in effect sizes from FATDF compared to standard FADTI only for sites that already showed the strongest associations with FADTI (Fig. 4c).

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Fig. 3 Density plots of the mean TDF and DTI FA in the full WM skeleton from HIV+ and seronegative control participants from each site a before and b after harmonization with ComBat. c The mean percent change between original and harmonized full WM FA values are listed. d In controls only, box plots of the mean TDF and DTI FA in the full WM before and after harmonization are shown to highlight differences across cohorts even in controls

3.3 Cross Scanner Variability Despite differences in manufacturer, field strength, spatial and angular resolution, FADTI and FATDF measures were both highly correlated across Resilience study scanning protocols in the CC (FATDF Pearson’s r = 0.98, FADTI r = 0.97), CR (FATDF Pearson’s r = 0.97, FADTI r = 0.99), and full WM (FATDF Pearson’s r = 0.96, FADTI r = 0.99). A two-tailed, paired T -test revealed a significantly greater percentage increase in FATDF values compared to FADTI values between older and updated protocols, only in the CR (P = 0.004) and full WM (P = 0.04).

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Fig. 4 a Effect sizes (Cohen’s d and standard error) from meta- and mega-analyses of TDF and DTI FA differences between HIV+ and seronegative control participants before (FATDF Mega FDR threshold P F D R = 0.002, Meta P F D R = 0.001; FADTI Mega P F D R = 0.002, Meta P F D R = 0.001) and after harmonization with ComBat (FATDF Mega P F D R = 0.017, Meta P F D R = 0.008; FADTI Mega P F D R = 0.005, Meta P F D R = 0.002) in the corona radiata (CR), corpus callosum (CC), and full WM. b Heterogeneity scores (I 2 ) reflect the percentage of total variance in effect sizes explained by heterogeneity between sites. c Effect sizes (absolute value) of TDF and DTI FA differences between HIV+ and seronegative control participants for each site, before and after harmonization in the CR

4 Discussion Here we evaluated the effects of harmonization and higher-order tensor diffusion MRI scalar measures in a large multi-site HIV study. Harmonization with ComBat allowed sites such as Brown and UCSF, which deviated most from the other four sites, to achieve distributions more in line with the other cohorts. We note that Brown was the only site with hepatitis C co-infection, while the UCSF study included the oldest participants (Table 1). Although the distributions were more aligned, the differences between these cohorts and the others were not completely washed out. ComBat appears to preserve biologically meaningful variance between sites; the heterogeneity scores (I 2 ) did not change drastically after harmonization. We found no major effects of ComBat harmonization on meta- and mega-analysis findings. We note that in [6], voxel-wise information was used to harmonize measures while we used ROI information; fewer features may be a possible limitation of this approach, and future analyses will evaluate harmonization at the voxel-wise level.

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Improvements in scalar dMRI indices may play a major role in our ability to find consistent diffusion parameters and related neurobiological substrates across sites, regardless of harmonization. While many HIV studies report inconsistent differences in FADTI , in addition to heterogeneity in the HIV populations studied and methodology, this may be due to established limitations of the FADTI measure itself. Time constraints are often placed on dMRI protocols in clinical settings. HARDI or multishell protocols, often used to overcome these limitations, are still rare in clinical population studies, although this may change in the near future. In large multi-site clinical studies, such as ENIGMA—where reliable reconstruction of many other proposed higher-order diffusion models may not be feasible—the TDF may be a suitable option. HIV-related CC microstructural differences are some of the most consistently reported findings [2], and both FATDF and FADTI detected similar effect sizes in this region with highly coherent WM organization. FATDF was able to detect more widespread differences in the full WM and larger effect sizes, in particular in the CR, a region with crossing fibers, which was not significant with FADTI . This suggests HIV may have more widespread and robust effects on WM microstructure than previously reported. Protocol, scanner, and field strength changes between the original and improved Resilience study dMRI acquisitions resulted in similar changes of FATDF and FADTI values in coherent CC WM, but FATDF changed significantly more than FADTI in the CR. The DTI model is limited in regions with crossing fibers regardless of the quality of DWI acquisition protocols; i.e., crossing fibers may result in microstructural measures falsely representing those of unhindered diffusion. FATDF , however, models such crossings even in relatively low angular resolution data [12], and may still be sensitive to biological variability in cases where the DTI model does not capture anisotropic diffusion. This suggests FATDF could be more sensitive to biological as well as protocol differences. This may explain the fact that in pairwise tests, Brown (with its high rate of patient comorbidities) remains significantly different from other sites after harmonization for FATDF , but not FADTI . This is further suggested by the higher I 2 estimate in FATDF effect size variance between sites, particularly in the CR where FADTI shows effectively no variability. The largest differences between HIV+ participants and controls were detected with FATDF in the CR in both metaand mega-analyses across sites, whereas no significant differences could be detected with FADTI . This suggests that differences and changes in FATDF between sites and protocols are capturing real biological effects and not just noise. Larger studies with dMRI data collected with various protocols from the same participants are necessary to draw further conclusions. Statistical approaches such as ComBat may be helpful for harmonizing measures derived in multi-cohort imaging studies, yet parallel efforts into improving diffusion metrics to more accurately and consistently represent the underlying white matter microstructure are also critical. Advances in both of these lines of research will ultimately help us uncover common neurobiological consequences of HIV infection, and allow us to dig deeper into modulators of disease effects on the brain in HIV+ individuals around the world.

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Acknowledgements Funding for ENIGMA is provided as part of the BD2K Initiative U54 EB020403 to support big data analytics, and by P41 EB015922. Work from each site was funded by: (1) UCLA: K23MH095661, Clinical and Translational Research Center Grants UL1RR033176 and UL1TR000124 (ADT); MH19535 (TK); (2) Serbia: Provincial Secretariat for Higher Education and Scientific Research 114-451-2730/2016-02; (3) UNSW: NHMRC APP568746 (LC); (4) Brown and ARCH: R01MH074368, the Lifespan/Tufts/Brown Center for AIDS Research P30 AI042853, P01AA019072 (RC); (5) UCSF: K23AG032872 (VV), R01AG048234, and R01AG032289; (6) Resilience: R01MH102151 (JA). This work was also supported by R01MH085604 Neuropathogenesis of clade C HIV in South Africa.

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Muti-shell Diffusion MRI Harmonisation and Enhancement Challenge (MUSHAC): Progress and Results Lipeng Ning, Elisenda Bonet-Carne, Francesco Grussu, Farshid Sepehrband, Enrico Kaden, Jelle Veraart, Stefano B. Blumberg, Can Son Khoo, Marco Palombo, Jaume Coll-Font, Benoit Scherrer, Simon K. Warfield, Suheyla Cetin Karayumak, Yogesh Rathi, Simon Koppers, Leon Weninger, Julia Ebert, Dorit Merhof, Daniel Moyer, Maximilian Pietsch, Daan Christiaens, Rui Teixeira, Jacques-Donald Tournier, Andrey Zhylka, Josien Pluim, Greg Parker, Umesh Rudrapatna, John Evans, Cyril Charron, Derek K. Jones and Chantal W. M. Tax Abstract We present a summary of competition results in the multi-shell diffusion MRI harmonisation and enhancement challenge (MUSHAC). MUSHAC is an open competition intended to stimulate the development of computational methods that reduce scanner- and protocol-related variabilities in multi-shell diffusion MRI data across multi-site studies. Twelve different methods from seven research groups have L. Ning (B) · S. C. Karayumak · Y. Rathi Brigham and Women’s Hospital, Boston, USA e-mail: [email protected] L. Ning · J. Coll-Font · B. Scherrer · S. K. Warfield · S. C. Karayumak · Y. Rathi Harvard Medical School, Boston, USA E. Bonet-Carne · F. Grussu · E. Kaden · S. B. Blumberg · C. S. Khoo · M. Palombo University College London, London, UK F. Sepehrband · D. Moyer University of Southern California, Los Angeles, USA J. Veraart New York University, New York City, NY, USA J. Coll-Font · B. Scherrer · S. K. Warfield Boston Children’s Hospital, Boston, USA S. Koppers · L. Weninger · J. Ebert · D. Merhof RWTH Aachen University, Aachen, Germany M. Pietsch · D. Christiaens · R. Teixeira · J.-D. Tournier King’s College London, London, UK A. Zhylka · J. Pluim Eindhoven University of Technology, Eindhoven, Netherlands G. Parker · U. Rudrapatna · J. Evans · C. Charron · D. K. Jones · C. W. M. Tax CUBRIC, School of Psychology, Cardiff University, Cardiff, UK D. K. Jones School of Psychology, Australian Catholic University, Melbourne, Australia © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_18

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been tested in this challenge. The results show that cross-vendor harmonization and enhancement can be performed by using suitable computational algorithms such as deep convolutional neural networks. Moreover, parametric models for multi-shell diffusion MRI signals also provide reliable performances. Keywords Diffusion MRI · Harmonisation · Spherical harmonics Deep learning · Parametric model

1 Introduction Multi-center clinical studies on mental disorders usually need to combine diffusion magnetic resonance imaging (dMRI) data acquired from different scanners to increase the statistical power and sensitivity of studies. But inter-scanner variability and different acquisition protocols could introduce inherent variability in dMRI data which is a major limitation in multi-center dMRI studies [1–3]. Therefore, there is a pressing need for reliable approaches to harmonise scanner- and/or protocol-related variability in dMRI data [4–8]. As a continuation of the single-shell diffusion MRI harmonisation challenge [9, 10], the multi-shell diffusion MRI harmonisation and enhancement challenge (MUSHAC) is an open competition that provides datasets from different scanners to enable the development new harmonisation algorithms for multi-shell dMRI data. It evaluates the performances of different algorithms using a test dataset from the same scanners based on several dMRI measures that are relevant in clinical studies. This work reports the preliminary evaluation results from MUSHAC. Data: The datasets are a subset of the benchmark database in [9, 10]. MUSHAC uses the data of 15 healthy volunteers who were scanned on a 3T Siemens Prisma and 3T Siemens Connectom Scanner. On both scanners, dMRI images were acquired with a ‘standard’ (ST) protocol and a ‘state-of-the-art’ (SA) protocol. The ST data from both scanners have an isotropic resolution of 2.4 mm, TE = 89 ms and TR = 7.2 s. Both ST dMRI data have 30 directions at b = 1200, 3000 s/mm2 . On the other hand, the Prisma-SA data has a higher isotropic resolution of 1.5mm, TE = 80 ms, TR = 7.1s and 60 directions at the same b-values while the Connectom-SA data has an even higher resolution of 1.2 mm with TE = 68 ms, TR = 5.4 s and 60 directions. Additional b = 0 s/mm2 images were also acquired with matched TE and TR across scanners. Moreover, structural MPRAGEs (e.g. T1-weighted anatomical MRI scans) were acquired on both scanners. The data from 10 randomly selected subjects were used as training data and the remaining 5 subjects were used for testing. Preprocessing: The b0 volumes were corrected for EPI distortions by applying FSL TOPUP on reversed phase-encoding pairs [11]. The data was corrected for eddy current induced distortions, subject motion, EPI distortions [12], and gradientnonlinearity distortions [13] with FSL TOPUP/eddy and in-house software kindly provided by Massachusetts General Hospital (MGH). Spatiotemporally varying

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b-vectors and b-values due to gradient nonlinearities of the Connectom scanner were made available [14–16]. All data were affinely registered to Prisma-ST using the corresponding fractional anisotropy (FA) maps with appropriate b-matrix rotation. Tasks: This competition included two tasks on multi-shell dMRI data harmonisation. Specifically, Task 1 is to predict Connectom-ST data using Prisma-ST data where both datasets have matching TE, TR and angular and spatial resolution. The second task includes the prediction of Prisma-SA data (Task 2a) and Connectom-SA data (Task 2b) given Prisma-ST data. Evaluation: For the test datasets, fractional anisotropy (FA) and mean diffusivity (MD) were obtained by fitting the diffusion tensor model to the low b-shell dMRI signals using the weighted least squares algorithm. The zeroth and second order rotationally invariant spherical harmonic (RISH) features [6], i.e. L0 and L2, were computed for both b-shells. Moreover, two multi-shell dMRI measures, including mean kurtosis (MK) [17] and the return-to-origin probability (RTOP) measure [18], were also computed using the DIPY [19] toolbox. All measures were computed from the predicted data of each subject and compared to the ground-truth measures derived from the acquired data, which were not released to the participants. The performances were evaluated in brain regions specified by brain masks excluding the cerebellum, obtained with the Geodesic Information Flow (GIF) algorithm [20]. Then, the percentage error (PE) and the absolute-value of PE (APE) were computed globally in a brain mask excluding cerebellum and regionally based on FreeSurfer regions excluding the cerebellum. The reference values for PE and APE in Task 1 were computed by comparing the dMRI measures from Connectom-ST and PrismaST data. Algorithms: Twelve different algorithms from seven research groups were evaluated for Task 1. Eight of these algorithms were also evaluated for Task 2a and Task 2b. Based on the similarities of the underlying methods, these algorithms can be grouped into three categories. The first category includes three interpolation-based algorithms, where Algorithm 1 uses spherical harmonic (SH) functions to interpolate signals along angular directions and Algorithms 2 and 3 interpolate the multi-shell dMRI signals using a parametric model for multi-shell dMRI signals. The second type of algorithms is based on regressions. In particular, Algorithm 4 uses linear regression based on RISH features, and Algorithms 5 and 6 are based on regression forest. The third type of algorithms use either deep convolutional neural networks (CNN), including Algorithms 7, 8, 10, 11, 12, or a fully connected feed-forward network, including Algorithm 9, to learn mappings between dMRI signals using SH representations. The twelve algorithms include but are not limited to the methods proposed in [8, 21, 22].

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2 Results 2.1 Scanner-to-Scanner Harmonisation (Task 1) Most algorithms could significantly reduce inter-scanner variabilities in the dMRI measures. All deep-CNN based approaches had similar performances in this task and are in general better than the other two types of algorithms. In particular, Fig. 1a, b illustrate the APE for the zero-order (L0) RISH features at the high b-shell and MK, respectively. The blue and red markers represent the median and mean values in global evaluations and the vertical bars represent the 20–80 percentiles of the distributions. The 80-percentile bars go beyond the displayed range of the plots. The two dashed horizontal lines show the reference median and mean values obtained by comparing the ground-truth measures from acquired data. Figure 2a illustrates the reference distribution of the PE in different brain regions between the acquired dMRI data. The frontal lobe and occipital lobe consistently have higher errors than other brain regions, which may be caused by stronger susceptibility distortions in these regions. On the other hand, Fig. 2b illustrates the region-wise PE corresponding to the output of Algorithm 12 which significantly reduced the errors in these ROI.

2.2 Spatial- and Angular Resolution Enhancement (Tasks 2a and 2b) Figure 3a, b show the APE of the L0 RISH features for the b = 3000 s/mm2 signals and the MK measure, respectively, for Task 2a. Figure 3c, d show the corresponding results for Task 2b. In this case, no reference lines are plotted since there is no ground-truth due to different spatial resolutions. It is interesting to note that the interpolation-based Algorithm 3 has consistently good performances in these tasks.

3 Discussion and Conclusion We present preliminary results on a comparison of the performance of several harmonisation algorithms that estimate mappings between multi-shell dMRI data acquired using different scanners and/or protocols. These algorithms include interpolation-based approaches, regression-based approaches and deep-learning based methods. Our results show that harmonisation algorithms could significantly

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(a) Region-wise PE of RTOP (Reference)

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Acknowledgements CMWT is supported by a Rubicon grant (680-50-1527) from the Netherlands Organisation for Scientific Research (NWO) and Wellcome Trust grant (096646/Z/11/Z). LN is supported in part by NIH grants R21MH115280 and R21MH116352. FG is funded by the Horizon2020-EU.3.1 CDS-QuaMRI grant (ref.: 634541) and by the Engineering and Physical Sciences Research Council (EPSRC ref.: EP/M020533/1 and EP/R006032/1). EBC is supported by Prostate Cancer UK (Grant PG14-018-TR2) and by the Engineering and Physical Sciences Research Council (EPSRC ref.: EP/M020533/1). DKJ was supported by MRC grant MR/K004360/1. Scan costs were supported by the National Centre for Mental Health (NCMH) with funds from Health and Care Support Wales and by the Wellcome Trust. JV is a Postdoctoral Fellow of the Research Foundation - Flanders (FWO; grant number 12S1615N). AZ has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 765148. SK was supported by the International Research Training Group 2150 of the German Research Foundation (DFG).

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9. Tax, C., et al.: Cross-vendor and cross-protocol harmonisation of diffusion tensor imaging data: a comparative study. ISMRM-ESMRMB. Number 0471 (2018) 10. Tax, C., et al.: Cross-scanner and cross-protocol diffusion mri data harmonisation: a benchmark database and evaluation of algorithms. submitted (2018) 11. Andersson, J.L., Skare, S., Ashburner, J.: How to correct susceptibility distortions in spinecho echo-planar images: application to diffusion tensor imaging. NeuroImage 20(2), 870–888 (2003) 12. Andersson, J.L., Sotiropoulos, S.N.: An integrated approach to correction for off-resonance effects and subject movement in diffusion MR imaging. NeuroImage 125, 1063–1078 (2016) 13. Glasser, M.F., Sotiropoulos, S.N., Wilson, J.A., Coalson, T.S., Fischl, B., Andersson, J.L., Xu, J., Jbabdi, S., Webster, M., Polimeni, J.R., Essen, D.C.V., Jenkinson, M.: The minimal preprocessing pipelines for the human connectome project. NeuroImage 80, 105–124 (2013) 14. Bammer, R., Markl, M., Barnett, A., Acar, B., Alley, M., Pelc, N., Glover, G., Moseley, M.: Analysis and generalized correction of the effect of spatial gradient field distortions in diffusionweighted imaging. Magn. Reson. Med. 50(3) 560–569 15. Sotiropoulos, S.N., Jbabdi, S., Xu, J., Andersson, J.L., Moeller, S., Auerbach, E.J., Glasser, M.F., Hernandez, M., Sapiro, G., Jenkinson, M., Feinberg, D.A., Yacoub, E., Lenglet, C., Essen, D.C.V., Ugurbil, K., Behrens, T.E.: Advances in diffusion MRI acquisition and processing in the human connectome project. NeuroImage 80, 125–143 (2013) 16. Rudrapatna, S.U., Parker, G.D., Roberts, J., Jones, D.K.: Can we correct for interactions between subject motion and gradient-nonlinearity in diffusion MRI? ISMRM. Number (1206 (2018)) 17. Jensen, J.H., Helpern, J.A., Ramani, A., Lu, H., Kaczynski, K.: Diffusional kurtosis imaging: the quantification of non-gaussian water diffusion by means of magnetic resonance imaging. Magn. Reson. Med. 53(6), 1432–1440 (2005) 18. Özarslan, E., Koay, C.G., Shepherd, T.M., Komlosh, M.E., Irfanoglu, M.O., Pierpaoli, C., Basser, P.J.: Mean apparent propagator (MAP) MRI: a novel diffusion imaging method for mapping tissue microstructure. NeuroImage 78, 16–32 (2013) 19. Garyfallidis, E., Brett, M., Amirbekian, B., Rokem, A., Van Der Walt, S., Descoteaux, M., Nimmo-Smith, I., Contributors, D.: Dipy, a library for the analysis of diffusion MRI data. Front. Neuroinformatics 8, 8 (2014) 20. Cardoso, M.J., Modat, M., Wolz, R., Melbourne, A., Cash, D., Rueckert, D., Ourselin, S.: Geodesic information flows: spatially-variant graphs and their application to segmentation and fusion. IEEE Trans. Med. Imaging 34(9), 1976–1988 (2015) 21. Scherrer, B., Schwartzman, A., Taquet, M., Sahin, M., Prabhu, S.P., Warfield, S.K.: Characterizing brain tissue by assessment of the distribution of anisotropic microstructural environments in diffusion-compartment imaging (DIAMOND). Magn. Reson. Med. 76(3), 963–977 (2016) 22. Blumberg, S.B., Tanno, R., Kokkinos, I., Alexander, D.C.: Deeper image quality transfer: training low-memory neural networks for 3D images. In: Frangi, A.F., Schnabel, J.A., Davatzikos, C., Alberola-López, C., Fichtinger, G. (eds.) Medical Image Computing and Computer Assisted Intervention—MICCAI 2018., pp. 118–125 (2018)

Part IV

Diffusion MRI Outside the Brain and Clinical Applications

Diffusion MRI Outside the Brain Rita G. Nunes , Luísa Nogueira , Andreia S. Gaspar , Nuno Adubeiro and Sofia Brandão

Abstract This manuscript provides an overview of recent developments in DiffusionWeighted Imaging (DWI) outside the brain, focusing on liver, breast, prostate, muskuloskeletal (MSK) and cardiac applications. A general introduction to crosscutting acquisition and image processing challenges is first provided. These often include short T2 relaxation times, the need to image a large field-of-view with the resulting complications in shimming the B0 field and achieving good fat suppression. Some of the strategies developed for dealing with motion, namely cardiac and respiratory motion are described. Specific sections are then presented for each of the aforementioned organs. A motivation for the clinical applicability of DWI is first provided, followed by specific image acquisition and processing considerations. Quantitative imaging is becoming standard in clinical practice, and the Apparent Diffusion Coefficient is routinely estimated in the liver, breast and prostate. Application of alternative signal models in these organs is being explored, including both the Supported by Fundação para a Ciência e a Tecnologia grants SFRH/BD/120006/2016, IF/00364/2013 and UID/EEA/5009/2013. R. G. Nunes (B) · A. S. Gaspar Institute for Systems and Robotics (LARSYS) and Department of Bioengineering, Instituto Superior Técnico, University of Lisbon, Lisbon, Portugal e-mail: [email protected] L. Nogueira · N. Adubeiro Department of Radiology, School of Health/Polytechnic of Porto (ESS-P.Porto), Porto, Portugal N. Adubeiro Institute of Biomedical Sciences Abel Salazar (ICBAS), University of Porto, Porto, Portugal S. Brandão Department of Radiology, Centro Hospitalar de São João - EPE/Faculty of Medicine, University of Porto, Porto, Portugal S. Brandão CESPU, CRL - Advanced Institute of Health Sciences, Gandra, Paredes, Portugal S. Brandão Associated Laboratory for Energy, Transports and Aeronautics (LAETA), Institute of Science and Innovation in Mechanical and Industrial Engineering (INEGI), Faculty of Engineering, University of Porto, Porto, Portugal © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_19

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Intravoxel Incoherent Motion and Diffusion Kurtosis models. Ongoing efforts are focused on evaluating the potential clinical added value of the extra parameters and on improving their repeatability. MSK and cardiac DWI have shown potential for assessing pathological changes in fiber architecture, but further validation is required to enable application in the clinical setting. Keywords Diffusion-weighted MRI · Liver · Breast · Prostate Muskuloskeletal system · Cardiac · Applications

1 Introduction The first clinically-established application of Diffusion-Weighted Imaging (DWI) was in the brain, for the detection of stroke [47]. Sensitizing MR images to the diffusion motion of water molecules has enabled to evaluate changes in the tissues before these become noticeable using conventional sequences. Despite the enormous potential of applying DWI to other organs, additional challenges had to be addressed. The next subsections will cover common issues in body DWI, and describe some of the details that have to be taken into account when optimizing this type of acquisition protocol, enabling DWI to be presently applied to many organs. Here we will focus on the liver, breast and prostate. However, DWI can also be applied to, for example, the kidneys, pancreas, spleen, bowels and uterus [94]; promising results have also been obtained in the placenta [87]. Examples of the application of DWI to more challenging imaging targets will also be provided including the muskuloskeletal (MSK) system and recent developments aiming to make cardiac DWI available in the clinic.

1.1 Diffusion Sensitization Modules The standard approach to sensitize images to diffusion is to apply a pair of strong gradients within a Spin-Echo (SE) module. Unfortunately, diffusion contrast results in reduced Signal-to-Noise Ratio (SNR) and also increases sensitivity to subject motion. When using the SE approach to achieve a high DW (b-value), the diffusion gradients need to be long, prolonging the echo time (TE) and inducing eddy currents (EC) which lead to significant spatial distortions. For organs with short T2 relaxation times, this strategy may not always be feasible, despite hardware developments enabling increasingly higher gradient amplitudes. A better alternative in that case may be to use the stimulated echo (STEAM) preparation. In STEAM the 180◦ refocusing pulse is replaced by two 90◦ pulses, and two diffusion gradients are then applied before the 2nd and after the 3rd 90◦ radiofrequency (RF) pulses. Higher b-values can be achieved by augmenting the mixing time, corresponding to the interval between the

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last two RF pulses, and during which only T1 -relaxation occurs. The advantage is that by not having to prolong TE, the SNR can be maintained. The use of shorter diffusion gradients may also be beneficial to reduce EC. Another option is to employ a doubly-refocused SE preparation module, which employs a set of four diffusion gradients instead of only a pair. By adjusting their duration, it is possible to minimize the resulting EC and hence reduce image distortions [80]. EC and susceptibility-related spatial distortions can be corrected for by registering the DW images to the b0 images and applying distortion correction using a B0 map [42]. Affine registration also minimizes the effect of subject motion, and the corresponding b-matrix correction is recommended [52].

1.2 Motion Minimization and Correction Strategies DWI is very vulnerable to subject motion, which is typically more severe outside the brain as a result of cardiac, respiratory and peristaltic movements. Motion during the diffusion preparation period introduces spurious phase terms in the k-space data, which can result in image artifacts when using multi-shot readouts [2]. Data inconsistencies can be avoided by using single-shot DW-echo planar imaging (EPI) sequences as the whole of k-space is covered after a single diffusion module. Singleshot EPI is therefore the most commonly used sequence and it provides the added benefit of enabling high data acquisition rates. To enable further acquisition speedups, DW-EPI with simultaneous slice excitation (SMS) [86] has recently been explored in body applications. No significant differences were found in prostate Apparent Diffusion Coefficient (ADC) estimates with SMS-acceleration versus standard single-shot EPI [99], nor in breast ADC while combining SMS with readout segmented (rs-)EPI [22]. The application of SMS acceleration to intravoxel incoherent motion (IVIM) analysis in the liver and the pancreas has also been investigated [76]. Although improvements in the SNR per time unit could be obtained by combining SMS with single-shot EPI, regardless of the acceleration factor (AF), image quality was comparable with an AF of 2, but scored poorer for AF of 3. Significant differences in diffusion parameters were found for an AF of 3 warranting further investigation [76]. In cases of severe motion, individual slice images may also be affected and display regions of signal dropout, which can be misinterpreted as being due to high diffusivity. An approach for reducing these effects is to modify the DW module to avoid phase accrual in the presence of constant velocity motion (nulling the first gradient moment) or even of constant acceleration (nulling the second moment). Using waveforms with multiple gradient lobes, sequence dead-times can be reduced and the desired gradient moments nulled, while minimizing TE. Diffusion gradient optimization accounting for hardware and sequence timing constraints has shown promise for liver and cardiac DWI, resulting in reduced artifacts and improved SNR [1]. Inter-scan motion also needs to be considered when performing quantitative diffusion measurements, as these require the combination of images acquired with varying

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levels of DW and/or gradient directions. In that case, co-registration of the images becomes necessary [42].

1.3 Other Common Acquisition Challenges An important issue in body imaging is that a large field-of-view (FOV) is often required to avoid aliasing artifacts. When using EPI readouts, this is problematic as longer readout windows are needed, resulting in increased image distortion, longer TE and lower SNR. When imaging a large heterogeneous FOV, shimming is also harder. Attaining a less homogeneous B0 field will in turn impact the success of fat suppression approaches, particularly relevant in body imaging. When the organ of interest occupies a small fraction of the FOV, one option is to use zoomed acquisitions, applying 2D RF pulse designs to excite a smaller region [81].

2 Liver Applications 2.1 Liver DWI in the Clinical Setting There has been a growing interest in applying DWI to detect and characterize focal liver lesions (FLL) and diffuse liver disease (e.g. fibrosis and nonalcoholic fatty liver disease) [3]. The presence of fat deposits is thought to restrict water diffusivity and incomplete suppression of the fat signal may further contribute to lower ADC measurements as fat is more viscous than water [63]. Since low b-value inherently suppresses intra-vascular signal, visual inspection of DW images enables to detect (small) FLL [104]. Quantitative DWI can aid in the differentiation between benign and malignant lesions, as typically ADC is lower in malignant lesions due to a higher tissue cellularity [26].

2.2 Technical Challenges in Liver DWI Diffusion quantification requires several images. As the liver changes position throughout the respiratory cycle, combining images with multiple b-values and/or multiple averages is challenging. Different strategies exist to deal with respiratory motion: breath-holding (BH), respiratory triggering (RT) and free-breathing (FB). BH and RT reduce image blurring, improving lesion detection, although the latter requires longer exams. Despite this, no significant differences were found between methods when quantifying the ADC in FLL [11]. However, Contrast-to-Noise Ratio (CNR) and SNR improvements

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were obtained with non-BH methods, as their higher acquisition rates enable more repetitions within the same scan times [11]. BH may not be sufficient to ensure consistent liver positions across acquisitions. Discarding spatially-inconsistent images can improve repeatability [10] while image registration may increase acquisition efficiency. In addition, liver DWI suffers from artifacts associated to cardiac motion, mostly spurious signal loss in the left liver lobe. Cardiac triggering helps, but similarly to RT, it prolongs exam times [55]. Image registration is particularly difficult in liver DWI: besides contrast variation with b-value and low SNR at high b-values, non-rigid tissue deformation also occurs [83]. To aid in motion correction, joint diffusion and motion parameter estimation has been proposed [49]. Optimization is done in an interleaved manner, keeping either the motion transformations or the diffusion parameters fixed and estimating the remaining parameters. Spatial regularization can be applied during diffusion estimation, as discussed below, and the spatial transformations can be constrained to be smooth. These approaches have shown promising results with lower ADC errors reported in synthetic data, and improved consistency observed in real datasets [83]. Similar strategies have been applied to the spleen [88] and bowel regions [49], where DWI faces similar challenges.

2.3 Application of Advanced Diffusion Models IVIM Model Since the liver is a highly-irrigated organ and as capillary microcirculation may be affected by pathology and/or confound microstructure diffusion estimates [54], there has been much interest in applying the IVIM model to this organ. IVIM requires the acquisition of multiple DWI (S) with varying levels of diffusion-weighting b to estimate the pseudo-diffusion coefficient D*, whose effect dominates at low b-values, the perfusion fraction f and the diffusion coefficient D Eq. 1 [51]. (1) S(b) = S(0)[(1 − f ) exp(−bD) + f exp(−bD ∗ )] Although the parameter D* is more susceptible to noise, it appears to be more sensitive compared to D in the evaluation of tumours or liver fibrosis [54]. A negative correlation has been found between D*, D, f and the degree of liver fibrosis, probably caused by the higher concentration of collagen fibers [54]. Unfortunately, bi-exponential fits are less stable in the presence of noise fluctuations compared to monoexponential fits, and this particularly impacts the repeatabilities of D* and f, which are poorer than that of D [54]. To improve fitting reliability, a large number of b-values and multiple averages can be obtained to ensure sufficient SNR at the cost of prolonged scanning times and higher motion sensitivity; several strategies exist to stabilize the solution without compromising scan efficiency. One option is to estimate the model parameters within a region of interest (ROI) and/or to use this estimate as an input for pixelwise fitting [69].

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Another strategy consist of applying the Bayesian formalism, more robust to signal fluctuations than least-square fitting approaches. These methods use prior knowledge on the parameter distributions [100], and can be designed to enforce spatial constraints (e.g. Gaussian “shrinkage” prior) [69]. Local homogeneity priors have been applied to limit variability between spatiallyconnected pixels. To minimize the corresponding energy functional, the application of graph cuts has been proposed; each iteration fuses the current solution and a new estimate, obtained from bootstrap re-sampling [23]. This strategy enabled to improve the repeatability of liver D* compared to independent voxelwise segmented least-squares fitting with the coefficient of variation decreasing from 47% to 17% [88]. Diffusion Kurtosis Imaging To increase sensitivity to diffusion restriction, higher b-values are required (≈1500-2000 s/mm2 ), in which case the standard monoexponential model is no longer adequate. Diffusion Kurtosis Imaging (DKI) enables to estimate the apparent kurtosis (K app ) which reflects the deviation of water diffusion displacement probability from the Gaussian form [82]. In DKI [39], the signal decay is given by Eq. 2. 1 ln[S(b)] = ln[S(0)] − bDapp + b2 Dapp 2 Kapp 6

(2)

where Dapp corresponds to the ADC corrected for the apparent kurtosis—term in K app . The kurtosis coefficient represents the deviation from free diffusion (when this parameter is null the DKI model reduces to the monoexponential diffusion model) and may reflect the presence of tissue barriers restricting water diffusion. Application of DKI has been explored in the liver, with viable lesions displaying significantly higher kurtosis values. The K app displayed a higher sensitivity than ADC to predict the post-treatment response in hypervascular hepatocellular carcinoma (85.7% compared to 79.6%) [29]. Similar diagnostic performance was reported for DKI metrics and ADC when staging liver fibrosis; when performing Receiver Operating Characteristic (ROC) analysis to identify stage 2 fibrosis or greater, an area-under-the curve (AUC) of 0.809 was obtained for kurtosis compared to 0.797 for the ADC [103]. In conclusion, non-invasive approaches for the characterization and surveillance of liver fibrosis, fat deposition or liver lesions may avoid unnecessary biopsies and increase the diagnostic accuracy of follow-up exams. Performing DWI in the abdominal region is challenging, and the trade-off between performing fast BH sequences or acquiring high quality data to assess structural indices may require deciding on a case-by-case basis.

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3 Breast Applications 3.1 Breast DWI in the Clinical Setting DWI is currently used as an adjunct tool in breast MRI. Guidelines to perform breast MRI include a dynamic contrast enhanced (DCE) acquisition, but gadolinium-based contrast agents are not always well-tolerated. Also, given recent reports of gadolinium deposition in the brain [18], this contrast agent should be avoided whenever possible. Furthermore, the higher sensitivity of DWI could potentially overcome mammography limitations in dense breasts, given the positive correlation between ADC and mammographic breast density [58]. For this reason, efforts to use DWI in breast cancer screening as a potential substitute for DCE are ongoing. As changes in tissue cellularity and cell membrane integrity affect water diffusivity, DWI is useful to identify malignant lesions which tend to present more compact environments and thus decreased diffusivity [45]—see Fig. 1. Qualitative (visual) DWI image analysis is still most frequently performed, but quantitative approaches are increasingly preferred. ROI analysis is the most used method to measure the ADC but its practical implementation remains under debate. Some authors suggest drawing the ROI directly on the ADC map, but this may introduce bias [71]. Instead, one can spatially match T2 - and T1 -weighted post-contrast images to the DWI and propagate the lesion ROI to the ADC map. Also, some researchers suggest considering all slices where the lesion is visible, while others advise selecting only the slice where it is more easily seen [75]. Choosing to encompass the whole lesion [74] or only its most solid part also affects the ADC. To allow comparisons, the same procedure should be consistently used.

3.2 Technical Challenges in Breast DWI Applying DWI in the breast has specific challenges, the first of which is the variable amount of fat, which can be challenging when optimizing fat suppression; this is further complicated by the breasts’ off-isocenter position, less optimal for B0 shimming [78]. To improve image quality, localized high order shimming, Short TI Inversion Recovery (STIR) or Spectrally Adiabatic Inversion Recovery (SPAIR)-based enable good fat suppression [68]. The other difficulties arise from the use of EPI based-sequences. EPI is prone to ghosting, chemical shift artifacts, geometric distortions and appearing of signal pile up regions due to magnetic susceptibility changes; the latter are especially problematic in structures near the air-breast interface. Applying parallel imaging (PI) is mandatory. PI shortens the readout window, minimizing distortions and chemical shift artifacts albeit at the cost of reduced SNR which may condition lesion detection [41]. On the other hand, artifacts in the DW images may impair correlating DCE-MRI and DWI data sets.

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Fig. 1 A 35-year-old woman with a malignant lesion (indicated by the arrows) in the outer quadrant of the left breast: a axial DCE, showing a spiculated 30 mm lesion (an heterogeneous and irregular mass); b sagittal DWI at b 50 s/mm2 and c b 1000 s/mm2 evidencing a hyperintense lesion; d the lesion appears hypointense on the ADC map. Mean ADC value of 0.816 × 10−3 mm2 /s. Histological results: invasive ductal carcinoma

To overcome these limitations, new acquisition strategies are being developed. An option is to use a reduced field-of-view (rFOV) excitation, instead of a bilateral large FOV. It enables higher resolution images with less artifacts, providing finer morphological detail [4, 16]. Another alternative is readout-segmented-EPI (rs-EPI), which reduces artifacts by shortening the inter-echo spacing [77]. rs-EPI has been reported to improve depiction of breast lesions and discrimination between lesion types [44]. However, both require longer exams. The effect of gadolinium on ADC values is also controversial. Nguyen et al. [65] investigated the impact of performing DWI before and after its administration, and found no differences in ADC values of malignant lesions, but increased ADC values in normal tissue when DWI followed DCE.

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3.3 Application of Advanced DWI Models Breast tissue exhibits a complex microstructure, and hence the assumption of isotropic free diffusion is not valid. Alternative models have been proposed to evaluate anisotropy, microvascularization and lesion heterogeneity for better characterization. Diffusion directionality can be assessed using Diffusion Tensor Imaging (DTI), through the fractional anisotropy (FA) index. Inconsistent values of FA have been found for malignant lesions and normal tissue [9, 19, 73]. Also, significantly lower values were reported in malignant lesions compared to normal tissue for the difference in eigenvalues λ1 - λ3 (absolute maximal anisotropy index) [25]. IVIM has also been applied to breast MRI. There have been reports of increased perfusion fraction in invasive breast cancers [5] and the joint use of IVIM and DCEMRI has been suggested to improve the specificity and accuracy of breast MRI. These were reported to improve from 70.2% and 82.8% to 85.1% and 87.5%, respectively [56]. Higher b-value regimes (>1500 s/mm2 ) have also been explored. Higher mean K app has been measured for malignant compared to benign lesions [67] and DKI suggested to further improve diagnostic accuracy (AUC of 0.849) beyond DCEMRI (AUC of 0.791) [53]. The use of an alternative model including DKI and IVIM contributions has shown promising results: combining thresholds for K app , Dapp and the perfusion fraction f enabled a minimum sensitivity of 94.7% and minimum specificity of 66.7% (considering two readers) [36]. Although lower, the sensitivity was not significantly different from that achieved using the recommended breast MRI protocol and BI-RADS assessment categories (100% sensitivity and 79.2% specificity). This approach may therefore become relevant for populations requiring alternatives to contrast-enhanced sequences (e.g. pregnant patients) [36, 37]. It has been shown that diffusion metrics in normal tissue and lesions show different dependence on the diffusion time [92]. Such measurements can be performed using a STEAM diffusion preparation module and varying the mixing time so as to prolong the interval between the two diffusion gradients. The longer diffusion times allow more chances for water molecules to experience hindrance due to barriers, such as cell membranes [92]. This dependence could be used to infer on microstructure complexity improving lesion characterization. In conclusion, the application of breast DWI in the clinical setting is already established. However, optimizing the technical parameters for DTI and DKI protocols may require adaptations to promote its wider use. Moreover, validation of their derived indices for translational medicine is currently under research. Understanding (ab)normal tissue microanatomy and cancer pathophysiology by means of these non-invasive parameters will require new paradigms, but their clinical value is undoubtable.

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4 Prostate Applications 4.1 Clinical Impact of Prostate DWI Prostate cancer (PCa) is the second most common cancer in men. Multiparametric MRI is the most used imaging modality in the diagnosis and staging of PCa. Recent guidelines (PI-RADS-MRI, version 2) indicate DWI (and the ADC value) as one of the most important sequences for aiding in PCa diagnosis stratification, patient follow-up and evaluation of therapeutic effectiveness [98]. DWI is particularly useful to evaluate the prostate peripheral zone (PZ), where the large majority of tumors (≈70%) occur. DWI enhances the value of MRI in PCa detection, with sensitivity and specificity between 71%-94% and 73%-100%, respectively [40].

4.2 Technical Challenges in Prostate DWI To fully harvest the potential of prostate DWI, sequence optimization is key. Guidelines aiming for standardization recommend a maximum TE of 90 ms and a minimum TR of 3.0 s to retain suitable SNR. Also, for diagnostic purposes, the slice thickness must not exceed 4 mm [98]. The selection of the coil is another critical aspect. Endorectal coils (ERC) are considered crucial at 1.5 T unless 16 (or more)-channel array coils are available. Despite increased artifacts, imaging at 3 T could be beneficial, as the extra signal can be used to increase spatial resolution or avoid the use of an ERC, improving patient comfort [98]. To enable the acquisition of thin slices, the use of an ERC at the highest available field strength may be required to achieve an adequate SNR, especially at very high b-values as established by the guidelines. DWI typically uses EPI readouts. As a consequence, images suffer from susceptibility artifacts that particularly affect the PZ; rFOV can reduce this artifact enabling to more easily distinguish between tumor and healthy tissues [46]. Several studies showed that using high b-values (>1000 s/mm2 ) could be beneficial; b 2000 s/mm2 and higher have been reported as useful for detection and localization, particularly for lesions in the transition zone [105]. As images suffer from low SNR and long exam times, synthesized images have been suggested with comparable diagnostic performance [30]. Figure 2 presents example DWI images from a patient with diagnosed PCa.

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Fig. 2 A 50-year-old man with PCa in the left PZ. a axial T2 -weighted showing a prostate with discretely enlarged dimensions, signal heterogeneity in its central region with some nodular areas, typical of adenomatous hyperplasia alterations; b b 1100 s/mm2 and c b 2000 s/mm2 , showing high signal intensity in the left PZ; d ADC map estimated from b 50–2000 s/mm2 , arrow indicates the hypointense lesion (mean ADC value was 0.7 × 10−3 mm2 /s). Prostatectomy results: PCa Gleason Score (GS) 4 + 3

4.3 Quantitative Prostate DWI Diffusion metrics can help in PCa characterization as potential biomarkers for disease prognosis, evaluation of tumor recurrence and therapeutic response [90]. The monoexponential model is the most often applied in the clinic, using two (or more) b-values for estimating ADC; adding diffusion sensitization values improves its estimates [38]. Similar ROI delineation strategies have been tested as in the breast: directly on the ADC map, placed on a single slice [38] or including the whole tumor [90]. Diagnostic performance, intra- and inter-reader reproducibility were reported

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not to be improved using whole-tumour instead of single-slice ROI [89]. The mean ADC within the ROI is usually used, but other histogram-derived metrics (e.g. 10% percentile) can be used [15]. The negative correlation between the degree of PCa malignancy (Gleason ScoreGS) and ADC helps tumor stratification and treatment planning, although some overlap in the ADC values between neighboring GS has been reported [91]. This may be partially due to variability in acquisition parameters among studies. Standardizing both acquisition parameters and post-processing is essential for a wider clinical validation of the ADC in the clinical setting.

4.4 Application of Advanced Diffusion Models DKI has also been applied in PCa with higher mean K app values and lower mean ADC in malignant lesions when compared to normal tissue [90, 95]. Few studies have compared the diagnostic performance of the monoexponential and DKI models in the prostate, with inconsistent results. Some reported an added value of DKI [57] while others reported similar diagnostic performance [90]. Nevertheless, histogramderived DKI metrics showed good reliability for GS stratification [96]. The IVIM model has also been used to study PCa [95], suggesting that the low ADC derives not only from changes in cellularity (affecting D) but partly originates from perfusion effects (f and D*). As biexponential fits are generally more sensitive to noise, producing less stable results, the IVIM maps display larger variability when compared to the ADC map. The parameter D* is prone to errors, particularly when f gets very small, which may explain the poor quality of this map. Diffusion anisotropy can differentiate benign prostatic hyperplasia from PCa in the central gland but displays limited utility for PZ PCa. Recent studies showed that FA could help distinguish PCa from normal glandular tissue [32] and assess tumor aggressiveness [93]. In conclusion, the greatest challenge of prostate DWI is characterization of PCa malignancy, reducing the need for biopsies. Alternative non-invasive biomarkers, such as the metrics derived from DTI acquisitions, DKI and IVIM models continue to be sought, with the aim to achieve improved diagnostic performance.

5 Muskuloskeletal Applications 5.1 Current Applications of DWI and DTI in the MSK System Clinically, ADC is used to differentiate benign from malignant MSK masses and to evaluate treatment response.

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Given the anisotropic nature of skeletal muscle, DTI and tractography enable characterizing properties such as fiber length or pennation angle. The principal eigenvector (ε1 ) is thought to align with local fiber orientation, but the morphological origin of ε2 and ε3 remains under study [33, 70]. Exercise or muscle injury impact ADC, Mean Diffusivity (MD) and FA, reflecting interstitial edema, cell swelling and loss of microstructural integrity. FA increase over time seems to be related with changes in radial diffusivity (λradial ) explained by a more densely packed parallel fiber alignment, while axial diffusivity (λaxial ) remains nearly unchanged from injury to repair [43].

5.2 Technical Challenges in MSK DWI Image Acquisition Optimizing the protocol implies considering the short T2 relaxation (≈40 ms at 3T) and small T2 /T1 ratio of skeletal muscles, using the shortest TE possible and typical maximum b-values of 400–500 s/mm2 to retain SNR (ideally higher than 25-30 in b0 images) [24]. Rician noise can cause inaccurate estimates, e.g. MD, λ2 and λ3 can be underestimated whilst λ1 and FA are overestimated. To evaluate measurement precision, Monte Carlo numerical simulations using wild or regular bootstrapping should be conducted [101]. When imaging large areas (e.g. the thighs), a larger FOV may compromise homogeneous fat suppression, affecting diffusion metrics. One of the most used fat-saturation methods is SPAIR combined with volume shimming. In muscular dystrophy, fatty tissue infiltration can be a confounder, and combining SPAIR and sliceselection gradient reversal (SSGR), or another saturation method, may be advisable [8, 35]. To evaluate microstructure (e.g. myofiber architecture, fiber diameter and λradial ), longer diffusion times (≈1000 ms) are needed. The STEAM diffusion preparation may be more adequate as it allows long diffusion times without prolonging TE [31], particularly relevant for short T2 . Image Post-Processing Tracking in muscular tissue is challenging as low FA values cause high uncertainty in ε1 and lead to early streamline termination. Also, prior to DTI estimation, images should be denoised; possible filters include linear minimum mean square error, local principal component (PC) analysis or anisotropic filters; the latter maintains sharp transitions between adjacent muscles [7]. The choice of tracking algorithm remains under debate. The axis of a long fusiform muscle is easy to define with deterministic algorithms. Seed points are often placed at the aponeurosis, and propagated until cut-off criteria are reached. Common tracking parameters are: 0.1 ≤ FA ≤ 0.5; max curvature angle 10◦ − 20◦ ; 15 mm ≤ fiber tract length ≤ 200 mm [6]. Lowering the maximum allowed curvature may be needed to avoid streamlines crossing muscle boundaries [33].

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Another challenge is the tendinous origin/insertion. Although well-depicted in anatomical images, partial volume effects in low resolution DWI may cause propagation errors in which the streamlines often do not represent the real ends of muscle fascicles. One possible solution is to treat these areas “separately”, through manual digitization and 3D mesh reconstruction. Then the eigenvectors of aponeurosis voxels can be replaced by those of neighboring voxels [13, 34].

5.3 From Diffusion to Microstructure Skeletal muscles consist of several compartments, displaying multi-exponential relaxation. These have been evaluated by acquiring several echoes at long diffusion times [102]. Multiple Echo diffusion Tensor Acquisition Technique (MEDITATE) with a Multiple-Modulation-Multiple-Echo (MMME)-magnetization preparation module enables sampling multiple stimulated or spin echoes with diffusion encoding to characterize time-dependent measurements [21], such as compartment fractions, fiber diameter, λradial and λaxial . Most studies have assumed Gaussian diffusion and monoexponential T2 decay. To see why this assumption is not correct, consider an experiment where a b-value of 200 s/mm2 is used. For diffusion times typical of this level of diffusion weighting, the root mean square displacement of water perpendicular to the muscle fiber would be around 10 µm, and hence of the order of the diameter of a muscle cell, which is typically between 10–100 µm. For this reason, signal contributions from intra- and extracellular structures should therefore be considered [21, 24]. Muscle volume, fiber length, pennation and physiological cross-sectional area influence its ability to generate force and are relevant to create realistic biomechanical models. The distribution of type 1 and type 2 fibers is important for monitoring the effects of training or treatment of muscular dystrophies in muscle strength. Higher FA values seem to indicate a higher proportion of type 1 fibers (slow-twitching), as this type of fibers is associated to lower values of λradial [84]. In conclusion, muscle DWI enables to characterize tissue microstructure, bringing new insights to sports practice or muscular disease. Although research focusing on optimizing acquisition protocols (due to the challenges of obtaining effective fat suppression and reduced TE), post-processing pipelines and algorithms is still needed, the major struggle is perhaps experimental validation of the in vivo parameters this technique provides.

6 Cardiac Applications 6.1 Technical Challenges in Cardiac DWI Cardiac DWI aims to characterize the anisotropic structure of the myocardium, to retrieve information on ventricular function and tissue remodeling.

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In cardiac DWI, water diffusion motion (in the order of μm) is overlaid with the cardiac cycle and muscle contraction, along with respiratory movements (in the order of cm); these contribute to the measured signal loss and cannot be separated from that resulting from diffusion. There is hence a need to apply strategies to overcome the influence of coherent motion, during scanning and/or post-processing stages [27, 62, 66]. Additional acquisition challenges include the short myocardium T2 relaxation time which limits TE, and B0 inhomogeneities in the heart’s vicinity which may impact image quality [28]. An obvious approach is to synchronize scanning with the electrocardiogram to acquire all the data within the same cardiac phase. Application of a STEAM diffusion preparation module, combined with an EPI readout strategy has been suggested in a dual-gated mode. The sequence then spans across two cardiac beats, applying monopolar diffusion gradients in each cardiac cycle with an interval of RR (one cardiac cycle), and the refocusing pulse is split into two 90◦ RF pulses, one synchronised with the first and the other with the second heartbeat [17, 79]. As the interval between RF pulses is long (=RR), the diffusion time is long but the diffusion gradients can be short in duration. To minimize sensitivity to cardiac motion, the gradients and RF pulses have to be applied at exactly the same cardiac phase, and the heart should be in the same position, which implies acquiring the data during BH. This method is sensitive to arrhythmias and has a long acquisition time (two cardiac cycles per image). An alternative is to use the SE diffusion preparation module where bipolar gradients are applied within the same heartbeat to create a velocity compensated sequence. As these sequences are performed within a single heartbeat, they can be combined with respiratory compensation techniques. Acceleration compensated sequences have also been developed; in this case, a diffusion prepared sequence is obtained by adding diffusion gradients to a T2 preparation module of a balanced Steady State Free Precession (bSSFP) sequence [64]. This acceleration-compensated method seems to be more adequate for FB while STEAM is more suitable for diastolic imaging [85]. Three dimensional information of the heart is clinically important since often the disease affects multiple myocardium regions. At present, achieving full coverage is prohibitive and DWI acquisition of as many as three long axis slices already requires long scan times. Extensive coverage of the human heart has been attempted, including 13 slices and 10 diffusion encoding directions, although at the cost of multiple BH [61]. For this reason, acceleration methods such as SMS are important. SMS has been combined with STEAM showing accurate results for an AF of 3 [50].

6.2 Application of Advanced Diffusion Models Application of IVIM to Diagnose Myocardial Ischemia The IVIM model has also been explored, providing improved fitting to in vivo cardiac DWI [14]. Measuring perfusion is useful to evaluate microcirculation of the myocardium in cases of ischemia [12]. IVIM is advantageous since it avoids the need for contrast agents;

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its downside is its sensitivity to motion. Application of IVIM was made feasible through the use of a post-processing algorithm for PC analysis filtering with temporal maximum intensity projection (MIP)—PCATMIP [72]. With this method, around 10 diffusion images can be acquired in FB with reduced cardiac motion. These images can then be registered to correct for inter-scan motion and a PC analysis performed for spatiotemporal filtering and improved SNR. A pixelwise temporal MIP can then identify the repetition with highest signal and remove others where signal loss occurred due to motion. Nevertheless, improvements have been observed by using a wavelet-based image fusion instead of PCATMIP to register DW images and reduce motion-related dropouts [97]. DTI and Myocardial Fiber (Dis)Arrangement Cardiac DTI was initially validated ex vivo, confirming that the heart is composed of two sets of crossing fiber populations as known from histology [48]. The healthy heart shows a continuous variation of myofiber orientation from the epicardium to the endocardium, with small local changes. Alterations in the continuity and orientation of the fibers provides useful information in pathologies such as myocardial infarction, hypertrophic cardiomyopathy (HCM), or diffuse myocardial disease [20, 59]. Cardiac DTI has been a topic of interest due to its capacity to retrieve complementary information on myocyte architecture and orientation, unattainable with other MRI modalities. Myocardial infarction is associated with fiber loss; DTI allows evaluating fiber architecture in the scar region, important to follow tissue remodeling especially when using tratography [60]. This may indicate the development of ventricular arrhythmias, and aid in treatment planning and prognosis. HCM is associated with tissue fibrosis, myocyte hypertrophy and loss of the parallel myocyte orientation (i.e. disarray). DTI may enable evaluation of the myocardial disarray, unfeasible with other MRI techniques such as late gadolinium enhancement, which is the current standard for evaluating microscopic scar [60]. This could be used to illustrate the arrhythmogenic substrate without the need for exogenous contrast agents in a near future. At present, clinical applications of cardiac DWI and DTI are still not fully developed. This is due to the specific challenges of cardiac imaging mostly related to motion, B0 inhomogeneity and the short T2 of the myocardium. Given the potentialities of cardiac diffusion, research on these topics is expected to continue, enabling increased applicability of this technique in the clinic.

7 Conclusions Including DWI in clinical protocols has resulted in increases in sensitivity and specificity and so the clinical potential of this sequence is undoubtable. Besides the known value of ADC, visual inspection of DWI is also applied as part of multiparametric estimation protocols in diagnostic and prognostic assessment. Obtaining sufficient SNR and artifact-free images is particularly challenging in body imaging. Some of the artifacts can be overcome with new hardware, optimized

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pulse sequences and reconstruction methodologies, as well as with advanced postprocessing techniques. The potential of DKI includes not just visual assessment of malignant lesion foci in ultra-high b-values but also quantification of the kurtosis indices, which can be used as biomarkers and should be seen as independent measures of tumor response to adjuvant therapies, beyond the analysis of gadolinium uptake which is currently the most frequently used approach in the clinical setting. The microstructural tissue properties assessed by diffusion imaging are expected to improve diagnostic specificity when combined with adequate biophysical modeling. In this regard, IVIM is increasingly applied in cancer due to the typical hypervascular and hypercellular features of tumours. Nevertheless, the biological significance of IVIM-related perfusion still needs to be fully understood. Application of DKI to MSK diseases is gaining interest, and has been shown to differentiate between neurogenic and systemic myopathies. Still, DTI applied to the MSK system and the myocardium needs to be further validated in the clinical setting and, for that purpose, translational research has the most relevant role. Ethics: Institutional review board approval was obtained for both breast (code CES 276/13) and prostate imaging (code 215/12 190/DEFI/195-CES).

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A Framework for Calculating Time-Efficient Diffusion MRI Protocols for Anisotropic IVIM and An Application in the Placenta Paddy J. Slator, Jana Hutter, Andrada Ianus, Eleftheria Panagiotaki, Mary A. Rutherford, Joseph V. Hajnal and Daniel C. Alexander

Abstract We develop a framework for calculating clinically-viable diffusion MRI (dMRI) protocols for anisotropic IVIM modelling. The proposed multi-stage framework combines previous approaches to dMRI protocol optimisation: first optimising b-values by minimizing Cramer-Rao lower bounds on parameter variances, and subsequently optimising gradient directions jointly to provide maximum angular coverage across all shells. This removes unnecessary measurements of closely spaced b-values with the same gradient directions, which encode very similar information, and hence reduces the total number of dMRI measurements. We applied the framework to establish an organ-specific, data-driven, set of optimised b-values and gradient directions for dMRI of the placenta. The optimised protocol leads to higher contrast-to-noise ratios in parameter maps compared to a naive protocol of comparable scan time. Applying this framework in other organs has the potential to reduce scanning times required for anisotropic IVIM modelling. Keywords Diffusion MRI · IVIM · Anisotropic IVIM · Placenta · Microstructural modelling · Experimental design · Cramer-Rao lower bound

P. J. Slator (B) · A. Ianus · E. Panagiotaki · D. C. Alexander Centre for Medical Image Computing and Department of Computer Science, University College London, London, UK e-mail: [email protected] J. Hutter · M. A. Rutherford · J. V. Hajnal Centre for the Developing Brain, King’s College London, London, UK J. Hutter · J. V. Hajnal Biomedical Engineering Department, King’s College London, London, UK © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_20

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1 Introduction The intravoxel incoherent motion (IVIM) model first proposed by Le Bihan [12], considers both diffusion within tissue and microcirculation within capillaries to be isotropic. Recently, models which consider the effect of coherent orientation, in both the microvascular and tissue structure, on the diffusion MRI (dMRI) signal are emerging. These include two-compartment models which allow for anisotropy in both fast(associated with perfusion) and slow- (associated with diffusion) attenuating signal components, and can be treated as anisotropic extensions to IVIM. These models, which we call anisotropic IVIM in this paper, are emerging as a useful tool in skeletal muscle [11], cancer [16], kidney [9, 13, 15], heart [1], and brain [7] imaging. The utility of dMRI data depends heavily on the choice of diffusion-encoding gradient directions and b-values, often termed the gradient table. The best choice of gradient table depends on the microstructural and microcirculatory properties of the tissue of interest. For anisotropic IVIM models, parameter estimation requires rich, and therefore long, dMRI acquisitions. In order for anisotropic IVIM to be clinically viable, shorter protocols are required. In this work we present a general framework for developing time-efficient dMRI protocols optimised for subsequent anisotropic IVIM model fitting. The framework takes advantage of a redundancy in current anisotropic IVIM protocols—the fact that dMRI measurements at closely spaced b-values with the same gradient direction typically encode very similar information. We demonstrate the proposed framework by calculating an optimised dMRI gradient table for placental imaging. dMRI is emerging as a promising modality for studying the placenta—dMRI-derived parameters have shown utility as biomarkers for fetal growth restriction [3, 5, 14, 19, 20] and pre-eclampsia [18]. These imaging biomarkers of placental function could benefit maternal and fetal health, and are therefore of great interest and importance. It particular, anisotropic IVIM models have shown promise in the placenta [17]. We scanned a single subject using the gradient table optimised for placental dMRI, and also a naive gradient table of comparable length. We find that the optimised protocol offers visual and quantitative improvements over the naive.

2 Methods 2.1 dMRI Protocols for Anisotropic IVIM Models Current protocols for IVIM MRI don’t usually consider the orientational dependence of the dMRI signal. Instead, they typically take a single measurement at each bvalue, or average measurements across the principal diffusion gradient directions (left-right, anterior-posterior, and superior-inferior). The protocols generally have a large number of closely spaced b-values spanning the perfusion (roughly b < 200 s mm−2 ) and diffusion (roughly b > 200 s mm−2 ) regimes in order to accurately

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Fig. 1 Visualisations of typical dMRI gradient tables for IVIM (top row) and anisotropic IVIM (bottom row). The IVIM gradient table has 3 diffusion gradient directions per b-value shell, and the anisotropic IVIM table has 15 directions per shell

estimate the IVIM model parameters—diffusivity, pseudo-diffusivity, and perfusion fraction. Figure 1 (top panel) visualises a typical IVIM protocol. When extending to anisotropic IVIM models a protocol which assesses the signal dependence on diffusion gradient direction is required. One approach is to add a single additional shell with high angular resolution, as in reference [11], although this only provides information on anisotropy at a single b-value. The most common approach in the literature is to incorporate a large number of gradient directions at each shell. For example Finkenstaedt et al. use 20 directions at 14 non-zero b-value shells [7], Notohamiprodjo et al. and Liu et al. use 20 directions at 9 non-zero b-values [13, 15], and Hilbert et al. use 30 directions at 5 non-zero b-values [9]. Clearly, this approach leads to very long scanning times. Figure 1 (bottom panel) shows diffusion gradient directions for a typical anisotropic IVIM protocol. Our aim was to develop a placental dMRI protocol suitable for anisotropic IVIM model fitting, which was feasible within a limited scanning time. We developed the following framework whilst working on placental dMRI, but our approach is generalisable to any anisotropic IVIM system.

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2.2 Proposed Framework Our proposed framework can be summarised as follows: 1. Optimise b-values given a fixed number of shells and gradient directions (following [2]) 2. Add additional high angular resolution shells at key b-values 3. Optimise gradient directions jointly across all shells (following [4]) The framework, particularly the estimation of b-values in steps 1 and 2, requires prior knowledge of the tissue of interest. In particular, we need to know an appropriate microstructural model for use in this tissue, and estimates of typical model parameter values. The crucial difference between this framework and existing dMRI protocols for anisotropic IVIM is step 3. Rather than using the same diffusion gradient directions for closely spaced b-values (as in Refs. [7, 9, 13, 15]) we use a set of directions jointly optimised across b-values. This takes advantage of the fact that dMRI measurements with the same diffusion gradient direction and proximal b-values are likely to be very similar. Prior to the optimisation we need to fix the number of number of shells and gradient directions in steps 1 and 2. In step 1 we aim to estimate the optimum bvalues for estimating the IVIM parameters, so we choose a large number of shells with few gradient directions per shell, for example 3 directions and 10 shells. The purpose of step 2 is to add additional angular resolution to assess anisotropy in both the perfusion and diffusion regimes. So, for example, we could use 2 shells and 15 gradient directions. These choices are ultimately constrained by the total number of acquisitions available. The choices we state above would lead to 60 (3 × 10 + 2 × 15) diffusion-weighted (i.e. b > 0) acquisitions in total. We now give more details on each of the steps of the proposed framework. In step 1 we follow the approach of Alexander [2]. This requires a choice of microstructural model, and knowledge of typical model parameters, to guide the optimisation. The output of this method is a set of b-values. However, since the optimisation method is based on estimated, fixed model parameter values, these b-values are often clustered around “ideal” values. As model parameters will take a range of values in in-vivo scans, it may be sensible to heuristically spread the b-values out in order to increase the robustness of the gradient table to parameter variation. In step 2 we increase angular resolution at selected b-values by adding diffusion gradient directions. In general, we recommend adding two additional shells—one relating to the perfusion regime and one the diffusion regime. To choose the optimum b-values for these additional shells, we use estimates of the diffusivity (D) and pseudo-diffusivity (D p ). The ideal b-values for these shells are then (following [10]) approximately equal to 1.11/D and 1.11/D p respectively. These b-values will most likely be close to a b-value from step 1, in this case it makes sense to simply add the additional gradient directions to this shell. Given these b-values, in step 3 we optimise diffusion gradient directions across shells following Caruyer et al. [4]. As opposed to standard protocols with the same

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gradient directions at every shell, this approach yields a set of directions with optimal coverage across the entire set of shells. For computational convenience we recommend optimising gradients directions separately for the shells from step 1, and the high angular resolution shells from step 2. It may also be convenient to further split the step 1 shell optimisation, by separately calculating optimal directions for the perfusion (b < 200 s mm−2 ) and diffusion (b > 200 s mm−2 ) regimes. Although it is possible, we believe there is little additional benefit to optimising across the full set of b-value shells.

2.3 Application to the Placenta We now apply our proposed framework to optimise a dMRI acquisition protocol for the placenta. We aim for a scan time of less than 10 min, which corresponds to around 60 dMRI volumes in our set up. As described in the previous section, step 1 in our optimisation framework requires a choice of microstructural model and parameter values. Slator et al. [17] proposed that the most descriptive models for the placenta were anisotropic extensions to IVIM, and in particular a zeppelinzeppelin model. This is a two-compartment model which allows for anisotropy in both fast- (associated with perfusion) and slow- (primarily associated with diffusion) attenuating signal components. The normalised zeppelin-zeppelin dMRI signal is given by     S(b, G) = f exp −bGT Dp G + (1 − f ) exp −bGT DG

(1)

where Dp and D are cylindrically symmetric diffusion tensors (i.e. zeppelins), representing perfusion and diffusion compartments respectively, f is the perfusion fraction, b is the b-value, and G is the gradient direction. Slator et al. also reported typical zeppelin-zeppelin model parameter values: f = 0.55, axial diffusivity of Dp = 0.44 mm2 s−1 , radial diffusivity of Dp = 0.03 mm2 s−1 , axial diffusivity of D = 5 × 10−3 mm2 s−1 , radial diffusivity of D = 1.1 × 10−3 mm2 s−1 . We fixed at these values in step 1 of the optimisation. We also fixed the zeppelin orientations, although the optimisation is invariant to these. We assumed an imaging scheme with 12 gradient shells and 3 measurements per shell. The b-values returned by this optimisation are given in Fig. 2a. In places, these values are closely spaced together. We increased robustness to different parameter values by spacing the b-values. We also added measurements at b = 1200 s mm−2 and b = 1600 s mm−2 so that the final protocol is capable of assessing the effect of diffusion restriction and kurtosis on the signal. We ended with the following choice of b-values: 5, 10, 20, 30, 50, 100, 200, 400, 600, 800, 1200, 1600 s mm−2 (Fig. 2b). To further increase protocol sensitivity, we added additional dMRI volumes at key b-values (step 2). We initially intended to add 2 shells with 15 diffusion directions each based on diffusivity and pseudo-diffusivity values. The typical diffusivity reported in the placenta and uterine wall is 1.4 × 103 mm2 s−1 [17], which implies a

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shell at b = 800 s mm−2 (i.e ≈ 1.11/1.4 × 10−3 mm2 s−1 , following [10]). We therefore made the choice to change the b = 800 s mm−2 shell from step 1 to a 15 direction shell. The pseudo-diffusivity varies significantly in different areas of the placenta and surrounding tissue. Slator et al. observed typical values of 0.03 mm 2 s−1 in the placenta, and 0.06 mm2 s−1 in the uterine wall [17]. We therefore chose to add additional volumes at both b = 18 s mm−2 s−1 (i.e. ≈1.11/0.06) and b = 36 s mm−2 (≈1.11/0.03). The b-value optimisation in step 1 yielded b-value estimates close to these. Our choice was to remove the b = 20 s mm−2 and b = 30 s mm−s shells, and

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Fig. 3 Visualisations of the proposed placental dMRI gradient table optimisation (top row), and a naive gradient table used in a previous, exploratory study (bottom row)

replace with a 8 direction shell at b = 18 s mm−2 , a 3 direction shell at b = 25 s mm−2 and a 7 direction shell at b = 36 s mm−2 . Finally we implemented step 3, optimising gradient directions jointly across all b-values following Caruyer et al. [4]. For computational convenience, we did this in four stages: for the 3 gradient direction shells with b ≤ 100 s mm−2 , the 3 direction shells with b ≥ 200 s mm−2 , the 15 direction shell at b = 800 s mm−2 , and the 15 directions in total at b = 18 s mm−2 and b = 36 s mm−2 . We present a visualisation of the b-values and gradient directions for the optimised protocol in Fig. 3, top panel.

2.4 Data Acquisition We scanned a single healthy pregnant subject, gestational age 34 weeks 1 day, using the optimised gradient table on a 3T Philips Achieva scanner with a 32-channel cardiac coil. Six interleaved b = 0 images were obtained (giving 65 dMRI volumes in total). The other acquisition parameters were as follows: 2 mm isotropic resolution, TE = 95 ms, TR = 7.5 s, FOV = 352 × 352 × 96 mm, 48 contiguous coronal slices, scan time 8 min 8 s. For comparison, we also scanned the same subject with the naive exploratory gradient table presented by Slator et al. in Ref. [17]. The other acquisi-

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tion parameters for this naive scan were: 2.2 mm isotropic resolution, TE = 129 ms, TR = 8 s, FOV = 352 × 352 × 77 mm, 35 contiguous coronal slices, 59 dMRI volumes, scan time 7 min 52 s. The two scans were performed immediately after each other. For comparison with the optimised protocol, we present a visualisation of this naive protocol in Fig. 3, bottom panel. All methods were carried out in accordance with relevant guidelines and regulations, the study was approved by the local Research Ethics Committee, and informed written consent was obtained prior to imaging.

2.5 Data Processing For both scans, placenta and uterine wall regions of interest (ROIs) were manually defined on the first b = 0 image. We estimated the SNR in each slice, following Dietrich et al. [6], and found values of around 15–20 for the naive and 25–30 for the optimised protocol. We fit the zeppelin-zeppelin model voxel-by-voxel to the normalised dMRI signal (hence avoiding T2 dependence within a single scan) assuming a Rician noise model with SNR set to 20. We also fit the diffusion tensor to the full set of dMRI volumes using the MRtrix3 function dwi2tensor [21], hence calculating apparent diffusion coefficient (ADC) and fractional anisotropy (FA) maps. To quantify differences between parameter maps derived from naive and optimised protocols we calculated contrast-to-noise ratios (CNRs), considering the contrast between the placenta and the uterine wall. Following Ref. [8] this is given by |μ p − μu | CNR =  σ p2 + σu2

(2)

where μ p , μu are the mean parameter values in the placenta and uterine wall ROIs respectively, and σu , σ p are the standard deviations. It should be noted that these statistics assume that tissue has a homogeneous structure—i.e. they do not consider the capability for sub-structure visualisation.

3 Results The optimised placenta dMRI gradient table offered clear qualitative improvements over the naive table of comparable length. Figure 4 displays b = 0 images, ADC maps, and FA maps for both protocols. Parameter maps appear improved for the optimised protocol: we observed patches which potentially reflect subplacental structures (e.g. Fig. 4, white circles). Optimised maps also more clearly delineate the boundary between the placenta and uterine wall (Fig. 4, white arrows). In agreement with previous observations [17], we see high ADC and high FA in the uterine wall. Figure 5 shows parameter maps derived from zeppelin-zeppelin model fits. The optimised

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Fig. 4 b = 0 image and DTI parameter maps for naive and optimised placental dMRI protocols. Circles illustrate a patch of high ADC and high FA. Arrows highlight the placental boundary

Fig. 5 Zeppelin-zeppelin parameter maps. Axial diffusivity relates to the diffusion compartment. Arrows indicate the boundary between the placenta and uterine wall

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Table 1 Parameter values in ROIs for naive and optimised dMRI protocols, given as mean ± standard deviation. Zeppelin axial diffusivity and FA refer to the diffusion compartment (i.e. D) Placenta Uterine wall Parameter Naive Optimised Naive Optimised ADC (mm2 s−1 ) FA Zeppelin axial diffusivity (mm2 s−1 ) Zeppelin FA Perfusion fraction

0.0027 ± 0.0069 0.64 ± 0.25 0.0019 ± 0.0012

0.0026 ± 0.0015 0.46 ± 0.21 0.0023 ± 0.0010

0.0041 ± 0.0084 0.76 ± 0.20 0.0029 ± 0.0016

0.0043 ± 0.0037 0.68 ± 0.19 0.0031 ± 0.0016

0.41 ± 0.35 0.42 ± 0.21

0.34 ± 0.24 0.40 ± 0.16

0.53 ± 0.37 0.55 ± 0.18

0.53 ± 0.32 0.52 ± 0.18

protocol maps show greater contrast between structures, e.g. the placental boundary in columns 1–4 (Fig. 5, white arrows), and less noisy appearance within homogeneous regions. The quantitative measurements mostly supported these qualitative improvements in parameter maps. The CNR for the placenta and uterine wall was generally improved in parameter maps derived from the optimised gradient table (Fig. 6). This suggests that the optimised protocol can produce parameter maps that better differentiate between these two distinct anatomical structures. The mean values of dMRI-derived parameters are closely matched across the two protocols (Table 1), aside from a reduction in FA for the optimised.

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4 Discussion and Conclusion Anisotropic IVIM models are emerging as useful tools in organs where vasculature has a coherent orientation [1, 7, 9, 11, 13, 15, 16]. However, the required gradient tables are long, which is a barrier to potential clinical applications. Here we present a framework for establishing dMRI protocols suited to anisotropic IVIM model fitting within clinically reasonable scanning times. By separately optimising b-values and gradient directions, we focus measurements at the ideal shells, and maximise angular resolution across these shells. We apply this framework to design a comprehensive, data-driven, organ-specific placenta dMRI gradient table. This protocol shows qualitative (Figs. 4 and 5) and quantitative (Fig. 6) improvements over a naive placental dMRI protocol of comparable length. However, the two dMRI protocols we compare have different echo times. This is not ideal, and it is not clear the extent to which the observed improvements are due to the different dMRI gradient tables, as opposed to the inevitable improvement in SNR (we calculated an approximately 40% lower SNR for the naive protocol). To satisfactorily answer this question, both further scanning (with matched echo times and voxel size) and a simulation study are necessary. A simulation study could furthermore aim to quantify the robustness of the gradient table to variations in tissue parameters, and also quantify the accuracy of parameter estimates, since the Cramer-Rao lower bound only considers the precision of these estimates. We split our b-value optimisation into two steps where the number of b-values and gradient directions were fixed. It is feasible to construct a framework for gradient table optimisation where the number of shells, and directions per shell, are completely free. However this would add complexity to the optimisation problem. We also optimisated diffusion gradient directions separately for different categories of shells. It is possible to jointly optimise across all shells, but again this would be a much more complicated calculation. Another potential limitation of our approach is that in step 1 and 2 we optimised based on single, fixed microstructural model parameter values. Clearly, the value of these parameters will change due to normal biological variability and disease, and the extent to which this affects the optimal gradient table is not clear. This variation in dMRI-derived metrics and relaxation times means that a truly “optimised” gradient table does not exist. This is especially the case in an organ as heterogenous as the placenta. In future work we could calculate a protocol by averaging the Cramer-Rao lower bound for multiple combinations of model parameters covering the observed range. Although further development and validation is necessary, the presented framework offers many advantages over current approaches. In particular, it offers enhanced angular coverage compared to standard protocols. Therefore, the framework presented in this paper can be immediately applied to develop shorter anisotropic IVIM gradient tables in other organs, the only required knowledge being typical IVIM parameter values within the tissue of interest. It is easily adaptable to different scan time limitations, through reducing the assumed number of dMRI volumes in steps 1 and 2 of the optimisation. Using this framework could help ensure that data

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acquired during clinically limited scanning times has the maximum possible utility for anisotropic IVIM model fitting. This subsequently increases the likelihood of translating anisotropic IVIM dMRI research into biomedical imaging applications.

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14. Moore, R., Strachan, B., Tyler, D., Duncan, K., Baker, P., Worthington, B., Johnson, I., Gowland, P.: In utero perfusing fraction maps in normal and growth restricted pregnancy measured using IVIM echo-planar MRI. Placenta 21(7), 726–732 (2000). https://doi.org/10.1053/plac.2000. 0567 15. Notohamiprodjo, M., Chandarana, H., Mikheev, A., Rusinek, H., Grinstead, J., Feiweier, T., Raya, J.G., Lee, V.S., Sigmund, E.E.: Combined intravoxel incoherent motion and diffusion tensor imaging of renal diffusion and flow anisotropy. Magn. Reson. Med. 73(4), 1526–1532 (2015). https://doi.org/10.1002/mrm.25245 16. Panagiotaki, E., Walker-Samuel, S., Siow, B., Johnson, S.P., Rajkumar, V., Pedley, R.B., Lythgoe, M.F., Alexander, D.C.: Noninvasive quantification of solid tumor microstructure using VERDICT MRI. Cancer Res. 74(7), 1902–1912 (2014). https://doi.org/10.1158/0008-5472. CAN-13-2511 17. Slator, P.J., Hutter, J., McCabe, L., Gomes, A.D.S., Price, A.N., Panagiotaki, E., Rutherford, M.A., Hajnal, J.V., Alexander, D.C.: Placenta microstructure and microcirculation imaging with diffusion MRI. Magn. Reson. Med., Dec 2017. https://doi.org/10.1002/mrm.27036 18. Sohlberg, S., Mulic-Lutvica, A., Lindgren, P., Ortiz-Nieto, F., Wikström, A.K., Wikström, J.: Placental perfusion in normal pregnancy and early and late preeclampsia: a magnetic resonance imaging study. Placenta 35(3), 202–206 (2014). https://doi.org/10.1016/j.placenta.2014.01. 008 19. Sohlberg, S., Mulic-Lutvica, A., Olovsson, M., Weis, J., Axelsson, O., Wikström, J., Wikström, A.K.: Magnetic resonance imaging-estimated placental perfusion in fetal growth assessment. Ultrasound Obstet. Gynecol. 46(6), 700–705 (2015). https://doi.org/10.1002/uog.14786 20. Song, F., Wu, W., Qian, Z., Zhang, G., Cheng, Y.: Assessment of the placenta in intrauterine growth restriction by diffusion-weighted imaging and proton magnetic resonance spectroscopy. Reprod. Sci. 24(4), 575–581 (2017). https://doi.org/10.1177/1933719116667219 21. Veraart, J., Sijbers, J., Sunaert, S., Leemans, A., Jeurissen, B.: Weighted linear least squares estimation of diffusion MRI parameters: strengths, limitations, and pitfalls. NeuroImage 81, 335–346 (2013). https://doi.org/10.1016/j.neuroimage.2013.05.028

Spatial Characterisation of Fibre Response Functions for Spherical Deconvolution in Multiple Sclerosis Carmen Tur , Francesco Grussu , Ferran Prados , Sara Collorone , Claudia A. M. Gandini Wheeler-Kingshott and Olga Ciccarelli

Abstract Brain tractography based on diffusion-weighted (DW) MRI data has been increasingly used to investigate crucial pathophysiological aspects of several neurological conditions, including multiple sclerosis (MS). The advent of fibre tracking methods based on constrained spherical deconvolution (CSD), which recovers the fibre orientation distribution function (fODF) by performing a single-kernel (or uniform-kernel) deconvolution of the measured DW signals with non-negativity Supported by ECTRIMS Post-doctoral Research Fellowships, Horizon2020-EU.3.1 CDSQUAMRI project (ref: 634541), EPSRC Platform Grant for medical image computing for nextgeneration healthcare technology (ref: EP/M020533/1), Guarantors of Brain Non-Clinical Postdoctoral Fellowships, Spinal Research, Craig H. Neilsen Foundation, Wings for Life, Multiple Sclerosis Society of Great Britain and Northern Ireland, the NIHR UCLH Biomedical Research Centre. The NMR unit where this work was performed is supported by grants from the Multiple Sclerosis Society of Great Britain and Northern Ireland, Philips Healthcare, and by the UCL/UCLH NIHR (National Institute for Health Research) BRC (Biomedical Research Centre. C. Tur (B) · F. Grussu · F. Prados · S. Collorone · C. A. M. G. Wheeler-Kingshott · O. Ciccarelli Queen Square MS Centre, Department of Neuroinflammation, Queen Square Institute of Neurology, Faculty of Brain Sciences, University College London, London, UK e-mail: [email protected] F. Grussu Department of Computer Science, Centre for Medical Image Computing, University College London, London, UK F. Prados Department of Medical Physics and Biomedical Engineering, Centre for Medical Image Computing, University College London, London, UK C. A. M. G. Wheeler-Kingshott Department of Brain and Behavioural Sciences, University of Pavia, Pavia, Italy C. A. M. G. Wheeler-Kingshott Brain MRI 3T Research Centre, IRCCS Mondino Foundation, Pavia, Italy O. Ciccarelli National Institute for Health Research Biomedical Research Centre, University College London Hospitals, London, UK © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_21

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constraints, has meant an important breakthrough. However, it is unclear whether using a uniform kernel deconvolution of the measured DW signals for the whole brain is appropriate, especially in pathology. In this study, our main aim was to explore the validity of using a uniform fibre kernel for spherical deconvolution in a cohort of 19 patients with a first inflammatory-demyelinating attack of the central nervous system suggestive of MS and 12 age-matched healthy controls. In particular, considering that the number of peaks is a key feature the fODF and is known to impact directly on downstream fibre tracking, we assessed the association between patientwise mean number of (fODF) peaks in the non-lesional white matter obtained with a uniform kernel and the bias or differences in the estimation of local diffusion properties when a uniform kernel (instead of a locally-fitted voxel-wise kernel) was used. Finally, in order to support our in-vivo results, we performed a simulation analysis to further assess the theoretical impact of using a uniform kernel. Our in-vivo results showed non-significant trends towards an influence of the bias in the estimation of the local diffusion properties when a uniform kernel was used on the number of peaks. In the simulation analysis, a clear association was observed between such bias and the number of peaks. All this suggests that the use of a uniform kernel to estimate the fODFs at the voxel level may not be adequate. However, we acknowledge that the approach followed here has some limitations, mainly derived from the methods used to estimate the voxel-wise local diffusion properties. Further investigations using larger in-vivo data sets and performing more comprehensive simulation analyses are therefore warranted. Keywords Fibre response function · Connectivity analysis · Multiple sclerosis

1 Introduction Brain tractography based on diffusion-weighted (DW) MRI data has been increasingly used to investigate crucial pathophysiological aspects of several neurological conditions [1]. The relatively recent incorporation of fibre tracking methods, based on constrained spherical deconvolution (CSD) [2] and other techniques, such as Q-ball imaging [3], has meant an important breakthrough in the study of brain connectivity. CSD allows an accurate characterisation of fibre orientation distribution functions (fODFs) especially in regions containing multiple fibre populations, such as the corona radiata. The CSD approach recovers the fODF by performing a singlekernel deconvolution of the measured DW signals with non-negativity constraints. Earlier fibre tracking methods, instead, based on the diffusion tensor model [4], were unable to identify crossing fibres, implying a much less accurate fibre reconstruction than currently used CSD-based methods. However, despite the clear advantages of the CSD approach, it is unclear whether using a single-kernel deconvolution of the measured DW signals for the whole brain is actually appropriate. This is especially relevant in pathological conditions where the degree of white matter (WM) damage may differ across different brain locations.

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In this study, we aimed to evaluate the validity of using a single fibre kernel for spherical deconvolution i.e. of using a uniform fibre response function for all WM locations across the brain. Specifically, we tested this in patients with a first clinical episode suggestive of multiple sclerosis (MS) or clinically isolated syndrome (CIS). Novel biomarkers are urgently needed to differentiate between those CIS patients who will develop MS and those who will not, and connectivity and CSD metrics could provide important information for this purpose. However, it is key to test the validity of model assumptions in the presence of pathology before clinical deployment, and this paper focuses on this. It is expected that strong differences between the local (voxel-dependent) fibre response and the assumed fibre response (uniform across all WM) for CSD can lead to erroneous estimations of the fODF. Therefore, our main aim was to test whether an association between salient characteristics of the fODF obtained with a WM-uniform fibre response and the difference between the local and assumed fibre response properties existed. Specifically, here we assessed the association between patientwise average number of peaks in the non-lesional WM obtained with a uniform kernel and the bias in the estimation of local diffusion properties when a uniform kernel for the whole brain (instead of a locally-fitted voxel-wise kernel) was used (SMT) [5]. The number of peaks is a key feature the fODF and is known to impact directly on downstream fibre tracking. As a secondary aim, we compared patients and controls in terms of mean number of peaks across all non-lesional WM voxels and assessed the impact of having used a uniform kernel on such comparison, considering the uniformvs-locally-fitted-kernel-derived bias as a confounder or as an effect modifier. Finally, we performed a simulation analysis to confirm our in-vivo results. Our results showed some evidence against the appropriateness of using a uniform kernel to estimate the fODFs at the voxel level, especially through the simulation analysis. This would suggest that using a uniform kernel may not be adequate to estimate fODF properties.

2 Methods 2.1 Subjects We included patients who consecutively attended an MS clinic within three months of their CIS (i.e. within three months of symptom onset). A group of age-matched healthy controls (HCs) was also included. All participants underwent an MRI scan at study baseline. Informed written consent was obtained for all participants.

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2.2 MRI Acquisition The scans used in this study were part of a previous multi-parameter clinical study [6]. All scans were performed using a 3T Achieva system (Philips Medical Systems, Best, Netherlands) with a 32-channel head coil. The scanner maximum gradient strength was 65 mT m−1 . The following types of MR images were acquired for all subjects: • Anatomical inversion-prepared 3D T1-weighted turbo field echo (resolution = 1 × 1 × 1 mm3 , TE/TR: 3.2/7.0 ms, TI: 836.46 ms); • Axial PD-/T2-weigted turbo spin echo (resolution = 1 × 1 × 1 mm3 ; TE1/TE2/TR = 15/85/3500 ms); • Clinically-feasible multi-shell diffusion-weighted (DW) (resolution = 2.5 × 2.5 × 2.5 mm3 ; TE/TR = 82 ms/12 s; 53 directions: b = 300 s/mm2 (8 directions), b = 711 s/mm2 (15 directions), b = 2000 s/mm2 (30 directions); 7 b = 0 s/mm2 images).

2.3 MRI Pre-processing Lesion Outline and Lesion Filling. In patients, T2-hyperintense WM lesions were manually outlined using a semi-automated edge-dissection tool (JIM v6.0, Xinapse systems, Aldwincle, UK, http://www.xinapse.com) by an experienced rater (SC). For this purpose, PD/T2-weighted images were used and current guidelines were followed [7]. From the subtraction of the PD and T2 images we computed a pseudoT1-weighted image [8]. We then used these pseudo-T1-weighted images to compute the transformation with the 3D T1-weighted images implementing a symmetric registration approach using NiftyReg software package (http://niftyreg.sf.net). We then applied the obtained transformations to the T2-weighted lesion masks to move them from native space to 3D T1-weighted space. Afterwards, we filled the 3D T1weighted images using a non-local patch-match lesion filling algorithm [9], part of NiftySeg software package (http://niftyseg.sf.net). The Geodesic Information Flows (GIF) method v2.0 [10] was then used to parcellate the T1 filled images. The GIF method v2.0 is freely available in NiftyWeb platform (http://cmictig.cs.ucl.ac.uk/ niftyweb/) [11]. Co-registration of 3D T1-weighted and DW Images. First, eddy current correction using FSL [12] to the DW images was performed. Subsequently, a geometric distortion correction of the DW images was carried out, using BrainSuite v.15b [13]. For this step, the original (non lesion-filled) 3D T1-weighted images were initially co-registered to the mean b = 0 images using a constrained non-rigid registration algorithm based on mutual-information; afterwards, the actual correction took place. All undistorted DW data as well as their final alignment with anatomical 3D T1 data were visually inspected and considered of good quality. Finally, the lesions identified in the PD/T2-weighted space were warped to each subject’s DWI space, as this

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Fig. 1 Estimated fODF in the corpus callosum in a patient (with a callosal lesion) and a control. In the top row there is T1-weighted image of a patient with a (T1-hypointense) lesion in the confluence of the corpus callosum and the corticospinal tract. The middle row shows the fODF map and the bottom row shows the details of a single fODF. Abbreviations: fODF: fibre orientation distribution function

allowed the individual characterisation of microstructural properties at the level of normal-appearing white matter (NAWM) and lesional WM by appropriately defined masks.

2.4 fODF Estimation Voxel-wise fODFs were estimated using the response function (RF) obtained through a multi-tissue algorithm [14] using MRtrix3 (v.10/2016) (see Fig. 1 for an example). The algorithm is a state-of-the-art fODF estimation technique, and employs a uniform fibre response kernel across the whole WM. Such a kernel is estimated from voxels containing coherent WM made of well-aligned fibres, whose position in the brain can be obtained as one of the outputs of the fODF estimation routines [15]. We recorded the position of such voxels, as this enables the comparison of the local (spatially-variant) versus uniform (non-spatially variant) fibre response kernels. Following fODF calculation, we estimated the number of fODF peaks within each voxel, for all WM voxels and all subjects, using the sh2peaks and peaks2amp commands in MRtrix3 (see Figs. 2 and 3 for a few examples in patients and controls). The sh2peaks command was based on a Newton search to find the local maxima (peaks) of the fODF in each voxel. With the peaks2amp command we converted the peak-directions image to an amplitudes image. Since the fODF is a function that describes the distribution of all possible fibre orientations within a voxel, the number of peaks of that distribution was taken as a proxy for the true number of WM fibre populations within that voxel. For this, in the WM fODF, we searched for a maximum of 3 peaks, based on previous publications [16]. The criterion to decide whether an observed peak in the distribution function could reflect a true WM fibre peak was

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based on its size (i.e. its amplitude, in the 3D representation of voxel-wise fODF, as shown in Fig. 1): through the peaks2amp function in MRtrix3, a voxel-wise estimation to the amplitudes of the biggest three fODF peaks was obtained (one image volume per peak); however, only the peaks whose amplitude was greater than 5% of the greatest peak amplitude within that voxel were considered as true peaks [16]. With an exploratory purpose, a threshold of 10% was also considered to investigate the consistency of our results.

2.5 Characterisation of Fibre Responses We estimated the local properties of the underlying fibre response kernel on a voxelby-voxel basis using microscopic diffusion imaging based on SMT [5]. In the SMT framework, the DW signal in each voxel is modelled as the convolution between the local fODF and the voxel-dependent fibre response kernel k(b, g, u), i.e.  A(b, g) = A0

p(u) k(b, g, u) d S 2

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u∈S 2

Above, A0 is the non-DW signal, b is the b-value, g is the gradient direction and the integral is performed over the unit sphere S 2 . The SMT method is based on the key property of the DW signal, such that its spherical mean at fixed b-value, i.e. quantity ¯ A(b) =

 A(b, g) d S 2

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g∈S 2

does not depend on the underlying fODF but only on the local kernel k(b, g, u). Therefore, k(b, g, u) can be estimated on a voxel-by-voxel basis from a collection of ¯ obtained at multiple b-values, which are estimated by averaging spherical means A(b) DW images obtained over multiple b-shells, i.e. ¯ A(b) ≈

N (b) 1  A(b, gn ) N (b) n=1

(3)

where N (b) is the number of DW images acquired at fixed b-value b. In this work, we use a first approximation for k(b, g, u) and modelled it as an axially-symmetric diffusion tensor, which is characterised by two independent parameters, i.e. the microscopic axial and radial diffusivity (μAD and μR D) as: k(b, g, u) = e−b g with D(u) = (μAD − μR D) uuT + μR D I.

T

D(u) g

(4)

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Fig. 2 Normal-appearing white matter and lesional white matter fODF maps in two patients. Abbreviations: fODF: fibre orientation distribution function; NAWM: normal-appearing white matter; WM: white matter

In practice, for the comparison of the local fibre kernel properties with respect to the uniform kernel properties used for fODF calculation by MRtrix3 we proceeded as follows: 1. We fitted the single-compartment SMT [5] voxel-by-voxel (for all WM voxels), obtaining microscopic fractional anisotropy (μF A) and microscopic axial, radial and mean diffusivities (μAD, μR D and μM D) as described above. 2. We calculated the difference, in each voxel, between the local diffusion metrics (μF A, μAD, μR D and μM D in turn) and the mean value of the same metric over the voxels selected by MRtrix3 to calculate the uniform response function (quantities δμF A = μF A – μF Amean ; δμAD = μAD – μADmean ; δμR D = μR D – μR Dmean ; δμM D = μM D – μM Dmean . 3. For each subject, the mean values of δμF A, δμAD, δμR D and δμM D across non-lesional white matter voxels were computed. 4. We tested whether δμF A, δμAD, δμR D and δμM D calculated in point 2 were associated with the mean number of fODF peaks across non-lesional voxels, as this would suggest that spatially variant characteristics of fibre responses not modelled by MRtrix may cause significant spurious peaks in the fODF. 5. Finally, we compared patients and controls in terms of the number of fODF peaks and assessed whether δμF A, δμAD, δμR D and δμM D played a role in such comparison, acting as confounders or modifying the relationship between number of fODF peaks and ‘patient status’ (see Sect. 2.6 for more details).

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Fig. 3 White matter fODF maps in two controls. Abbreviations: fODF: fibre orientation distribution function; WM: white matter

2.6 Statistical Analysis Association Between δμFA, δμAD, δμRD, δμMD and Number of Peaks in Nonlesional WM Tissue. We tested whether there was a significant association between δμF A and number of peaks through Pearson’s correlation analyses (at 5% significance level). A significant association would be suggestive of an influence of the method used to estimate the response function and the results obtained with regard to the number of peaks. Similar analyses were run for δμAD, δμR D and δμM D. Comparison Between Patients and Controls in Terms of Number of Peaks in Non-lesional WM Tissue, Assessing the Role of δμFA, δμAD, δμRD, δμMD as Potential Confounders. We first compared patients and controls in terms of mean number of peaks in non-lesional WM tissue through univariable linear regression models, considering the variable number of peaks as the dependent variable and the binary variable patient status as the only explanatory variable. Afterwards, these models were re-run adjusting for age, gender and lesion load. For this purpose, controls and patients without lesions were assigned a lesion load equal to zero, as previously done [17]. We then assessed the association between δμF A, δμAD, δμR D and δμM D and ‘patient status’ (i.e. patient/control) also through univariable linear regression models where δμF A, δμAD, δμR D and δμM D were considered, in turn, as the dependent variable, and the binary variable ‘patient status’ as the only explanatory variable. These models provided us with the estimated unadjusted differences between patients and controls in terms of δμF A, δμAD, δμR D and δμM D. Afterwards, the same models were re-run adjusting for age, gender and lesion load, in order to obtain the adjusted differences between groups. Finally, we run again the regression models with mean number of peaks across non-lesional WM voxels as the dependent variable and ‘patient status’ as explanatory variable, including δμF A, δμAD, δμR D and δμM D, in turn, as covariates. This series of regression models allowed us to investigate whether potential differences between patients and controls in mean number of peaks could be explained by an unequal distribution of δμF A, δμAD, δμR D and δμM D between patients and controls.

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Comparison Between Patients and Controls in Terms of Number of Peaks in Non-lesional WM Tissue, Assessing the Role of δμFA, δμAD, δμRD, δμMD as Potential Effect Modifiers: Interaction Analysis. Linear regression models similar to those explained above were fitted but this time also including an interaction term between ‘patient status’ and δμF A, δμAD, δμR D, δμM D (in turn) among the covariates. These analyses allowed us to assess whether a potential relationship between number of peaks and patient status depended on the amount of bias between local and assumed voxel-wise diffusion properties, i.e. δμF A, δμAD, δμR D, δμM D. Similarly, these models allowed us to assess whether an eventual association between δμF A, δμAD, δμR D, δμM D and number of peaks depended on whether the subject was a patient or a control. All statistical analyses were carried out in STATA/SE 14.2.

2.7 Simulation Study We ran computer simulations in Python using DiPy 0.14.0 to corroborate our in vivo findings. Specifically, we studied the impact on fODF reconstruction of using deconvolution kernels whose properties are different from those of the underlying fibres. For this, we correlated the reconstructed number of fODF peaks with the difference between the diffusion properties of the fibre and those of the kernel, i.e. quantities δμF A, δμAD, etc., similarly to our in-vivo analysis. The number of peaks is a salient feature of the fODF, which impacts directly on downstream processing such as tractography. Therefore, assessing its dependence on the deconvolution settings provided important insight into the accuracy of fODF reconstruction methods. For our simulations, we considered a simple scenario, where the DW signal originated from a single fibre described as an axially symmetric diffusion tensor with axial diffusivity (μAD) of 1.7 µm2 /ms and radial diffusivity (μR D) of 0.3 µm2 /ms. We synthesised the DW signal according to the same multi-shell protocol used in vivo, and added Rician noise at signal-to-noise ratio of 30 at b = 0. Afterwards, we performed spherical deconvolution with DiPy (harmonic order of 6; regularisation weight of 1) and detected the number of fODF peaks (threshold on peak amplitude: 5%; search on 2,450 points uniformly distributed over the sphere). We iterated the process for 5,000 times, using a different noise instantiation and a different deconvolution kernel for each iteration. Each time, the deconvolution kernel was described as a diffusion tensor, with μAD drawn at random within the uniform distribution [0; 3] μm 2 /ms and with μR D drawn within the uniform distribution [0; μR D]. Finally, we correlated the detected number of peaks with δμF A, δμAD, etc. using a non-parametric Spearman’s correlation test.

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3 Results 3.1 Descriptive Statistics Nineteen patients (ten male) and 12 HCs (four male) were finally included. Patients’ mean age (standard deviation, SD) was 36 (2.03) years and HCs’ mean age was 35 (2.19) years. Patients’ median lesion load was 3.10 mL (range: 0.03–14.69 mL).

3.2 Association Between δμF A, δμ A D, δμR D, δμM D and Number of Peaks in Non-lesional WM Tissue Although no significant associations between δμF A, δμAD, δμR D, δμM D and number of peaks in non-lesional WM tissue were observed (Table 1), the Pearson’s correlation coefficients reached moderate levels of strength, especially in controls.

3.3 Comparison Between Patients and Controls in Terms of Number of Peaks in Non-lesional WM Tissue, Assessing the Role of δμF A, δμ A D, δμR D, δμM D as Potential Confounders The univariable linear regression model built to assess differences between patients and controls in terms of number of peaks showed that patients had significantly greater number of fODF peaks than controls in non-lesional tissue (p = 0.007). This result remained unchanged after adjusting for age, gender and lesion load (Table 2). When we assessed the differences between patients and controls in terms of mean

Table 1 Association between delta di f f usion metrics and number o f peaks (Pearson’s correlation coefficients and p-values). Abbreviations: 95% CI: 95% Confidence Interval; δμAD: delta microscopic axial diffusivity; δμM D: delta microscopic mean diffusivity; δμF A: delta microscopic fractional anisotropy; δμR D: delta microscopic radial diffusivity Delta diffusion metric All subjects Patients Controls δμF A (dimensionless units) δμAD (μm 2 /ms) δμR D (μm 2 /ms) δμM D (μm 2 /ms)

−0.1639, p = 0.378

−0.0322, p = 0.896

−0.2523, p = 0.429

−0.1903, p = 0.305 0.1954, p = 0.292 −0.1377, p = 0.460

−0.0914, p = 0.710 0.1692, p = 0.489 −0.0364, p = 0.882

−0.2725, p = 0.392 0.5429, p = 0.068 −0.1525, p = 0.636

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Fig. 4 In this picture the associations between the estimated number of peaks and the delta diffusion metrics are depicted. Although no significant results were observed, a trend towards an association was observed in some cases Table 2 Unadjusted and adjusted differences between patients and controls in number of peaks. Abbreviations: 95% CI: 95% Confidence Interval; δμAD: delta microscopic axial diffusivity; δμM D: delta microscopic mean diffusivity; δμF A: delta microscopic fractional anisotropy; δμR D: delta microscopic radial diffusivity Model specification Difference between patients and controls (95% CI), p-value Unadjusted Adjusted for age, gender and lesion load Adjusted for δμF A, age, gender and lesion load Adjusted for δμAD, age, gender and lesion load Adjusted for δμR D, age, gender and lesion load Adjusted for δμM D, age, gender and lesion load

0.084 (0.025–0.143), p = 0.007 0.086 (0.021–0.152), p = 0.012 0.078 (0.013–0.144), p = 0.021 0.077 (0.014–0.141), p = 0.019 0.093 (0.029–0.157), p = 0.006 0.077 (0.011–0.144), p = 0.024

number of peaks adjusting for delta diffusion metrics, we found that the original results were also almost unchanged (Fig. 4). Finally, when we assessed the association between δμF A, δμAD, δμR D, δμM D and ‘patient status’ through univariable linear regression models we did not find any significant result (Table 3). These results remained virtually unchanged after adjusting for age, gender and lesion load.

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Table 3 Differences between patients and controls in delta diffusion metrics. (*) Adjusted for age, gender and lesion load. Abbreviations: 95% CI: 95% Confidence Interval; δμAD: delta microscopic axial diffusivity; δμM D: delta microscopic mean diffusivity; δμF A: delta microscopic fractional anisotropy; δμR D: delta microscopic radial diffusivity Delta diffusion metric Unadjusted differences Adjusted differences (*) (95% CI), p-value (95% CI), p-value δμF A (dimensionless units) δμAD (μm 2 /ms) δμR D (μm 2 /ms) δμM D (μm 2 /ms)

−0.009 (−0.050 to 0.032), p = 0.646 −0.030 (−0.170 to 0.110), p = 0.668 −0.009 (−0.030 to 0.011), p = 0.363 −0.016 (−0.060 to 0.028), p = 0.459

−0.019 (−0.062 to 0.024), p = 0.371 −0.057 (−0.200 to 0.086), p = 0.422 −0.008 (−0.031 to 0.016), p = 0.516 −0.024 (−0.068 to 0.020), p = 0.274

3.4 Comparison Between Patients and Controls in Terms of Number of Peaks in Non-lesional WM Tissue, Assessing the Role of Delta Diffusion Metrics as Potential Effect Modifiers: Interaction Analysis None of the interaction terms added to the models shown in Table 2 yielded significant results, indicating that delta diffusion metrics did not act as potential effect modifiers in relation to the differences in number of peaks between patients and controls (data not shown).

3.5 Results of the Simulation Study The simulation study showed strongly significant associations between the number of peaks and the delta diffusion metrics (see Fig. 5 for the case of true number of peak equal to 1). Importantly, the direction of the Spearman’s correlation coefficients was the same as that seen in the in-vivo results.

4 Discussion and Conclusions Our results, based on both the in-vivo and simulation analyses, suggest that the use of a uniform kernel may not be appropriate to estimate fODF characteristics. Whereas the evidence mainly came from the simulation analysis, similar trends, with similar size and direction of correlation coefficients, were observed in the in-vivo part. However, the in-vivo results did not reach significance, possibly because the in-vivo measures

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Fig. 5 Correlation analysis between voxel-wise number of (detected) peaks and delta diffusion metrics for an scenario where the true number of peaks was equal to 1. This figure shows that the mis-specification of the kernel (represented by the delta diffusion metrics) implied a misspecification (i.e. overestimation) of the number of peaks. Abbreviations: δμAD: delta microscopic axial diffusivity; δμM D: delta microscopic mean diffusivity; δμF A: delta microscopic fractional anisotropy; δμR D: delta microscopic radial diffusivity

represented the mean across all non-lesional WM voxels in the brain for each subject, which meant that (1) correlation analyses were only carried out with 31 independent data points (since we had 31 participants in the study); (2) some effects may have been averaged across the brain and therefore diluted. Importantly, our results showed that the greater the difference between the true (local) and the assumed (uniform for the whole brain) response function or kernel, the greater the overestimation of the number of fODF peaks. Since the estimated number of peaks determines the number of modelled crossing fibres, the use of a uniform kernel instead of a locally-fitted one may imply the modelling of spurious fibres, which is still a common flaw of diffusion-based tractography methods [18]. Of note, although our study was the first one investigating the appropriateness of using a uniform kernel versus a locally-fitted one for spherical deconvolution in MS, several authors had already focused on the estimation of an appropriate voxel-wise kernel to obtain accurate predictions of the fODF [19, 20].

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In our cohort, we found that patients had significantly higher fODF number of peaks than controls. A possible explanation for these results is that the higher mean number of peaks in patients than controls reflected the presence of microscopic changes in the WM beyond the visible inflammatory activity. Multi-modal imaging and post-mortem validation may be able to disentangle whether this is caused by microstructural changes such as demyelination rather than inflammation. It is known that CSD methods are very sensitive to errors in the chosen fibre response and when the measured signal deviates from the actual fibre response, more false peaks are likely to be observed [21]. However, its seems clear that these differences between patients and controls in the number of peaks were unlikely to be due to a greater misspecification of the kernel in patients than in controls as a result of having used a uniform kernel to determine the number of fODF peaks. On the other hand, further analyses with greater and more heterogeneous cohorts, e.g. with different MS phenotypes, not only patients with a first demyelinating attack, are probably needed to understand the reasons behind such differences between patients and controls. Importantly, this study has some potential limitations which need to be taken into account. For instance, other methods, such as a the use of a multi-compartmental model [22], to characterise the microscopic diffusion tensor metrics could have probably been more accurate. For this reason, further studies using these other models should be also carried out to confirm our results. Additionally, we do not have any evidence for the quality of the model fit (to obtain microscopic diffusion tensor metrics) being homogeneous across the brain, or across groups, i.e. in patients and controls, which can also be seen as a limitation of our study and requires further analyses too. Another consideration refers to the fact that we did not compute locally the SMT-derived metrics (i.e. microscopic diffusion tensor metrics) using the assumed response function and the locally-obtained diffusion signal. Instead, since the assumed fibre response was the same for the whole brain, we computed, for each one of the microscopic diffusion tensor metrics, the mean across the voxels chosen by MRtrix to estimate the response function. It is possible that if we had obtained locally SMT-derived metrics using the assumed response function and the locallyobtained diffusion signal, we would have obtained more accurate estimates of the impact of using a uniform kernel. Furthermore, an fODF could have been obtained from the SMT model and the comparison between local and assumed fODFs would have been more accurate than the comparison through the SMT-derived microscopic diffusion tensor metrics. A further point to be taken into account is that in this study we did not perform a reproducibility scan-re-scan analysis. Additionally, we did not evaluate the ability of number of peaks to predict which CIS patients would convert to MS in the mid/long term, which was beyond the scope of this study. However, biomarkers that are useful in clinical practice need to be methodologically and biologically robust. Therefore, further investigations are needed to assess these two important aspects of the metric number of peaks. In conclusion, our results showed some evidence against the appropriateness of using a uniform kernel to obtain the fODFs at the voxel level, which can lead to potentially serious biases in brain tractography studies.

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Edge Weights and Network Properties in Multiple Sclerosis Elizabeth Powell, Ferran Prados, Declan Chard, Ahmed Toosy, Jonathan D. Clayden and Claudia Gandini A. M. Wheeler-Kingshott

Abstract Graph theory is able to provide quantitative parameters that describe structural and functional characteristics of human brain networks. Comparisons between subject populations have demonstrated topological disruptions in many neurological disorders; however interpreting network parameters and assessing the extent of the damage is challenging. The abstraction of brain connectivity to a set of nodes and edges in a graph is non-trivial, and factors from image acquisition, post-processing and network construction can all influence derived network parameters. We consider here the impact of edge weighting schemes in a comparative analysis of structural brain networks, using healthy control and relapsing-remitting multiple sclerosis subjects as test groups. We demonstrate that the choice of edge property can substantially

E. Powell (B) Department of Medical Physics and Biomedical Engineering, University College London, London, UK e-mail: [email protected] E. Powell · F. Prados · D. Chard · A. Toosy · C. G. A. M. Wheeler-Kingshott Faculty of Brain Sciences, Queen Square MS Centre, UCL Institute of Neurology, University College London, London, UK F. Prados Department of Medical Physics and Bioengineering, Centre for Medical Image Computing, University College London, London, UK D. Chard National Institute for Health Research Biomedical Research Centre, University College London Hospitals, London, UK J. D. Clayden Developmental Imaging and Biophysics Section, Great Ormond Street Institute of Child Health, University College London, London, UK C. G. A. M. Wheeler-Kingshott Department of Brain and Behavioural Sciences, University of Pavia, Pavia, Italy C. G. A. M. Wheeler-Kingshott Brain MRI 3T Center, IRCCS Mondino Foundation, Pavia, Italy © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_22

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affect inferences of network disruptions in disease, ranging from ‘primarily intact connectivity’ to ‘complete disruption’. Although study design should predominantly dictate the choice of edge weight, it is important to consider how study outcomes may be affected. Keywords Graph theory · Network · Edge weight · Graph property · Connectivity · Permutation · Multiple sclerosis

1 Introduction The brain is a complex network of grey matter nuclei densely interconnected by axon bundles. Communication between these cortical regions is the foundation of brain function, and damage to the connecting axon bundles is thought to cause a range of neurological disorders [1, 2]; techniques for estimating axonal damage and characterising grey matter interactions could therefore offer considerable opportunities in advancing our understanding of brain structure and function in neurological diseases. The application of graph theory to this network representation of the human brain is a valid step towards mapping connectivity: cortical regions and their pairwise interactions can be considered as nodes (vertices) and edges, with edge weights describing some structural or functional property of the connection between regions. Graph theoretical analysis has consistently demonstrated non-random features in human brain graphs, including small-world organisation and modularity [3, 4], as well as network disruptions in a variety of neurological disorders [5, 6]. Moreover, it is possible to characterise aspects of network topology at different levels, for example at both local (nodal) and global (whole network) levels. The technique is not without its challenges, though. The choice of vertices and edges inherently dictates the derived network properties, even to the extent that parcellation strategy [7] and tractography algorithm [8] have a demonstrable effect. Naturally, edge weights also influence derived network measures [9]. Appropriate edge weighting strategies are widely discussed [10, 11]. For the structural connectome—defined as spatially distinct cortical regions connected by axon bundles reconstructed using diffusion tensor imaging (DTI)—typical edge weights include the number of streamlines (NSL) connecting pairwise regions, the mean fractional anisotropy (FA) of the tract, and binary values. All are valid weighting schemes: the NSL and FA offer some suggestion of connection ‘integrity’ or ‘efficiency’, while binary graphs provide analytical simplicity. Network parameters are known to vary as a result of the chosen weighting scheme [12]; however the extent to which this variability may affect a comparative analysis of networks between groups is not obvious. The purpose of this study was to evaluate the impact of the edge weighting scheme on intergroup network differences in order to generalise results and highlight possible confounding factors. We perform an analysis of global and local network properties

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using healthy control (HC) and relapsing-remitting multiple sclerosis (RRMS) subject groups as test sets, and implement a robust permutation-based approach for statistical hypothesis testing that is novel in this context.

2 Methods 2.1 Participants Twenty seven HC subjects (16 female; mean age 37 ± 12 years) and 33 RRMS patients (24 female; mean age 40 ± 10 years; median EDSS score 2.0) were recruited; written consent was obtained for all subjects. No significant age or gender differences were observed between groups ( p = 0.27 and p = 0.28 respectively).

2.2 MRI Acquisition and Pre-processing Images were acquired using a 3T MRI system (Philips Healthcare, Best, Netherlands) with a 32-channel head coil. Diffusion-weighted image (DWI) volumes were acquired with diffusion weighting along 61 directions and b = 1200 s/mm2 , plus 7 volumes with b = 0 s/mm2 , with resolution 2 × 2 × 2 mm3 , TR = 24000 ms and TE = 68 ms. A 3D T1 -weighted fast field echo with resolution 1 × 1 × 1 mm3 , TR = 6.9 ms, TE = 3.1 ms and inversion time TI = 824 ms was also acquired on each participant. All DWI were corrected for eddy current [13], motion and susceptibility distortions [14] in native space. The T1 -weighted images were registered [14] to the corrected DWI and parcellated [15] into structurally-defined sub-regions. Estimates of fibre orientation were obtained from a fit of the ball-and-sticks model to the DWI [16]; a maximum of 3 fibres were modelled per voxel. Probabilistic tractography [17] was performed using 1000 streamline seeds in each white matter voxel. The DTI-derived average FA for each reconstructed tract was also calculated [17].

2.3 Network Reconstruction Network vertices were defined as the 98 cortical regions identified by the anatomical parcellation. For brevity, each vertex was assigned a number from 1 to 98 such that the set of vertices was given by V = {v1 , v2 , ..., vn }, with n = 98. Correspondences with anatomical regions are given in Table 1. Association matrices were generated [17] for each subject and masked to remove any edges absent in more than N subjects, where N = max (NHC , NRRMS ) + 2 and

284 Table 1 Correspondences between anatomical regions and vertex numbers

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Left

Anatomical region

1

2

Anterior cingulate gyrus

3

4

Anterior insula

5

6

Anterior orbitofrontal cortex

7

8

Angular gyrus

9

10

Calcarine cortex

11

12

Central operculum

13

14

Cuneus

15

16

Entorhinal cortex

17

18

Frontal operculum

19

20

Frontal pole

21

22

Fusiform gyrus

23

24

Gyrus rectus

25

26

Inferior occipital gyrus

27

28

Inferior temporal gyrus

29

30

Lingual gyrus

31

32

Lateral orbitofrontal cortex

33

34

Middle cingulate gyrus

35

36

Medial frontal cortex

37

38

Middle frontal gyrus

39

40

Middle occipital gyrus

41

42

Medial orbitofrontal gyrus

43

44

Medial postcentral gyrus

45

46

Medial precentral gyrus

47

48

Medial superior frontal gyrus

49

50

Middle temporal gyrus

51

52

Occipital pole

53

54

Occipital fusiform gyrus

55

56

Pars opercularis

57

58

Pars orbitalis

59

60

Posterior cingulate gyrus

61

62

Precuneus

63

64

Parahippocampal gyrus

65

66

Posterior insula

67

68

Parietal operculum

69

70

Postcentral gyrus

71

72

Posterior orbitofrontal cortex

73

74

Planum polare

75

76

Precentral gyrus

77

78

Planum temporale

79

80

Subcallosal area

81

82

Superior frontal gyrus

83

84

Supplementary motor cortex

85

86

Supramarginal gyrus

87

88

Superior occipital gyrus

89

90

Superior parietal lobule

91

92

Superior temporal gyrus

93

94

Temporal pole

95

96

Pars triangularis

97

98

Transverse temporal gyrus

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(c) NSL-weighted network

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(b) FA-weighted network

(d) NSLcor-weighted network

Fig. 1 Example network types generated for a single subject. a Binary network, b Network weighted using average tract FA, c Network weighted using the NSL, d Network weighted using the NSL corrected for tract length

NHC , NRRMS denote HC and RRMS group sizes respectively. Masking in this way ensured that any given edge was present in at least two subjects within a group and aided the statistical analysis. All vertices remained connected for all subjects. Edges were weighted using four different metrics commonly reported: the NSL, the NSL corrected for tract length (NSLcor ), the mean tract FA, and a simple binary weight. The correction for tract length was implemented as the product of the NSL connecting two regions and the average length of those streamlines. An example of each association matrix generated for a single subject is provided in Fig. 1.

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2.4 Statistical Analysis Global and local network properties were calculated [17] for each subject and edge weighting scheme. The global metrics evaluated were efficiency, mean shortest path and modularity; the local properties were efficiency, clustering coefficient, node strength and betweenness centrality. Significant differences in global network properties between HC and RRMS groups were identified using a t-test ( p < 0.05). Significant intergroup differences in local network properties were determined using a permutation-based approach. This strategy enabled the null distribution of p values to be empirically derived whilst taking multiple comparisons into consideration, from which a corrected p value could be estimated. Separate null distributions of p values were generated for each network type and property. At each permutation, then, group labels were randomly reallocated to create new ‘HC’ and ‘RRMS’ groups; original group sizes were preserved in each of the 1000 samples generated. At each node the mean local network properties were compared between sample groups using a t-test. The minimum p value obtained across all node comparisons at each permutation was added to the null distribution for a given weighting scheme and network property; the procedure was then that of a step-down ‘minimum p’ approach [18]. The family-wise Type I error rate was subsequently controlled for in the strong sense.

3 Results 3.1 Intergroup Differences in Global Network Properties Global efficiency was significantly lower ( p < 0.05) in RRMS patients relative to HC subjects in networks weighted using FA, NSLcor and NSL edge properties; the mean shortest path was correspondingly greater ( p < 0.05) in the RRMS population across the same networks. Binary networks exhibited no intergroup differences in global efficiency and mean shortest path; however modularity was significantly greater in the HC group ( p < 0.05) (Table 2).

Table 2 Significant global network differences between HC and RRMS subject groups Edge weight Global efficiency Mean shortest path Modularity FA NSLcor NSL Binary

HC > MS, p < 0.001 HC < MS, p = 0.002 – HC > MS, p < 0.001 HC < MS, p = 0.008 – HC > MS, p = 0.022 HC < MS, p = 0.017 – – – HC > MS, p = 0.047

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These results are consistent with published findings [19, 20]. Of note here is that inferences of intergroup differences in global network properties were unaffected by the choice of edge property for weighted networks, but were substantially different between weighted and binary networks.

3.2 Intergroup Differences in Local Network Properties Local Efficiency Lower nodal efficiencies were observed in the RRMS cohort across all weighted networks, which is consistent with published reports [19]; no alterations were detected in binary networks (Fig. 2a). However, the set of vertices with different efficiency properties between groups was highly inconsistent across the weighted network types. FA-weighted networks exhibited the greatest intergroup differences, with lower efficiency in 97 out of 98 nodes in the RRMS cohort ( p < 0.05, corrected). In NSLcor -weighted networks only 78 nodes demonstrated alterations between groups ( p < 0.05, corrected), while in NSL-weighted networks the proportion was lower still at just 12 nodes ( p < 0.05, corrected). The variation in findings between the NSLcor - and NSL-weighted networks was particularly striking. It is possible here that additional uncertainties in NSL-weighted networks, resulting from inherent biases in probabilistic tractography towards tract length [21], could be driving the discrepancies. Clustering Coefficient Lower clustering coefficients were observed in the RRMS group in all weighted networks (Fig. 2b), in line with previous studies [19]. The extent of the alterations was again highly variable between the weighted network types: substantially more intergroup differences were identified in FA-weighted networks (54 out of 98 nodes; p < 0.05, corrected) than in NSLcor - and NSL-weighted networks (5 and 4 nodes respectively; p < 0.05, corrected). No differences were observed in binary networks. Nodal Strength Lower nodal strengths were found in the RRMS group in FAweighted, NSLcor -weighted and binary networks (Fig. 2c), consistent with published findings [19]. The proportion of nodes exhibiting different strength properties between the HC and RRMS groups once more varied across the network types, with 14 out of 98 nodes ( p < 0.05, corrected) identified in FA-weighted networks, 4 nodes ( p < 0.05, corrected) in NSLcor -weighted networks and 1 node ( p < 0.05, corrected) in binary networks. No intergroup differences were observed in NSLweighted networks. Betweenness Centrality Minimal intergroup differences were observed in betweenness centrality: only 2 nodes in FA-weighted networks displayed significantly greater ( p < 0.05, corrected) betweenness centrality in the RRMS group (Fig. 2d). No other networks indicated any differences.

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(a) Local efficiency

(c) Node strength

(b) Clustering coefficient

(d) Betweenness centrality

Fig. 2 Significant intergroup differences ( p < 0.05, corrected) in local network properties. a Local efficiency, b Clustering coefficient, c Node strength, d Betweenness centrality. Each numbered segment corresponds to a node, as specified in Table 1. In each sub-figure the concentric rings correspond to specific network types: the outermost ring (ring 1) corresponds FA-weighted graphs; ring 2 to NSLcor -weighted networks; ring 3 to NSL-weighted networks; ring 4 (innermost ring) to binary networks. The colour indicates whether the network property is significantly greater in HC subjects (blue) or RRMS patients (red)

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4 Discussion We have explicitly demonstrated the impact of the edge weighting scheme on a comparative analysis of network properties using HC subjects and RRMS patients with very mild disease severity as example data sets. While graph theoretical analyses performed over different network types will be naturally incongruent to a degree, the disparities presented here are striking. Given any one of the network types in isolation, as is commonly reported, the assessment of damage to the structural connectome of these RRMS patients would be substantially different, with potential conclusions ranging from ‘intact connectivity’ to ‘complete disruption despite the mild disability’. In FA-weighted networks, for example, reductions were observed in the local efficiency of almost every node and in the clustering coefficient of more than half the nodes. Further, regions of the default mode network (DMN)—which is important for high level function and prone to impairment in MS [22]—such as the precuneus and posterior cingulate gyrus displayed significantly reduced nodal strength. From these findings it may be inferred that the networks of these RRMS patients were substantially damaged despite the relatively mild disability levels (the median EDSS was just 2.0, indicating no major motor, visual, sensory or cognitive disabilities), and that the graph properties were in fact sensitive to subtle MS pathology. In NSLcor -weighted networks, on the other hand, the clustering coefficient, node strength and betweenness centrality were unaffected in the majority of nodes, and core DMN nodes in particular showed no changes; only local efficiency appeared to indicate any alterations. It may be concluded here, then, that networks in this RRMS cohort were only moderately disrupted, and potentially reflected their relative lack of clinical disability. Evidently, interpretations of network analyses must be made with caution: graph theoretical metrics may be sensitive to subtle alterations between groups but they lack biological specificity. Factors known to systematically bias estimates of structural connectivity range from head motion [23] and low signal-to-noise ratio [24] during acquisition to the parcellation strategy and tractography algorithm adopted during image processing. This study highlights for the first time how outcomes in intergroup comparative studies across different network types may be affected by the combined influence of biological alterations and factors unrelated to inter-regional connectivity. Specifically, comparisons of local network properties between groups may be particularly prone to variations across network types. The global metrics evaluated here were comparatively less sensitive to weighting scheme; however the utility of global properties is limited by a lack of specificity to potentially more meaningful local alterations. The relative absence of intergroup differences both locally and globally in binary networks is likely to reflect the similarity in edge set between groups, and the applied edge threshold is likely to be influential here. A complete analysis of individual edges was beyond the scope of this work, but could be considered in future studies.

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The specificity of local network properties to biological alterations may be improved to a degree by including appropriate confounding variables, such as estimates of head motion, as covariates in the statistical analysis [23]. The permutation testing implemented here can be easily extended to include covariates. Moreover, it is a powerful and robust approach for controlling multiple correlated comparisons that is novel in this context, and would be beneficial in comparative studies of the same nature. In particular, it could be interesting to compare local network properties in alternative subject populations, such as those with more severe pathology: substantial microstructural changes may then outweigh confounding factors and result in more consistent outcomes. Ultimately, the choice of edge weight is largely dependent on the study design in question. It is important to consider, though, that the nuances of each weighting scheme influence the derived network parameters, which may in turn substantially impact outcomes in intergroup comparative studies.

5 Conclusions Network-based approaches offer important contributions towards analysing the connections that form the basis of brain structure and function; however the interpretation of graph theoretical properties remains challenging. We demonstrate, using HC and RRMS subjects as test populations, that the choice of edge weight in intergroup comparisons is non-trivial and can substantially affect inferences of network disruptions in disease. Study design should primarily drive the choice of weighting scheme, but potential confounding factors and interpretation pitfalls must be considered. Acknowledgements The NMR unit where this work was performed is supported by grants from the Multiple Sclerosis Society of Great Britain and Northern Ireland, Philips Healthcare, and supported by the UCL/UCLH NIHR (National Institute for Health Research) BRC (Biomedical Research Centre).

References 1. 2. 3. 4.

Geschwind, N.: Disconnexion syndromes in animal and man: Part I. Brain 88, 237–294 (1965) Geschwind, N.: Disconnexion syndromes in animal and man: Part II. Brain 88, 237–294 (1965) Bassett, D.S., Bullmore, E.: Small-world brain networks. Neuroscientist 12(6), 512–523 (2006) van den Heuvel, M.P., Sporns, O.: Rich-club organization of the human connectome. J. Neurosci. 31(44), 15775–15786 (2011) 5. Bassett, D.S., Bullmore, E., Verchinski, B.A., Mattay, V.S., Weinberger, D.R., MeyerLindenberg, A.: Hierarchical organization of human cortical networks in health and schizophrenia. J. Neurosci. 28(37), 9239–9248 (2008) 6. He, Y., Dagher, A., Chen, Z., Charil, A., Zijdenbos, A., Worsley, K., Evans, A.: Impaired small-world efficiency in structural cortical networks in multiple sclerosis associated with white matter lesion load. Brain 132(12), 3366–3379 (2009)

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7. Zalesky, A., Fornito, A., Harding, I.H., Cocchi, L., Yücel, M., Pantelis, C., Bullmore, E.T.: Whole-brain anatomical networks: does the choice of nodes matter? Neuroimage 50(3), 970– 983 (2010) 8. Bastiani, M., Shah, N., Goebel, R., Roebroeck, A.: Human cortical connectome reconstruction from diffusion weighted MRI: the effect of tractography algorithm. Neuroimage 62(3), 1732– 1749 (2012) 9. Rubinov, M., Sporns, O.: Complex network measures of brain connectivity: uses and interpretations. Neuroimage 52, 1059–1069 (2010) 10. Fornito, A., Zalesky, A., Breakspear, M.: Graph analysis of the human connectome: promise, progress, and pitfalls. Neuroimage 80, 426–444 (2013) 11. Bullmore, E.T., Sporns, O.: Complex brain networks: graph theoretical analysis of structural and functional systems. Nature Rev. Neurosci. 10, 186–198 (2009) 12. Cheng, H., Wang, Y., Sheng, J., Kronenberger, W.G., Mathews, V.P., Hummer, T.A., Saykin, A.J.: Characteristics and variability of structural networks derived from diffusion tensor imaging. Neuroimage 61(4), 1153–1164 (2012) 13. Andersson, J., Sotiropoulos, S.N.: An integrated approach to correction for off-resonance effects and subject movement in diffusion MR imaging. Neuroimage 125, 1063–1078 (2015) 14. Bhushan, C., Haldar, J.P., Joshi, A.A., Leahy, R.M.: Correcting susceptibility-induced distortion in diffusion-weighted MRI using constrained nonrigid registration. In: APSIPA ASC, pp. 1–9 (2012) 15. Cardoso, M., Modat, M., Wolz, R., Melbourne, A., Cash, D., Rueckert, D., Ourselin, S.: Geodesic information flows: spatially-variant graphs and their application to segmentation and fusion. IEEE Trans. Med. Imaging. 34(9), 1976–1988 (2015) 16. Behrens, T., Berg, H., Jbabdi, S., Rushworth, M.F., Woolrich, M.: Probabilistic diffusion tractography with multiple fibre orientations: what can we gain? Neuroimage 34(1), 144–155 (2007) 17. Clayden, J., Mu, S., Storkey, A., King, M., Bastin, M., Clark, C.: TractoR: magnetic resonance imaging and tractography with R. J. Stat. Softw. 44(8), 1–18 (2011) 18. Westfall, P., Young, S.: Resampling-Based Multiple Testing: Examples and Methods for p-value Adjustment. Wiley (2003) 19. Shu, N., Duan, Y., Xia, M., Schoonheim, M.M., Huang, J., Ren, Z., Sun, Z., Ye, J., Dong, H., Shi, F.D., Barkhof, F., Li, K., Liu, Y.: Disrupted topological organization of structural and functional brain connectomes in clinically isolated syndrome and multiple sclerosis. Sci. Rep. 6 (2016) 20. Llufriu, S., Martinez-Heras, E., Solana, E., Sola-Valls, N., Sepulveda, M., Blanco, Y., MartinezLapiscina, E.H., Andorra, M., Villoslada, P., Prats-Galino, A., Saiz, A.: Structural networks involved in attentional function in multiple sclerosis. Neuroimage Clin. 13, 288–296 (2017) 21. Zalesky, A., Fornito, A.: A DTI-derived measure of cortico-cortical connectivity. IEEE Trans Med. Imaging 28(7), 1023–1036 (2009) 22. Gamboa, O.L., Tagliazucchi, E., Von Wegner, F., Jurcoane, A., Wahl, M., Laufs, H., Ziemann, U.: Working memory performance of early MS patients correlates inversely with modularity increases in resting state functional connectivity networks. Neuroimage 94, 385–395 (2014) 23. Baum, G.L., Roalf, D.R., Cook, P.A., Ciric, R., Rosen, A.F.G., Xia, C., Elliott, M.A., Ruparel, K., Verma, R., Tunç, B., Gur, R.C., Gur, R.E., Bassett, D.S., Satterthwaite, T.D.: The impact of in-scanner head motion on structural connectivity derived from diffusion MRI. Neuroimage 173, 275–286 (2018) 24. Jones, D.K., Knösche, T.R., Turner, R.: White matter integrity, fiber count, and other fallacies: the do’s and don’ts of diffusion MRI. Neuroimage 73, 239–254 (2013)

Part V

Tractography and Connectivity Mapping

Measures of Tractography Convergence Daniel C. Moyer, Paul Thompson and Greg Ver Steeg

Abstract In the present work, we use information theory to understand the empirical convergence rate of tractography, a widely-used approach to reconstruct anatomical fiber pathways in the living brain. Based on diffusion MRI data, tractography is the starting point for many methods to study brain connectivity. Of the available methods to perform tractography, most reconstruct a finite set of streamlines, or 3D curves, representing probable connections between anatomical regions, yet relatively little is known about how the sampling of this set of streamlines affects downstream results, and how exhaustive the sampling should be. Here we provide a method to measure the information theoretic surprise (self-cross entropy) for tract sampling schema. We then empirically assess four streamline methods. We demonstrate that the relative information gain is very low after a moderate number of streamlines have been generated for each tested method. The results give rise to several guidelines for optimal sampling in brain connectivity analyses. Keywords Tractography · Cross entropy · Simulation

1 Introduction It is common practice in brain imaging studies to estimate the trajectories of white matter fascicles, the bundles of axons that connect different gray matter regions in D. C. Moyer · P. Thompson (B) Imaging Genetics Center, Stevens Institute for Neuroimaging and Informatics, University of Southern California, Los Angeles, CA 90007, USA e-mail: [email protected] D. C. Moyer e-mail: [email protected] D. C. Moyer · G. V. Steeg Information Sciences Institute, Marina del Rey, CA 90292, USA © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_23

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the brain. It is difficult to observe the fascicles directly in vivo; instead, researchers currently rely on diffusion imaging—a variant of standard anatomical MRI—and the “tube like” structure of most major neural pathways to reconstruct the brain’s major fiber bundles. We observe measurements of water diffusion in brain tissue, fit local diffusion models, and then reconstruct the trajectories by curve fitting. This process is known as tractography, and reconstructed fibers are referred to as tracts.1 The number of possible curves is combinatorially large, and grows rapidly as image resolution increases. Closed form likelihoods for sets of tracts (tractograms) have not yet been found even under simple local models. Thus, tract sampling is performed, where only a subset of the possible tracts are recovered. Sampling is usually performed in an ad-hoc manner, and post-hoc analyses usually treat tracts as direct observations instead of realizations of global trajectories from local models. The concept that tracts are drawn from a distribution is rarely acknowledged, and the issue of convergence is almost never addressed. A simple question remains unanswered: how many tracts should we sample? Even in the most general sense there is no consensus on this topic. Recent works have used 100,000 [11], 500,000 [10] and 10,000,000 [18] tracts for scans of isotropic resolution 2 mm (with 64 directions), 1.25 mm (90 directions), and 2.5 mm (60 directions), respectively. The pragmatic tractography user might then think to simply take the maximum of these options to ensure convergence; for small studies this may be feasible, but given the trajectory of imaging studies—now performed in biobanks of over 10,000 scans—this choice clearly will not scale to the regimes demanded by current and future clinical studies2 using current technology. The answer to whether or not such large tractogram sizes are required will dictate practical planning in large studies. It is useful then to understand empirically the sample complexity of these phenomena. If we choose a sample size too small our tractograms will be sensitive random fluctuations; if instead we choose a sample size too large, we could be wasting very large amounts of storage and computation. The trade-off between accuracy and cost forms a curve of solutions to the (ill-posed) question of “how many tracts?” In the present work we provide a method for measuring this trade-off. We leverage cross entropy as a relative measure of accuracy, applying our estimator to a test set of subjects under several different tractography methods.

1 It

is important to distinguish between white matter fibers (fascicles) and observed “tracts.” In this paper we use “tracts” to denote the 3D-curves recovered from diffusion-weighted imaging via tractography algorithms. 2 The largest study to date aims to have 100,000 subjects participating in the imaging cohort [15]. By our back-of-the-envelope estimate, 100,000 subjects with 10,000,000 tracts each would require about 1.5 petabytes of disk space, just for the tractograms.

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1.1 Relevant Prior Work Surprisingly literature on the number of samples required is sparse. In the documentation of Probtrackx [2] the number of posterior samples is taken to be large (5000 per vertex) specifically to ensure convergence. This solution is not feasible for most other methods, as Probtrackx does not usually keep computed trajectories, but is the most direct reference to tractogram convergence that we found in the literature. In Cheng et al. [3] the authors address the convergence of network measures. While this is a similar issue, it is seen through the lens of graph theory. In a network context the absolute (objective) number of tracts strongly influences the downstream measures. Thus, the authors advise against larger numbers of seeds in order to avoid spurious fibers. Similarly, Prasad et al. [17] investigate effect of the number of streamlines used on a group-wise comparison task. The authors note that larger numbers of streamlines tend to decrease p-values in nodal degree comparisons. Similar to this, Maier-Hein et al. [14] provide a tractography challenge in which the measure of success depends in part on how few false positives were identified. This work is an important if not critical step for the community to address issues raised by others (e.g., Thomas et al. [20]), but its evaluation design runs counter to our premise here that tractography is a simulation. It is equivalent to taking the support of the distribution as its characterizing quality. Jordan et al. [12] propose a method for measuring streamline reproducibility. While they do not provide as much theoretical analysis nor do they identify the estimation method employed (leading to non-standard choices for their empirical work), the method at its core is the same as described here.

2 Framework and Methods 2.1 Minimal Surprise as a Stopping Condition Diffusion MRI does not provide direct observations of whole white matter fascicles; the unit of observation is a q-space measurement of diffusion propensity for given gradient angles, which is resolved and processed into quantized voxel measurements. Researchers simulate fascicle trajectories from fitted local (voxel-wise) models, which have parameters estimated from the diffusion signal (for reference and review, see Sotiropoulos and Zalesky [19]). Some of these simulations are deterministic given a seed point, while others are stochastic. In both cases, however, seed points are not bijective with tracts. If the choice of seed point is random, it is then reasonable to view the tractography generation process as sampling from a distribution P defined over the space of possible tracts, stratified by seed points. The space of possible tracts has at minimum a qualitative notion of similarity. While there is no agreed upon metric, various distances and similarities have been proposed [8, 16, 23]. Choosing a reasonable metric, a simplistic estimator Pˆ of tract

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distributions can be constructed using Kernel Density Estimation (KDE). These estimators enjoy concentration results3 even in our general context [4] with minimal constraints on metric choice or kernel shape. The construction of distribution estimates Pˆ lead us to a naïve condition for assessing convergence: we should stop sampling when Pˆ stops gaining information (“stops learning”). Pˆ is a representation of our beliefs about P. A sample from P containing information not in Pˆ (i.e., contrary to our beliefs) is then qualitatively surprising. As we update our beliefs Pˆ we learn about P; when the surprise no longer decreases we have stopped updating our beliefs, so we have stopped gaining information, which is exactly our stopping condition. Thus we should sample until we have minimal surprise. Before moving on, it should be made explicit: clearly Pˆ will be misspecified. Without a natural metric, it is not clear if an estimator class could ever be guaranteed to include P in every case. With this in mind, our provided method is left openended toward choice of metric, kernel shape, smoothing parameter, etc. Further, for different datasets, the convergence rate of Pˆ will be different. Higher resolution data will provide more information, providing more intrinsic information in P, meaning Pˆ has more surprise to overcome. The point of this paper is not to estimate P exactly, nor to establish objective numerical standards for the number of tracts for every case, but to provide a method to investigate the complexity of tract generation. It is our belief that with reasonable choices a conservative estimate can be made to the number of samples required.

2.2 Quantitative Measures of Surprise Kullback-Leibler divergence (KL divergence) is a classic measurement for the differences between distributions. While it has numerous interpretations and applications, in the information theory context it measures the “extra information” required to encode samples drawn from P using the optimal encoding for Pˆ [5]. This is often used as a pseudo-distance, as “close” distributions have similar optimal encodings. KL divergence has the following analytic form, with 0 log 0 = 0 by convention:  ˆ ˆ K L[ P | Pˆ ] = E[log P(x) − log P(x)] = P(x)[log P(x) − log P(x)]d x (1)     ˆ dx . (2) = P(x) log (P(x)) d x − P(x) log P(x)      Negative Entropy

Cross Entropy

ˆ Following For any P the Negative Entropy term is clearly fixed, regardless of P. our previous intuition, this corresponds to the intrinsic surprise of P, the number of bits required to encode P under the optimal encoding. The Cross Entropy term Pˆ converges as the number of samples used to construct it increases. This does not guarantee that Pˆ converges to P.

3 In short, we are asserting that

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ˆ encoding in place of P. It is thus our corresponds to the induced surprise of using P’s desired measure of surprise; measuring Cross Entropy is the focus of our empirical method. Both have units of bits of information (or nats, depending on the base of the log).

2.3

Fixed-KDE Cross Entropy

We construct a simple estimator of Cross Entropy using Kernel Density Estimation N be samples from a distribution P, and let d(xi , x j ) be a metric. (KDE). Let {xi }i=1 The (unnormalized) squared exponential kernel function for distance d is defined as k(xi , x j ) = exp(−γd(xi , x j )).

(3)

Using this function, we can then define an unnormalized Gaussian KDE: N ˆ i) = 1 k(xi , x j ) Q(x N j=1

(4)

We use Q(xi ) to emphasize that this distribution is off by a normalizing constant, N ˆ i ) to denote that this is an estimate of Q based on {xi }i=1 . We assume there and Q(x

exists a finite normalization constant, Z = Q(x)d x < ∞. An approximation of P the distribution of tracts is then N 1 ˆ 1 ˆ P(x) = Q(x) k(x, x j ) = Z Z N j=1

(5)

Using Eq. 5, we can form an approximation to the Cross Entropy up to a constant. ˆ Cross entropy: H[ P(x); P(x)] = −



ˆ P(x) log P(x)dx

(6)



ˆ P(x) log Q(x)dx + Const (7) ⎤ ⎡ N 1 ⎣ 1 k(xi , x j )⎦ + Const ≈− log N x N j=1 =−

i

(8)

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2.4 Calculating Tract Sample Cross Entropy Several metric spaces have been proposed [8, 16] for streamlines. These allow us to apply our KDE estimator to the streamline distributions. We use the Mean Direct Flip distance of Garyfallidis et al. [8] as the distance metric in Eq. 3. We then estimate the Cross-Entropy of samples to tract distributions in these spaces using Eq. 8. While Eq. 8 is always off by an unknown constant, the rate of information gain is preserved. This allows us to measure the marginal value (in bits) of increasing sample sizes. Further, for fixed γ the objective information content is comparable across distributions. As tracts are simulated and each observation is independent, many issues are simplified in our approximation. In particular, we separate the {xi } and {x j } samples, letting the number of xi be large (∼106 ) no matter the choice of N (the number of x j ) in order to ensure a good estimate of the outer expectation. We can further separate the kernel computation step and the summation step, which allows for fast bootstrap estimates as well as faster computation for increasing sizes of N . Again, this coincides with Jordan et al. [12], who use a KDE-like form to assign a weight (density) to each streamline. The authors do not identify this as a KDE (and accordingly use a non-standard power-law kernel), but otherwise produce the same method.

2.5 Scale Considerations Note that each KDE used fixed and un-tuned bandwidth γ1 . Thus, our KDE converges to P ∗ K instead of P, where K (x) = Z −1 k(x, 0). We believe this to be a feature and not a bug: first, though the tractography simulation is often conducted with high numerical precision (using, e.g., floating point representation), the trajectories of streamlines are determined in part by the diffusion measurements. These measurements are ill-resolved relative to machine precision. Second, later analyses of streamline distributions are rarely conducted at a local scale. Fixing γ allows for “convergence” to be contextualized to a particular spatial scale, albeit after the choice of a tract metric. Thus, if the practitioner does not need to resolve small scale differences, a corresponding choice of γ can be made.

3 Empirical Assessment We apply the Cross Entropy estimator to 44 test subjects from the publicly available Human Connectome Project dataset [22]. We used the minimally preprocessed T1-weighted (T1w) and diffusion-weighted images (DWI) rigidly aligned to MNI 152 space. Briefly, the preprocessing of these images included gradient nonlinearity correction (T1w, DWI), motion correction (DWI), eddy current correction (DWI),

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and linear alignment (T1w, DWI). We used the HCP Pipeline FreeSurfer [6] protocol, running the recon-all pipeline. Tractography was conducted using the DWI in 1.25-mm isotropic MNI 152 space. Three separate types of tractography were conducted, using three separate local models and two different tracking methods. All three were performed using the Dipy [9] implementation. The local models were as follows: 1. Constrained Spherical Deconvolution (CSD) [21] with a harmonic order of 8 2. Constant Solid Angle (CSA) [1] with a harmonic order of 8 3. Diffusion Tensors (DTI) (see, e.g., [13]) Probabilistic streamline tractography was performed using Dipy’s probabilistic direction getter for CSD and CSA local models, while deterministic tracking was completed using EuDX [7] for DTI and the Local Tracking module for CSD. For each local model streamlines were then seeded at 2 million random locations (total) in voxels labeled as likely white matter by FSL’s FAST [24] and propagated in 0.5 mm steps, constrained to at most 30◦ angles between segments. Each set of tracts was split into two equally sized subsets. Cross-entropy was then calculated using Eq. 8 for five separate length-scale choices γ = { 41 , 21 , 1, 2, 4}, using 1000 streamline intervals to allow for parallelization and efficient storage. This portion of the procedure was written in C++ with OpenMP.4 In the top row of Fig. 1 we plot the cross-entropy (surprise) as a function of the number of samples when sampling at random from within the white matter mask. The curve for γ = 0.25 is nearly flat, while for γ = 0.5 and γ = 1 the curves quickly bottom out after a short steep learning period. As the length scale increases, the learning rate becomes quite slow. At γ = 4 the curves begin to bottom out at 250,000 samples. However, for each of these scales, no changes occur after 500,000 samples. If there is relevant signal in these samples that is not contained on average in the previous half-million samples, it has a very small relative amount of information about the distribution of tracts. In general for each length scale both deterministic methods have less entropy than the probabilistic methods. Further, DTI converges (flattens) much faster than the other methods. These results match our prior intuitions about these methods. Probabilistic CSD has the largest information content, though we condition this in that in general noise adds information, albeit useless information. Thus, this is not alone a measure of quality, though it does differentiate the method as certainly sampling a tract distribution unlike the others. In the bottom row of Fig. 1 we plot the incremental (marginal) entropy rate. For each length scale for each method the marginal information gained per thousand streamlines after the initial learning period is very low; this is unsurprising, but raises questions about the utility of sampling past several million tracts. In order to build intuition for qualitative differences between values of γ, we plot in Fig. 2 the spatial extent of the kernel function centered on one streamline from the inferior longitudinal fasciculus for each of the five length scales tested. While 4 Please

contact the authors to access the code.

Fig. 1 Top row: we plot the self cross-entropy (surprise) of each method as a function of the number of tracts, conditioned on the length scale γ (given by the plot colors). Bottom row: we plot the incremental (marginal) entropy gain per 1000 streamlines, also conditioned on the length scale. In both row we eschew plotting the x-axis to the full one million samples as the unplotted section is essentially flat

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Fig. 2 We plot a streamline from the inferior longitudinal fasciculus, as well as each tract above 5% of the maximum kernel amplitude for each given γ. At γ = 4 only the target tract itself is left

in general the spatial extent is difficult to visualize, we can demonstrate its support practically by plotting all sampled tracts above a threshold, in this case 5% of the function maximum. As can be seen in the figure, for low values of γ the spatial extent is quite large, while for larger values of γ the function is extremely peaked. These curves are artificially smooth due to the binning of the number of samples by thousands; while in each plot the curves are monotonic, this is an overall trend (at the scale of thousands of samples) and not true locally (at the scale of single samples). We performed a second experiment using the same tracking parameters, but seeding only in a single voxel at a time. For each voxel in the white matter we generated two tractographies (one train set, one test set), seeding only in that voxel 100 times. We then computed the cross-entropy between the two tractographies as a function of the number of samples in the training set. In Fig. 3 we plot the marginal change in entropy over the white matter mask for one subject, both at the start of the process and the end. For many regions the sampling process converges very quickly. However, for certain regions the sampling process converges quite slowly (albeit without considering samples from other voxels). This suggests at surface level high variance in complexity of the white matter when aggregating into voxels. Further, this provides evidence that a weighted seeding scheme might be more efficient than gridded or uniformly random schema.

4 Discussion Our results suggest that in general the number of tracts required to closely approximate the underlying distribution is low using the methods we tested here. This can only be reduced by conditioning tracts or seeds on specific anatomic priors (e.g., seeding only in the corpus callosum or in a specific region). Further, the Human Connectome Project [22] data are widely regarded as examples of the state of the art. It is our hypothesis that other datasets may have less signal content, leading to sharper (more steep) curves and even faster convergence (as more signal content would, we hypothesize, better resolve complex crossing-fiber regions). Our theoretical analysis and method is also inherently limited: we cannot differentiate between “valid” signal and spurious systemic error. By sampling to the scales

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Fig. 3 Here are displayed slices of the white matter of one subject, centered on (60, 85, 61) (MNI). Colors denote the average change in entropy per marginal tract in each voxel, conditioned on seeding in that voxel. The left column shows the average change in the first five tracts seeded in each voxel, while the right column shows the average for tracts 96–100. Most regions converge almost immediately, but specific regions converge very slowly. Note that these measurements do not account for similarity to tracts seeded in other voxels

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described in this paper researchers can accurately recover the streamline distribution (up to the chosen spatial scale). If this distribution includes “false” modes, i.e. modes that not from a corresponding biophysical structure, we will also consistently recover those false modes. Larger sample sizes in simulations do nothing to guard against false signal; modes of both types will be reinforced, while random fluctuations will attenuate. Consider a tractography method that when queried ten times generates complete noise5 nine times and generates a real tract one time on average. Clearly this method has high noise (90% useless!) and would not do well in major tractography challenges (e.g., [14]), but, we claim, this hypothetical method would still have value. Since the number of real tracts is finite and the number of possible noise tracts is comparatively large, increasing the sample size will lead to each real tract having more corresponding samples than the noise tracts. Given enough time we may recover the set of real tracts. Further consider a tractography method that when queried ten times generates complete noise eight times, generates a real tract one time, and generates a false but non-random tract the remaining time on average. Like the first hypothetical method, this second method produces 90% useless tracts. However, increasing sample size will not lead to vanishing error in this second case due to the false mode (the nonrandom false tract). Clearly, no method is purely in the first case; all methods likely have false modes. However, we posit that understanding the number, location, and size of these false modes is just as important as measuring false positive rate. It is illustrative that F1score is unable to differentiate between these two hypothetical methods. We believe such considerations should be taken into account, in that valuable methods might be slow to converge, and thus currently ignored. Further, we believe that in undersampled regimes the assessment of signal peaks is inherently inaccurate. It is important to note that “oversampling” only carries computational drawbacks. Aside from the time and disk space costs, there is no danger in sampling more under i.i.d. conditions, e.g. 10,000,000 tracts [18]. Further, most if not all downstream analyses will not be completely information efficient (this would be akin to an electrical machine being completely energy efficient). Thus, sampling further may be required for later methods applied to the tractogram. Our results are lower bounds, and not meant to be the end-all-be-all for the number of tracts to sample. Again however, “undersampling”, i.e. using a number of samples below the start of the converged regime, is dangerous, and risks producing spurious results without other diagnostic warning. By the data processing inequality, the amount of information in the tractogram is the maximum amount of information about the tract generating distribution available to downstream methods.

5 The

definition of complete noise is actually tricky, but we could use a proxy of “drawing a tract length uniformly at random between 1 and the number of voxels, and filling its sequence with points drawn uniformly from the domain of the image”.

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5 Conclusion In this paper we have investigated the empirical sample complexity of tractography distributions. We have provided a general methodology for estimating and comparing the information content of streamline methods, along with an example analysis of the Human Connectome Project dataset using four different tracking pipelines. Acknowledgements This work was supported by NIH Grant U54 EB020403 and DARPA grant W911NF-16-1-0575, as well as the NSF Graduate Research Fellowship Program Grant Number DGE-1418060.

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20. Thomas, C., et al.: Anatomical accuracy of brain connections derived from diffusion mri tractography is inherently limited. PNAS 111(46), 16574–16579 (2014) 21. Tournier, J.D., et al.: Resolving crossing fibres using constrained spherical deconvolution: validation using diffusion-weighted imaging phantom data. NeuroImage 42(2), 617–625 (2008) 22. Van Essen, D.C., et al.: The WU-Minn human connectome project: an overview. NeuroImage 80, 62–79 (2013) 23. Wassermann, D., et al.: Unsupervised white matter fiber clustering and tract probability map generation: applications of a gaussian process framework for white matter fibers. NeuroImage 51(1), 228–241 (2010) 24. Zhang, Y., Brady, M., Smith, S.: Segmentation of brain MR images through a hidden Markov random field model and the expectation-maximization algorithm. IEEE Trans. Med. Imaging 20(1), 45–57 (2001)

Brain Connectivity Measures via Direct Sub-Finslerian Front Propagation on the 5D Sphere Bundle of Positions and Directions Jorg Portegies, Stephan Meesters, Pauly Ossenblok, Andrea Fuster, Luc Florack and Remco Duits Abstract We propose a novel connectivity measure between brain regions using diffusion-weighted MRI. This connectivity measure is based on optimal sub-Finslerian geodesic front propagation on the 5D base manifold of (3D) positions and (2D) directions, the so-called sphere bundle. The advantage over spatial front propagations is that it prevents leakage at omnipresent crossings. Our optimal fronts on the sphere bundle are geodesically equidistant w.r.t. an asymmetric Finsler metric, and can be computed with existing anisotropic fast-marching methods. Comparisons to ground truth connectivities provided by the ISBI-HARDI challenge reveal promising results, both quantitatively and qualitatively. We also apply the connectivity measures to real data from the Human Connectome Project. Keywords Diffusion MRI · Brain connectivity · Fast-marching · Sphere bundle · Geodesic fronts · Finsler geometry

1 Introduction We propose a connectivity measure between regions of the brain, based on diffusionweighted MRI (dMRI) data. It is based on a quasi1 -distance defined through shortest paths in the joint space of positions and directions. Such paths are geodesics with respect to a highly anisotropic Finsler function (also known as ‘Finsler metric ’). Although they can be recovered [9] from the quasi-distance that we compute numer1 ‘Quasi’

pertains to its asymmetrical nature.

Jorg Portegies, Stephan Meesters and Remco Duits: Joint main authors. J. Portegies · S. Meesters · P. Ossenblok · A. Fuster · L. Florack · R. Duits (B) Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands e-mail: [email protected] S. Meesters Academic Center for Epileptology Kempenhaeghe and Maastricht University Medical Center, Maastricht, The Netherlands © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_24

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ically via anisotropic Fast Marching (FM) [26, 27], we do not require explicit tractography or streamline counting. Tractography methods are commonly included in dMRI processing pipelines to model white matter anatomy (fiber tracts) in the brain. Tractograms can be used to define a measure for axonal fiber connectivity in terms of fiber counts between any two regions in the brain. Here we advocate that a connectivity measure should be derived directly from (wavefront propagations on) the 5D sphere bundle represented by dMRI data, rather than by counting streamlines in a tractogram. For geodesic tractography , a global optimization problem is formulated based on a functional inferred from the dMRI data. Several functionals have been proposed, both for DTI and HARDI [11, 12, 19, 24, 25, 30, 37]. By virtue of their global nature, functional methods are more robust to noise than streamline methods. However, two issues must be dealt with, viz. proper handling of crossings and computational efficiency. Current front propagation methods typically take much longer than nongeodesic streamline tractography methods (cf. [6, 40]). To this end we consider geodesic wavefront propagation on the sphere bundle, i.e. the base manifold R3 × S 2 of positions and directions, using the fiber orientation distribution function (FOD) obtained by constrained spherical deconvolution (CSD). CSD typically perfoms well in HARDI-challenges [4] and is included in neuroimaging software packages like DIPY and MRtrix [40]. Our connectivity measure relies on the quasi-distance mathematically scrutinized by Duits et al. [9]. Its four main characteristics are as follows: i. It includes a penalization of both length and curvature. Long and/or highly curved curves yield higher distances. ii. This penalization is based on the FOD. Curves poorly supported by the FOD yield higher distances. iii. It is defined on the 5D sphere bundle. This avoids leaking fronts at crossings, cf. Fig. 1 for an illustration on the (2+1)D sphere bundle. iv. It is induced by an asymmetric Finsler function defined on the sphere bundle. This allows for strictly forward geodesic front propagation, recall Fig. 1. Theoretical Novelty. Our geodesic front propagation method for connectivity fits in the general contextual image processing paradigm used for denoising, fiber enhancement, and sharpening [5, 8, 28, 32, 34, 35, 39, 43]. It is similar to the pioneering work by Péchaud et al. [31]. There are, however, three major differences with our approach: i. In their definition of geodesic length [31], the anisotropy of the metric depends on the FOD, with isotropic behavior for low FOD amplitude. This was necessary to split numerical computations into a slow and fast part based on fast marching, respectively Dijkstra’s algorithm. However, in regions with low FOD values, we want to impose the same ‘stiffness’ of the curve for correct front propagation. Therefore, we propose a single, direct and accurate method that only computes optimal geodesic wavefronts.

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Fig. 1 Conventional geodesic wavefront propagation in the image domain (in red) typically leaks at crossings, whereas wavefront propagation in the domain Rd × S d−1 (in green) does not suffer from this complication. For simplicity we depict the case d = 2 in this figure. For dMRI the same idea applies to the case d = 3. A minimum intensity projection over orientation gives optimal fronts in the image. The cost for moving through the orange parts is lower than elsewhere, and is computed from the FOD. The ‘leakage problem’ is absent both for propagating symmetric sub-Riemannian spheres (left) and for propagation of asymmetric Finsler spheres (right)

ii. It has now become feasible to do the entire computation using anisotropic FM (thanks to work by Jean-Marie Mirebeau [27], as an efficient alternative [36] to existing PDE models [2, 33]). The high efficiency of anisotropic FM comes along with only a limited and reasonable loss in accuracy [9, 33, 36], due to a special design of anisotropic acute stencils [26] that guarantee causality. iii. Our connectivity measure for source and sink regions S1 , S2 ⊂ R3 × S 2 exploits asymmetric (sub)-Finslerian quasi-distances instead of (sub)-Riemannian distances, symmetrized afterwards by switching the role of S1 and S2 and taking the minimum. The driving asymmetric Finsler function, as opposed to a symmetric Riemannian metric tensor, recall Fig.1, only allows for forward propagation. As a result it avoids highly unnatural cusps and it improves stability w.r.t. directions in source and sink regions, cf. [9, Def. 2, Thms. 2 and 3, Fig. 15] for proofs, visualizations of cusps, and further details. The novelty over existing geometric path optimization approaches is as follows: i. Our geodesic wavefront propagation extends both Riemannian [11] as well as . Finslerian wavefront propagation [24, 25, 37] to the larger base manifold M = R3 × S 2 , i.e. the sphere bundle. This avoids overlapping fronts [37] (recall Fig.1) and accounts for curvature. ii. We consider asymmetric Finsler functions and quasi-distances to enforce a spatially forward propagation, cf. Fig.1, right column. Our framework also includes highly anisotropic symmetric Riemannian metrics for bi-directional propagation,

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see the middle column in Fig. 1. It admits infinite spatial anisotropy,constrain3 n i ∂x i . ing spatial motion at (x, n) ∈ R3 × S 2 to the vector field n · ∇R3 = i=1 The latter relies on asymmetric sub-Finslerian or symmetric sub-Riemannian geometry. For convergence results, see [9, Thm. 2]. Experimental Contributions. After introducing our novel connectivity measure (accounting for crossings, stability and optimality) and its efficient implementation in Sect. 2, we demonstrate its use on two datasets: i. The ISBI HARDI Challenge 2013 dataset [4] provides a synthetic image simulating a complex configuration of fiber bundles with ground truth connections. We show that in the majority of cases, our connectivity agrees with true connectivity. ii. In human brain data from the Human Connectome Project (HCP) we estimate structural connectivity between the anterior nucleus of the thalamus and various cortical and sub-cortical regions. Results are generally in line with anatomical expectations.

2 Theory We build on optimal path theory [9], using CSD [41] to construct an FOD image. . We denote the sphere bundle by M = R3 × S 2 , and the FOD scalar field by f : 3 2 + R ×S →R .

2.1 Measuring Curvature-Penalized Path Length To obtain the length of a path γ : [0, 1] → M we construct a Finsler function F0 : ˙ ∈ T M → R+ on the tangent bundle T M over M. For p = (x, n) ∈ M and p˙ = (˙x, n) Tp M it is defined by:  ˙ 2 ) if x˙ = λn with λ ≥ 0 . C 2 (p)(ξ 2 |˙x · n|2 + n ˙ = F0 (p, p) +∞ otherwise. 2

(1)

The cost function C : M → R+ will be explained in more detail later. The parameter ξ > 0 balances the relative costs of spatial translation and in-place rotation. The length of a Lipschitzian curve γ : [0, 1] → M then equals  1 . F0 (γ (t), γ˙ (t)) dt, (2) L(γ ) = 0

and the (quasi-)distance between two points p, q ∈ M is defined as

Brain Connectivity Measures via Direct Sub-Finslerian . . .

. d(p, q) = inf{L(γ ) | γ (0) = p, γ (1) = q, γ ∈ Lip([0, 1], M)}.

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(3)

The distance is a specific solution (see [9]) to the sub-Finslerian eikonal PDE : 

∇ S d−1 U (q)2 + ξ −2 ((n · ∇Rd U (q))+ )2 = C(q) and U (p) = 0,

(4)

. . where U (q) = d(p, q), p fixed, and (a)+ = max{a, 0}. This sub-Finslerian PDEsystem is a limit ( ↓ 0) of the anisotropic Finslerian eikonal PDE system: 

∇ S d−1 U (q)2 + ξ −2 ((n · ∇Rd U (q))+ )2 +  2 n ∧ ∇Rd U (q)2 = C(q) . U (q) = 0 if q ∈ A = {p} .

(5)

We will admit source regions A ⊃ {p} and fix the anisotropy parameter  = 0.1 as justified in previous work [9, 36]. The cost function C above is a transformation of the FOD field f that reflects the following conditions: • Moving through an isotropic region should be expensive. To achieve this, we use on each position an L1 normalization on the sphere, i.e. . f 1 (y, n) =  S2

f (y, n) . f 1 (y, n) and f 2 (y, n) = ∈ [0, 1]. || f 1 ||∞ f (y, n)dσ (n)

• To further increase the cost for motion through isotropic regions, we define an additional position-dependent (orientation-independent) cost Ciso :  M max f 2 (y, n) ≤ m ∈ [0, 1] n∈S 2 Ciso (y) = 1 otherwise, with M = 5, m = 0.4. We optionally include a white matter mask in Ciso (y). • Finally, we use the following cost function: C(y, n) = Ciso (y)

1+σ with p > 1, σ > 0. 1 + σ f 2 (y, n) p

(6)

Increase of p sharpens the cost-function. Increase of σ > 0 favors alignment with the ODF data-term over curve-stiffness.

2.2 Connectivity Measures Connectivity between Points. Distance does not give information about connectivity directly, as it scales with the physical length of a path. We therefore define the connectivity κ between two points p, q ∈ M by a scale invariant ratio:

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. κ(p, q) =

1 0

FC≡1 (γ ∗ (t), γ˙ ∗ (t)) dt , dFC (p, q)

(7)

with γ ∗ the shortest path w.r.t. FC , with C as above. This brings out the effect of the data along a path irrespective of its (Euclidean) length. Note that κ = 1 if C = 1 along the entire geodesic γ ∗ , i.e., if γ ∗ is perfectly supported by the data. Connectivity between Regions. We define the connectivity 0 ≤ K (A, B) ≤ 1 between regions A, B ⊂ M as follows: K (A, B) = min{k(B, A), k(A, B)}, with 1 1 ∗ ∗ 1  0 FC ≡1 (γ ∗∗ (t), γ˙ ∗∗ (t)) dt . 1  0 FC ≡1 (γ (t), γ˙ (t)) dt + , k(A, B) = |B| min d(a, b) |A| min d(b, a) b∈B

a∈A

a∈A

with γ ∗ and γ ∗∗ the minimizers of the denominators.

b∈B

(8)

2.3 Fast Marching to Compute Distances Propagation of geodesically equidistant wavefronts starting from a source point p ∈ M is governed by the eikonal Eq. (4). With a numerical method called anisotropic FM [26, 27] one computes the geodesic distance to all other points q ∈ M in a single pass. Along with the geodesic distance (including the FOD-based cost) the curve length (excluding the FOD-based cost) can be computed without having to compute the actual optimizing curves, cf. [3]. Therefore, for each of the two terms in the connectivity measure (8) we need to run the FM solver only once. The singular nature of the sub-Finsler metric results in numerical problems for FM solvers, which is why we need to approximate it with a highly anisotropic genuine Finsler metric, i.e. we solve (5) instead of (4). In theory we have convergence [9, Thm. 2]. In practice, this approximation is very close [9, 36].

3 Experiments and Results Results for the ISBI HARDI Challenge 2013 Dataset. The ISBI HARDI Challenge 2013 dataset is a digital phantom that was designed for the ISBI 2013 Reconstruction Challenge. It represents a spherical volume that contains 27 bundles, and simulates dMRI with 64 uniformly distributed gradient directions with a b-value of 3000 s/mm2 . Since there are 27 bundles, each with a head (H) and a tail (T), we have 54 points on the boundary of the sphere (where the centerlines of these bundles cross the boundary). We assume these points to be known, and we run the FM algorithm 54 times, each time with a different point as seed (source) point and all other points as end points. This gives us 2809 geodesics that we indicate with γi X → jY , i, j ∈ {1, . . . , 54},

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Fig. 2 Visualization of the ISBI dataset from different viewpoints. The two left-most images show the full dataset, the other four images show a selection of the bundles. The black lines inside these bundles indicate the geodesics obtained with FM

X, Y ∈ {H, T }, where the cases i = j ∧ X = Y are excluded. The results of the FM and geodesic tracking are shown in Fig. 2. We see that the geodesics follow the anatomy of the dataset. We compute the connectivity between each seed point and all end points, see Boxplot of Fig. 3 for the results. For example, choosing point ‘1 H ’ as seed point (leftmost sample in the figure), we obtain 53 connectivity values with all other end points. The median, lower and upper quartile and maximum and minimum are computed from these values and plotted in the figure. The value of geodesic γ1 H →1T (true connection) is indicated with the red dot. Ideally, this has the highest connectivity value. If we rank the geodesics by their connectivity values, then in 21 out of 54 cases the true geodesic receives the highest rank. The average rank by connectivity is 3.5 (1 being optimal). However, the method performs poorly on the two rather challenging cases γ3 H →3T and γ3T →3 H (if we exclude these, the average rank becomes 2.7). Results for the HCP Dataset. The computation of the newly developed structural connectivity measure was evaluated for the connections of the anterior nucleus of the thalamus (ANT), and various cortical and subcortical atlas regions of healthy volunteers. The rationale for choosing the ANT as seed region was that this cortical structure (within the Papez circuit) may play a central role for epilepsy patients in

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Fig. 3 Boxplot showing the sample median (white bar), 25% and 75% (gray box), max. and min. of connectivity values between the seed point (x-axis) and end points. Red dots indicate connectivity between seed and true end point. Since in most cases the red dot lies well above the gray box our connectivity measure is fairly distinctive

the propagation of seizure activity, cf.[10, 14, 22]. Clinical trials have demonstrated that the DBS treatment is effective in reducing the median seizure frequency by 56% [10]. In a first preliminary step towards being able to assess why half of the patients are non-responders to the treatment, the proposed brain connectivity measures are applied to estimate the structural connectivity between the ANT and various cortical and subcortical atlas regions. The HCP provides neuroimaging data from healthy subjects (N = 10) containing T1-weighted and T2-weighted anatomical brain scans as a well as a diffusionweighted imaging (DWI) scan acquired by a multi-shell HARDI protocol (3 shells, 128 directions). Details on the scanning protocol can be found in work by Van Essen et al. [42]. The standard data preprocessing pipeline of the HCP project was utilized in which the usual corrections are applied [13]. The Morel atlas provides a collection of segmented thalamic nuclei in MNI space [17, 23, 29]. To segment the ANT for each individual subject, the anterior, medial and ventral subnuclei of the ANT, provided in the Morel atlas, were merged and a non-linear transformation to subject space was done by use of FSL [20]. Furthermore, the Harvard-Oxford atlas [7] was employed for segmenting various cortical and subcortical targets and was nonlinearly transformed to subject space by the same procedure as the Morel atlas. The multi-shell CSD model [21, 41] available in the MRtrix software package [40] was used to compute a fiber orientation density (FOD) function of each DWI scan. The FOD function is subsequently aligned to the anatomical T1-weighted MRI via linear

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coregistration in FSL. We used the FOD function p → f (p) to define the cost function p → C(p) according to (6), with ( p, σ ) = (3, 20), and the available gray-matter map to define the factor Ciso (y). The bending-stiffness parameter was set to ξ = 0.1. The FM algorithm is initialized with lifted seeds, i.e. points in M = R3 × S 2 , using the sh2peaks function in MRtrix [40] to obtain the directional arguments. As the full symmetric connectivity measure is computationally demanding, requiring a FM pass for each included atlas region, structural connectivity is computed only using the asymmetric connectivity metric (see r.h.s. of (8)): . 1  k(A, B) = |B| b∈B

1 0

FC≡1 (γ ∗ (t), γ˙ ∗ (t)) dt , min d(a, b)

(9)

a∈A

Here A denotes the ANT seed region and B the target cortical or subcortical brain region. The results of the connectivity analysis applied to healthy volunteers is presented in Fig. 5. The geodesic distance map is plotted (using our Visualization Tool available at https://goo.gl/D5Q7dE) relative to the ANT seed regions and target atlas regions. The average connectivity value is listed for several cortical and subcortical regions, relative to the ANT as seed region, computed for the left hemisphere (Fig.5, bottom). From anatomical and connectivity studies it is known that the ANT has extensive connections to the cingulate cortex and hippocampus [15, 16, 18, 38], to visual and frontal cortices [18] as well as a relation to the medial temporal lobe [1]. Indeed, it can be observed that the cingulate gyrus and the calcarine sulcus have a high connectivity (Fig. 5, top left). Furthermore, the frontal medial, temporal and insular cortices appear to have moderate connectivity values, where the inferior frontal gyrus and accumbens show lowest values.

4 Conclusion We proposed a new connectivity measure based on asymmetric sub-Finslerian (or symmetric Riemannian) wavefront propagation on the sphere bundle. It is capable of handling crossings in dMRI/HARDI data, uses only front propagation of optimal geodesics (without spatial cusps [9]), and can be computed with an accurate, direct anisotropic FM method (without the operator splitting in [31]). We tested our connectivity measures on the synthetic data of the ISBI HARDI Challenge 2013. Comparisons to ground truth connectivities reveal promising results, both quantitatively (Fig. 3 in this article), and qualitatively (Fig. 4). We also tested our connectivity measures on HCP human brain data, where the structural connectivity results from the ANT region to surrounding ROI’s are in line with anatomical expectations (Fig. 5). These findings are promising in support of deep-brain stimulation for epilepsy treatment. However, in contrast to the phantom study, ground

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Fig. 4 We visualize the geodesics γi H → jT and γi T → j H , with color indicating connectivity-based rank, green (red) corresponds to high (low) rank. Akin to Fig. 3, many of the ‘true’ geodesics receive a higher connectivity value than ‘false’ geodesics

Fig. 5 Average connectivity value listed for a selection of cortical and sub-cortical regions from the Harvard-Oxford atlas, relative to the ANT as seed region, computed for the left hemisphere of 10 healthy volunteers. The anatomical locations of several regions is shown for a projection of the minimal geodesic distance map (top-left) and the T1-weighted anatomical image (top-right)

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truth connectivity values are absent. Therefore, although the results on the HCP-data seem to indicate feasibility, this application should be further investigated. Acknowledgements Data provided in part by the HCP, WU-Minn Consortium (PI’s: D. Van Essen and K. Ugurbil; 1U54MH091657). We thank S. Mariën for co-developing the rching Tool, available at https://goo.gl/D5Q7dE (Downloads section). We thank J.M. Mirebeau for his Hamiltonian fastmarching C++-code, available at https://www.math.u-psud.fr/~mirebeau. The research leading to these results has received funding from the European Research Council under the European Community’s 7-th Framework Programme (FP7/2007-2013) / ERC grant Lie Analysis, agr. nr. 335555.

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Inference of an Extended Short Fiber Bundle Atlas Using Sulcus-Based Constraints for a Diffeomorphic Inter-subject Alignment Nicole Labra Avila, Jessica Lebenberg, Denis Rivière, Guillaume Auzias, Clara Fischer, Fabrice Poupon, Pamela Guevara, Cyril Poupon and Jean-François Mangin Abstract We present a new framework for the creation of an extended atlas of short fiber bundles between 20 and 80 mm length. This method uses a Diffeomorphic intersubject alignment procedure including information of cortical foldings and forces the accurate match of the sulci that have to be circumvented by the U-bundles. Then, a clustering is performed to extract the most reproducible bundles across subjects. First results show an increased number of U-bundles consistently mapped in the general population compared with previous atlases created from the same database. Future analysis over this new extended Brain atlas may improve our understanding of the relationship between the folding pattern and the U-bundle variability. The ultimate aim will be the possibility to detect abnormal configurations induced by developmental issues. Keywords White matter · U-fiber · U-bundles · Short fiber bundles · dMRI · Diffusion MRI · Brain atlas · Bundle atlas · Diffeomorphic alignment · Sulcus base alignment · Cortical folding pattern · Sulci

This project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under Grant Agreement No. 785907 (HBP SGA2). N. L. Avila (B) · D. Rivière · C. Fischer · F. Poupon · J.-F. Mangin UNATI, CEA/DRF/Neurospin, Gif-sur-Yvette, France e-mail: [email protected]; [email protected] N. L. Avila Université Paris Sud, Paris Saclay, France J. Lebenberg Université Paris Diderot, Sorbonne Paris Cité, UMR-S 1161 INSERM, Paris, France G. Auzias Institut de Neurosciences de la Timone, Aix-Marseille University, Marseille, France P. Guevara Universidad de Concepción, Concepción, Chile C. Poupon UNIRS, CEA/DRF/Neurospin, Gif-sur-Yvette, France © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_25

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1 Introduction At a macroscopic level, information transfer in the human brain arises from both short-range connections between adjacent areas and long-range connections between distant areas. While the backbone of long-range pathways is known for a long time, knowledge on short-range connections in the human brains is sparse in the anatomical literature. The core structure of the short-range connections is a set of U-shaped fiber bundles that circumvent the Cortical folds while remaining in the superficial part of white matter under the cortical mantle. Diffusion MRI is the first technique allowing the large-scale study of their organization. Hence, several research projects have recently proposed maps of the largest U-bundles inferred from tractography. The most extended atlases dedicated to U-fiber bundles inferred today include about one hundred bundles reproducible across groups of subject [13, 18], but much larger atlases are still to come [19]. This timely research program, however, is still in its infancy because of the challenges raised by the variability of the Cortical folding pattern. While the core part of the long pathways travelling through deep white matter is barely influenced by Cortical folding, the U-fiber organization is dependent on the Cortical folding pattern. Furthermore, the impact of the large variability of these Folding pattern is an open issue. In previous works, the Bundle atlases have been inferred using non supervised clustering after alignment of a population of tractograms. This spatial normalization was either based on affine transformations [13], nonlinear registration of the tensor fields [18] or groupwise alignment of the tractograms [19]. None of these approaches is dealing with the accurate matching of the Folding patterns across subjects, which is a challenge for most of the spatial normalization approaches. In this paper, we propose to perform the alignment with a Diffeomorphic strategy imposing explicit matching of the most stable cortical Sulci. The objective of our framework is two fold: (1) increasing the number of U-bundles consistently mapped in the general population; (2) improving our understanding of the relationship between the Folding pattern and the U-bundle variability.

2 Materials and Methods Datasets We used the same HARDI database of 77 healthy subjects as in previous works [2, 13, 18]. Acquisitions were obtained using a Siemens Magnetom TrioTim 3T MRI, 12-channel head coil. The protocol included a high resolution T1-weighted acquisition (echo time = 2.98 ms, repetition time = 2,300 ms, 160 sagittal slices, 1.0 × 1.0 × 1.1 mm) and a DW-EPI sequence along 41 directions (2.0 × 2.0 × 2.0 mm, b = 1,000 s/mm2 , plus one b = 0 image, echo time = 87 ms, repetition time = 14,000 ms, 60 axial slices).

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Preprocessing Tractography First, we processed the data using the Connectomist 2.0 software [7]. The data was preliminary corrected for all sources of artifacts, including spatial distortions relative to morphological scans [6]. Then, the analytical Q-ball model of Descoteaux et al. [8] was computed to obtain ODF fields. A Streamline regularized deterministic tractography was performed across the entire T1-based brain mask after removal of the folding negative skeleton, to prevent spurious trajectories through CSF. As 8 seeds per voxel were used to reconstruct the white matter trajectories, the massive tractograms contain around eight million streamlines per subject. For this reason, each individual tractogram was then compressed to a few thousand fiber clusters [11], each represented by a centerline of 51 equidistant tridimensional points. Sulci recognition For each subject, 125 cortical Sulci were automatically identified using the Morphologist toolbox [10, 15] of BrainVISA [4, 5]. The same procedure was applied to the non linear ICBM 152 atlas. In each brain, the labelling of the Sulci was manually corrected by two raters, to initialize a process aiming at improving the understanding of the folding variability using reproducible U-bundles. Diffeomorphic Registration The subjects and the ICBM atlas were aligned using a two steps method proposed by Auzias et al. [3] and Lebenberg et al. [14] which combines an approach called DISCO (DIffeomorphic Sulcal-based COrtical deformation) [3] with the intensitybased method known as DARTEL distributed with the SPM software [1]. First, DISCO algorithm generates a population level template space from a set of 43 reliable Sulci defined in each brain. Then, this DISCO template space is used to initialize DARTEL. This robust registration framework improves the alignment of the main Folding patterns and also the grey matter maps overlaps, improving the Inter-subject alignment of the Short fiber bundles circumventing Sulci. The target space was the MNI space represented by the non linear ICBM 152 template [16]. Processing An histogram of the centerlines lengths in native space from all the subjects was computed first as shown in Fig. 1. A peak is reached in both hemispheres around 20 mm and then a skewed pattern appears with peaks and local minima. This may suggest a structure in bundle geometry. In order to explore this hypothesis and, to facilitate the clustering processing computational and timing resources, we have divided centerlines into subsets grouped by ranges of length. The subsets created were: (1) Centerlines between 20 and 35 mm length (2) Centerlines between 35 and 50 mm (3) Centerlines between 50 and 65 mm (4) Centerlines between 65 and 80 mm. We discarded centerlines over 80 mm considered as “long”. Registration of centerlines Subsets of centerlines obtained after compression and histogram analysis were brought to the sulcus-based template in MNI space obtained previously with DISCO + DARTEL. Figure 2 shows a comparison between DISCO + DARTEL registration and the Talairach affine registration [13] and the DTI-TK non linear registration [18]. The new method significantly improves the alignment of centerlines around Cortical foldings, allowing the visualization of the Sulci path

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(a) Left-hemisphere.

(b) Right-hemisphere.

Fig. 1 Histogram of centerline lengths for left-hemisphere and right-hemisphere

Fig. 2 Comparison between DISCO + DARTEL registration with the Talairach affine registration and the DTI-TK linear registration method. The new registration method improves significantly the alignment of centerlines around cortical Sulci

among centerlines which was not possible neither with the Talairach nor DTI-TK methods. Clustering Each subset of centerlines was split into two test groups containing the centroids of 39 and 38 subjects respectively, as shown in Fig. 3. A clustering algorithm inspired by DBSCAN [9] was applied over each group to detect reproducible fiber bundles. DBSCAN algorithm is a simple algorithm which enables the efficient extraction of clusters with arbitrary shapes, an important characteristic to deal with the various kind of U-fiber bundles. The algorithm implemented is described in Algorithm 1 and has some changes relative to the original, mainly to optimize and speed up the process. First, the measure of distance used to compare two centerlines is described in Fig. 4. Because the centerlines could be traversed with different directions depending on how they are stored in the database, first we computed the traditional Euclidean distance d E between each pair of corresponding points in centerlines i and j in the direct and inverse orientation, as shown in Fig. 4a and defined by Eq. 1.

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Fig. 3 Process to obtain the new atlas of Short fibers. First, an histogram of all centerlines lengths for all the subjects was performed. Then subsets of centerlines from all the subjects aggregating all subjects were defined with length windows. Centerlines of each subset were brought to the sulcusbased template in MNI space. A clustering based on DBSCAN was executed over each subset. Finally, clusters obtained from all the subsets were joined and manually filtered

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Fig. 4 Similarity measure of the algorithm. a Shows both possibilities for pairwise distance computed between centerlines. b Shows the radio of neighborhood of each centroid used in the algorithm. Maximum distances between centerline A and centerline B and C are in the radio of A. They are considered as similar to A, unlike centerline D

d Orientation = min

 n 

(d E (i k , jk )) ,

k=1

n 

 (d E (i k , jn−k ))

with n = 51

(1)

k=1

We calculated the sum of all the distances in each direction choosing the orientation associated to the minimum value to proceed further with the comparison between centerlines. Then, we obtained the similarity measure, defined by 2, between each i centerline respect to all the other centerlines in the dataset. d M E (S, C) = max (d E (i k , jk )) with n = 51 k

(2)

In order to avoid the creation of a giant distance matrix and all the computational resources involved, we stored only the similarity distances of the closest neighbors in increasing order of neighborhood. We have empirically chosen the number of neighbors as 100, creating a matrix of similarity distances of size N x100, where N = Number of centerlines in the dataset. Because of the heterogeneity of the intersubject bundle geometry, clusters do not all have the same density and properties and these differences increase when the size of the centerlines decreases. This turns

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Algorithm 1 DBSCAN-based algorithm implemented 1: Compute distance matrix and the corresponding index matrix for every i 2: centerline in the subset and its N closest neighbors n(i, j) with N = 100, and 3: j = 1, 2 ... N in increasing order. 4: while there are differences in labellization with respect to previous iteration do 5: for each i in the group do 6: if i does not belong to any cluster yet then 7: - Create new cluster ci 8: - Assign a radius rci of neighborhood: For the first iteration use an 9: arbitrary value (we used 1.3 times the distance to the nearest centerline). 10: - After, use the rci associated to i in the previous iteration. 11: - Look for all the neighbors n(i, j) of i whose distance dn(i, j) < rci 12: in the group. Label them as belonging to ci and actualize rci with 13: equation 3 14: - Repeat lines 11 to 13 with the neighborhood of every new neighbor that 15: belongs to ci recursively until there is no more neighbors to evaluate. 16: - Repeat all the process from line 7 to 15 until every i has a label. 17: end if 18: end for 19: Compare labelisation of previous iteration 20: end while 21: Erase clusters made up for centerlines of less than 50% of the subjects in the group.

Algorithm 2 Algorithm implemented to compare similarity between clusters of subatlases 1: for every cluster Ci in test group 1 do 2: for every cluster C j in test group 2 do 3: for every centerline i in cluster C I do 4: for every centerline j in cluster C J do 5: Compute the similarity measure di, j between i and j following 6: equations 1 and 2 and compare with radius rC I and radius rC J 7: If di, j < rC I and di, j < rC J , then i and j are similar. If i is similar 8: to 5% of centerlines in C j and the 50% of the centerlines in C I reach 9: this condition, then C I and C J are the same. 10: end for 11: end for 12: end for 13: end for

the extraction of Short fibers into a very challenging process, where tuning specific parameters to extract specific clusters could be necessary. For this reason, we also incorporated a specific radius of neighborhood for each cluster rC that evolves through iterations until convergence. This radius is not applied to all the clusters but over every centerline, as shown in Fig. 4b. This allows us to extract bundles along sulci without divisions, preserving the shape around Cortical foldings. The actualization of the radius depends on a decreasing learning rate and is defined by Eqs. 3 and 4. (3) rci = rci + a ∗ (di, j − rci )

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with: a = 1.0/(1.0 + K ∗ j) with K = constant and j = 1, ...n

(4)

Only clusters with centerlines that belong to more than 50% of the subjects of each test group were kept. The results of both test groups of subjects were compared with Algorithm 2 in order to construct a sub-atlas with the clusters present in both groups. Finally, each sub-atlas obtained in each subset of length were joined in order to get the complete atlas with bundles between 80 and 20 mm length.

3 Results As we divided the bundle of centerlines in sub-groups by lengths, we have assigned different colours for a better visualization as is shown in Fig. 5. In the range 65– 80 mm, we obtained 27 clusters for left hemisphere and 20 for right hemisphere. Clusters are visualized in different shades of red. For the range of 50–65 mm we obtained 32 and 33 clusters for the left and for the right hemispheres respectively. The corresponding colors are in shades of blue. The clusters of these two subsets are not superficially located because most of them are passing under several Sulci. On the other hand, bundles in the next two ranges are more superficial to the cortex. We have found 45 clusters in the left side and 51 in the right side for the range 35–50 mm length and 39 clusters for the left and 52 for the right hemisphere in range of 20–35 mm. In total, we obtained 153 clusters in the left hemisphere and 156 in the right side as shown in Fig. 6. Unlike previous atlases [13, 18], the centerline clusters show strong relationship with the cortical Sulci. Most of the clusters tend to circumvent one sulcus in order to create a pathway between areas separated by a fold. The accurate alignment of the Folding pattern across subjects allows disentangling bundles sharing one extremity. While the exact connecting point of such bundles in the cortical mantle is currently beyond reach, our atlas provides a skeleton of their core organization.

4 Discussion and Conclusions This paper presents a first attempt at performing a study of the Short fiber bundle organization while taking into account the variability of the Cortical folding pattern. First, we have shown the effectiveness of the alignment of tractograms using the DISCO + DARTEL approach. Thanks DARTEL to the incorporation of sulcus-based constraints, this new alignment facilitates the identification of bundles circumventing Sulci and enables to extend the knowledge of short range connectivity geometry. After clustering, 309 clusters have been found. In constrast with the previous atlases created from the same database [13, 18], it is an increase of around 100 clusters for

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Fig. 5 Clusters in the range of 20–80 mm length

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Fig. 6 All the clusters in the range of 20–80 mm length

each hemispheres. An extended comparison between atlases will be performed to quantify the impact of the new alignment approach in the atlas creation. Furthermore, this new method allows us to obtain the new Short fibers atlas in a reasonable time and it works with computational resources of standard machines. Future work will aim at exploring the impact of the Folding pattern polymorphisms on the organization of the U-bundles. The ultimate aim will be the modeling of the normal variability of these two facets of the cortex morphology, to get the possibility to detect abnormal configurations induced by developmental issues.

References 1. Ashburner, J., et al.: A fast diffeomorphic image registration algorithm. Neuroimage 38(95), 113 (2007) 2. Assaf, Y., et al.: The CONNECT project: combining macro- and micro-structure. Neuroimage 80, 273–282 (2013) 3. Auzias, G., et al.: Diffeomorphic brain registration under exhaustive sulcal constraints. IEEE Trans. Med. Imaging 30(6), 1214–1227 (2011) 4. Brainvisa Homepage. http://brainvisa.info/web/index.html. Last accessed 6 July 2018 5. Cointepas, Y., et al.: BrainVISA: software platform for visualization and analysis of multimodality brain data. In: OHBM. Presented at the OHBM, Brighton (2001) 6. Dubois, J., et al.: Correction strategy for diffusion-weighted images corrupted with motion: application to the DTI evaluation of infants white matter. Magn. Reson. Imaging 32(8), 981–992 (2014) 7. Duclap, D., et al.: Connectomist-2.0: a novel diffusion analysis toolbox for BrainVISA. In: 29th ESMRMB, Lisbonne, Portugal (2012) 8. Descoteaux, M., et al.: Regularized, fast and robust analytical Q-ball imaging. Magn. Reson. Med. 58, 497–510 (2007)

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9. Ester, M., et al.: A density-based algorithm for discovering clusters a density-based algorithm for discovering clusters in large spatial databases with noise. In: Proceedings of the 2nd International Conference on Knowledge Discovery and Data Mining KDD 1996, Portland, Oregon, pp. 226–231 (1996) 10. Fischer, C., et al.: Morphologist 2012: the new morphological pipeline of BrainVISA. In: OHBM. Presented at the OHBM, Beijing, China (2012) 11. Guevara, P., et al.: Robust clustering of massive tractography datasets. Neuroimage 54(3), 1975–1993 (2011) 12. Guevara, P., et al.: Automatic fiber bundle segmentation in massive tractography datasets using a multi-subject bundle atlas. Neuroimage 61(4), 1083–1099 (2012) 13. Guevara, M., et al.: Reproducibility of superficial white matter tracts using diffusionweighted imaging tractography. Neuroimage 147, 703–725 (2017) 14. Lebenberg, J., et al.: A framework based on sulcal constraints to align preterm, infant and adult human brain images acquired in vivo and post mortem (2018) 15. Mangin, J.-F., et al.: A framework to study the cortical folding patterns. Neuroimage 23(1), 129–138 (2004) 16. Mazziotta, J., et al.: A probabilistic atlas and reference system for the human brain: international consortium for brain mapping (ICBM). Philos. Trans. R. Soc. Lond. Ser. B 356, 1293–1322 (2001) 17. O’Donnell, L., et al.: Automatic tractography segmentation using a high-dimensional white matter atlas. IEEE Trans. Med. Imaging 26(11), 1562–1575 (2007) 18. Roman, C., et al.: Clustering of whole brain white matter short association bundles using HARDI data. Front. Neuroinformatics 11, 73 (2017) 19. Zhang, F., et al.: Whole brain white matter connectivity analysis using machine learning: an application to autism. Neuroimage 172, 826–837 (2018)

Resolving the Crossing/Kissing Fiber Ambiguity Using Functionally Informed COMMIT Matteo Frigo, Isa Costantini, Rachid Deriche and Samuel Deslauriers-Gauthier

Abstract The architecture of the white matter is endowed with kissing and crossing bundles configurations. When these white matter tracts are reconstructed using diffusion MRI tractography, this systematically induces the reconstruction of many fiber tracts that are not coherent with the structure of the brain. The question on how to discriminate between true positive connections and false positive connections is the one addressed in this work. State-of-the-art techniques provide a partial solution to this problem by considering anatomical priors in the false positives detection process. We propose a novel model that tackles the same issue but takes into account both structural and functional information by combining them in a convex optimization problem. We validate it on two toy phantoms that reproduce the kissing and the crossing bundles configurations, showing that through this approach we are able to correctly distinguish true positives and false positives. Keywords Tractography · False positives · Diffusion MRI · Resting state functional MRI

1 Introduction The increasing interest in connectomics shed light on the intrinsic limitations of tractography based connectomes [6, 13, 19]. Tractography itself is an ill-posed problem that aims at reconstructing the white matter fiber tracts from diffusion MRI (dMRI) data and is shown to generate a significant number of false positive connections between brain regions [13]. This is due to the fact that whenever two bundles of axons are organized in a crossing or kissing configuration, in-vivo tractography is not able to distinguish their actual trajectories from diffusion data. This implies the presence of many spurious entries in the structural connectome obtained from a tractogram. Moreover, specificity of brain networks has been demonstrated to be twice M. Frigo (B) · I. Costantini · R. Deriche · S. Deslauriers-Gauthier ATHENA Project Team, Inria Sophia-Antipolis Mediterranée, Université Côte D’Azur, Nice, France e-mail: [email protected] © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_26

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as important as sensitivity when estimating properties of the brain networks [19]. The re-establishment of an equilibrium between highly sensitive and highly specific networks is a challenge in brain connectomics and is the goal of this work. The task of extracting only true positive connections from a tractogram is also called tractography optimization or tractogram filtering [8, 9]. Several tools to perform this task have been proposed in the last years, including SIFT2 [17], LiFE [15] and COMMIT [7]. While the first has the goal of matching a tractogram with the fiber densities estimated using the spherical deconvolution diffusion model, the two last are designed in such a way that they identify a subset of streamlines that explains the acquired diffusion MRI through a linear forward model. This model takes into account only the intra-cellular compartment in the case of LiFE and more complex tissue microstructure models in the case of COMMIT. Recent studies showed that anatomical priors can be taken into account while detecting false positives [8]. Daducci et al. [5] analyzed how the intrinsic bundle structure of both the brain and tractograms can be used to tackle the false positives issue, while Girard et al. [10] exploited microstructural information to solve the kissing bundles problem. Still, both these approaches showed some limitations. What we propose in this work is to overtake the purely structural-based approach and consider both structural and functional information in the formulation of the tractography optimization problem. We do this by showing that the injection of functional priors coming from restingstate functional MRI (rs-fMRI) gives an effective answer to the false positives issue. The functional information is exploited in the form of static functional connectivity and it plays the role of promoting the involvement of bundles that connect highly correlated cortical regions in the fitting of the dMRI signal. We encapsulate all of this within the COMMIT framework, which provides a flexible tool for defining a convex optimization problem that, for a given tractogram, simultaneously promotes sparsity among the bundles, takes into account the considered functional information and selects the streamlines that are sufficient to explain the diffusion MRI signal. We call this novel framework Functionally Informed COMMIT (FIC). This formulation is tested on TVB [16] and Phantomas [3] based synthetic phantom datasets that reproduce the two problematic configurations of crossing and kissing bundles, which are the source of most of the biases in connectivity estimation.

2 Theory As the name suggests, tractography optimization implies the solution of a minimization problem. The functional to be minimized takes into account both the structural information coming from dMRI and the functional information given by rs-fMRI. The space in which we want to optimize it is the space of streamlines, hence we want to assign to each of them a coefficient xi ≥ 0. This coefficient quantifies the contribution of streamline i to the definition of the dMRI signal, which means that if xi = 0 then the i-th streamline does not explain the diffusion signal and will be marked as a false positive. The fitting of the dMRI signal is tackled by means of

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the COMMIT framework. In particular, we use the provided linear forward model for diffusion MRI that predicts the signal generated by a tractogram through a given microstructure model. The one we are going to consider during this work is the StickZeppelin-Ball model [14]. This leads to the definition of the optimization problem as 1 (1) x ∗ = argmin Ax − y22 + Ω(x) n x∈R+ 2 where n is the number of streamlines, y ∈ Rm is the vectorized dMRI signal, A ∈ Rm×n is the linear operator that projects streamline weights on the diffusion data according to the microstructure model and Ω : Rn → R is a regularization term that for Ω(x) = 0 returns the classical formulation of COMMIT, which will be used as a first reference for our results. Regularization term. Our objective is to promote sparsity in a structured way, namely we want to promote the preservation of the streamline bundles, instead of individual streamlines, that are sufficiently significant to explain the diffusion MRI signal. This leads to the definition of the following regularization term Ω(x) = λ



wg xg 2

(2)

g∈G

where G is a collection of groups of indexes (i.e. streamline bundles) forming a partition of [1, n], wg is a positive weight associated to each group g, xg is the restriction of x ∈ Rn to the entries indexed by g and λ ≥ 0 is the regularization parameter. This Ω is a classical choice [1] in structured sparsity regularization and is also known as 1,2 regularization. The groups can be both decoupled and overlapping, leading to different strategies in the solution of the optimization problem. See [11] for an exhaustive discussion on the mathematical details of this formulation and [8] for an application of this concept in tractography optimization. A common choice for the specific weight associated to each group is Ng−1/2 , where Ng is the cardinality of group g. This normalization avoids preserving a group just because it involves a smaller number of variables. In the context of our problem this means that we try to avoid penalizing bigger bundles more than smaller bundles. This choice of wg is called group sparsity and will be used as a second reference to compare our results. Notice that group sparsity is purely based on a structural prior, hence it does not take into account the fact that a bundle could connect two regions that are more or less functionally correlated. We can quantify the functional correlation between cortical regions i and j by means of the functional correlation coefficient ci, j . If bundle g connects regions i and j, we associate to it the coefficient cg = ci, j . How to compute this functional correlation coefficient is the question addressed in Sect. 3.1. Our objective is to favor more bundles that are associated to high functional correlation than the ones associated to low functional correlation. We designed a strategy for determining the weight wg that reads as follows:

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s = 0.1 s = 0.5 s = 1.0 s = 1.5

80 60

wg

Fig. 1 Each line represents the weight wg as a function of cg for Ng = 1 and four different values of the shape parameter s. This plot suggests that our weighting strategy gives high control over the low functional correlation case

40 20 0 0.2

1 wg =  Ng



0.4

cg

0.6

0.8

   1 − cg2 1 exp −1 . cg2 2s 2

1

(3)

The function of cg defined in Eq. (3) enjoys the property of being equal to zero when cg = 1 and going asymptotically to infinity when cg → 0+ . The undefined parameter s controls the speed with which wg goes to infinity. In particular it affects more the behavior in the case of low functional correlation than in the case of high functional correlation by acting as a smooth threshold parameter, as shown in Fig. 1. Combining the COMMIT optimization problem defined in Eq. (1) together with the designed joint structural-functional prior offered by Eqs. (3) and (2), we are now able to give the mathematical formulation of FIC. Definition 1 (Functionally Informed COMMIT ) Let S be a set of n fiber tracts and let G be the bundle structure over these fiber tracts. Moreover, let A and y be as in Eq. 1, λ > 0, Ng equal to the cardinality of bundle g, s > 0 and let cg > 0 be the functional correlation coefficient between the regions connected by bundle g. The solution of the following minimization problem is called FIC filtering of S.  1 1  x = argmin Ax − y22 + λ Ng x∈Rn+ 2 g∈G ∗



   1 − cg2 1 exp − 1 xg 2 . cg2 2s 2

(4)

Theorem 1 The FIC filtering of a tractogram exists, is unique and can be obtained in polynomial time. Proof The target functional of Eq. (4) is strictly convex and lower semi-continuous, hence its solution exists and is unique. Moreover it can be split in a smooth part with L-Lipschitz continuous gradient and a non-smooth part for which we are able

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to compute the proximal operator [11], hence the solution can be approximated in polynomial time [2].

3 Methods The two studied datasets were both generated with Phantomas [3] for the structural part (dMRI) and The Virtual Brain (TVB) [16] for the functional part (rs-fMRI). The diffusion MRI signal was simulated for a geometry of 94 × 94 × 94 voxels of (2.0 mm)3 size in 64 sampling points uniformly distributed [4] on a single shell at b = 1500 [s/mm2 ]. The phantom consists of two streamline bundles either crossing or kissing in the center of the image. This is an extremely challenging configuration for tractography because voxels in the crossing/kissing region are identical for both configuration. See Figs. 2 and 3 for a visual description of the simulated phantoms. On each of these datasets we ran a probabilistic anatomically constrained tractography algorithm seeding from the white matter. For each of the cortical regions we used TVB [16] to simulate the BOLD time series with repetition time (TR) of 1 s, obtaining a 15 min time series for each region. The chosen noise model is Gaussian and additive with standard deviation equal to 5 · 10−4 , which corresponds to the one suggested in [18].

3.1 Dynamic Functional Connectivity The wanted functional correlation coefficient corresponds to the dynamic functional connectivity (dFC) between two regions and is estimated using sliding-window correlation. The window is defined by its length, its shape and the offset time. As far as the window length is concerned, we set it to 100 s following the strategy described in [12]. For the window shape we consider a rectangular box which gives unitary weight to each involved time point. The window offset is set to 1 TR and the functional correlation between each pair of windows is estimated using the Pearson correlation coefficient. Finally, the maximal correlation among all the windows is elected to represent the wanted dFC coefficient.

3.2 Parameters Choice The experiments were run for the following values of parameters λ and s: λ = [1.0, 1.5, 2.0, 2.5] , s = [0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5] .

(5) (6)

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Fig. 2 Kissing. The shown results were computed for λ = 2.5 and s = 1. Subplot a shows the ground truth geometry of the phantom, where each bundle has a distinctive color. Subplots b, c, and d show every bundle belonging to the filtered tractogram. Those marked as FP should not be present after the tractography optimization while those marked as TP should be. Bundles connecting regions 0 ↔ 1 and 2 ↔ 3 were not generated by the tractography algorithm

4 Results For each phantom we show the ground truth geometry and the solutions obtained with classical COMMIT, group sparsity (GS) and FIC in Figs. 2 and 3. The ideal result preserves only the bundles that belong to the ground truth. It can be noticed how COMMIT and GS are not able to distinguish between true positive (TP) and false positive (FP) bundles, while FIC correctly detects the FPs and returns a properly filtered tractogram. This holds true for both the kissing bundles configuration and the crossing bundles configurations. For the result given by each combination λ and s we computed two types of connectivity matrix. The first one is a classical streamline count and is indicated as T . The second one is what we call the FIC connectome, which is defined as Hi, j = xg∗ 2 , where bundle g connects regions i and j and x ∗

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Fig. 3 Crossing. The shown results were computed for λ = 2.5 and s = 1. Subplot a shows the ground truth geometry of the phantom, where each bundle has a distinctive color. Subplots b, c, and d show every bundle belonging to the filtered tractogram. Those marked as FP should not be present after the tractography optimization while those marked as TP should be. Bundles connecting regions 0 ↔ 1 and 2 ↔ 3 were not generated by the tractography algorithm

is the solution obtained with FIC. Matrix T is normalized by the total number of streamlines while H is normalized by its maximal entry. Both of them share the same sparsity pattern, which reads as follows: ⎡

0 ⎢0 Gc = ⎢ ⎣∗ 0

0 0 0 ∗

∗ 0 0 0

⎤ 0 ∗⎥ ⎥ 0⎦ 0

⎡

0 ⎢0 Gk = ⎢ ⎣0 ∗

0 0 ∗ 0

⎤ 0∗ ∗ 0⎥ ⎥ 0 0⎦ 00

(7)

where G c is the sparsity pattern of the crossing configuration, while G k is the sparsity pattern of the kissing configuration. Due to different kind of normalization we expect the positive entries of G(T ) to be equal to 0.25 and the positive entries of G(H ) to be equal to 1.0. The mean and the standard deviation of the resulting connectivity

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matrices are reported in Eq. 8 for the FIC connectome and Eq. 9 for the streamline count of the crossing phantom. The analogue for the kissing phantom is shown in Eqs. 10 and 11 respectively. ⎡

⎤ 0. 0. 1. ± 0. 0. ⎢ 0. 0. 0. 0.776 ± 0.02⎥ ⎥ μ(Hc ) = ⎢ ⎣1. ± 0. ⎦ 0. 0. 0. 0. 0.776 ± 0.02 0. 0.

(8)

⎡

⎤ 0. 0. 0.241 ± 0.001 0. ⎢ 0. 0. 0. 0.259 ± 0.001⎥ ⎥ μ(Tc ) = ⎢ ⎣0.241 ± 0.001 ⎦ 0. 0. 0. 0. 0.259 ± 0.001 0. 0.

(9)

⎡

⎤ 0. 0. 0. 0.99 ± 0.007 ⎢ ⎥ 0. 0. 0.99 ± 0.007 0. ⎥ μ(Hk ) = ⎢ ⎣ ⎦ 0. 0.99 ± 0.007 0. 0. 0.99 ± 0.007 0. 0. 0.

(10)

⎡

⎤ 0. 0. 0. 0.255 ± 0.001 ⎢ ⎥ 0. 0. 0.245 ± 0.001 0. ⎥ μ(Tk ) = ⎢ ⎣ ⎦ 0. 0.245 ± 0.001 0. 0. 0.255 ± 0.001 0. 0. 0.

(11)

5 Conclusions We proposed an evolution of the the recently proposed [5] bundle sparsity formulation of COMMIT that takes into account the functional information coming from rs-fMRI. The results obtained on two simple but very problematic phantoms prove that the FIC approach is effective for the detection of false positive connections in tractograms. Moreover these results are stable with respect to the two parameters to be tuned. We expect that the more complex configurations given by real data will need a deeper study of these parameters. Finally, we introduced a new concept of connectome that can’t be considered neither purely structural nor purely functional but inherits features both from structural and functional information, hence it represents a first step towards the construction of a joint structural and functional connectome. Acknowledgements The authors would like to thank Rebecca Bonham-Carter for the help in simulating resting state networks. This work received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (ERC Advanced Grant agreement No 694665: CoBCoM–Computational Brain Connectivity Mapping).

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References 1. Bach, F., Jenatton, R., Mairal, J., Obozinski, G., et al.: Structured sparsity through convex optimization. Stat. Sci. 27(4), 450–468 (2012) 2. Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009) 3. Caruyer, E., Daducci, A., Descoteaux, M., Houde, J.C., Thiran, J.P., Verma, R.: Phantomas: a flexible software library to simulate diffusion mr phantoms. In: ISMRM (2014) 4. Caruyer, E., Lenglet, C., Sapiro, G., Deriche, R.: Design of multishell sampling schemes with uniform coverage in diffusion MRI. Magn. Reson. Med. 69(6), 1534–1540 (2013) 5. Daducci, A., Barakovic, M., Girard, G., Descoteaux, M., Thiran, J.P.: Reducing false positives in tractography with microstructural and anatomical priors. In: ISMRM 2018-International Society for Magnetic Resonance in Medicine (2018) 6. Daducci, A., Dal Palú, A., Descoteaux, M., Thiran, J.P.: Microstructure informed tractography: pitfalls and open challenges. Front. Neurosci. 10, 247 (2016) 7. Daducci, A., Dal Palù, A., Lemkaddem, A., Thiran, J.P.: Commit: convex optimization modeling for microstructure informed tractography. IEEE Trans. Med. Imaging 34(1), 246–257 (2015) 8. Frigo, M., Barakovic, M., Thiran, J.P., Daducci, A.: Hierarchical tractography optimisation. In: CoBCoM 2017-Computational Brain Connectivity Mapping Winter School Workshop, p. 1 (2017) 9. Frigo, M., Gallardo, G., Costantini, I., Daducci, A., Wassermann, D., Deriche, R., DeslauriersGauthier, S.: Reducing false positive connection in tractograms using joint structure-function filtering. In: OHBM 2018-Organization for Human Brain Mapping (2018) 10. Girard, G., Fick, R., Descoteaux, M., Deriche, R., Wassermann, D.: Axtract: microstructuredriven tractography based on the ensemble average propagator. In: International Conference on Information Processing in Medical Imaging, pp. 675–686. Springer (2015) 11. Jenatton, R., Mairal, J., Obozinski, G., Bach, F.R.: Proximal methods for sparse hierarchical dictionary learning. In: ICML, pp. 487–494 (2010) (No. 2010, Citeseer) 12. Leonardi, N., Van De Ville, D.: On spurious and real fluctuations of dynamic functional connectivity during rest. Neuroimage 104, 430–436 (2015) 13. Maier-Hein, K.H., Neher, P.F., Houde, J.C., Côté, M.A., Garyfallidis, E., Zhong, J., Chamberland, M., Yeh, F.C., Lin, Y.C., Ji, Q., et al.: The challenge of mapping the human connectome based on diffusion tractography. Nat. Commun. 8(1), 1349 (2017) 14. Panagiotaki, E., Schneider, T., Siow, B., Hall, M.G., Lythgoe, M.F., Alexander, D.C.: Compartment models of the diffusion MR signal in brain white matter: a taxonomy and comparison. Neuroimage 59(3), 2241–2254 (2012) 15. Pestilli, F., Yeatman, J.D., Rokem, A., Kay, K.N., Wandell, B.A.: Evaluation and statistical inference for human connectomes. Nat. Methods 11(10), 1058 (2014) 16. Sanz Leon, P., Knock, S.A., Woodman, M.M., Domide, L., Mersmann, J., McIntosh, A.R., Jirsa, V.: The virtual brain: a simulator of primate brain network dynamics. Front. Neuroinformatics 7, 10 (2013) 17. Smith, R.E., Tournier, J.D., Calamante, F., Connelly, A.: Sift2: enabling dense quantitative assessment of brain white matter connectivity using streamlines tractography. Neuroimage 119, 338–351 (2015) 18. The virtual brain: the resting state network scripting tutorial (2018) 19. Zalesky, A., Fornito, A., Cocchi, L., Gollo, L.L., van den Heuvel, M.P., Breakspear, M.: Connectome sensitivity or specificity: which is more important? Neuroimage 142, 407–420 (2016)

Voxel-Wise Clustering of Tractography Data for Building Atlases of Local Fiber Geometry Irene Brusini, Daniel Jörgens, Örjan Smedby and Rodrigo Moreno

Abstract This paper aims at proposing a method to generate atlases of white matter fibers’ geometry that consider local orientation and curvature of fibers extracted from tractography data. Tractography was performed on diffusion magnetic resonance images from a set of healthy subjects and each tract was characterized voxel-wise by its curvature and Frenet–Serret frame, based on which similar tracts could be clustered separately for each voxel and each subject. Finally, the centroids of the clusters identified in all subjects were clustered to create the final atlas. The proposed clustering technique showed promising results in identifying voxel-wise distributions of curvature and orientation. Two tractography algorithms (one deterministic and one probabilistic) were tested for the present work, obtaining two different atlases. A high agreement between the two atlases was found in several brain regions. This suggests that more advanced tractography methods might only be required for some specific regions in the brain. In addition, the probabilistic approach resulted in the identification of a higher number of fiber orientations in various white matter areas, suggesting it to be more adequate for investigating complex fiber configurations in the proposed framework as compared to deterministic tractography. Keywords Tractography · Fiber clustering · Brain atlases

I. Brusini (B) · D. Jörgens · Ö. Smedby · R. Moreno Department of Biomedical Engineering and Health Systems, KTH Royal Institute of Technology, Stockholm, Sweden e-mail: [email protected] D. Jörgens e-mail: [email protected] Ö. Smedby e-mail: [email protected] R. Moreno e-mail: [email protected] I. Brusini Department of Neurobiology, Care Sciences and Society, Karolinska Institutet, Stockholm, Sweden © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_27

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1 Introduction Diffusion magnetic resonance imaging (dMRI) is a noninvasive imaging method that is sensitive to the diffusion process of water molecules in biological tissues. In the brain white matter (WM), water diffusion is much faster along neural fiber bundles than perpendicularly to them [1]. For this reason, voxel-wise information on the orientation of groups of neuronal axons [19], and thus on the structural connectivity between brain regions, can be estimated from dMRI. Moreover, by using high angular resolution diffusion imaging (HARDI), it is possible to extract complex fiber configurations (e.g. crossing fibers) and construct fiber trajectories by applying tractography methods. Once tractography is performed, it is possible to cluster the obtained fibers into bundles. Fiber clustering is not a new research topic and many proposals have been made, based on the definition of a similarity measure between whole fiber trajectories [2, 6, 10, 13]. However, to the best of our knowledge, an approach that has not been explored so far is performing the clustering of fibers separately for each voxel. By using this approach, a description of the local geometry of fibers can be obtained without focusing directly on the whole fiber trajectory as an entity to be clustered. In this work, we propose a methodology for generating atlases that represent voxelwise geometrical features of tractography pathways by using local fiber clustering methods. In general, curvature of a 3D trajectory is an orientation-dependent feature that is linked to the Frenet–Serret frames, also called TNB frames [5]. In our approach, we first cluster curvature and TNB frames per voxel in each subject with the aim of grouping fiber tracts that locally show similar orientation and curvature. In a second step, the centroids of the identified clusters of all subjects are clustered in order to create a population-level atlas. The main purpose of creating atlases of voxel-wise fiber geometries is to support the investigation of fiber configurations in small WM regions, as well as provide possible priors for tractography algorithms for improving their accuracy.

2 Methods 2.1 Dataset The data used for the present work are part of the 500 Subjects + MEG2 data release (November 2014) of the Human Connectome Project (HCP) [20, 21]. In this data release, scans from 431 subjects include multishell HARDI data consisting of 3 × 90 gradient directions at b-values 1000, 2000 and 3000 s/mm2 and 18 b0 acquisitions. The subjects are all healthy adults of either gender and 81 of them were randomly chosen for the present work. The preprocessed diffusion data were affinely registered to the ICBM-152 template [11] as described by [22], obtaining images with voxel size of 1 × 1 × 1 mm.

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2.2 Fiber Orientation Distributions and Tractography The fiber orientation distribution (FOD) of each subject was reconstructed using the constrained spherical deconvolution (CSD) algorithm by Tournier et al. [16, 19], which is already implemented in the software package MRtrix (www.mrtrix.org). CSD was applied combining the shells at b = 1000 s/mm2 and b = 3000 s/mm2 . A maximum spherical harmonic order lmax = 8 was used. Fiber trajectories were obtained by applying both a deterministic and a probabilistic tractography algorithm compiled in MRtrix: SD_STREAM [18] and iFOD2 [17], respectively. Default values were set for most of the parameters of the tractography, such as the step-size (0.5 mm for iFOD2 and 0.1 mm for SD_STREAM), the FOD amplitude cutoff (equal to 0.1) for terminating streamlines, or the maximum angle between successive steps (90◦ × step-size/voxel-size). The maximum length of any streamline was set to 250 mm and the number of modeled streamlines per subject was 300 000.

2.3 Geometrical Feature Extraction For each tract in each subject, its tangent vector T, binormal vector B and angle of curvature Θ were extracted at equidistant positions Pi along its extent. Let v1 and v2 be the vectors corresponding to two successive track segments (delimited respectively by the pairs of points Pi−1 , Pi and Pi , Pi+1 ), the geometrical features in Pi were computed as: Pi+1 − Pi−1 , (1) T= ||Pi+1 − Pi−1 || B=

v1 × v2 , ||v1 × v2 ||

(2)



 v1 × v2  Θ = arctan . v1 · v2

(3)

Thus, a triplet (T, B, Θ) was obtained for every Pi . T and B are two of the five elements describing the TNB frame of the curve. The other three elements are the normal unit vector N, the curvature κ and the torsion τ . N is going to be employed later on for fiber clustering, Θ was used instead of κ, whereas the assumption τ = 0 was made. To compare the curvature information computed from the two tested tractography algorithms, it is necessary to have the same step-size, which was different in the two methods (0.1 mm for the deterministic vs. 0.5 mm for the probabilistic). The issue was approached by appropriately sampling the pathways: in the deterministic case, Pi−5 and Pi+5 were used instead of Pi−1 and Pi+1 .

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Fig. 1 Two low-curvature tracts are similar if they have (almost) the same tangent, for any direction of N (left). Instead, if the tracts share the same tangents and their curvature is high, parallel vectors N yield low distances while opposite ones yield large distances (right)

The final voxel-wise characterization of a given tract was computed as the mean of all the triplets associated to the tract’s points in that specific voxel.

2.4 Local Distance Between Fibers Once each tract had been characterized by its local geometrical features, this voxelwise information could be used to cluster similar fibers. Given two tracts described by their triplets F1 = (T1 , B1 , Θ1 ) and F2 = (T2 , B2 , Θ2 ), the distance between them was defined as a modification of the geodesic distance in the space of rotations SO(3): ⎧ −1 ⎪ ⎨2 cos (| T1 · T2 |) + λ | Θ1 − Θ2 |, if min(Θ1 , Θ2 ) < Θth . D(F1 , F2 ) = 2 cos−1 ( |T1 ·T2 |+|B1 ·B22 |+|N1 ·N2 |)−1 )+ ⎪ ⎩ otherwise. +λ | Θ1 − (N1 · N2 ) · Θ2 |, (4) N1 = B1 × T1 and N2 = B2 × T2 are the normal vectors of the two tracts. λ is a parameter to control the importance of the difference in curvature in the computation of the distance and it was empirically set to λ = 2 in the experiments of the present work. Finally, Θth is an angle threshold to distinguish between low and high curvature cases. If one (or both) tracts have low curvature, the distance between frames is mainly given by that between their tangents, since the normal and binormal vectors become unstable. On the other hand, if both curves are significantly bent, the orientation of N is much more important. This concept is summarized in Fig. 1. In order to choose the value of Θth , the curvature angles of several probabilistic datasets (in which higher values of curvature were found as compared to deterministic data) were analyzed. Observing that approximately half of the tracts have a Θ below 10◦ and half above 10◦ , Θth was set to 10◦ in the experiments.

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2.5 Agglomerative Hierarchical Clustering The pairwise distances between the tracts were used to perform agglomerative hierarchical clustering. This technique partitions the data by grouping at each step the most similar (according to a given distance measure) individuals or groups of individuals [9]. Thus, each partitioning step could be described by the data being grouped together and their single linkage distance (i.e. the distance between the two closest pairs of individuals, where only pairs consisting of one individual from each group are considered). Let n be the number of tracts in a voxel, then the optimal number of clusters (between 1 and n − 1) was chosen by implementing an extended version of the Davies–Bouldin criterion, which searches for the clustering solution with the minimum ratio of within-cluster and between-cluster distances [7]. A distance threshold dth was set and the last partition having distance lower than dth (consisting of K clusters) was searched. If K > 2 clusters, the original Davies–Bouldin criterion was applied to all the clustering solutions between 2 and K . Instead, if K = 1 or K = 2, the criterion was not applied and the number of clusters was set to K . This extension was needed for two reasons. First, by applying only the original criterion, the option of having only one cluster was not taken into account. Second, the Davies–Bouldin criterion alone is very selective with tractography data, which are known to be inherently noisy, so the threshold dth allowed to get more inclusive clusters. In this work, dth was set to 60◦ , due to reasonable results obtained from preliminary tests on synthetic data and, subsequently, on real data in regions known for having a single fiber population.

2.6 Atlas Creation The clustering algorithm was applied to each voxel of each subject, creating individual maps. These maps include the centroids of each identified cluster, which were obtained from a cluster’s elements with the following procedure: 1. A random reference tract was chosen and all the tangent vectors pointing in an opposite direction were flipped together with their binormal vectors. 2. The tangents were averaged and normalized to obtain the centroid’s tangent Tc . 3. The curvature angles were averaged, obtaining Θc . 4. The binormal vectors were averaged. This average was projected onto the plane perpendicular to Tc and then normalized, obtaining Bc . Each centroid Ci (i = 1, . . . , N , where N is the total number of clusters in the voxel) was stored in the map together with a score si defined as: si =

ni − 1 , di2 + 1

(5)

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where n i is the number of elements in the i-th cluster and di is the within-cluster distance (average distance between the cluster’s elements and its centroid). The higher si , the more reliable the cluster representation is assumed to be. Outlier rejection was then performed by eliminating clusters having di higher than the 95 percentile of the within-cluster distance distribution (estimated from a subset of four subjects) and clusters containing less than 5 tracts and/or less than 10% of the total tracts in the voxel. A new voxel-wise clustering, similar to the one described in Sect. 2.5, was performed on the centroids from all the individual maps to create the final atlas. The average number of elements to be clustered per voxel was approx. 32, compared to approx. 17 elements (tracts) per voxel that had to be clustered for the creation of the individual maps. As an increase in the number of elements was expected to lead to a decrease of the computed single-linkage distances between them, dth was consequently decreased to 30◦ . In addition, the previously computed centroids were likely to be less noisy than the original tractography data, so a lower dth was considered to be more reasonable for the second clustering step. Moreover, the new centroids were now computed using a weighted average (instead of the simple average) with the scores si as weights. In this way, the most reliable individual results had a higher influence in the final atlas. Finally, for each voxel of the atlas, the a posteriori probability Pn of having n clusters could be estimated by looking at the number of clusters obtained in each of the 81 individual maps in the given voxel.

3 Results 3.1 Individual Maps The first clustering step described in Sect. 2.5 was applied to each subject, creating 81 individual maps of centroids for each of the two tractography methods. In probabilistic data, in each subject the maximum number of identified clusters was either 4 or 5. Instead, using deterministic data, the maximum number was either 3 or 4. Moreover, the amount of voxels with more than one cluster (i.e. the estimated multi-fiber voxels) was always higher in the maps derived from probabilistic data than in the deterministic ones. The number of clusters identified in one slice of one subject is shown in Fig. 2 as an example. To check if the number of clusters and their orientations matched the fiber orientations estimated with the CSD algorithm, some centroids were visually compared to the FOD in their corresponding voxels. A large portion of voxels having one main fiber orientation resulted in only one cluster in both the deterministic and probabilistic case. Moreover, where the FOD had more than one relative maximum, it was often possible to detect more than one centroid having tangents aligned to the FOD’s maxima, especially from the probabilistic dataset (see Fig. 3). However, it has to be

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(b) Deterministic tractography.

Fig. 2 Visualization of the number of clusters obtained in one slice of one random subject applying the proposed clustering method to probabilistic tractography data (a) and deterministic tractography data (b). The underlying image is the fractional anisotropy (FA) map of the subject

noted that an exact correspondence between FOD’s relative maxima and clustering solutions could not always be found.

3.2 Distribution of the Number of Clusters As described in Sect. 2.6, for each voxel of the ICBM-152 template the a posteriori probability Pn of the number of clusters was estimated. One important aspect of Pn is the estimated probability of exhibiting one cluster: the lower it is, the higher are the chances of having multiple orientations. Such probability was computed for every voxel for both the probabilistic and deterministic cases. In various WM regions the probability of finding a single orientation is higher in the deterministic dataset compared to the probabilistic one (see Fig. 4). The difference between the probabilistic and deterministic cases can be observed in the final atlases as well (Fig. 5). Not only the pattern of the number of clusters appears more heterogeneous in the probabilistic atlas, but also the regions with a high number of clusters (especially ≥3) are more present and often larger. On the other hand, various brain structures (e.g. the corpus callosum or the corticospinal tract) show the same clustering solution in both atlases.

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Fig. 3 Comparison of the clustering solutions (from the iFOD2 dataset) in six voxels with the respective FOD. The TBN frames of the clusters’ centroids (T in red, N in green and B in blue. These colors are not related to those used for representing the FOD on the left.) are plotted with different line styles (solid, dashed or dotted), one for each cluster in the voxel. In a the centroids discriminate a region with crossings (the two clusters in the middle) from the adjacent two regions with one main orientation. In b a more complex configuration is presented with one voxel having a total of three clusters

4 Discussion In the present work, geometrical features of tractograms were extracted from a set of healthy subjects and used to cluster fibers at a voxel level, generating brain atlases of orientation of WM fibers. Traditional methods for clustering fibers usually use their complete trajectories [2, 6, 10, 13]. While TNB frames have been proposed to describe fiber bundles locally as a postprocessing after clustering complete fiber

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(a) Probabilistic tractography

(b) Deterministic tractography Fig. 4 Voxel-wise a posteriori probability of having one cluster, estimated from both probabilistic (a) and deterministic (b) tractography in three slices. The above representations were obtained using only the voxels with FA higher than 0.1 in the ICBM FA atlas (JHU-ICBM-FA-1 mm), which is underlying in the figure. It is possible to observe, especially in (a), various regions of the brain in which the probability value decreases, suggesting the presence of multiple orientations. Some of these regions, which may be involved in crossings with other structures, are portions of the corticospinal tract (CST), anterior corona radiata (ACR), posterior thalamic radiation (PTR), superior longitudinal fasciculus (SLF), superior corona radiata (SCR)

trajectories [6], the implementation of a voxel-wise clustering technique based on TNB frames is, to the best of our knowledge, new. The implemented clustering method, together with the definition of a new local distance measure between fiber tracts, can be used for studying geometrical features of neural fibers. The proposed methodology was applied on both probabilistic and deterministic tractography data and differences between the two could already be observed from the individual maps. Due to their generally higher number of multifiber voxels (as highlighted in Fig. 4), the results of the probabilistic maps are in agreement with previous studies suggesting that probabilistic tractography is more suitable for identifying multiple fiber orientations [17, 18]. Moreover, a good agreement between the number of fiber clusters and FOD peaks was obtained. However, it was also noted that our method, which uses global information (i.e. tractograms) to estimate local orientations, can disagree with purely local orientation measurements. Indeed, in our results, multi-fiber voxels were underrepresented compared to other

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(a) Probabilistic tractography.

(b) Deterministic tractography.

Fig. 5 Number of clusters identified in four slices of the probabilistic (a) and deterministic (b) atlases. The underlying image is the ICBM FA atlas (JHU-ICBM-FA-1 mm)

previous studies [12, 15], especially in the individual subjects’ maps (Fig. 2). This could be related to the number of modeled streamlines, which might not be large enough to capture all possible fibers, especially those not belonging to major fiber bundles and therefore associated to lower FOD peaks. In the current work, two population-level atlases (one probabilistic and one deterministic) were analyzed, one per tractography algorithm. In certain brain regions with a few identified fiber orientations (mainly one or two, with a few cases of three) the clustering is very similar in the probabilistic and deterministic case. For example, similarities were found in the corpus callosum (one fiber cluster), in the intersection of the corticospinal tract and the pontine crossing tract (two clusters), and in the intersection of the corona radiata, the superior longitudinal fasciculus and the corpus callosum (three clusters). However, in other areas the probabilistic atlas shows more fiber orientations and more heterogeneous patterns, suggesting again that a higher number of directions, and thus of 3D fiber configurations, is taken into account when probabilistic tractography is used. On the other hand, in both atlases, small and peripheric brain regions showing more than three fiber clusters per voxel appear to be less plausible (see Fig. 5). This issue could be approached by improving the preprocessing pipelines and employing more sophisticated tractography methods, which could potentially reduce the between-subject variability in those areas. While the Track Orientation Density (TOD) method [8] was not proposed for creating population atlases, it potentially could be adapted for characterizing neural tracts in a voxel-wise manner as an alternative to our method. The main difference between the two approaches is that the TOD is a continuous representation of the

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tract main directions on the 5D spatio-angular domain, while the proposed method is based on a discrete cluster representation. Using a discrete representation can be advantageous, for example, as a prior for improving fiber tracking. In our method the optimal number of clusters is automatically selected in every voxel (cf. Sect. 2.5), while a thresholding of the TOD results might be needed to select the most representative tract orientations. Alternatively, in [4], kernel regression estimation of voxel-wise fiber orientations was proposed where a local model is fitted to the diffusion signal. This approach has been used for interpolating and fusing tractography data for constructing population-based atlases. Unlike this approach, our method considers curvature information in the clustering, which is not available in the diffusion signal. Our ongoing research includes to assess differences in the atlases when a large number of streamlines per subject is used (e.g. 3 million). In addition, since the analyses were performed using fixed parameters (e.g. dth and Θth ), further investigations will be carried out to optimize the relation between the choice of parameters and the total number of fiber tracts. Moreover, a previous study has also shown that local curvature estimations tend to be sensitive to the choice of the tractography algorithm and its settings [3]. For this reason, the influence of the curvature term in the atlases will be assessed in details. Finally, following the approach in [14], we plan to compare the use of our method with TOD maps as priors for improving bundle-wise tracking.

5 Conclusion This paper proposes a methodology to create atlases of WM fibers by clustering orientation and curvature of tractography data at a voxel level. The method was applied to both probabilistic and deterministic data. Both atlases agreed in most regions where the actual structure of WM is well known, while disagreeing in regions with complex fiber structures. This suggests that more advanced tractography methods might only be required for some specific regions in the brain. Moreover, several areas showed heterogeneous patterns in the number of clusters (especially in the probabilistic atlas), indicating the difficulties of finding a universally valid representation of fiber structures at a voxel level in those areas. Implementing a voxel-wise method makes it possible to investigate fiber structures at a local level, without relating them to entire bundles, as well as to analyze how specific WM sites vary across subjects. Moreover, the atlases created with our proposed method could be suitable for being used as priors for tractography algorithms, since the centroids of each cluster can provide voxel-wise information of the main fiber orientations.

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Acknowledgements Data were partially provided by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University.

References 1. Alexander, D.C.: Modelling, fitting and sampling in diffusion MRI. In: Laidlaw, D., Weickert, J. (eds.) Visualization and Processing of Tensor Fields, pp. 3–20. Springer (2009) 2. Brun, A., Knutsson, H., Park, H.J., Shenton, M.E., Westin, C.F.: Clustering fiber traces using normalized cuts. In: Proceedings of the International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI), vol. LNCS 3216-I, pp. 368–375 (2004). https:// doi.org/10.1007/b100265.Clustering 3. Brusini, I., Jörgens, D., Smedby, Ö., Moreno, R.: Influence of tractography algorithms and settings on local curvature estimations. In: Proceedings of OHBM Annual Meeting (2017) 4. Cabeen, R.P., Bastin, M.E., Laidlaw, D.H.: Kernel regression estimation of fiber orientation mixtures in diffusion MRI. NeuroImage 127, 158–172 (2016) 5. do Carmo, M.P.: The local theory of curves parametrized by arc length. In: Differential Geometry of Curves and Surfaces, pp. 16–26. Prentice Hall (1976) 6. Corouge, I., Gouttard, S., Gerig, G.: Towards a shape model of white matter fiber bundles using diffusion tensor MRI. In: International Conference on Engineering in Medicine and Biology Society (EMBS), vol. 2, pp. 344–347. IEEE (2004). https://doi.org/10.1109/ISBI. 2004.1398545 7. Davies, D.L., Bouldin, D.W.: A cluster separation measure. IEEE Trans. Pattern Anal. Mach. Intell. 1(2), 224–227 (1979). https://doi.org/10.1109/TPAMI.1979.4766909 8. Dhollander, T., Emsell, L., Van Hecke, W., Maes, F., Sunaert, S., Suetens, P.: Track orientation density imaging (TODI) and track orientation distribution (TOD) based tractography. NeuroImage 94, 312–336 (2014) 9. Everitt, B., Landau, S., Leese, M., Stahl, D.: Hierarchical clustering. In: Cluster Analysis, pp. 71–110. Wiley (2011). https://doi.org/10.1002/9780470977811.ch4 10. Garyfallidis, E., Brett, M., Correia, M.M., Williams, G.B., Nimmo-Smith, I.: Quickbundles, a method for tractography simplification. Front. Neurosci. 6(175) (2012) 11. Grabner, G., Janke, A.L., Budge, M.M., Smith, D., Pruessner, J., Collins, D.L.: Symmetric atlasing and model based segmentation: an application to the hippocampus in older adults. In: Proceedings of International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI), vol. 9, pp. 58–66 (2006). https://doi.org/10.1007/11866763_8 12. Jeurissen, B., Leemans, A., Tournier, J.D., Jones, D.K., Sijbers, J.: Investigating the prevalence of complex fiber configurations in white matter tissue with diffusion magnetic resonance imaging. Hum. Brain Mapp. 34(11), 2747–2766 (2013). https://doi.org/10.1002/hbm.22099 13. O’ Donnell, L., Westin, C.F.: White matter tract clustering and correspondence in populations. In: Proceedings of International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI), vol. 8-I, pp. 140–147 (2005) 14. Rheault, F., St-Onge, E., Sidhu, J., Chenot, Q., Petit, L., Descoteaux, M.: Bundle-specific tractography. In: Computational Diffusion MRI, pp. 129–139. Springer International Publishing (2018) 15. Schultz, T.: Learning a reliable estimate of the number of fiber directions in diffusion MRI. Med. Image Comput. Comput. Assist. Interv. 15, 493–500 (2012) 16. Tournier, J.D., Calamante, F., Connelly, A.: Robust determination of the fibre orientation distribution in diffusion MRI: non-negativity constrained super-resolved spherical deconvolution. NeuroImage 35(4), 1459–1472 (2007). https://doi.org/10.1016/j.neuroimage.2007.02.016

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17. Tournier, J.D., Calamante, F., Connelly, A.: Improved probabilistic streamlines tractography by 2nd order integration over fibre orientation distributions. Proc. Int. Soc. Magn. Reson. Med. 18, 1670 (2010) 18. Tournier, J.D., Calamante, F., Connelly, A.: MRtrix: diffusion tractography in crossing fiber regions. Internat. J. Imaging Syst. Technol. 22(1), 53–66 (2012). https://doi.org/10.1002/ima. 22005 19. Tournier, J.D., Calamante, F., Gadian, D.G., Connelly, A.: Direct estimation of the fiber orientation density function from diffusion-weighted MRI data using spherical deconvolution. NeuroImage 23(3), 1176–1185 (2004). https://doi.org/10.1016/j.neuroimage.2004.07.037 20. Van Essen, D.C., Ugurbil, K., Auerbach, E., Barch, D., Behrens, T.E.J., Bucholz, R., Chang, A., Chen, L., Corbetta, M., Curtiss, S.W., Della Penna, S., Feinberg, D., Glasser, M.F., Harel, N., Heath, A.C., Larson-Prior, L., Marcus, D., Michalareas, G., Moeller, S., Oostenveld, R., Petersen, S.E., Prior, F., Schlaggar, B.L., Smith, S.M., Snyder, A.Z., Xu, J., Yacoub, E.: The human connectome project: a data acquisition perspective. NeuroImage 62(4), 2222–2231 (2012). https://doi.org/10.1016/j.neuroimage.2012.02.018 21. WU-Minn Consortium Human: Connectome Project: WU-Minn HCP 500 Subjects. Data Release: Reference Manual (2014) 22. Zappalà, S.: Evaluation of the inter-subject variability of white matter tract curvature and its influence on finite element simulations of an anisotropic model of the brain tissue. Master’s thesis. KTH Royal Institute of Technology, Stockholm, Sweden (2016)

Obtaining Representative Core Streamlines for White Matter Tractometry of the Human Brain Maxime Chamberland , Samuel St-Jean , Chantal M. W. Tax and Derek K. Jones

Abstract Diffusion MRI infers information about the micro-structural architecture of the brain by probing the diffusion of water molecules. The process of virtually reconstructing brain pathways based on these measurements is called tractography. Various metrics can be mapped onto pathways to study their micro-structural properties. Tractometry is an along-tract profiling technique that often requires the extraction of a representative streamline for a given bundle. This is traditionally computed by local averaging of the spatial coordinates of the vertices, and constructing a single streamline through those averages. However, the resulting streamline can end up being highly non-representative of the shape of the individual streamlines forming the bundle. In particular, this occurs when there is variation in the topology of streamlines within a bundle (e.g., differences in length, shape or branching). We propose an envelope-based method to compute a representative streamline that is robust to these individual differences. We demonstrate that this method produces a more representative core streamline, which in turn should lead to more reliable and interpretable tractometry analyses. Keywords Tractography · Tractometry · Bundle envelope · Core streamline · Diffusion MRI

1 Introduction Tractography derived from diffusion MRI infers information about the structural architecture of the brain. In most studies, diffusion MRI metrics (e.g., fractional M. Chamberland (B) · C. M. W. Tax · Derek K. Jones Cardiff University Brain Research Imaging Centre (CUBRIC), School of Psychology, Cardiff University, Cardiff, UK e-mail: [email protected] S. St-Jean Center for Image Sciences, Department of Radiology, University Medical Center Utrecht, Utrecht, The Netherlands Derek K. Jones Mary McKillop Institue for Health Research, Australian Catholic University, Victoria, Australia © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_28

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anisotropy (FA)) are often collapsed to a single scalar value per bundle [1]. Recently, a trend towards tract profiling [2, 3] and direction-specific measurements within a voxel has emerged. Along-tract analysis is a technique that maps a given metric over the course of a bundle. The term tractometry was originally introduced by Bells et al. [4] and the technique has been refined over the years by various groups [2, 3, 5–7]. It can be used to characterise areas of the brain with abnormal properties in patients [8–10]. At the core of tractometry lies the concept of a representative streamline [11] which is used to project metrics along the course of a given bundle. This is typically done by resampling all streamlines forming the bundle to n points and by averaging their spatial coordinates in a point-wise fashion [2, 3, 11]. This technique will produce a reasonable estimate of the average trajectory when there is very little branching and dispersion between the streamlines forming the pathway (Fig. 1, left). However, it is known that streamlines within a given bundle can vary in length and orientation, making it inappropriate to directly average their coordinates [11]. Indeed, if the underlying streamlines are even slightly dispersed from each other, the resulting representative streamline obtained by simply averaging the coordinates can end up running outside of the shape of the bundle (Fig. 1, right). Not only does this representation become anatomically implausible, but also it can directly hamper further steps down the tractometry pipeline (e.g., when averaging metrics along different sections of the pathway). A common solution to overcome this problem is to perform tractometry only within the compact portion of the pathway by excluding data from the extremities, which tend to include fanning [2, 12]. This approach greatly helps to (1) quickly obtain a representative streamline and (2) mitigate variability between

Fig. 1 Point-wise streamline averaging illustrated for two scenarios: pruned bundles (left) and fanning bundles (right). The mean streamline (purple) runs outside of the shape of the bundle when streamlines within a bundle slightly diverge from each other. Bundles: cingulum (Cg) and arcuate fasciculus (AF)

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subjects since bundles are essentially reduced to a simpler representation. However, cutting both extremities inherently limits the benefits conferred by state-of-the-art tractography techniques that can recover fanning and branching portions of white matter fasciculi. Here, we propose a technique to generate a representative streamline that is robust to multiple streamline lengths, arching configurations and orientations that naturally occur within a bundle.

2 Theory and Methods 2.1 Acquisition and Processing Multi-shell high angular resolution diffusion MRI data were acquired on a Siemens Prisma scanner (TR = 4500 ms, TE = 80 ms, with b-values of 1200, 3000, 5000 s/mm2 , 60 diffusion directions per shell and 15 non diffusion weighted images at a voxel size of 1.5 mm isotropic). Correction for subject motion and distortions caused by eddy currents were performed using FSL eddy and topup [13]. Next, fibre orientation distributions functions (fODF) were computed using multi-shell multi-tissue constrained spherical deconvolution [14]. Tractography was performed using FiberNavigator [15], followed by manual bundle extraction of the corticospinal tract (CST), fornix (Fx), cingulum (Cg), arcuate fasciculus (AF) and inferior frontooccipital fasciculus (iFOF). The main goal of the dissection plan was to preserve the characteristic anatomy, including fanning and branching, of each bundle [16]. Tractography parameters were set as follows: min. fODF amplitude: 0.1, step size: 0.5 mm, max. angular threshold: 45◦ , min./max. streamline length: 30/200 mm with 1.5 million seeds covering the whole brain.

2.2 Proposed Representative Streamline Extraction Algorithm The proposed method starts by averaging the top 5% longest streamlines from a bundle of interest to get a coarse approximation of the bundle’s core, defined by C = { pi ∈ R3 |i = 1, . . . , n} where pi is a 3D point of the representative streamline. This core C will serve as guidance for the propagation of a convex hull envelope along the entirety of the bundle. Next, we generate an orthogonal plane Pi at each point pi ∈ C using the normal vector ni formed by pi and pi+1 . Then, for each streamline from the bundle of interest, we find all line segments intersecting the current orthogonal plane Pi up to a distance threshold of t mm (which is interactivelydefined in our implementation). If multiple points from a streamline intersect a plane Pi (e.g., a spurious streamline doubling back on itself), only the closest point to C is preserved so that the actual shape of the bundle is represented by its 3D envelope.

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Fig. 2 Example of a cross-section generated along the core of a bundle from a normal plane (Pi ). The black dots represent in-plane streamlines with their convex hull Hi in dark blue. On the right, the representative streamline (light blue) is obtained by linking the centre of mass of each convex-hull (light blue dot)

We then compute a 2D convex hull Hi ⊂ Pi for each group of points found in the previous step as illustrated in Fig. 2. Finally, the centres of mass of each hull Hi are linked together by fitting a 4th order b-spline curve comprising k knots, where k is a user-defined parameter. This step ensures that the representative streamline is located at the centre since there is no guarantee that this is the case using the initial approximation of C defined using the longest streamlines. The proposed framework is integrated within FiberNavigator, where all parameters are accessible to the user.

2.3 Label Maps Generation We compared our technique with conventional point-based resampling (n = 50) computed using the mean distance flip algorithm, accounting for streamline direction [9]. Distance maps were then generated by computing the minimum Euclidean distance between each point of the bundle and the core representative streamline. A transfer function was used to visually map sections of the bundle assigned to respective locations along the core. A unique and smooth colour grading from one end of the bundle to the other indicates a correct assignment along C, which in turn reflects how the diffusion metrics are averaged locally.

3 Results 3.1 Bundles Without Branching and Dispersion Figure 3 shows results on tubular-shaped bundles (e.g., CST, Fx, iFOF) with little dispersion between the streamlines’ starting and ending points. One can observe that

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Fig. 3 Label maps and representative streamlines illustrated for tubular-shaped bundles with little dispersion at the end points (e.g., directly connecting two brain regions) show agreement between the two methods (square: resampling method, circle: proposed method)

the core streamline (i) has a length comparable to the rest of the streamlines within each bundle; (ii) lies inside the boundaries defined by all the streamlines. The colour grading also shows a gradual labelling of the streamlines’ points along the core.

3.2 Bundles with Branching and Dispersion Figure 4 shows results on more complex bundles having different streamline lengths. In this example, the Cg consists of sub-components that connect the posterior cingulate cortex (PCC) to the medial prefrontal cortex (red streamlines) and the PCC to the parahippocampal gyrus (green streamlines). The complex configuration of the bundles inherently leads to an unrepresentative streamline when using the conventional point-based averaging (white arrows), as well as incorrect assignment of different sections along the bundle (indicated by the repeated green sections). The proposed cross-section method recovers an anatomically representative pathway which stays within the shape of the bundle. In the second example, the AF aggregates multiple sub-components with various arching streamlines that project to different areas of the lateral cortex. The representative streamline extracted from traditional averaging appears shorter than the full course of the bundle (white streamline). The inferior temporal aspect of the bundle (dark blue) is also incorrectly averaged and collapsed to the first point of the representative streamline. The last panel shows that the representative streamline traverses the full length of the bundle when using the proposed technique, which is also supported by the unique colour grading of the label map. Figure 5 shows FA profiles computed along a tubular-shaped tract (Fx) and a fanning-shaped tract (AF). The Fx profiles appear similar in both resampling and cross-sections methods, except for a small shift induced by the larger anterior extent of the proposed method. The AF shows large differences between the two techniques

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in terms of profiling (red star), mostly due to the sub-optimality of the mean streamline as the representative streamline.

4 Discussion and Conclusion We have shown that taking an average streamline as the representative pathway of a bundle can lead to non-representative results in the presence of tract branching and dispersion. To address this problem, we used orthogonal planes throughout the bundle to derive a representative core streamline that traverses its entire centre of mass and therefore, allowing for a more interpretable tractometry. Generating core streamlines based on convex-hulls has been applied previously in the creation of 3D meshes for visualisation [17], as well as for extracting skeleton streamlines for connectivity analysis using an atlas [18]. Here, we additionally show that this approach produces more representative core streamlines for tractometry analyses in various bundle shapes. A drawback to the current approach is that it requires that at least some streamlines run from one end of the bundle to the other. Otherwise, the crosssection propagation may halt prematurely and thus affect the computation of the representative streamline. In addition, we assume that the input bundles are already pruned from streamline loops and undesired false positives. Since tract morphology varies between subjects, truncation and resampling streamlines based on a length criterion [3] or number of points [2, 9] may inadvertently discard important information that is specific to the pathway of interest. This loss of information is inherent to the truncation approach and should be minimised when assigning diffusion metrics to a representative pathway to preserve the

Fig. 4 Complex fanning bundles with dispersed end-points reveal an astray mean streamline when using a conventional resampling approach to compute the core C, as well as incorrect label mapping (white arrows)

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Fig. 5 FA profiles illustrated for two representative streamline extraction algorithms. Left: Similar profiles are obtained for a tubular-shaped bundle (Fx). Right: Different profiles are obtained for a branching bundle (AF). The red star shows the location on the bundle where the maximum difference between the two methods occurs. Outlines of the bundles are shown for anatomical reference

full extent of a bundle as much as possible [11]. We showed that this inevitably leads to discrepancies in tract profiling when assigning diffusion metrics to a representative streamline. An alternative approach to truncation and tract averaging could be to remove the need for a representative streamline by matching geometrical properties of streamlines between subjects [12, 19]. Yet, the performance of those techniques still requires investigation for complex white matter configurations (e.g., fanning and branching). Nevertheless, truncation can also be useful to reduce potential issues associated with tractography and could still be applied in a post-processing step to our technique, once the representative streamline has been generated. Finally, a potential improvement to the proposed method would be to recursively generate multiple convex hulls, allowing the algorithm to recover various sub-branches in fanning bundles. This could help in achieving a simplified—yet still anatomically accurate—representation of the core of a bundle, which is the centrepiece for the future design of tractometry pipelines. Acknowledgements M.C. is supported by the Postdoctoral Fellowships Program from the Natural Sciences and Engineering Research Council of Canada (PDF-502385-2017). S.SJ. is funded by the Fonds de recherche du Quebec Nature et technologies (FRQNT). C.M.W.T. is supported by a Rubicon grant from the Netherlands Organisation for Scientific Research (680-50-1527). This work was also supported by a Wellcome Trust Investigator Award (096646/Z/11/Z), a Wellcome Trust Strategic Award (104943/Z/14/Z), and by the Engineering and Physical Sciences Research Council (EP/M029778/1).

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References 1. Jones, D.K., Catani, M., Pierpaoli, C., Reeves, S.J., Shergill, S.S., O’Sullivan, M., Maguire, P., Horsfield, M.A., Simmons, A., Williams, S.C., Howard, R.J.: A diffusion tensor magnetic resonance imaging study of frontal cortex connections in very-late-onset schizophrenia-like psychosis. Am. J. Geriatr. Psychiatry. 13(12), 1092–1099 (2005) 2. Yeatman, J.D., Dougherty, R.F., Myall, N.J., Wandell, B.A., Feldman, H.M.: Tract profiles of white matter properties: automating fiber-tract quantification. PloS One 7(11), e49790 (2012) 3. Colby, J.B., Soderberg, L., Lebel, C., Dinov, I.D., Thompson, P.M., Sowell, E.R.: Along-tract statistics allow for enhanced tractography analysis. Neuroimage 59(4), 3227–3242 (2012) 4. Bells, S., Cercignani, M., Deoni, S., Assaf, Y., Pasternak, O., Evans, C., Leemans, A., Jones, D.: Tractometry–comprehensive multi-modal quantitative assessment of white matter along specific tracts. In: Proceedings ISMRM. Vol. 678 (2011) 5. Jones, D.K., Travis, A.R., Eden, G., Pierpaoli, C., Basser, P.J.: PASTA: pointwise assessment of streamline tractography attributes. Magn. Reson. Med. 53(6), 1462–1467 (2005) 6. Corouge, I., Fletcher, P.T., Joshi, S., Gouttard, S., Gerig, G.: Fiber tract-oriented statistics for quantitative diffusion tensor mri analysis. Med. Image Anal. 10(5), 786–798 (2006) 7. De Santis, S., Drakesmith, M., Bells, S., Assaf, Y., Jones, D.K.: Why diffusion tensor mri does well only some of the time: variance and covariance of white matter tissue microstructure attributes in the living human brain. Neuroimage 89, 35–44 (2014) 8. Dayan, M., Monohan, E., Pandya, S., Kuceyeski, A., Nguyen, T.D., Raj, A., Gauthier, S.A.: Profilometry: a new statistical framework for the characterization of white matter pathways, with application to multiple sclerosis. Hum. Brain Mapp. 37(3), 989–1004 (2016) 9. Cousineau, M., Jodoin, P.M., Garyfallidis, E., Cote, M.A., Morency, F.C., Rozanski, V., GrandMaison, M., Bedell, B.J., Descoteaux, M.: A test-retest study on Parkinson’s PPMI dataset yields statistically significant white matter fascicles. NeuroImage Clin. 16, 222–233 (2017) 10. Groeschel, S., Tournier, J.D., Northam, G.B., Baldeweg, T., Wyatt, J., Vollmer, B., Connelly, A.: Identification and interpretation of microstructural abnormalities in motor pathways in adolescents born preterm. NeuroImage 87, 209–219 (2014) 11. O’Donnell, L.J., Westin, C.F., Golby, A.J.: Tract-based morphometry for white matter group analysis. Neuroimage 45(3), 832–844 (2009) 12. Glozman, T., Bruckert, L., Pestilli, F., Yecies, D.W., Guibas, L.J., Yeom, K.W.: Framework for shape analysis of white matter fiber bundles. Neuroimage 167, 466–477 (2018) 13. Andersson, J.L., Sotiropoulos, S.N.: An integrated approach to correction for off-resonance effects and subject movement in diffusion mr imaging. Neuroimage 125, 1063–1078 (2016) 14. Jeurissen, B., Tournier, J.D., Dhollander, T., Connelly, A., Sijbers, J.: Multi-tissue constrained spherical deconvolution for improved analysis of multi-shell diffusion mri data. NeuroImage 103, 411–426 (2014) 15. Chamberland, M., Whittingstall, K., Fortin, D., Mathieu, D., Descoteaux, M.: Real-time multipeak tractography for instantaneous connectivity display. Front. Neuroinformatics 8, 59 (2014) 16. Rojkova, K., Volle, E., Urbanski, M., Humbert, F., Dell’Acqua, F., Thiebaut de Schotten, M.: Atlasing the frontal lobe connections and their variability due to age and education: a spherical deconvolution tractography study. Brain Struct. Funct. 221(3), 1751–1766 (2016) 17. Enders, F., Sauber, N., Merhof, D., Hastreiter, P., Nimsky, C., Stamminger, M.: Visualization of white matter tracts with wrapped streamlines. In: IEEE Visualization 2005 (VIS 2005), p. 7. IEEE (2005) 18. Duda, J.T., McMillan, C., Grossman, M., Gee, J.C.: Relating structural and functional connectivity to performance in a communication task. Int. Conf. Med. Image Comput. Comput. Assist. Interv. (MICCAI) 13, 282–9 (2010) 19. Parker, G.D., Lloyd, D., Jones, D.K.: The best of both worlds: Combining the strengths of TBSS and tract-specific measurements for group-wise comparison of white matter microstructure. In: International symposium on magnetic resonance in medicine (Singapore), Vol. 2036 (2016)

Improving Graph-Based Tractography Plausibility Using Microstructure Information Matteo Battocchio, Gabriel Girard, Muhamed Barakovic, Mario Ocampo, Jean-Philippe Thiran, Simona Schiavi and Alessandro Daducci

Abstract Tractography is a unique tool to study neurological disorders for its ability to infer the major neural tracts from diffusion MRI data, thus allowing the investigation of the connectivity of the brain in-vivo. Tractography has seen a remarkable interest over the years and a large number of algorithms have been proposed. The choice of which method to use in a given application is usually a trade-off between its computational complexity and the quality of the reconstructions. So-called “shortest path ” methods represent an interesting option, as they are computationally efficient, robust to noise and their formulation is very flexible. However, they also come with limitations. For instance, the reconstructed streamlines tend to be collapsed and to share part of their path, especially in regions with highly curved fiber bundles. This can introduce voxels with incorrectly high or low streamline density , which does not correspond to the underlying fiber geometry. To mitigate this problem, we propose an iterative procedure that uses microstructure information and provides feedback to the shortest path tractography algorithm about the plausibility of the reconstructions. We evaluated our method on a synthetic phantom and show that the spatial distribution of streamlines is in closer agreement with the ground truth. Keywords Diffusion MRI · Tractography · Graph model · Shortest path · Microstructure informed tractography · Brain connectivity

M. Battocchio (B) · M. Ocampo · S. Schiavi · A. Daducci Computer Science Department, University of Verona, Verona, Italy e-mail: [email protected] G. Girard · M. Barakovic · J.-P. Thiran · S. Schiavi · A. Daducci Signal Processing Laboratory (LTS5), École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland J.-P. Thiran Department of Radiology, University Hospital Center (CHUV) and University of Lausanne (UNIL), Lausanne, Switzerland © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_29

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1 Introduction Tractography is a class of algorithms whose aim is to reconstruct the major neural tracts in the white matter (WM) using diffusion weighted magnetic resonance imaging (DW-MRI) data [10]. These techniques offer a powerful tool to investigate non invasively the major brain network organization in both healthy subjects and patients [3]. Tractography algorithms can be classified into three main categories: line propagation, energy minimization and shortest path . Each approach has its strengths and limitations. Here we provide a brief summary; for a more detailed review see [10] and references therein. Line propagation algorithms, e.g. [11], are the most intuitive and computationally efficient algorithms: the idea consists in reconstructing WM pathways (or streamlines) one-by-one by following the principal diffusion directions extracted from DWMRI and using an integration process from a starting position. The main drawback of these approaches is the error propagation that might occur in the computation of the streamlines due, for example, to imaging noise. To overcome this issue, global energy minimization approaches have been proposed, e.g. [13]. As opposed to previous methods, the full tractogram, i.e. the entire set of streamlines, is reconstructed at once by solving large-scale inverse problems to find those streamlines that best explain the data. Results obtained with these approaches outperform those of line propagation [7], but the price to pay is a significant increase in complexity and computational burden. Shortest path tractography, e.g. [8, 9, 12], addresses the reconstruction from a different perspective: streamlines are reconstructed by seeking for paths of least hindrance to diffusion that connects two regions, defined as the shortest path with respect to a given metric. This yields a good compromise between accuracy and efficiency. For instance, in [14], voxels are set as nodes of a graph which are connected to their immediate spatial neighbours by edges. Their weights represent probabilities of transition that are based on the coherence between the corresponding Orientation Distribution Functions (ODFs) estimated from DW-MRI. Reconstructing streamlines that connect different brain regions can then be seen as finding the shortest path between the corresponding nodes. This is an attractive approach for two reasons: (i) it is very efficient as optimized graph search algorithms exist and (ii) the possibility to define ad-hoc metrics makes this formulation flexible and allows for adding additional information into the tracking process. Nonetheless, a major drawback of these approaches is that many reconstructed streamlines tend to be collapsed and to share part of their path, especially in regions with highly curved fiber bundles. This can introduce voxels with incorrectly high or low streamline density , which does not correspond to the underlying fiber geometry [1]. In this article, we present an iterative procedure that uses microstructure information to provide feedback to the tractography algorithm about the plausibility

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Fig. 1 A schematic illustration of the proposed iterative procedure to provide feedback to the tracking algorithm about the plausibility of the current reconstructed streamlines

of the reconstructed streamlines with the aim of mitigating the main limitation of graph-based tractography approaches. In the following sections, we first describe the details of the algorithm and then report results based on numerical simulations to highlight the potential of the proposed method.

2 Methods Our proposed method uses an iterative procedure that alternates between graph-based tractography and evaluation of the plausibility of the reconstructed streamlines. The procedure is illustrated in Fig. 1. An initial estimate of the graph is constructed from the DW-MRI signal by capturing the structural coherence between neighbouring voxels. Then, the algorithm loops over the following steps: • STEP1: graph-based streamline tractography ; • STEP2: evaluation of the tractogram plausibility; • STEP3: graph’s weights update to improve the evaluation. This loop is stopped when the root mean square error (RMSE) between the reference streamline density (computed from the DW-MRI data) and the one computed by COMMIT [5] reaches a plateau. In other words, it stops when new iterations would not add streamlines that improve the WM volume coverage. We use DSI-studio (http://dsi-studio.labsolver.org) to compute the ODFs from the DW-MRI signal. The ODFs are then used to compute the graph’s transition probabilities with the MITTENS software [4].

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2.1 STEP1: Graph-Based Streamline Tractography In the graph, WM voxels are represented by nodes that are connected to their immediate spatial neighbours through weighted edges. These weights are based on estimates that the WM structure continues from one voxel into its neighbour. The probability that a voxel is connected to any other voxel is defined as the product of the negative logarithm of transition probabilities along the shortest path connecting these two voxels [14]. This can be computed efficiently using Dijkstras algorithm [6], which is implemented in the MITTENS software [4].The streamlines are constructed as the sequences of nodes corresponding to shortest paths between regions of interest (ROIs).This is made by running Dijkstra’s algorithm [6], iterating over each node from a ROI and looking for the shortest path that connects it with any node of the other ROIs. We then apply a smoothing function based on splines that divides each streamline with the same fixed number of points and then performs an interpolation using a user-defined percentage of them. In this study we interpolate the streamlines using 25% of the points.

2.2 STEP2: Evaluation of the Tractogram Plausibility After reconstructing the streamlines with STEP1, the plausibility of the tractogram is evaluated using the Convex Optimization Modeling for Microstructure Informed Tractography (COMMIT ) framework [5]. In a nutshell, COMMIT expresses tractography and tissue microstructure in a unified framework using convex optimization. N ×N ×N ×N Given a tractogram F, the acquired DW-MRI image I ∈ R+x y z d , composed of Nd measurements over Nv = N x N y Nz voxels, can be modelled as I = A(F) + η, where A : F → I is an operator modelling the signal contribution of each fiber to every voxel and η is the noise. Notably, if the trajectories and compartment signal patterns are known a priori, A can be efficiently implemented as a linear operator. The signal in each voxel can be seen as a linear combination of the diffusion arising from the streamlines intersecting the voxel, in addition to local contributions from other tissues; then, the joint problem can be expressed as a system of linear equations: y = Ax + η ,

(1)

where y ∈ R+Nd Nv contains the measured data and A ∈ R Nd Nv ×Nc is the linear operator implementing a generic multi-compartment model for the contribution of the tracts

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to the signal in each voxel. These contributions x ∈ R+Nc can be estimated by solving a non-negative least-squares problem: argmin ||Ax − y||22 such that x ≥ 0 .

(2)

In our case, we used a simple model which accounts for only the restricted (intraaxonal) water pool as represented by the streamlines, and can adequately provide the relative density distribution map and the RMSE needed for the stopping criterion. The set of streamlines is then filtered according to the weights given by COMMIT , discarding the ones with null contribution to the signal. This is based on the assumption that either a streamline contributes notably to the signal explanation or the following iterations will add more streamlines that will reduce the contributions of the previous ones.

2.3 STEP3: Graph’s Weights Update to Improve the Evaluation The third step is to update the graph using the voxel-wise fitting error. In particular, we focus on the streamlines density distribution of the set of candidate fibers in relation with the reference streamlines density obtained by processing the DW-MRI data. The graph is modified to improve the density coverage of the WM structure, promoting or penalizing paths passing through voxels with a streamlines density different from the reference one. In the proposed framework, we iterate over all the edges of the graph, updating their weights using the dissimilarity between the corresponding voxels streamline density and the ones of the reference density. For each edge, the updating function takes as input the original weight (computed based on the ODFs and geometric constraints) and the average between the differences in streamlines density of the nodes at its ends, and it assigns a new weight. The aim of this new value is to promote or penalize the passing of the streamlines through that specific voxel. This is emphasized using the parameter γ (see Algorithm 1). Moreover, the value of γ influences the streamline-tracking and the RMSE’s rate of decrease which determines the method’s rate of convergence. To summarize, if we let G be the voxel-graph built starting from the diffusion data, densit y_map the density map computed by COMMIT from the set of streamlines retrieved, GT _densit y_map the reference phantom density map, ROIs the regions among which we want to estimate the streamlines, E the set of edges E = e1 , e2 , ..., en of the graph G, W the set of the corresponding weights W = we1 , we2 , ..., wen , Nodes the set of nodes and W  the set of weights of ¯ the update-function has been implemented as described in the updated graph G, Algorithm 1.

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Input: (G, GT _densit y_map, densit y_map) Result: G¯ : W  ∝ W | (densit y_map - GT _densit y_map) foreach n ∈ N odes do node_densit y = densit y_map[ n] GT _n_densit y = GT _densit y_map[ n] n_densit y_di f f = node_densit y - GT _n_densit y foreach neighbour of n do average_node_densit y = (n_densit y_di f f + neighbour _densit y_di f f ) · 21 scalar _ f actor = w_n(n, neighbour ) + average_node_densit y if node_densit y == GT _n _densit y then continue end if node_densit y > GT _n _densit y then wn (n, neighbour ) = scalar _ f actor + scalar _ f actor ·γ else wn (n, neighbour ) = scalar _ f actor + scalar _ f actor · γ1 end end end

Algorithm 1: Procedure to update the weights of the graph’s edges using microstructure information.

To ensure that the reconstructed streamlines are consistent with the underlying anatomy, we always update the weights of the initial graph, whose edges encode the coherence between neighbouring voxels.

2.4 Testing Dataset To evaluate quantitatively our algorithm we used the simulated dataset prepared for the IEEE International Symposium on Biomedical Imaging (ISBI) 2013 Reconstruction Challenge [2]. The phantom consists of 27 known ground-truth fiber bundles, mimicking challenging branching, kissing, crossing structures at angles between 30◦ and 90◦ , with various bundle curvature and size (see Fig. 2a)). The normalized streamlines density of the ground truth is shown in Fig. 2b, and was obtained computing the number of fibers passing through each voxel and dividing the resulting image by the maximum value.

3 Results and Discussion Figure 3 shows that the spatial distribution of the streamlines reconstructed with our method is in closer agreement with the underlying ground truth streamline density. The process of flattening the density corresponding to bottlenecks created by the

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Fig. 2 Simulated dataset used for validation: a 3D view of the bundles geometry, b corresponding normalized streamline density, c mask used for tractography

original shortest path algorithm implies that the streamlines do not follow necessarily the most direct trajectory but the one which, after the graph update step, has the lowest sum of the edge weights along the path that connects the two regions, reflecting the reference WM pathways distribution. In the first column of Fig. 3, we can notice how some of the reconstructed streamlines are characterized by local changes of direction. This might be due to a fixed γ value for all nodes when updating the edge weights. This implies that is some cases we are promoting or penalizing excessively the passage through a node, causing local direction changes in streamlines. These erroneous streamlines can be discarded with the COMMIT evaluation. However, the use of the streamline density in the evaluation limits the filtering of all erroneous streamlines. In future work, we plan to use the full DW-MRI signal in our framework to update the edge weights, not solely optimizing for streamline density but also local streamline orientations. The decrease in the absolute error of the streamline density shown in the third column Fig. 3 corresponds to a more uniform coverage of the WM volume obtained by the streamlines reconstructed with our proposed approach. We can notice how our algorithm is able to retrieve paths that were poorly reconstructed without the microstructure information while flattening the overall density (see the red arrows).

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Fig. 3 Comparison between: a ground truth, b shortest path tractography without microstructure information and c our proposed method

4 Conclusions As all the streamline-tracking methods, graph-based shortest path tractography presents limitations. In this article, we introduced a method to overcome the problem of bottlenecks and consequent incorrect streamline density . This is possible with the graph data structure that allows an easy integration of local microstructure properties estimation. This guides the algorithm to provide a more realistic WM density representation. In future work, we will test the combination of graph-based tractography and microstructure imaging on more realistic DW-MRI simulation and in-vivo data.

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References 1. Bastiani, M., Shah, N.J., Goebel, R., Roebroeck, A.: Human cortical connectome reconstruction from diffusion weighted MRI: the effect of tractography algorithm. NeuroImage 62(3), 1732– 1749 (2012) 2. Caruyer, E., Daducci, A., Descoteaux, M., Houde, J.c., Thiran, J.p., Verma, R.: Phantomas: a flexible software library to simulate diffusion MR phantoms. In: International Symposium on Magnetic Resonance in Medicine (ISMRM 2014)). Milan, Italy (2014) 3. Ciccarelli, O., Catani, M., Johansen-Berg, H., Clark, C., Thompson, A.: Diffusion-based tractography in neurological disorders: concepts, applications, and future developments. Lancet. Neurol. 7(8), 715–727 (2008) 4. Cieslak, M., Brennan, T., Meiring, W., Volz, L.J., Greene, C., Asturias, A., Suri, S., Grafton, S.T.: Analytic tractography: a closed-form solution for estimating local white matter connectivity with diffusion MRI. NeuroImage 169, 473–484 (2018) 5. Daducci, A., Dal Palù, A., Lemkaddem, A., Thiran, J.P.: COMMIT: convex optimization modeling for microstructure informed tractography. IEEE Trans. Med. Imaging 34(1), 246–257 (2015) 6. Dijkstra, E.W.: A note on two problems in connexion with graphs. Numerische Math. 1(1), 269–271 (1959) 7. Fillard, P., Descoteaux, M., Goh, A., Gouttard, S., Jeurissen, B., Malcolm, J., RamirezManzanares, A., Reisert, M., Sakaie, K., Tensaouti, F., Yo, T., Mangin, J.F., Poupon, C.: Quantitative evaluation of 10 tractography algorithms on a realistic diffusion MR phantom. NeuroImage 56(1), 220–234 (jan 2011) 8. Iturria-Medina, Y., Canales-Rodríguez, E., Melie-García, L., Valdés-Hernández, P., MartínezMontes, E., Alemán-Gómez, Y., Sánchez-Bornot, J.: Characterizing brain anatomical connections using diffusion weighted MRI and graph theory. NeuroImage (jul, 2007) (2008) 9. Jbabdi, S., Bellec, P., Toro, R., Daunizeau, J., Plgrini-Issac, M., Benali, H.: Accurate anisotropic fast marching for diffusion-based geodesic tractograph. Int. J. Biomed. Imaging (2008) 10. Jeurissen, B., Descoteaux, M., Mori, S., Leemans, A.: Diffusion MRI fiber tractography of the brain. NMR Biomed., e3785 (2017) 11. Mori, S., Crain, B.J., Chacko, V.P., Van Zijl, P.C.M.: Three-dimensional tracking of axonal projections in the brain by magnetic resonance imaging. Ann. Neurol. 45(2), 265–269 (1999) 12. Parker, G.J.M., Wheeler-Kingshott, C.A.M., Barker, G.J.: Estimating distributed anatomical connectivity using fast marching methods and diffusion tensor imaging. IEEE Trans. Med. Imaging 21(5), 505–512 (2002) 13. Reisert, M., Mader, I., Anastasopoulos, C., Weigel, M., Schnell, S., Kiselev, V.: Global fiber reconstruction becomes practical. NeuroImage 54(2), 955–962 (2011) 14. Zalesky, A.: DT-MRI fiber tracking: a shortest paths approach. IEEE Trans. Med. Imaging 27(10), 1458–1471 (2008)

Deterministic Group Tractography with Local Uncertainty Quantification Andreas Nugaard Holm, Aasa Feragen, Tom Dela Haije and Sune Darkner

Abstract While tractography is routinely used to trace the white-matter connectivity in individual subjects, the population analysis of tractography output is hampered by the difficulty of comparing populations of curves. As a result, analysis is often reduced to population summaries such as TBSS, or made pointwise with similar interaction of remote and nearby tracts. As an easy-to-use alternative, we propose population-wide tractography in MNI space, by simultaneously considering diffusion data from the entire population, registered to MNI. We include voxel-wise quantification of population variability as a measure of uncertainty. The group tractography algorithm is illustrated on a population of subjects from the Human Connectome Project, obtaining robust population estimates of the white matter tracts. Keywords Tractography · Population analysis · Uncertainty quantification

1 Introduction Tractography is the process of integrating local directional info obtained by diffusion weighted imaging (DWI) to obtain a measure of the connectivity between points or regions in the brain. In population analysis, tractography is typically followed by a group aggregation step, where tractography results from multiple subjects are pooled and summarized in a way that enables intuitive and interpretable analysis of differences between populations. Current approaches to this problem rely on summarizing the individual subjects’ tractography results to make them comparable in a common A. N. Holm (B) · A. Feragen · T. Dela Haije · S. Darkner Department of Computer Science, University of Copenhagen, Copenhagen, Denmark e-mail: [email protected] A. Feragen e-mail: [email protected] T. Dela Haije e-mail: [email protected] S. Darkner e-mail: [email protected] © Springer Nature Switzerland AG 2019 E. Bonet-Carne et al. (eds.), Computational Diffusion MRI, Mathematics and Visualization, https://doi.org/10.1007/978-3-030-05831-9_30

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coordinate system. Such approaches typically discard information, and most methods also do not distinguish the interactions between nearby and remote tract estimates, leading to reduced confidence and overestimated uncertainty. To counter these problems, we propose group tractography , where the diffusion information from multiple subjects is aggregated in a common coordinate system prior to tractography. Related Work. The, perhaps, simplest approach to group analysis of tractography, is to aggregate the tractography output into heatmaps, which can be transported to standard space using image registration and analyzed using classical statistics or machine learning by concatenating the registered images into vectors. While simple, this approach has drawbacks: registration error, low resolution and individual variation can easily cause estimates of the same trajectory in two different subjects, to lie very close to each other, but not on top of each other. Although, intuitively, two such nearby trajectories should lead to a rather confident estimate of the trajectory, the heatmap will instead return a fat, unconfident estimate of the tract. In other words, the resulting estimates should expect to have low confidence and overestimated uncertainty. A second approach to group analysis, for instance used in TBSS [15], is to summarize tracts, understood as collections of estimated trajectories, using a single prototype trajectory. Such trajectories can be attributed with properties such as fractional anisotrophy (FA) or mean diffusivity (MD) along the curve. While such prototypes are easy to compare in population analysis, they also discard potentially useful information. Recent work includes various approaches to clustering and registering tracts [5, 11, 18]. While these algorithms move closer to solving the underlying problem, these solutions are also more difficult: Both variation in tractography performance and anatomical differences may lead to non-corresponding tracts or clusters in different subjects. Such issues are particularly dire in patients with diseases or pathologies. We contribute a simple and efficient algorithm for performing group tractography in standard space, along with local quantification of uncertainty in the population diffusion data on which the tractography depends. We provide an implementation and illustration of performance on data from the Human Connectome Project [4], which shows that the algorithm gives robust population estimates of the white matter tracts.

2 Methodology Diffusion MRI measures the directional diffusion of water at different locations (voxels) in the brain. Under the hypothesis that water diffuses primarily along white matter fibers, this allows us to estimate their pathways. In this paper, voxel-wise diffusion is summarized in a diffusion tensor characterized by its eigenvalues (describing the shape) and eigenvectors (describing the orientation). Below, we will refer to the eigenvector corresponding to the maximal eigenvalue as the principal direction.

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The eigenvalues also define the fractional anisotropy (FA) of the diffusion tensor, which quantifies the directional isotropy of its diffusion. Tractography can be performed in multiple ways, and methods are typically organized in two categories, namely deterministic and probabilistic tractography. We will focus on deterministic streamline tracking [13], which follows the tracts along the principal direction using Euler’s method. Our implementation of Euler’s method uses nearest neighbor interpolation. Defining t as the iteration/time, xt as the previous point, vt as the interpolated principal direction in xt and h as the step size, the point at time t + 1 is estimated as: xt+1 = xt + h

vt . |vt |

(1)

As the diffusion tensor is reflection symmetric, the principal eigenvector vt is only defined up to sign. To ensure that we stay consistent in the direction we are iterating, we update the sign of the principal direction vt using the sign of the inner product between the current and previous diffusion direction in Euler’s method: vt ← sign(vt−1 , vt )

vt . |vt |

This ensures that we take steps of an angle at most 90 degrees.

Algorithm 1: Group streamline tracking - Forward Euler scheme

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Input: p, V, FA Output: x t = 1, x1 = p initialize V0 repeat xc = nearest(xt ) Vt = {V(xc , i) | i = 1, . . . , N } for i = 1 : N do v vt,i ← sign(vt−1,i , vt,i ) |vt,i t,i |

if angle(vt−1,i , vt,i ) > 45 or FA(xc , i) < 0.2 then Vt ← Vt \ vt,i end end |Vt | vˆ t = |V1t | i=1 vt,i

vˆ t ← |ˆvvˆt | t if angle(ˆvt−1 , vˆ t ) > 45 then break end xt = xt−1 + h vˆ t t ←t +1 until |Vt | = 0 or t > max iterations;

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Deterministic Group Tractography. We propose to extend the streamline tracking algorithm to operate on an entire population. This can be obtained by adapting the principal direction in Euler’s method. Instead of applying the step vt in subject space, we propose to transfer all subjects’ principal directions into MNI-space, where we estimate a population dependent principal direction, defined as: N 1  vt,i vˆ t = sign(vt,i , vt−1,i ) , N i=1 |vt,i |

(3)

where i is the subject index, N is the population size and vt,i is the principal direction of the i’th subject at the point xt in MNI-space at time t. Euler’s method for group tractography thus becomes: xt+1 = xt + h

vˆ t . |ˆvt |

(4)

Euler’s method is flexible, therefore vˆ t can be swapped for various other estimates. While iterating through Euler’s method, we introduce two standard stopping criteria: the FA and the angle between the previous and current principal direction, with thresholds set as 0.2 and 45 degrees, respectively. The stopping criteria are applied to each subject individually, If a subject exceeds one of the thresholds, it will be excluded from the estimation of vˆ t , and it will not be used any further in the tracking procedure. Pseudo code for the group streamline tracking is detailed in algorithm 1. The pseudo code illustrates the process of estimating a single tract, starting at the given seed p. Here, we define the function xc = nearest(x) which returns the voxel center xc closest to x. The array V consists of the principal directions V(xc , i) at the MNIspace voxel center xc for subject i. The array FA is similarly defined, such that FA(xc , i) is the FA at the MNI-space voxel center xc for subject i. At any time t the set Vt denotes the population of principal directions vt,i at the current point xt . We start tracking at the seed point p, and use its principal direction to take the first step. After initialization, we find the new nearest point and apply our stopping criteria, to check if we want to terminate. If not we iterate until the stopping criteria are met or until a given amount of iterations are reached. After termination of Euler’s method we return the estimated tract. Local Uncertainty Quantification for Principal Directions. We quantify how well our subject-wise principal directions align in each voxel, as a measure of variation in the diffusion data. For this purpose we use the von Mises-Fisher (vMF) distribution, defined on the unit sphere, with probability density function f (x; μ, κ) = cd (κ)eκμ x , t

where

(5)

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Fig. 1 The population’s (normalized) principal directions at two example locations in the brain illustrate how κˆ distinguishes between different levels of directional variation. Voxels sampled inside (left) and outside (right) the corpus callosum show low and high directional variation, respectively, along with their associated κˆ values

cd (κ) =

 κ 2k+d  1 κ d/2−1 , and I (κ) = . d (2π )d/2 Id/2−1 (κ) Γ (d + k + 1)k! 2 k≥0

(6)

The vMF distribution is parametrized by μ and κ, where μ is the mean direction, and κ quantifies the concentration of the vectors on the unit sphere, see Fig. 1. As an analytical formula for κˆ is not generally possible, we will use a fast approximation of κˆ [2]: κ(x ˆ c) = ||



r¯ (xc )d − r¯ (xc )3 , 1 − r¯ (xc )2

(7)

V(x ,i)||

c i where r¯ (xc ) = and d is dimensionality. While this is not the most accurate N approximation of κˆ possible, better approximations need iterative methods, which are infeasible due to the large number of voxels. As high values of κˆ indicate high concentration, we interpret this as low uncertainty in the “population principal direction” used for group streamline tractography. This is illustrated for two example voxels in Fig. 1.

3 Experimental Validation Data. We illustrate group tractography using 10 subjects from The Human Connectome Project 1 (HCP). All subjects contain diffusion data, T1w images with a 1.25 mm resolution, and the b-values goes up to 3000. For the diffusion tensor model, we used FSL’s DTIFit with default settings. 1 https://db.humanconnectome.org/.

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Fig. 2 The heat map shows the uncertainty at each voxel, applied as an overlay on the MNI-atlas. White = low, red = high

Experimental Setup. The first step of the pipeline is to register all subjects into the common template space given by the MNI-atlas [6]. Registration was performed using FSL 2 [9, 16, 17] on the T1 weighted images (1.25 mm isotropic voxels), linear registration [8, 10] was used to create an initial transformation matrix for non-linear registration [1]. Having registered T1 images to MNI-space, the resulting warps were used to reorient the principal directions to match the newly registered data with the FSL function vecreg. Following registration, we provide our extended streamline tracking algorithm with the 10 subjects, to create the group tractography in MNIspace as described in Sect. 2. The step size h in Euler’s method was set to 0.25 voxels. Results. The κ-map in Fig. 2 shows the uncertainty between the population in MNIspace. We observe that the variation between subjects is low near corpus callosum, and high outside corpus callosum, as expected The heat map in Fig. 3 is the result of full-brain tractography, seeding in every MNI-space voxel which corresponds to white or gray matter in at least one subject. As the algorithm discards subjects that meet stopping criteria during tracking, we visualize how many subjects are used in any voxel (Fig. 4). As expected, the number of subjects used drops near the cortex, where individual anatomical variation is high. Finally we visualize the performance of group streamline tractography on the cortico-spinal tract (CST). From the full brain tractography shown in Fig. 3 we extract those streamlines starting in the brain stem (extracted from the Harvard-Oxford subcortical atlas). We randomly selected 10000 of the resulting streamlines, and visualized them using ExploreDTI [12] in Fig. 5.

4 Discussion and Conclusion We have proposed group tractography as an alternative to aggregating subject-wise tractography results into a population representation. Moreover, as part of the group 2 https://fsl.fmrib.ox.ac.uk/fsl/fslwiki/FSL.

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Fig. 3 The heat map shows the number of unique streamlines crossing each voxel, inserted as an overlay on the MNI-atlas. White = many, red = few. The counts were thresholded at 500, with max value 1394

Fig. 4 The heat map shows the number of subjects active in tracking at any voxel, inserted as an overlay on the MNI-atlas. Yellow = many, red = few

Fig. 5 Visualization of single subject streamline tractography (left) and group streamline tractography (right) in the CST, using a seed region defined by the brain stem

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tractography algorithm, we make local estimates of uncertainty in the population of principal directions used for tractography. We have demonstrated the use of group tractography on a population of HCP subjects. Our current implementation applies deterministic streamline tracking from DTI, and is therefore both simple and efficient, but also limited in its expressivity. While the idea extends to more complex models and probabilistic tractography, this also poses several challenges which are in themselves currently unsolved. The most important open problem would be the groupwise aggregation of fODFs: Should one estimate subject-wise fODFs and aggregate these into an uncertain fODF estimate in MNI space? In order to do this, we would need to transfer fODF models via registration, and then estimate their distributions, which are both nontrivial tasks. Alternatively, one could register the diffusion data to MNI space and estimate fODFs there. This would, however, require fODF interpolation, which for general fODFs typically leads to inflated uncertainty [3]. Extension of group tractography to more advanced diffusion models is thus an open problem for future work. Our current uncertainty estimates are local: They quantify the variation in the population’s principal directions at any voxel in MNI space. This is certainly useful, as it quantifies the uncertainty in the data going into tractography—but it does not quantify the uncertainty in the tractography results themselves. In particular, once a streamline hits an area with high uncertainty, one may expect that uncertainty to be propagated throughout the tracking. A common way to visualize tractography output is through heat maps, which does visualize propagated uncertainty. However, as pointed out in the introduction, heat maps do not quantify uncertainty per se, but rather give blurred-out, low-confidence estimates in the presence of variation. This effect could be avoided by quantifying the uncertainty in the actual streamline trajectories. While some uncertainty estimates of estimated tract trajectories exist [7, 14], these do not apply to streamline tracking algorithms, and make strong assumptions on the form of the uncertainty. Another open problem is thus to quantify the uncertainty in estimated trajectories. Finally, while our end goal is population analysis, our current algorithm only returns a population summary. Future work this also includes utilizing these summaries for interpretable detection of population differences, for instance in hypothesis testing. Acknowledgements Data were provided [in part] by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University. This research was supported by Center for Stochastic Geometry and Advanced Bioimaging, funded by a grant from the Villum Foundation, as well as a grant from the Lundbeck Foundation. TDH was covered by a block stipend from the Villum Foundation.

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Index

A Agglomerative hierarchical clustering, 349 Anisotropic fast marching, 310 Anisotropic IVIM, 251–254, 261, 262 Anisotropy, 184 Artificial neural network, 113 Axon diameter, 92

B Bayesian models, 111 Biomarker, 237, 238, 243 Blind source separation, 70 Brain atlas, 323 Brain development, 133 BrainVISA, 325 Breast, 227–229, 233–235, 237, 243 Bundle atlases, 324

C Cardiac, 227–229, 231, 240–242 Challenge, 177, 180 ComBat, 203, 205, 208, 210, 212, 213 COMMIT, 369–371, 373 Computational methods, 105 Connectivity, 281, 282, 289 Connectivity measure, 309 Connectome, 335, 342 Connectomist 2.0, 325 Constant solid angle, 301 Constrained spherical deconvolution (CSD), 56, 194, 197–199, 301, 347, 350, 361 Convolutional neural network, 115, 219 Cortical folding, 324, 325, 329 Cortical folding pattern, 324, 330 Cortical folds, 324

Cramer-Rao lower bound, 251, 256, 261 Cross entropy, 295

D DARTEL, 325, 326, 330 Data pooling, 157, 159, 160, 163–166, 169 Davies-Bouldin criterion, 349 Deep autoencoder, 116 Deep learning, 105, 124, 173, 174, 180, 194, 196–200 Deep neural network, 114 Demyelination, 266, 278 Denoising, 43 Deterministic streamline tracking, 379 Deterministic tractography, 347, 350, 351, 353, 354 Diffeomorphic, 324 Diffeomorphic inter-subject alignment, 323 DIffeomorphic Sulcal-based COrtical deformation (DISCO), 325, 326, 330 Diffusion, 194–196, 200, 227–232, 235, 237–243 Diffusion-attenuated signal, 184 Diffusion imaging, 173, 174 Diffusion magnetic resonance imaging (DMRI), 29, 70, 124 Diffusion MRI, 77, 133, 157–161, 163, 166– 169, 205, 212, 217, 218, 324, 335– 337, 339, 359, 361, 367, 368, 370, 371 Diffusion tensor, 31 Diffusion tensor imaging (DTI), 204–210, 213 Diffusion-weighted, 184, 265, 266, 268, 270, 273 Diffusion weighted imaging, 19, 43, 377

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388 Diffusion-weighted magnetic resonance imaging (DW-MRI), 105, 194–196, 200 Dipy, 301 Dispersion, 93 Distance in SO(3), 348

E Edge weight, 281, 282, 286, 289, 290 Eikonal PDE, 313 Encoding waveforms, 3 Enhancement, 217, 218, 221, 222 Ensemble average diffusion propagator (EAP), 30, 31, 36 Experimental design, 251

F False positives, 335, 336 Fiber classification, 149, 152 Fiber clustering, 346, 350, 352–354 Fiber distribution function, 59 Fiber orientation distribution (FOD), 71, 347, 350, 352, 353 Fiber orientation distribution function (fODF), 78, 184, 265–267, 269–274, 276–278, 361 FiberNavigator, 361, 362 Finsler function, 309 First year of life, 134 Folding pattern, 324, 325, 330, 332 Fractional anisotropy (FA), 351, 353 Functional correlation, 337–339 Functionally Informed COMMIT, 336 Funk-Radon transform (FRT), 32

G Gaunt coefficient, 80 Generative adversarial network, 117 Geodesic tractography, 310 Global tractography, 186 Graph-based deep learning, 133 Graph CNN, 134 Graph property, 281, 282, 286–290 Graph theory, 281, 282, 289 Greedy iterative fashion, 22 Group streamline tracking, 380 Group tractography, 378

H HARDI database, 324

Index Harmonization, 157, 159, 160, 163–169, 173–175, 177–181, 194, 200, 205, 208–210, 212, 213, 217–222 High angular resolution diffusion imaging (HARDI), 194, 200, 346 Histology, 95, 97, 194, 195, 197 HIV, 203–206, 208, 212, 213 Human connectome project, 300, 378

I ICBM 152, 325 Infant, 184 Information theory, 4 Inter-scanner, 194, 200 Inter-subject alignment, 325 Intravoxel incoherent motion (IVIM), 252, 253 ISBI 2013 phantom, 372 ISBI 2013 reconstruction challenge, 314

J Joint optimization problem, 22

K Kernel density estimation, 298 Kullback-Leibler divergence, 298

L Learning, 4, 6 Linear regression, 109 Liver, 227–232 Local quantification of uncertainty, 378 Longitudinal harmonization, 184 Longitudinal prediction, 134

M Machine learning, 105 Mapping fucntions, 184 Maximum likelihood, 21 Maximum mean discrepancy, 124 Mean direct flip distance, 300 Measurement space, 4 Mega-analysis, 163–165 Meta-analysis, 163–165 Method of moments, 184 Metric, 4 Microstructural modelling, 254, 255, 261 Microstructure, 93 Microstructure imaging, 58

Index Microstructure informed tractography, 370 MNI space, 325, 327 Morphologist, 325 Multi-center, 218 Multi-compartment, 60 Multi-compartment constrained spherical deconvolution, 60 Multi-compartment spherical mean, 60 Multi-fiber voxels, 350, 351, 353 Multiple sclerosis, 281, 283, 285–290 Multi-shell, 94 Multi-shell dMRI, 218–220 Multi-shot EPI, 20 Multi-site, 160, 163–166, 204, 205, 208, 212, 213 Multi-site study, 173, 174, 179 Muskuloskeletal, 227, 228, 238 Mutual information, 5

N Navigator-free, 20 Neighborhood resolved fiber orientation distributions (NRFOD), 145, 147, 153 Network, 281–283, 285–290 Neural networks, 143, 145 Neurite orientation dispersion and density imaging (NODDI), 127 Nonnegative matrix factorization, 70 Null Space, 194, 196–200 Number of peaks, 266, 267, 269–278

O Orientation distribution function (ODF), 32, 59

P Parametric model, 219 Perfusion, 252–255 Permutation, 283, 286, 290 Phantom, 336, 339–342 Phase inconsistencies, 20 Placenta, 252, 253, 256, 261 Probabilistic streamline tractography, 301 Probabilistic tractography, 347, 350, 351, 353, 354 Prostate, 227–229, 236–238, 243 Pulsed Gradient Spin-Echo, 57

Q Q-ball model, 325

389 q-space learning, 124

R Random forest, 110 Registration, 382 Reliability, 165, 168 Reorient the principal directions, 382 Resampling, 360, 362–364 Research reproducibility, 56 Residual graph convolution, 136 Restoration, 43 Return-to-axis probability (RTAP), 30–32 Rigid–motion, 20 Rotation invariant features (RIF), 78

S Sample complexity, 296 Shortest path, 367, 368, 370, 373, 374 Short fiber, 327, 330, 332 Short fiber bundles, 323, 325 SHResNet, 175, 177–181 Simulation, 14 Single-shot echo planar imaging, 19 Sparsity penalization, 72 Spatial-angular regularization, 51 Spherical, 174–177 Spherical deconvolution, 265–267, 273, 277, 278 Spherical harmonics, 45, 59, 78, 94, 219 Spherical mean, 56, 70, 94, 185 Spherical moments, 184 Spherical variance, 185 SPM software, 325 Streamline density, 367–369, 371–374 Streamline regularized deterministic tractography, 325 Streamlines, 144–152 Structural connectivity, 315 Sulci, 324–326, 330 Super-resolution reconstruction, 43 Supervised classification, 143, 144, 146, 152 Support vector machines, 112 Surprise, 295 Synthesis, 124

T Tensor distribution function (TDF), 203, 205, 207, 209, 210, 213 Tissue architecture, 124 Tissue microstructure, 77 Tissue-type segmentation, 70

390 TNB frame, 346, 347, 352 Total variation, 46 Tractography, 144–146, 148, 162, 166–169, 184, 265, 266, 273, 277, 278, 295, 335, 339, 359, 361, 365, 367–370, 373, 374 Tractography limitations, 372, 374 Tractography optimization, 336, 337, 340, 341 Tractometry, 359, 360, 364, 365 Training set, 143–148, 152 U U-bundle, 323–325, 332

Index U-fiber, 324, 326 Uniform kernel, 266, 267, 271, 276–278 U-shaped fiber bundles, 324

V Validation, 97 Validation set, 147, 148 Von Mises-Fisher, 380

W Waveform sets, 4 White matter, 92, 143, 144, 324