Registration, atlas generation, and statistical analysis of high angular resolution diffusion imaging based on riemannian structure of orientation distribution functions 1

Registration, Atlas Generation, and Statistical Analysis of High Angular Resolution Diffusion Imaging based on Riemannian Structure ofOrientation Distribution Functions Jia Du Department

Registration, Atlas Generation, and Statistical Analysis of High Angular Resolution Diffusion Imaging based on Riemannian Structure of

Orientation Distribution Functions

Jia Du Department of Bioengineering National University of Singapore

A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS OF

PhilosophiæDoctor (PhD)

June 18, 2013

1 Reviewer:

2 Reviewer:

3 Reviewer:

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I hereby declare that the thesis is my original work and it has been written

by me in its entirety I have duly acknowledged all the sources of

information which have been used in the thesis

This thesis has also not been submitted for any degree in any university

previously

Jia Du,June 18, 2013

I would like to express my gratitude to my advisor, Dr Anqi Qiu, forgiving me professional guidance, unending encouragement and full researchsupport I appreciate all her contributions of time and ideas over the course

of my Ph.D studies I am also deeply grateful to our collaborator, Dr.Alvina Goh, in the Department of Mathematics, National University ofSingapore I thank her for the technical support and constructive advicethroughout this work

I would also like to thank my lab mates and friends who have been giving

me advice and support during those four years: Dr Jidan Zhong, JordanBingren Bai, Ta Anh Tuan, Hock Wei Soon, Dr Sergey Kushnarev, YanboWang, Kelei Chen, Jiajing Li, Mengqiao Dai, and Liuya Min

- 1033 ,2012.

• Jia Du, Alvina Goh, Anqi Qiu, Bayesian Estimation of White Matter Atlas from High Angular Resolution Diffusion Imaging IEEE Transactions on Image Pro-

cessing, submitted

• Jia Du, Alvina Goh, Sergey Kushnarev, Anqi Qiu, Geodesic Linear Regression

on Orientation Distribution Function with its Application to Aging Study

neering, 2011 (oral presentation)

• Jia Du, A Pasha Hosseinbor, Moo K Chung, Andrew L Alexander, Anqi Qiu, Diffeomorphic Metric Mapping of Hybrid Diffusion Imaging based on BFOR

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Signal Basis, Information Processing in Medical Imaging (IPMI), 2013.

• Jia Du, Alvina Goh, Anqi Qiu, Bayesian Atlas Estimation from High Angular Resolution Diffusion Imaging (HARDI), Geometric Science of Information (GSI),

1.1 Motivation 1

1.2 Research Challenges and Thesis Contributions 6

1.2.1 Registation 6

1.2.2 Atlas Generation 10

1.2.3 Statistical Analysis 12

2 Background 15 2.1 Riemannian Manifold of ODFs 15

2.2 Large Deformation Diffeomorphic Metric Mapping 18

2.2.1 Diffeomorphic Metric 18

2.2.2 Conservation Law of Momentum 20

3 Diffeomorphic Metric Mapping of High Angular Resolution Diffusion Imag-ing based on Riemannian Structure of Orientation Distribution Functions 23 3.1 Affine Transformation on Square-Root ODFs 25

3.2 Diffeomorphic Group Action on Square-Root ODFs 29

3.3 Large Deformation Diffeomorphic Metric Mapping for ODFs 32

3.3.1 Gradient ofJ with respect to m t 34

3.3.1.1 Derivation of the gradient ofE x with respect toφ1 38

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3.3.2 Euler-Lagrange Equation for LDDMM-ODF 41

3.3.3 Numerical Implementation 41

3.4 Synthetic Data 44

3.5 HARDI Data of Children Brains 47

3.5.1 Comparison of LDDMM-FA, LDDMM-DTI and LDDMM-ODF 47 3.5.2 Comparison with existing ODF registration algorithm 52

3.5.3 Computational complexity of LDDMM-FA, LDDMM-ODF and LDDMM-Raffelt 54

3.6 Summary 55

4 Bayesian Estimation of White Matter Atlas from High Angular Resolution Diffusion Imaging 59 4.1 General Framework of Bayesian HARDI Atlas Estimation 61

4.2 The Shape Prior of the Atlas and the Distribution of Random Diffeo-morphisms 63

4.3 The Conditional Likelihood of the ODF Data 65

4.4 Expectation-Maximization Algorithm 67

4.4.1 Derivation of Update Equations ofσ2 andm0in EM 70

4.5 Results 71

4.5.1 HARDI Atlas Generation 73

4.5.2 Convergence and Effects of Hyperatlas Choice of the HARDI Atlas Estimation 74

4.5.3 Aging HARDI atlases 81

4.5.4 Comparison with existing method 82

4.6 Summary 83

5 Geodesic Regression of Orientation Distribution Functions with its Appli-cation to Aging Study 85 5.1 Geodesic Regression on the ODF manifold 86

5.1.1 Least-Squares Estimation and Algorithm 89

5.1.1.1 Derivation of the Least-Squares Estimation 90

5.1.2 Statistical Testing 94

5.2 Experiments 95

5.2.1 Experiments on Synthetic ODF Data 95

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5.2.2 Experiments on Real Human Brain Data: Aging Study 101

5.2.2.1 Image Acquisition and Preprocessing 1015.2.2.2 Geodesic Regression of ODFs and Aging Effect 1035.3 Summary 106

6 Conclusion and Future Work 111

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Progress in the diffusion weighted magnetic resonance imaging (DW-MRI)techniques over the last two decades has enabled neuroscientists to imagethe human brain white matter in-vivo and from large populations Diffusiontensor imaging (DTI), which models the axonal orientations of neuronsusing a three-dimensional ellipsoid tensor, has become one of the mostpopular methods to study the white matter micro-structure for identifyingneuropathology of mental illnesses and understanding fundamental neu-roscience questions on brain connections However, a major shortcoming

of DTI is that it can only reveal one dominant axonal orientation at eachlocation while between one to two thirds of the human brain white matterare thought to contain multiple axonal bundles crossing each other Recentadvances in DW-MRI, such as high angular resolution diffusion imaging(HARDI), address this well-known limitation of DTI by modeling the waterdiffusion with an orientation distribution function (ODF) that can capturemultiple axonal orientations at a voxel For both scientific and clinicalapplications, it is necessary to develop methods to represent, compare andmake correct inferences from the rich information provided by HARDIdata However, the main challenge arises from the complexity of HARDIdata as the existing analysis frameworks based on scalar images or DTI areunable to handle such data

The main contribution of this thesis is providing an HARDI-based analysisframework for the studies of white matter similarities and differencesacross large populations Under a unified Riemannian manifold of ODF,the framework includes three components: registration, atlas generationand statistical analysis Firstly, we propose a novel ODF-based registrationalgorithm, which seeks an optimal diffeomorphism between ODFs of two

subjects in a spatial volume domain and at the same time, locally reorients

an ODF in a manner such that it remains consistent with the surroundinganatomical structure Next, we develop a Bayesian probabilistic model

to estimate the ODF atlas for a specified population, which serves as acommon space to eliminate statistical bias introduced during the atlasselection In addition, to perform statistical inference on ODFs in thecommon space, we develop an algorithm that allows for geodesic regression

on directly the manifold of ODFs and thus, avoid the loss of potentialinformation during dimension reduction or feature extraction from ODFs.Finally, we apply this framework to examine the effects of normal aging in

a large group of healthy subjects spanning the adult age range The resultsshow that the ODF-based framework is able to detect age-related changes

in the white matter regions where fibers cross, and thus, offer new insightsinto the understanding of white matter microstructure deterioration duringnormal aging

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List of Tables

4.1 Convergence of the atlas quantified through the√

ODF metric squarebetween the atlases estimated at the current and previous iteration and

σ2in each iteration of the atlas estimation 755.1 Numbers of voxels with age-related significance in each regressionsout of14881 voxels in the white matter mask after the correction formultiple comparisons 105

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LIST OF TABLES

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List of Figures

1.1 Illustration of DTI versus HARDI The colors in panel (b) indicts thefractional anisotropy (FA) of each tensor, where blue stands for low FAvalue and red for high value Similar to the shape of ODF, the colors

in panel (c) also indices the relative values of ODF in each direction,where blue stands for low ODF value and red for high value 51.2 The ODF-based analysis framework for the HARDI-based studies ofwhite matter similarities and differences across large populations 72.1 Illustration of the manifold of square-root ODFs (picture token andmodified from [1])) 162.2 Flow equation 192.3 Mapping one shape to another via the group action of diffeomorphictransformation On the right, the diffeomorphisimφ is shown on the

square grid 192.4 Geodesic specified by initial momentum 213.1 The role of Chapter 3 in the ODF-based analysis framework 243.2 Illustration of affine transformation on square-root ODFs (Similar tothe shape of ODF, the colors of ODF also indices the relative values ofODF in each direction, where blue stands for low ODF value and redfor high value.) 25

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LIST OF FIGURES

3.3 Examples of local affine transformations on an isotropic ODF in thefirst row, an ODF with a single orientation fiber in the middle row,and an ODF with crossing fibers in the bottom row From left toright, three types of affine transformations,A, on the ODFs are demon-

strated: in panel (a), a rotation with angle θ z, where A = [cos θ z −

sin θ z 0; sin θ z cos θ z 0; 0 0 1]; in panel (b), a vertical shearing withfactor ρ y, where A = [1 0 0; −ρ y 1 0; 0 0 1]; and in panel (c), avertical scaling with factorς y whereA = [1 0 0; 0 ς y 0; 0 0 1] 303.4 Illustration of diffeomorphic group action on square-root ODFs 323.5 The first and second rows respectively illustrate the original HARDIand their enlarged images Compared to the image on panel (a), theimage on panel (b) has the same ODFs but a different ellipsoidal imageshape, while the image on panel (c) shows different ODFs but thesame circular image shape Panels (d) and (e) show the deformations(grid) and the corresponding momenta (arrows), calculated using∇ φ1E

in Eq (3.19), for mapping the image on panel (a) to panels (b) and(c), respectively Panels (f) and (g) show the deformations and thecorresponding momenta, calculated using the gradient in our previouswork [2], for mapping the image on panel (a) to panels (b) and (c),respectively 453.6 Comparison between the LDDMM-ODF and LDDMM-DTI algorithms.Panels (a, b) respectively show the template and target HARDI and theirenlarged images, where the ODF or diffusion tensor at each locationcontains two crossing fibers with equal orientation distribution Panel (c)illustrates the template HARDI image transformed via the deformationgiven in panel (d), the result of the LDDMM-ODF algorithm Panel (e)illustrates no deformation found via the LDDMM-DTI algorithm andthus the template HARDI image remains 46

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LIST OF FIGURES

3.7 Panels (a-d) respectively show the maps of mean symmetrized Kullback–Leibler (sKL) divergence of the ODFs between the template and the sub-jects deformed via affine, LDDMM-FA, LDDMM-DTI, and LDDMM-ODF For each voxel, the red (sKL=0.5) indicts the difference betweenthe template and the deformed subjects is large, while the blue (sKL=0)indicts the two corresponding ODFs are equal 503.8 Panels (a-d) show the maps of mean squared error of spherical har-monics coefficients (MSE of SH) of the ODFs between the templateand the subjects deformed via affine, LDDMM-FA, LDDMM-DTI, andLDDMM-ODF respectively For each voxel, the red (MSE=0.1) indictsthe difference between the template and the deformed subjects is large,while the blue (MSE=0) indicts the two corresponding ODFs are equal 513.9 sKL and MSE of SH cumulative distributions across the whole brainimage and averaged over all 25 subjects are shown in blue for affine,cyan for LDDMM-FA, yellow for LDDMM-DTI, and red for LDDMM-ODF, respectively 523.10 Panels (a-h) show the maps of mean symmetrized Kullback–Leibler(sKL) divergence of the ODFs between the template and the subjectsdeformed via affine, LDDMM-FA, LDDMM-DTI, and LDDMM-ODFfor the three major white matter tracts of the corpus callosum (CC)and bilateral corticospinal tracts (CST-left, CST-right) For each voxel,the red (sKL=0.5) indicts the difference between the template and thedeformed subjects is large, while the blue (sKL=0) indicts the twocorresponding ODFs are equal 563.11 sKL averaged over all25 subjects are shown for the corpus callosum(CC) and bilateral corticospinal tracts (CST-left, CST-right) when affine(blue), LDDMM-FA (cyan), LDDMM-DTI (yellow), or LDDMM-ODF(red) are applied 573.12 Dice overlap ratios averaged over all 25 subjects deformed by affine(blue), LDDMM-FA (cyan), LDDMM-DTI (yellow), or LDDMM-ODF(red) 57

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LIST OF FIGURES

3.13 Spatial distribution of mean squared error of spherical harmonics ficients across all subjects For each voxel, the red (MSE=0.1) indictsthe difference between the template and the deformed subjects is large,while the blue (MSE=0) indicts the two corresponding ODFs are equal 583.14 Illustration of three fiber tract masks, CST-left, CST-right and CC 583.15 Mean squared error of spherical harmonics coefficients (MSE of SH) inthe region where the CC and CST intersects each other 584.1 The role of Chapter 4 in the ODF-based analysis framework 614.2 Illustration of the general framework of the Bayesian HARDI atlasestimation 634.3 The evolution ofψ0(s, x) over the optimization of the atlas estimation.

coef-Panels from left to right showψ0(s, x) before the optimization, at the

first, fifth, and tenth iterations, respectively The intensity indicates the

√

ODF metric of each voxel with respect to the spherical ODF Thelarger the value, the more anisotropic the ODF is 744.4 Illustration of the branching and crossing bundles in the estimatedatlases over the entire population group Panels (a,d,g) show the ODFfield in the coronal, axial, and sagittal views In each row, the secondand third panels show two zoom-in regions for branching and crossingbundles corresponding to the anatomy on the first panel 774.5 The evolution of the average diffeomorphic metric between individualsubjects and the estimated atlas, with the standard deviation shown bythe error bars 784.6 Influences of the hyperatlas on the estimated atlas Two HARDI datasets(panels (a, c)) were respectively used as the hyperatlas in the Bayesianatlas estimation, which generated the atlases shown in panels (b, d).Panel (e) shows the√

ODF metric square between the two hyperatlases

on (a, c), while panel (f) shows that between the atlases on (b, d) 79

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LIST OF FIGURES

4.7 The cumulative distributions of the√

ODF metric square between the two hyperatlases (Figure 4.6 (a, c)) and between the two estimated atlases (Figure 4.6 (b, d)) are respectively shown in the dashed and solid lines For each voxel, the red (metric=0.1) indicts the difference between the template and the deformed subjects is large, while the blue

(metric=0) indicts the two corresponding ODFs are equal 80

4.8 Comparison of HARDI atlases respectively generated from young and old adults In each row, the last three columns show three zoom-in regions for branchzoom-ing and crosszoom-ing bundles correspondzoom-ing to the anatomy given on the first panel 82

4.9 Comparison between Bayesian and averaged atlases 84

5.1 The role of Chapter 5 in the ODF-based analysis framework 86

5.2 Geodesic regression on manifoldΨ . 88

5.3 Illustration of synthetic ODFs for single (a) and crossing fibers (b) 96

5.4 Illustration for synthetic √ ODF data, regression result and ground truth under four levels of noise (M = 0.1, 0.5, 1.0, 2.0): In each panel, each column shows the ODFs at x i = 0, 0.2, 0.4, 0.6, 0.8, 1.The first five rows illustrate the synthetic ODFs, while the next row shows the regression result The bottom row shows the ground truth for the geodesic regression 97

5.5 Evaluation of the geodesic regression accuracy using synthetic√ ODF Panels (a) and (b) show the plots of the mean square error of ˆψ and ˆξfor estimated geodesic regression at four noise levels (M = 0.1, 0.5, 1, 2) against the number of observationsn respectively . 98

5.6 Consistency of results under different metrics including the geodesic distance, the L2 norm of spherical harmonics coefficients, and the symmetric Kullback-Leibler divergence between ODFs, 99

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