Statistical Models for Diffusion Weighted MRI Data

Team Background Diffusion Anisotropy and White Matter Connectivity Interpolation and Smoothing of Diffusion Tensors Non-Euclidean metrics on Ω (SPDMs) Models for raw data and DT estimation References Statistical Models for Diffusion Weighted MRI Alexey Koloydenko Department of Mathematics Royal Holloway University of London 1st December 2008 Koloydenko Statistical Models for Diffusion Weighted MRI Data

Team Prof. Ian L. Dryden Statistics, Nottingham U. Dr. Paul Morgan Nottingham U. Medical U. South Carolina Prof. Dorothee Auer Academic Radiology, Nottingham U. Dr. Christopher Tench Clinical Neurology, Nottingham U. PhD candidate Diwei Zhou Marie-Curie Medical Imaging Fellow, Statistics, Nottingham U. Other Marie-Curie Medical Imaging Fellows Nottingham U.

Background MRI is a variety of relatively non-invasive methods to examine microstructure of various types of matter. All MRI modalities, or weightings (proton density, T 1, T 2, FLAIR, diffusion), are based on common principles:

Basic Principles Nuclei (e.g. proton=nucleus of H as in H 2 O) can be collectively programmed by imposition of external electromagnetic disturbances such as a very strong static magnetic field and radio-frequency signals. Once a radio-frequency disturbance is removed, the engaged nuclei begin to relax (in various ways), echoing R-F signals and gradually forgetting their program. The echos are received and processed. Since the targeted nuclei (most commonly protons) are identical regardless of their location and other factors, any observed difference in their response is attributed to characterisitics of the surrounding environment (e.g. density, flow). Inverting such relationships forms a basis for inference about the environment.

The MRI modalities differ by ways in which they assess the collective response of the nuclei to the program. Indeed, most modalities assess collective behaviour of the nuclei in the course of relaxation. Diffusion weighting encodes (local) diffusion rates (of water molecules) [Stejskal & Tanner, 1965] and is one of the more recent MRI modalities [Basser et al., 1994] used in clinical practice.

Diffusion MRI probes matter locally in predefined directions q i = (q 1i, q 2i, q 3i ) T RP 2, i = 1, 2,..., N estimating distribution of X, displacement of water molecules within a material over a fixed time interval t. In structureless environments, water diffuses equally in all directions, resulting in rotationally invariant (isotropic) p(x), pdf of X. Microstructure (e.g. white matter axons, or nerve fibres) creates anisotropy as the molecules prefer to diffuse along instead of across the denser matter (e.g. fibre walls). Thus, p(x) can reveal directionality of the material in addition to the material s density.

[Alexander, 2006]

Diffusion Tensor A standard ( single compartment ) model for diffusion X N(0, 2tD), t - diffusion time, D - diffusion tensor Multicompartment models - mixtures of Gaussians K p(x) = a k f (x; 0, 2tD k ), k=1 K k=1 a k = 1 a 1,..., a K > 0 Diffusion tensors are non-negative definite symmetric matrices, can be conveniently identified with ellipsoids via λ 1 0 0 D = V 0 λ 2 0 V T. 0 0 λ 3

What is Diffusion MRI good for? Axons (fibres), brain s main connectors, form white matter bundles. Various developmental and pathological conditions of the brain are expressed through white matter organization and integrity. DMRI can provide mapping of the connectivity of the brain. In particular, DMRI aims to improve diagnosis of stroke, epilepsy, and multiple sclerosis.

Measures of Connectivity are usually measures of anisotropy, derived from D: Eigenvectors 3 0 0 D = 0 2 0 e 1 = (1, 0, 0) T e 2 = (0, 1, 0) T V = e 1 e 2 e 3 = I 0 0 1 e 3 = (0, 0, 1) T Invariant characteristics Mean diffusivity MD = tr(d) 3 = 2 Volume ratio VR = 3/4 1 MD 3 3P (λ j MD) 2 Fractional anisotropy FA = j=1 = det(d) 3 2 3P λ 2 j j=1 0.4629 < 1

Raw DMRI data FA maps probing a single direction q i a single DT model fitted to data from 30 directions

Interpolation and smoothing Need for interpolation and smoothing In group comparison studies, arrays of FA values within a certain region of interest (ROI) are compared. Multiple instances of ROI need to be coregistered. DWMRI resolution is relatively low, e.g. voxel= 1.25mm 1.25mm 2.5mm solutions: Interpolation carried out on FA maps may be naive, whereas (additional) interpolation carried out on raw data would be difficult and may be unnecessary. Diffusion tensors D? Significance: Conclusions (e.g. psychiatric) might depend on the method used [Chao et al., 2008].

How to interpolate Ds? How to average Ds? Let Ω be the set of 3 3 non-negative definite symmetric real matrices D. If D 1, D 2..., D N Ω and w 1, w 2,..., w N are non-negative weights with w i = 1, then wi D i Ω. However, these (Euclidian) averages might be problematic: interpolating paths {wd 1 + (1 w)d 2, w [0, 1]} are prone to swelling (Figure 1, row 1), (and extrapolation can lead out of space).

Figure 1: Euclidian (1), Cholesky (2), Procrustes (3), Riemannian* (4), Log-Euclidian (5), Root-Euclidian (6).

Other metrics, averages, and statistics I Given a distance function d on Ω Ω and a probability distribution of D with density f, ED = arg inf d(d, D ) 2 f (D)dD D Ω is known as the Fréchet mean. Some relevant non-euclidian distances Log-Euclidian [Arsigny et al., 2007] (Figure 1, row 5) d L (D 1, D 2 ) = log(d 1 ) log(d 2 ) Weighted average - geometric: exp ( w i log(d i ))

Other metrics, averages, and statistics II Affine invariant Riemannian metric [Batchelor et al., 2005, Bio et al., 2006, Fletcher & Joshi, 2007, Moakher, 2005, Schwartzman et al., 2008] (Figure 1, row 4) ( ) d R (D 1, D 2 ) = log D 1/2 1 D 2 D 1/2 1 Cholesky-based distance [Wang et al., 2004] (Figure 1, row 2): chol(d)chol(d) T = D, where chol(d) is lower triangular, positive diagonal d C (D 1, D 2 ) = chol(d 1 ) chol(d 2 ) Weighted average - ( i=1 w ichol(d i )) ( i=1 w ichol(d i )) T

Other metrics, averages, and statistics III Square root-based distance [Dryden et al., 2008] (Figure 1, row 6) d 1/2 (D 1, D 2 ) = D 1/2 1 D 1/2 2. distance based on Procrustes size-and-shape [Dryden et al., 2008], [Le, 1988], [Kendall, 1989] (Figure 1, row 3) d S (D 1, D 2 ) = min L 2 L 1 R, R O(3) where LL T = D (e.g. L = chol(d), or L = D 1/2 ). The minimizer R = UV T, where U, V are as in SVD L T 2 L 1 = V ΛU T

Relevant consequences Interpolation between D 1 and D 2 via wl 1 R + (1 w)l 2 For N > 2, sample Fréchet means ˆD arg min D Ω N d S (D i, D) 2 i=1 can be found using the Generalized Procrustes Algorithm [Gower, 1975]. More generally, the same algorithm works with general weights w 1, w 2,..., w N which is also useful for smoothing. Confidence bounds can be computed and effects tested by mapping the data to the tangent space of Ω at ˆD.

Team Background Diffusion Anisotropy and White Matter Connectivity Interpolation and Smoothing of Diffusion Tensors Non-Euclidean metrics on Ω (SPDMs) Models for raw data and DT estimation References Figure 2: FA maps obtained from healthy volunteer data, single compartment DTI estimation based on a Bayesian approach [Zhou et al., 2008] before (left) and after (right) Procrustes based interpolation with 3:1 upsampling and smoothing. Koloydenko Statistical Models for Diffusion Weighted MRI Data

d F (D 1, D 2 ) = inf R O(3),β R L 2 L 2 βl 1R, is the scale invariant version of d S leading to a new measure of anisotropy 3 PA(D) = ( λ j λ) 2 3/2d F (I, D) = 3 j=1 2 3 λ j j=1 similarly to FA = 3 2 3 (λ j λ) 2 j=1 3 λ 2 j j=1 0 PA 1.,

Team Background Diffusion Anisotropy and White Matter Connectivity Interpolation and Smoothing of Diffusion Tensors Non-Euclidean metrics on Ω (SPDMs) Models for raw data and DT estimation References Figure 3: Original FA map (top left), FA (top right) and PA (bottom left) maps after the interpolation and smoothing, close-ups revealing anticipated separation of cingulum and corpus callosum. Koloydenko Statistical Models for Diffusion Weighted MRI Data

Team Background Diffusion Anisotropy and White Matter Connectivity Interpolation and Smoothing of Diffusion Tensors Non-Euclidean metrics on Ω (SPDMs) Models for raw data and DT estimation References Models for raw data and DT estimation Noiseless DMRI signal S(q) = S(0) p(x) exp(iqx)dx, R 3 S(0) is signal with no diffusion-weighting, q = control paramater q/ q R 3. Spherical schemes fix control parameter : q q/ q S 2 Under the zero-advection assumption p(x) = p( x) S(q) = S(0) p(x) cos(qx)dx R 3 S( q) = S(q) q RP 2 Koloydenko Statistical Models for Diffusion Weighted MRI Data

hemi-truncated icosahedron provides 30 axial directions "uniformly" covering RP 2.

Additive and independent Gaussian noise on Re(S) and Im(S) = 0 Rician noise on S, Gaussian. Gaussian diffusion assumption: hence p(x) = S(q) = K a k f (x; 0, 2tD k ) k=1 K a k exp(q T D k q) k=1 Given directions q 1,..., q N, the usual model for data: S = µ + ε, Var(ε) = σ 2 I, sometimes misleadingly referred to as the "Gaussian mixture model".

Gaussian mixture model: K f q (s; θ) = a k f (s; s e tqt D k q, σ 2 ), k=1 f (s; µ, σ 2 ) - univariate normal pdf. Same mean response but var[s(q)] = σ 2 + (s ) 2 (e tqt D 1 q e tqt D 2 q ) 2 α(1 α), where K = 2, α = a 1. Different data generation mechanism, still plausible EM-based estimation of parameters and "hidden states"

D 1 = ( 5.5 0 4.5 ) 0 2 0 = 4.5 0 5.5 ( 1 0 1 2 2 0 1 0 1 0 1 2 2 ) ( 10 0 0 0 2 0 0 0 1 ( ) 1 0 1 2 2 0 1 0 1 2 0 1 2 ) D 2 = 5 0 0 0 1 0 0 0 10 Figure 4: Hidden state missclassification errors estimated from the simulated data (left) and theoretical errors (right) based on the true parameters.

Figure 5: MC simulation of 1000 ML estimates of D 1. Rays through the origin trace 95% empirical confidence ellipses (CE) around the intrinsically averaged axial directions (top left): principal (red), secondary (green), and ternary (cyan). Planes tangent to S 2 at the mean principal, v 1 (bottom left), secondary, v 2 (top right), and ternary, v 3 (bottom right), directions preserve the distances from the point of tangency to the individual estimates; the planar 95% CEs are mapped back to the sphere via the exponential map.

Team Background Diffusion Anisotropy and White Matter Connectivity Interpolation and Smoothing of Diffusion Tensors Non-Euclidean metrics on Ω (SPDMs) Models for raw data and DT estimation References Conclusions The non-euclidean analysis inspired by DT MRI naturally extends to other applications where covariance matrices need to be processed, and possibly beyond since SPDMs are very common (e.g. quantum states) Hidden MRF models on discrete hemispheres naturally extend independent mixture models "Using a new MRI method, researchers at University X suggest that condition Y might have to do with neurobiological changes of kind Z in the white matter fibres" Koloydenko Statistical Models for Diffusion Weighted MRI Data

Hidden MRF, neighbourhoods on discrete hemispheres

References I Alexander, D. 2006. Visualization and image processing of tensor fields. Springer. Chap. An introduction to computational diffusion MRI: the diffusion tensor and beyond. Arsigny, Vincent, Fillard, Pierre, Pennec, Xavier, & Ayache, Nicholas. 2007. Geometric means in a novel vector space structure on symmetric positive-definite matrices. Siam journal on matrix analysis and applications, 29(1), 328 347. Basser, P. J., Mattiello, J., & LeBihan, D. 1994. Mr diffusion tensor spectroscopy and imaging. Biophysical journal, 66(1), 259 267. Batchelor, P. G., Moakher, M., Atkinson, D., Calamante, F., & Connelly, A. 2005. A rigorous framework for diffusion tensor calculus. Magnetic resonance in medicine, 53(1), 221 225.

References II Bio, Théme, Pennec, Xavier, Pennec, Xavier, Fillard, Pierre, Fillard, Pierre, Ayache, Nicholas, Ayache, Nicholas, & Epidaure, Projet. 2006. A riemannian framework for tensor computing. International journal of computer vision, 66, 41 66. Chao, Tzu-Cheng, Chou, Ming-Chung, Yang, Pinchen, Chung, Hsiao-Wen, & Wu, Ming-Ting. 2008. Effects of interpolation methods in spatial normalization of diffusion tensor imaging data on group comparison of fractional anisotropy. Magnetic resonance imaging, In Press, Corrected Proof,. Dryden, Ian L., Koloydenko, A., & Zhou, Diwei. 2008. Non-euclidean statistics for covariance matrices, with applications to diffusion tensor imaging. Annals of applied statistics, submitted.

References III Fletcher, P.T., & Joshi, S. 2007. Riemannian geometry for the statistical analysis of diffusion tensor data. Signal processing, 87(2), 250 262. Moakher, Maher. 2005. A differential geometric approach to the geometric mean of symmetric positive-definite matrices. Siam journal on matrix analysis and applications, 26(3), 735 747. Schwartzman, Armin, Dougherty, Robert F., & Taylor, Jonathan E. 2008. False discovery rate analysis of brain diffusion direction maps. Ann. appl. stat, 2(1), 153 175. Stejskal, E. O., & Tanner, J. E. 1965. Spin Diffusion Measurements: Spin Echoes in the Presence of a Time-Dependent Field Gradient. The journal of chemical physics, 42(1), 288 292.

References IV Wang, Z., Vemuri, B.C., Chen, Y., & Mareci, T. 2004. A constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex DWI. Ieee trans. on medical imaging, 23(8). Zhou, Diwei, Dryden, Ian, Koloydenko, Alexey, & Li, Bai. 2008 (March). A Bayesian method with reparameterization for diffusion tensor imaging. Page 69142J of: Reinhardt, Joseph M., & Pluim, Josien P. W. (eds), Proceedings of spie medical imaging 2008: Image processing, vol. 6914.