Multivariate Statistical Analysis of Deformation Momenta Relating Anatomical Shape to Neuropsychological Measures

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1 Multivariate Statistical Analysis of Deformation Momenta Relating Anatomical Shape to Neuropsychological Measures Nikhil Singh, Tom Fletcher, Sam Preston, Linh Ha, J. Stephen Marron, Michael Wiener, and Sarang Joshi Scientific Computing and Imaging Institute (SCI) The University of Utah June 17, 2010

2 Riemannian metric and atlas construction Geodesics Geodesic computation and deformation momenta Multivariate statistical analysis Partial Least Squares and SVD ADNI data Implementation details Evolving the mean along LV directions (Deforming brain) Evolving the mean along LV directions (Log Jacobians)

3 Given a Large collection of anatomical images of subjects with detailed Neuropsychological assessments how does one relate anatomical variation to Neuropsychological variables. Driving problem: The ADNI database currently has 313 Mild Cognitive Impairment (MCI) subjects each with detailed Neuropsychological evaluations. Each Neuropsychological evaluation is a real score associated with various examinations such as Clinical Dementia Rating scale, Audio Verbal Learning test (immediate and delayed),.

4 motivation... Figure: image space and clinical response space

5 motivation... Conventionally, anatomical variation has been studied by generating transformations between each subject in the population and the pre-selected template. We study anatomical variation by simultaneously generating transformations between entire population and a common reference coordinate system in large deformation diffeomorphic setting (LDDMM)

6 motivation... Earlier studies on characterization on neuroanatomical changes: statistical analysis of deformation maps using associated jacobians of transformations (deformation based morphometry) or directly by the analysis of displacement maps. We present a multivariate analysis of diffeomorphic transformations of the whole brain: relating complex anatomical changes observed in the population with neuropsychological responses such as clinical measures of cognitive abilities, audio-verbal learning and logical memory.

7 all in all... The purpose of this study is to extract and identify shape deformation patterns in brain anatomy that relate to observed clinical scores depicting cognitive abilities. We do a global statistical analysis of brain anatomy without any segmentation or a priori region of interest.

8 Review and notations... Riemannian metric and atlas construction Geodesics Geodesic computation and deformation momenta Multivariate statistical analysis Partial Least Squares and SVD Images are modelled as real-valued L 2 functions on domain Ω R 3 Diffeomorphic transformations φ Diff V (Ω) are elements of subgroup of diffeomorphisms Diff(Ω), φ : Ω Ω that are generated by flows of smooth, time indexed velocity fields, v(t, y) : (t [0, 1], y Ω) R 3 The function φ v (t, x) given by the solution of the ODE dy dt = v(t, y) with the initial condition y(0) = x defines a diffeomorphism of Ω.

9 Riemannian metric and atlas construction Geodesics Geodesic computation and deformation momenta Multivariate statistical analysis Partial Least Squares and SVD Riemannian metric: Inducing an energy using a Sobolev norm with partial differential operator L on v s. The distance between the identity transformation and a diffeomorphism ψ is defined as the minimization d(id, ψ) 2 = min { 1 } Lv(t, ), v(t, ) dt : φ v (1, ) = ψ( ) 0 The distance between any two diffeomorphism is defined as d(φ, ψ) = d(id, ψ φ 1 ).

10 Riemannian metric and atlas construction Geodesics Geodesic computation and deformation momenta Multivariate statistical analysis Partial Least Squares and SVD Atlas construction: The deformation φ is defined as the optimal time-varying velocity field ˆv, based on the minimum energy criteria: ˆv = 1 argmin v: φ t=v t(φ t) 0 Lv(t, ), v(t, ) 2 dt + 1 σ 2 Ω I 0 φ 1 I 1 2 dx Given a collection of anatomical images {I i, i = 1,, N}, the minimum mean squared energy atlas construction problem is that of jointly estimating an image Î and N individual deformations: 1 {Î, ˆφ i } = argmin I,φ i N N i=1 Ω I φ 1 i I i 2 dx + d(id, φ i ) 2

11 Riemannian metric and atlas construction Geodesics Geodesic computation and deformation momenta Multivariate statistical analysis Partial Least Squares and SVD Geodesics: generalization of straight line In euclidean space - shortest path is a straight line. In Riemannian manifold - shortest smooth curve segment Figure: geodesics - flat euclidean space and a curved manifold

12 Energy minimization Riemannian metric and atlas construction Geodesics Geodesic computation and deformation momenta Multivariate statistical analysis Partial Least Squares and SVD If γ : [a, b] M with γ(a) = x and γ(b) = y, a variation of γ keeping endpoints fixed is a family of curves. length functional, L(γ) = b a γ (t) M dt, where γ (t) T γ(t) M and the norm is given by Riemannian metric at γ(t) energy functional, E(γ) = b a γ (t) 2 M dt. Geodesic: critical path for E (critical path for L) Initial velocity determines the evolution of the curve, γ

13 Notion of momentum Riemannian metric and atlas construction Geodesics Geodesic computation and deformation momenta Multivariate statistical analysis Partial Least Squares and SVD We interpret 1 2 v 2 V as kinetic energy v 2 V = Lv, v Analogous to classical physics, we interpret Lv as momentum.

14 Riemannian metric and atlas construction Geodesics Geodesic computation and deformation momenta Multivariate statistical analysis Partial Least Squares and SVD The joint minimizer of the atlas construction problem estimates an atlas image Î while simultaneously solving the N LDDMM image matching problems. The Euler-Lagrange equations associated with the LDDMM problem coincides with the Euler-Lagrange equations of geodesics on the group of diffeomorphisms. The geodesic equations are completely determined via the initial momenta Lv 0 and are in direction of the gradient of deforming image (Younes et. al). Thus at the minimizer, for each of the N image matching problems the initial velocity is given by the equation Lv i (0, x) = a0 i (x) Î (x).

15 Riemannian metric and atlas construction Geodesics Geodesic computation and deformation momenta Multivariate statistical analysis Partial Least Squares and SVD The joint minimizer of the atlas construction problem estimates an atlas image Î while simultaneously solving the N LDDMM image matching problems. The Euler-Lagrange equations associated with the LDDMM problem coincides with the Euler-Lagrange equations of geodesics on the group of diffeomorphisms. The geodesic equations are completely determined via the initial momenta Lv 0 and are in direction of the gradient of deforming image (Younes et. al). Thus at the minimizer, for each of the N image matching problems the initial velocity is given by the equation Lv i (0, x) = a0 i (x) Î (x).

16 Riemannian metric and atlas construction Geodesics Geodesic computation and deformation momenta Multivariate statistical analysis Partial Least Squares and SVD The joint minimizer of the atlas construction problem estimates an atlas image Î while simultaneously solving the N LDDMM image matching problems. The Euler-Lagrange equations associated with the LDDMM problem coincides with the Euler-Lagrange equations of geodesics on the group of diffeomorphisms. The geodesic equations are completely determined via the initial momenta Lv 0 and are in direction of the gradient of deforming image (Younes et. al). Thus at the minimizer, for each of the N image matching problems the initial velocity is given by the equation Lv i (0, x) = a0 i (x) Î (x).

17 Riemannian metric and atlas construction Geodesics Geodesic computation and deformation momenta Multivariate statistical analysis Partial Least Squares and SVD The joint minimizer of the atlas construction problem estimates an atlas image Î while simultaneously solving the N LDDMM image matching problems. The Euler-Lagrange equations associated with the LDDMM problem coincides with the Euler-Lagrange equations of geodesics on the group of diffeomorphisms. The geodesic equations are completely determined via the initial momenta Lv 0 and are in direction of the gradient of deforming image (Younes et. al). Thus at the minimizer, for each of the N image matching problems the initial velocity is given by the equation Lv i (0, x) = a0 i (x) Î (x).

18 Riemannian metric and atlas construction Geodesics Geodesic computation and deformation momenta Multivariate statistical analysis Partial Least Squares and SVD The joint minimizer of the atlas construction problem estimates an atlas image Î while simultaneously solving the N LDDMM image matching problems. The Euler-Lagrange equations associated with the LDDMM problem coincides with the Euler-Lagrange equations of geodesics on the group of diffeomorphisms. The geodesic equations are completely determined via the initial momenta Lv 0 and are in direction of the gradient of deforming image (Younes et. al). Thus at the minimizer, for each of the N image matching problems the initial velocity is given by the equation Lv i (0, x) = a0 i (x) Î (x). The quantity a0 i (x) Î (x) is referred to as the initial momenta.

19 Riemannian metric and atlas construction Geodesics Geodesic computation and deformation momenta Multivariate statistical analysis Partial Least Squares and SVD Evolution equations and geodesic shooting... Each of the i = 1,, N geodesic equations evolve according to Lv i (t) = a i (t) Î (t) (1) da i (t, ) dt dî (t) dt + (a i (t)v i (t)) = 0 (2) = Î (t) T v i (t) (3) Equation (3) is the infinitesimal action of the velocity field v i on the image, while (2) is the conservation of momenta.

20 intuition... Riemannian metric and atlas construction Geodesics Geodesic computation and deformation momenta Multivariate statistical analysis Partial Least Squares and SVD Figure: image shape space and clinical response space

21 Riemannian metric and atlas construction Geodesics Geodesic computation and deformation momenta Multivariate statistical analysis Partial Least Squares and SVD Traditionally, Partial Least Squares (PLS) has been used to characterize pertinent directions between independent variable and dependent variable in a high dimensional multivariate regression setting. (introduced to neuroimaging community by Bookstein, 1994) We adapt the PLS methodology for the purpose of extracting and identifying deformation patterns in brain anatomy that relate to k observed clinical measures y i R k depicting cognitive and neuropsychological responses of each of the i = 1,, N subjects. The anatomical variation in the collection of I i is captured by the initial scalar momenta maps (a i (x)) at the atlas Î. These momenta maps govern the deformation of the atlas along the geodesic in the group diffeomorphism towards the respective individual images I i.

22 Precisely... Riemannian metric and atlas construction Geodesics Geodesic computation and deformation momenta Multivariate statistical analysis Partial Least Squares and SVD We find directions â in the momenta space, defined at the atlas in terms of deformation momenta a i s, and directions ŷ in the clinical response space, defined by y i s that explain their association in the sense of their common variance. We propose to extract these directions such that initial momenta when projected on to â and the corresponding clinical responses when projected on to ŷ have maximum covariance. We call these projections as latent variables, l a and l y respectively. To find the anatomical variation that covaries maximally with clinical responses, we perform PLS analysis between the scalar momenta fields a i and the response space y i.

23 Problem... Riemannian metric and atlas construction Geodesics Geodesic computation and deformation momenta Multivariate statistical analysis Partial Least Squares and SVD The PLS problem is an optimization given by: max cov( â, a i, ŷ, y i ) subject to â = 1, ŷ = 1 (4) The subsequent directions are found by removing the component extracted (deflating the data) both in momenta space and the clinical response space as: a i = a i â, a i and y i = y i ŷ, y i

24 Riemannian metric and atlas construction Geodesics Geodesic computation and deformation momenta Multivariate statistical analysis Partial Least Squares and SVD Problem... The solution to the above maximization problem (4) is the SVD of the covariance matrix of the dependent and independent variables. The corresponding direction vectors â s and ŷ s are the respective left and right singular vectors. The successive latent variables l a s and l y s are computed once by a single SVD.

25 Riemannian metric and atlas construction Geodesics Geodesic computation and deformation momenta Multivariate statistical analysis Partial Least Squares and SVD Statistical significance The statistical significance of the directions extracted by PLS analysis can be assessed using the projected data (the latent variables) l a s and l y s. We use non-parametric permutation tests for calculating the significance of the regression of l y s on l a s and use the R 2 (the proportion of variance explained in l y s by l a s) as the test statistics. The distribution of the R 2 statistic under the null hypothesis is calculated by randomly reordering the momenta and clinical response association and then recalculating the new SVD and its associated R 2 each time. The significance of a particular latent variable is measured by the p-value from the empirical distribution.

26 ADNI data Implementation details 313 Mild Cognitive Impairment (MCI) subjects from ADNI Images T1 weighted, bias field corrected and N3 scaled structural Magnetic Resonance Images (MRI) Neuropsychological measures Alzheimer s Disease Assessment Scale modified cognitive battery (adas-cog) Clinical Dementia Rating scale, Sum of Boxes (cdr.sb) Rey Audio Verbal Learning Test immediate recall (avlt.imm) Rey Audio Verbal Learning Test 30 min delayed recall (avlt.del) Logical Memory test of the Wechsler Memory Scale-Revised immediate recall (logic.imm) Logical Memory test of the Wechsler Memory Scale-Revised 30 min delayed recall (logic.imm)

27 Preprocessing the MRI ADNI data Implementation details Skull stripping, registration to talairach coordinates using freesurfer. Tissue-wise intensity normalization for white matter, gray matter and cerebrospinal fluid (CSF) using the expectation maximization (EM) based segmentation. The piecewise polynomial histogram matching.

28 ADNI data Implementation details The atlas was constructed with the 313 MCI subjects on the GPU cluster and the associated initial momenta fields a i were computed. Each p dimensional a i (i = 1,, 313, p = ) represents a row of a large 313 p matrix X of momenta maps. The corresponding k dimensional clinical outcome y i (i = 1,, 313 and k = 6) populates the rows of the matrix Y of clinical outcomes.

29 ADNI data Implementation details The atlas was constructed with the 313 MCI subjects on the GPU cluster and the associated initial momenta fields a i were computed. Each p dimensional a i (i = 1,, 313, p = ) represents a row of a large 313 p matrix X of momenta maps. The corresponding k dimensional clinical outcome y i (i = 1,, 313 and k = 6) populates the rows of the matrix Y of clinical outcomes.

30 ADNI data Implementation details The atlas was constructed with the 313 MCI subjects on the GPU cluster and the associated initial momenta fields a i were computed. Each p dimensional a i (i = 1,, 313, p = ) represents a row of a large 313 p matrix X of momenta maps. The corresponding k dimensional clinical outcome y i (i = 1,, 313 and k = 6) populates the rows of the matrix Y of clinical outcomes.

31 ADNI data Implementation details The atlas was constructed with the 313 MCI subjects on the GPU cluster and the associated initial momenta fields a i were computed. Each p dimensional a i (i = 1,, 313, p = ) represents a row of a large 313 p matrix X of momenta maps. The corresponding k dimensional clinical outcome y i (i = 1,, 313 and k = 6) populates the rows of the matrix Y of clinical outcomes. The PLS was then performed on X and Y data matrices.

32 ADNI data Implementation details The atlas was constructed with the 313 MCI subjects on the GPU cluster and the associated initial momenta fields a i were computed. Each p dimensional a i (i = 1,, 313, p = ) represents a row of a large 313 p matrix X of momenta maps. The corresponding k dimensional clinical outcome y i (i = 1,, 313 and k = 6) populates the rows of the matrix Y of clinical outcomes. The PLS was then performed on X and Y data matrices. The significance tests for the extracted momenta direction and the clinical response directions was performed using 100,000 permutations.

33 Statistical significance Table: Significance test permutations LV R p-value adas-cog cdr.sb avlt.imm avlt.del logic.imm logic.del *LV - latent variable

34 Evolving the mean along LV directions (Deforming brain) Evolving the mean along LV directions (Log Jacobians) Figure: (sagittal) evolving the mean atlas along LV1 t= -1, -0.5, 0, 0.5, 1 Geodesic evolution of the atlas Î along the â direction obtained from PLS for LV 1

35 Evolving the mean along LV directions (Deforming brain) Evolving the mean along LV directions (Log Jacobians) Figure: (axial) evolving the mean atlas along LV1 t= -1, -0.5, 0, 0.5, 1 Figure: (coronal) evolving the mean atlas along LV1 t= -1, -0.5, 0, 0.5, 1

36 Evolving the mean along LV directions (Deforming brain) Evolving the mean along LV directions (Log Jacobians) Figure: LV1 log jacobians overlayed on atlas Figure: LV6 log jacobians overlayed on atlas

37 The shape deformation patterns in anatomical structures show up evidently as a result of the PLS analysis of the momenta. latent variable 1 deformation patterns expansion of lateral ventricles and CSF clinical response direction increasing adas-cog and cdr.sb (measures of increasing cognitive degeneration) shrinkage of cortical surface decreasing AVLT and logical scores (measures of audio verbal learning and logical memory) shrinkage of the hippocampus, shrinkage of cortical and subcortical gray matter characteristic of disease progression in AD and related dementia

38 Latent Variable 6 The highly statistically significant LV6 explains an altogether independent set of anatomical deformation patterns that relate to corresponding patterns in audio-verbal learning scores and memory scores (immediate and delayed recall). The LV6 mainly explains deformations for learning and logical memory, owing to high absolute weights for AVLT and logic scores but very low weights to adas.cog and cdr.sb. The deformation patterns in anatomy show almost invarying hippocampal region.

39 Comparison to PLS ROI Analysis Figure: Scores of LV 1 of PLS of Momenta (Left) r= and Scores of 1 LV of PLS of ROI Volumes r=0.33

40 Comparison to PLS ROI Analysis Figure: Scores of LV 6 of PLS of Momenta (Left) r= and Scores of LV 6 of PLS of ROI Volumes r=0.2423

41 Thanks! Shashidhar Reddy, Ryan Russon, Benjamin Galvin Richard King, Norman Foster