Multivariate Statistical Analysis of Deformation Momenta ... · Nikhil Singh, Tom Fletcher, Sam Preston, Linh Ha, J. Stephen Marron, Michael Wiener, and Sarang JoshiMultivariate Statistical

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Technical BackgroundData configuration and computational framework

Significance testsVisualizing the Geodesic Shooting

Comments

Multivariate Statistical Analysis of DeformationMomenta Relating Anatomical Shape to

Neuropsychological Measures

Nikhil Singh, Tom Fletcher, Sam Preston, Linh Ha, J. StephenMarron, Michael Wiener, and Sarang Joshi

Scientific Computing and Imaging Institute (SCI)The University of Utah

June 17, 2010

Nikhil Singh, Tom Fletcher, Sam Preston, Linh Ha, J. Stephen Marron, Michael Wiener, and Sarang JoshiMultivariate Statistical Analysis of Deformation Momenta Relating Anatomical Shape to Neuropsychological Measures

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Motivation

Technical BackgroundRiemannian metric and atlas constructionGeodesicsGeodesic computation and deformation momentaMultivariate statistical analysisPartial Least Squares and SVD

Data configuration and computational frameworkADNI dataImplementation details

Significance tests

Visualizing the Geodesic ShootingEvolving the mean along LV directions (Deforming brain)Evolving the mean along LV directions (Log Jacobians)

CommentsNikhil Singh, Tom Fletcher, Sam Preston, Linh Ha, J. Stephen Marron, Michael Wiener, and Sarang JoshiMultivariate Statistical Analysis of Deformation Momenta Relating Anatomical Shape to Neuropsychological Measures

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I Given a Large collection of anatomical images of subjects withdetailed Neuropsychological assessments how does one relateanatomical variation to Neuropsychological variables.

I Driving problem: The ADNI database currently has 313 MildCognitive Impairment (MCI) subjects each with detailedNeuropsychological evaluations.

I Each Neuropsychological evaluation is a real score associatedwith various examinations such as Clinical Dementia Ratingscale, Audio Verbal Learning test (immediate and delayed),· · · .


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motivation...

Figure: image space and clinical response space


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motivation...

I Conventionally, anatomical variation has been studied bygenerating transformations between each subject in thepopulation and the pre-selected template.

I We study anatomical variation by simultaneously generatingtransformations between entire population and a commonreference coordinate system in large deformationdiffeomorphic setting (LDDMM)


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motivation...

I Earlier studies on characterization on neuroanatomicalchanges: statistical analysis of deformation maps usingassociated jacobians of transformations (deformation basedmorphometry) or directly by the analysis of displacementmaps.

I We present a multivariate analysis of diffeomorphictransformations of the whole brain: relating complexanatomical changes observed in the population withneuropsychological responses such as clinical measures ofcognitive abilities, audio-verbal learning and logical memory.


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all in all...

I The purpose of this study is to extract and identify shapedeformation patterns in brain anatomy that relate to observedclinical scores depicting cognitive abilities.

I We do a global statistical analysis of brain anatomy withoutany segmentation or a priori region of interest.


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Riemannian metric and atlas constructionGeodesicsGeodesic computation and deformation momentaMultivariate statistical analysisPartial Least Squares and SVD

Review and notations...

I Images are modelled as real-valued L2 functions on domainΩ ⊂ R3

I Diffeomorphic transformations φ ∈ DiffV(Ω) are elements ofsubgroup of diffeomorphisms Diff(Ω), φ : Ω→ Ω that aregenerated by flows of smooth, time indexed velocity fields,v(t, y) : (t ∈ [0, 1], y ∈ Ω)→ R3

I The function φv (t, x) given by the solution of the ODEdydt = v(t, y) with the initial condition y(0) = x defines adiffeomorphism of Ω.


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Riemannian metric: Inducing an energy using a Sobolev norm withpartial differential operator L on v ’s. The distance between theidentity transformation and a diffeomorphism ψ is defined as theminimization

d(id , ψ)2 = min

∫ 1

0〈Lv(t, ·), v(t, ·)〉dt : φv (1, ·) = ψ(·)

The distance between any two diffeomorphism is defined asd(φ, ψ) = d(id , ψ φ−1).


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Atlas construction: The deformation φ is defined as the ‘optimal’time-varying velocity field v , based on the minimum energy criteria:

v = argminv :φt=vt(φt)

∫ 1

0〈Lv(t, ·), v(t, ·)〉2dt +

1

σ2

∫Ω‖I 0 φ−1 − I 1‖2dx

Given a collection of anatomical images I i , i = 1, · · · ,N, theminimum mean squared energy atlas construction problem is thatof jointly estimating an image I and N individual deformations:

I , φi = argminI ,φi

1

N

N∑i=1

∫Ω||I φ−1

i − I i ||2dx + d(id , φi )2


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Geodesics: generalization of “straight” line

I In euclidean space - shortest path is a straight line.

I In Riemannian manifold - shortest smooth curve segment

I

Figure: geodesics - flat euclidean space and a curved manifold


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Energy minimization

I If γ : [a, b]→ M with γ(a) = x and γ(b) = y , a variation of γkeeping endpoints fixed is a family of curves.

I length functional, L(γ) =∫ ba ||γ

′(t)||Mdt, whereγ′(t) ∈ Tγ(t)M and the norm is given by Riemannian metricat γ(t)

I energy functional, E (γ) =∫ ba ||γ

′(t)||2Mdt.I Geodesic: critical path for E (critical path for L)I Initial velocity determines the evolution of the curve, γ


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Notion of momentum

I We interpret 12 ||v ||

2V as kinetic energy

I ||v ||2V = 〈Lv , v〉I Analogous to classical physics, we interpret Lv as momentum.


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I The joint minimizer of the atlas construction problemestimates an atlas image I while simultaneously solving the NLDDMM image matching problems.

I The Euler-Lagrange equations associated with the LDDMMproblem coincides with the Euler-Lagrange equations ofgeodesics on the group of diffeomorphisms.

I The geodesic equations are completely determined via theinitial momenta Lv0 and are in direction of the gradient ofdeforming image (Younes et. al).

I Thus at the minimizer, for each of the N image matchingproblems the initial velocity is given by the equationLv i (0, x) = ai

0(x)∇I (x).

I The quantity ai0(x)∇I (x) is referred to as the initial momenta.


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0(x)∇I (x).



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0(x)∇I (x).



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0(x)∇I (x).



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0(x)∇I (x).



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Evolution equations and geodesic shooting...

Each of the i = 1, · · · ,N geodesic equations evolve according to

Lv i (t) = ai (t)∇I (t) (1)

dai (t, ·)dt

+ ∇ · (ai (t)v i (t)) = 0 (2)

dI (t)

dt= ∇I (t)T v i (t) (3)

Equation (3) is the infinitesimal action of the velocity field v i onthe image, while (2) is the conservation of momenta.


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intuition...

Figure: image shape space and clinical response space


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I Traditionally, Partial Least Squares (PLS) has been used tocharacterize pertinent directions between independent variableand dependent variable in a high dimensional multivariateregression setting. (introduced to neuroimaging community byBookstein, 1994)

I We adapt the PLS methodology for the purpose of extractingand identifying deformation patterns in brain anatomy thatrelate to k observed clinical measures y i ∈ Rk depictingcognitive and neuropsychological responses of each of thei = 1, · · · ,N subjects.

I The anatomical variation in the collection of I i is captured bythe initial scalar momenta maps (ai (x)) at the atlas I . Thesemomenta maps govern the deformation of the atlas along thegeodesic in the group diffeomorphism towards the respectiveindividual images I i .


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Precisely ...

I We find directions a in the momenta space, defined at theatlas in terms of deformation momenta ai ’s, and directions yin the clinical response space, defined by y i ’s that explaintheir association in the sense of their common variance.

I We propose to extract these directions such that initialmomenta when projected on to a and the correspondingclinical responses when projected on to y have maximumcovariance. We call these projections as latent variables, laand ly respectively.

I To find the anatomical variation that covaries maximally withclinical responses, we perform PLS analysis between the scalarmomenta fields ai and the response space y i .


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Problem ...

The PLS problem is an optimization given by:

max cov(〈a, ai 〉, 〈y , y i 〉) subject to ‖a‖ = 1 , ‖y‖ = 1 (4)

The subsequent directions are found by removing the componentextracted (deflating the data) both in momenta space and theclinical response space as:

ai = ai − 〈a, ai 〉 and y i = y i − 〈y , y i 〉


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Problem ...

I The solution to the above maximization problem (4) is theSVD of the covariance matrix of the dependent andindependent variables.

I The corresponding direction vectors a’s and y ’s are therespective left and right singular vectors.

I The successive latent variables la’s and ly ’s are computed onceby a single SVD.


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Statistical significance

I The statistical significance of the directions extracted by PLSanalysis can be assessed using the projected data (the latentvariables) la’s and ly ’s.

I We use non-parametric permutation tests for calculating thesignificance of the regression of ly ’s on la’s and use the R2

(the proportion of variance explained in ly ’s by la’s) as the teststatistics.

I The distribution of the R2 statistic under the null hypothesisis calculated by randomly reordering the momenta and clinicalresponse association and then recalculating the new SVD andits associated R2 each time.

I The significance of a particular latent variable is measured bythe p-value from the empirical distribution.


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ADNI dataImplementation details

313 Mild Cognitive Impairment (MCI) subjects from ADNI

Images Neuropsychological measures

T1 weighted,bias fieldcorrected andN3 scaledstructuralMagneticResonanceImages (MRI)

Alzheimer’s Disease Assessment Scale modifiedcognitive battery (adas-cog)Clinical Dementia Rating scale, Sum of Boxes(cdr.sb)Rey Audio Verbal Learning Test immediate re-call (avlt.imm)Rey Audio Verbal Learning Test 30 min delayedrecall (avlt.del)Logical Memory test of the Wechsler MemoryScale-Revised immediate recall (logic.imm)Logical Memory test of the Wechsler MemoryScale-Revised 30 min delayed recall (logic.imm)


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Preprocessing the MRI

Skull stripping, registration to talairach coordinates usingfreesurfer.Tissue-wise intensity normalization for white matter, gray matterand cerebrospinal fluid (CSF) using the expectation maximization(EM) based segmentation.The piecewise polynomial histogram matching.


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I The atlas was constructed with the 313 MCI subjects on theGPU cluster and the associated initial momenta fields ai werecomputed.

I Each p dimensional ai (i = 1, · · · , 313 , p = 144×192×160)represents a row of a large 313× p matrix X of momentamaps.

I The corresponding k dimensional clinical outcome y i

(i = 1, · · · , 313 and k = 6) populates the rows of the 313× 6matrix Y of clinical outcomes.

I The PLS was then performed on X and Y data matrices.

I The significance tests for the extracted momenta directionand the clinical response directions was performed using100,000 permutations.


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Statistical significance

Table: Significance test - 100000 permutations

LV 1 2 3 4 5 6

R2 0.5010 0.5059 0.4515 0.4618 0.4356 0.5271

p-value 0.0669 0.1033 0.2289 0.3435 0.2775 0.0050

adas-cog 0.4373 0.0220 0.0973 -0.7900 0.4102 0.0800

cdr.sb 0.2069 0.9639 -0.1050 0.1077 -0.0266 -0.0688

avlt.imm -0.4738 0.2175 0.4403 -0.2855 -0.2802 0.6118

avlt.del -0.4484 0.1256 0.5166 0.0003 0.4574 -0.5542

logic.imm -0.4325 0.0582 -0.5045 -0.5228 -0.3517 -0.3976

logic.del -0.3916 0.0626 -0.5141 0.0972 0.6478 0.3865

*LV - latent variable


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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 I along the a direction obtainedfrom PLS for LV 1


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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


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Figure: LV1 log jacobians overlayed on atlas

Figure: LV6 log jacobians overlayed on atlasNikhil Singh, Tom Fletcher, Sam Preston, Linh Ha, J. Stephen Marron, Michael Wiener, and Sarang JoshiMultivariate Statistical Analysis of Deformation Momenta Relating Anatomical Shape to Neuropsychological Measures

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The shape deformation patterns in anatomical structures show upevidently as a result of the PLS analysis of the momenta.

latent variable 1

deformation patterns clinical response direction

expansion of lateral ventriclesand CSF

increasing adas-cog and cdr.sb(measures of increasing cogni-tive degeneration)

shrinkage of cortical surface decreasing AVLT and logicalscores (measures of audio verballearning and logical memory)

shrinkage of the hippocampus,shrinkage of cortical and sub-cortical gray matter

characteristic of disease progres-sion in AD and related dementia


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Latent Variable 6

I The highly statistically significant LV6 explains an altogetherindependent set of anatomical deformation patterns thatrelate to corresponding patterns in audio-verbal learningscores and memory scores (immediate and delayed recall).

I The LV6 mainly explains deformations for learning and logicalmemory, owing to high absolute weights for AVLT and logicscores but very low weights to adas.cog and cdr.sb.

I The deformation patterns in anatomy show almost invaryinghippocampal region.


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Comparison to PLS ROI Analysis

Figure: Scores of LV 1 of PLS of Momenta (Left) r=0.7078 and Scoresof 1 LV of PLS of ROI Volumes r=0.33


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Comparison to PLS ROI Analysis

Figure: Scores of LV 6 of PLS of Momenta (Left) r=0.7260 and Scoresof LV 6 of PLS of ROI Volumes r=0.2423


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Thanks!

I Shashidhar Reddy, Ryan Russon, Benjamin Galvin

I Richard King, Norman Foster


Multivariate Statistical Analysis of Deformation Momenta ... · Nikhil Singh, Tom Fletcher, Sam Preston, Linh Ha, J. Stephen Marron, Michael Wiener, and Sarang JoshiMultivariate Statistical

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