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Putting the Tensor back in Tensor-Based Morphometry Ged Ridgway 1, Sebastien Ourselin 1, Brandon Whitcher 2, Derek Hill 1 and Nick Fox 3 1.Centre for Medical Image Computing University College London, UK 2.GlaxoSmithKline Clinical Imaging Centre, London, UK 3.Dementia Research Centre, Institute of Neurology, UCL Funding: EPSRC CASE Studentship sponsored by GSK

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Movember update Thanks very much to everyone who has donated The total so far is £230 – more than enough for purple! If the total breaks £250 you can look forward to a Mauve Mo for the whole of next week ;-) Over £400 and Ill try to keep it for a fortnight!

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Summary of methodology Tensor-Based Morphometry –Voxel-wise statistical analysis of data derived from non-rigid registration transformations Typically –Scalar Jacobian determinant –Cross-sectional data –Uncorrected statistics (sometimes False Discovery Rate) Here –Multivariate strain tensors –Longitudinal MRI –Permutation-based correction for Family-Wise Error

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Clinical application Dementia –Neurological disorders impairing brain functions –Alzheimers Disease most common form (10% > 65) Longitudinal MR Imaging –Correlates with clinical and histological data –Non-invasive imaging of pre-clinical changes Inference of significant regional patterns –Between groups and/or over time –Correlations to clinical scores, etc.

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Baseline T1 (coronal slice)

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Follow-up (rigidly regd)

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Follow-up – Baseline

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After non-rigid registration (F3D)

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Non-rigid displacement magnitude

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Jacobian determinant (log2)

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Remarks/motivation Difference images provide poor localisation of change –Though note regularisation affects non-rigid localisation Local displacements less clinically relevant than local changes in shape –However, Jacobian determinant loses much information Serial data requires both longitudinal registration and inter-subject spatial normalisation for group stats –How should the transformations be combined? Multivariate SPM/SnPM...

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Transformed source overlaid with its original edges

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Close-up with displacement vectors overlaid

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The Jacobian matrix

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Visualisation of effects of local Jacobian matrices

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Interpretation of the Jacobian Physical transformations have |J| > 0 However –Eigenvalues can be negative or complex –Might not have full set of independent eigenvectors The Jacobian matrix is a linear transformation –Includes scaling, rotations, skews (translation irrelevant) Interpretation can be aided by decomposing the matrix into simpler components

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The singular value decomposition J = XSZ –X and Z have orthonormal columns –S is diagonal, and if |J| > 0 then all s i > 0 It is possible to ensure that X and Z are proper rotations with det=1 S produces stretches along the axes –Sandwiched between rotations, allows arbitrary J

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XS ZSZ

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Pure strain Diagonal J – very simple physical interpretation Sym. Pos. Def. matrices similar to diagonal ones –Correspond to scalings along rotated axes No extra rotations/skews – pure strain –I.e. principal stretches and principal axes –Easily illustrated with ellipsoid –Eig and SVD coincide: J=XSZ, with X=Z and X=X -1

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Pure strain

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The polar decomposition The SVD satisfies XX=I and ZZ=I –J = XSZ = X(ZZ)SZ = (XZ)(ZSZ) –J = XSZ = XS(XX)Z = (XSX)(XZ) So any Jacobian matrix can be written as the product of a rotation and an SPD matrix –J = RU = VR

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E.g. pure skew V R

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U R

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U V

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Lagrangian and Eulerian Tensors Tensors derived from the right stretch tensor U are Lagrangian Those from the left stretch tensor V are Eulerian Relates to fixed/deforming frame of reference Morphometry typically involves multiple moving source images registered to a single fixed atlas –J=RU means multiple U can be analysed in atlas space Next slide illustrates the strain ellipses for U...

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Note U generalises scalar TBM since |U| = |J| Principal strains or directions may be of interest too Nonlinearly transformed versions of U often used...

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Different types of strain Each of U and V leads to a family of different tensors –E.g. E(m) = (U m – I) / m Different definitions of strain may be of interest –Engineering strain, natural/logarithmic strain, etc. U = ZSZ and J = XSZ give U=(JJ) 1/2 –H = logm(U) has eigenvalues log(s i ) –Known as the Hencky strain tensor –Statistically nice, log(s i )~N gives log(s i s j )~N etc.

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Groups, manifolds and metrics Physical deformations have |J| > 0 Positive numbers form a group under multiplication 0.5 and 2 are equally far from the identity –Suggests d(a,b) = ||log(a/b)|| = ||log(a)-log(b)|| –This metric gives rise to the geometric mean –(log|J| also more likely to be normally distributed)

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Groups, manifolds and metrics Jacobian matrices with positive determinant also form a group under matrix multiplication They lie on a curved manifold However, no affine-invariant Riemannian metric exists –d(A,B) = ||logm(AB -1 )|| can violate triangle inequality Woods (2003): –Semi-Riemannian, pseudo-metric, Karcher mean –Deviations from mean

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Groups, manifolds and metrics SPD matrices also lie on a curved manifold

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Groups, manifolds and metrics SPD matrices also lie on a curved manifold Two natural Riemannian metrics exist –Batchelor/Moakher/Pennec: Affine-invariant d(A,B) = ||logm(A 1/2 B -1 A 1/2 )|| Iterative equation to find implicitly-defined Frechet mean –Arsigny: Log-Euclidean d(A,B) = ||logm(A) – logm(B)|| logm(A) can be vectorised Simple closed form expressions for mean, etc.

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Engineering/Physics vs Maths Continuum mechanics can lead from the Jacobian to the Hencky tensor via strains More abstract maths can lead to matrix logarithms of symmetric positive definite matrices like JJ –Desire to have |tensor| = |J| then gives U –Only difference to H is whether vectorisation just takes unique elements (Ashburner) or also scales off- diagonal elements so ||H|| = ||h|| (Lepore)

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Fractional and geodesic anisotropy In diffusion tensor imaging, level of directionality often of interest, e.g. relative and fractional anisotropy The distance metric on tensors allows a more rigorous definition of this –Anisotropy is measured by the distance between the tensor and its nearest isotropic counterpart Euclidean distance gives FA Geodesic Anisotropy uses Riemannian metric GA = ||H – I tr(H)/2|| F

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tanh(GA)

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Other measures Jacobian is gradient of the transformation field Are div(u) or curl(u) of interest? First, note contained in J –div(u) = tr(du/dx) = tr(J – I) –curl(u) has same elements as skew-sym J – J div(u) AKA volume dilatation –Proposed as part of Chung et als unified approach

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Left: log|J| Right: curl(u) Left: div(u) Right: GA

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1 1+b 1+a1 x2x2 x1x1

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1 1+b 1+a1 Right Top x2x2 x1x1 Ω Ω

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Longitudinal TBM Registration within-subject typically performed first Inter-subject spatial normalisation follows For scalar Jacobian determinant can simply resample More complicated for vector or tensor fields… Related problem in Diffusion Tensor Imaging –Except there microscopic, here macroscopic

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SourceReference a b c d (a)Macroscopic transformation of anatomy, according to image registration (b)Microscopic properties of water diffusion preserved (c)Macroscopic compression assumed to represent shorter tracts rather than shorter diffusion scale; hence smaller number of same shape ellipses (d)Orientation of ellipse transformed according to anatomical transformation, e.g. preserving principle direction of diffusion (PPD) [Alexander et al. 2001]. Continuity of fibre tracts should be preserved.

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Source A Source BReference The finite strain reorientation fails to reorient deformation vectors for the changes that occur in anisotropic scaling and/or shearing.

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b a d c Source Time 0 Time 1 Reference r0r0 s0s0 s1s1 r1r1 Serial deformation is also macroscopic, and hence transformation in reference space is conjugate to that in source space; compression of source onto reference also compresses longitudinal change.

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Spatial smoothing Intra-subject registration over time highly accurate Inter-subject spatial normalisation much less precise –Spatial smoothing required Different methods have been developed to reduce the danger of expansion and contraction cancelling out –E.g. VBM, VCM More work is needed, especially for multivariate TBM

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Statistical analysis Similar to VBM, TBM performs voxel-wise tests –Interpretable spatial pattern of significant findings While VBM and determinant-based TBM are mass- univariate, strain tensors require multivariate statistics –Random field theory not yet well established for Wilks L –Assumptions of normality may be more questionable –Estimation of covariance matrices may be unreliable Non-parametric permutation testing is appealing

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Parametric stats vs. Resampling-based Assumptions –Gaussian error, … Increase in variance due to restrictions ~ F Properties of random field => Pr(max>crit|H0) Assumptions –Exchangeable under null Arbitrary statistic –Including multivariate Resampling-based distribution of max(stat) => Pr(max>crit|H0) 95 th percentile

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Permutation testing Track permutations image-wise maxima –p-values corrected for Family-Wise (image-wise) Error Empirical null distr. shouldnt include active voxels –Track secondary (etc) maxima and locations –allows more sensitive step-down procedure Belmonte and Yurgelun-Todd (2001) IEEE TMI 20:243-8 Computationally intensive –But amenable to parallel implementation

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Results 36 probable Alzheimers Disease patients 20 age- and gender-matched control subjects Baseline and 12 month repeat scans Standard clinical T1-weighted images Between- then within-subject registration 8mm FWHM Gaussian smoothing

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s(log(det)) 1(1) s:smoothd s(displ) 3(6) GA(s(H)) 1(1) eig(s(H)) 3(6) s(H) 6(21) s(J) 9(45) As left, but abs(log(p)) for p < 0.05

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Log-Euc analysis of U, colour-coded by tr(sm(H))

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det LE J detLE2 FDR FWE

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div disp curl Some evidence for complementarity of the different measures

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Conclusions Most prominent findings were expansion of fluid spaces; contraction of hippocampal and temporal area Multivariate measures notably more powerful –Have potential to identify patterns of morphometric difference overlooked by conventional analysis Geodesic Anisotropy was found to be the best of several orientational measures including curl(u) and the major eigenvector of the strain tensor

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Further work Developments in visualisation and interpretation of the multivariate results would be helpful Log-Euc strain tensor method should be compared to Aff-invar, and to Woods semi-riemannian approach Between- then within-subject registration should be compared to Raos method Smoothing should be replaced with more sophisticated approaches

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References Moakher, M. A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices SIAM Journal on Matrix Analysis and Applications, 2005, 26, 735 A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices Batchelor, P. G.; Moakher, M.; Atkinson, D.; Calamante, F. & Connelly, A. A rigorous framework for diffusion tensor calculus Magn Reson Med, 2005, 53, A rigorous framework for diffusion tensor calculus Arsigny, V.; Fillard, P.; Pennec, X. & Ayache, N. Log-Euclidean metrics for fast and simple calculus on diffusion tensors Magn Reson Med, 2006, 56, Log-Euclidean metrics for fast and simple calculus on diffusion tensors Arsigny, V.; Commowick, O.; Pennec, X. & Ayache, N. A log-Euclidean framework for statistics on diffeomorphisms. MICCAI, 2006, 9, A log-Euclidean framework for statistics on diffeomorphisms. Chung, M. K.; Worsley, K. J.; Paus, T.; Cherif, C.; Collins, D. L.; Giedd, J. N.; Rapoport, J. L. & Evans, A. C. A unified statistical approach to deformation-based morphometry. Neuroimage, 2001, 14, Ashburner, J. Computational Neuroanatomy PhD Thesis University College London, 2000 Lepore, N.; Brun, C.; Chou, Y. Y.; Chiang, M. C.; Dutton, R. A.; Hayashi, K. M.; Luders, E.; Lopez, O. L.; Aizenstein, H. J.; Toga, A. W.; Becker, J. T. & Thompson, P. M. Generalized tensor-based morphometry of HIV/AIDS using multivariate statistics on deformation tensors. IEEE Trans Med Imaging, 2008, 27, Rao, A.; Chandrashekara, R.; Sanchez-Ortiz, G.; Mohiaddin, R.; Aljabar, P.; Hajnal, J.; Puri, B. & Rueckert, D. Spatial transformation of motion and deformation fields using nonrigid registration Medical Imaging, IEEE Transactions on, 2004, 23,

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