Xavier Pennec Asclepios team, INRIA Sophia- Antipolis Mediterrane, - PowerPoint PPT Presentation

Xavier Pennec Asclepios team, INRIA Sophia- Antipolis – Mediterranée, France Work presented here is taken from the PhD of Marco Lorenzi Statistical Computing on Manifolds for Computational Anatomy 4: Analysis of longitudinal deformations Infinite-dimensional Riemannian Geometry with applications to image matching and shape analysis

Statistical Computing on Manifolds for Computational Anatomy Simple Statistics on Riemannian Manifolds Manifold-Valued Image Processing Metric and Affine Geometric Settings for Lie Groups Analysis of Longitudinal Deformations X. Pennec - ESI - Shapes, Feb 10-13 2015 2

Statistical Computing on Manifolds for Computational Anatomy Simple Statistics on Riemannian Manifolds Manifold-Valued Image Processing Metric and Affine Geometric Settings for Lie Groups Analysis of Longitudinal Deformations  Measuring Alzheimer’s disease (AD) evolution  Parallel transport of longitudinal trajectories  From velocity fields to AD atrophy models X. Pennec - ESI - Shapes, Feb 10-13 2015 3

Alzheimer’s Disease  Most common form of dementia  18 Million people worldwide  Prevalence in advanced countries  65-70: 2%  70-80: 4%  80 - : 20%  If onset was delayed by 5 years, number of cases worldwide would be halved X. Pennec - ESI - Shapes, Feb 10-13 2015 4

Longitudinal structural damage in AD baseline 2 years follow-up Ventricle’s expansion Widespread cortical thinning Hippocampal atrophy X. Pennec - ESI - Shapes, Feb 10-13 2015 6

Measuring Temporal Evolution with deformations Geometry changes (Deformation-based morphometry) Measure the physical or apparent deformation found by deformable registration Quantification of apparent deformations Time i Time i+1 X. Pennec - ESI - Shapes, Feb 10-13 2015 7

Atrophy estimation for Alzheimer’s Disease Established markers of anatomical changes Local: TBM (Paul Thompson, UCLA) Global: BSI / KNBSI (N. Fox, UCL) Intensity flux through brain surface Local volume change: Jacobian SIENA (S.M. Smith, Oxford) (determinant of spatial derivatives matrix) percentage brain volume change X. Pennec - ESI - Shapes, Feb 10-13 2015 8

Atrophy estimation from SVFs = • Integrate Jac( f ) (~ TBI)  Volume change • Integrate log(Jac( f ))  Flux-like (~ BSI) • Calibrate to obtain “equivalent ”volume changes X. Pennec - ESI - Shapes, Feb 10-13 2015 9

Groupwise analysis: deformation-based morphometry  Register subjects and controls to atlas  Spatial normalization of Jacobian maps  Statistical discrimination between groups   det ( ( )) x Group 1 Atlas Group 2 [ N. Lepore et al, MICCAI’06] Population X. Pennec - ESI - Shapes, Feb 10-13 2015 10

Longitudinal deformation analysis in AD  From patient specific evolution to population trend (parallel transport of deformation trajectories)  Inter-subject and longitudinal deformations are of different nature and might require different deformation spaces/metrics Patient A ? ? Template Patient B PhD Marco Lorenzi - Collaboration With G. Frisoni (IRCCS FateBenefratelli, Brescia) X. Pennec - ESI - Shapes, Feb 10-13 2015 11

Statistical Computing on Manifolds for Computational Anatomy Simple Statistics on Riemannian Manifolds Manifold-Valued Image Processing Metric and Affine Geometric Settings for Lie Groups Analysis of Longitudinal Deformations  Measuring Alzheimer’s disease (AD) evolution  Parallel transport of longitudinal trajectories  From velocity fields to AD atrophy models X. Pennec - ESI - Shapes, Feb 10-13 2015 12

Parallel transport of deformation trajectories j v M SVF setting • v stationary velocity field j id • Lie group Exp(v) non-metric geodesic wrt Cartan connections Transporting trajectories: LDDMM setting • v time-varying velocity field Parallel transport of initial • Riemannian exp id (v) metric tangent vectors geodesic wrt Levi-Civita connection • Defined by intial momentum LDDMM: parallel transport along geodesics using Jacobi fields [Younes et al. 2008] X. Pennec - ESI - Shapes, Feb 10-13 2015 • [Lorenzi et al, IJCV 2012] 14

From gravitation to computational anatomy: Parallel transport along arbitrary curves Infinitesimal parallel transport = connection  g ’ ( X ) : TM  TM A numerical scheme to integrate symmetric connections: Schild’s Ladder [Elhers et al, 1972]  Build geodesic parallelogrammoid P( A) P’ N  Iterate along the curve P N A’ P’ 1 P 1   P 2 C   A P’ 0 A P’ 0 P 0 P 0 X. Pennec - ESI - Shapes, Feb 10-13 2015 16

Schild’s Ladder Intuitive application to images P SL ( A) Inter-subject registration T’ 0 T 0 A P’ 0 P 0 time [Lorenzi, Ayache, Pennec: Schild's Ladder for the parallel transport of deformations in time series of images , IPMI 2011 ] X. Pennec - ESI - Shapes, Feb 10-13 2015 17 [Lorenzi et al, IPMI 2011]

Pole Ladder: an optimized Schild’s ladder along geodesics P( A) T’ 0 T 0 C geodesic A’ A’ A P’ 0 P 0 Number of registrations minimized X. Pennec - ESI - Shapes, Feb 10-13 2015 18

Efficient Pole Ladder with SVFs Numerical scheme  Direct computation:  Using the BCH: [Lorenzi, Ayache, Pennec: Schild's Ladder for the parallel transport of deformations in time series of images , IPMI 2011 ] X. Pennec - ESI - Shapes, Feb 10-13 2015 19

Left, Right and Sym. Parallel Transport along SVFs Numerical stability of Jacobian computation X. Pennec - ESI - Shapes, Feb 10-13 2015 22

Parallel Transport along SVFs X. Pennec - ESI - Shapes, Feb 10-13 2015 23

Analysis of longitudinal datasets Multilevel framework Single-subject, two time points Log-Demons (LCC criteria) Single-subject, multiple time points 4D registration of time series within the Log-Demons registration. Multiple subjects, multiple time points Schild’s Ladder [Lorenzi et al, in Proc. of MICCAI 2011] X. Pennec - ESI - Shapes, Feb 10-13 2015 24

Atrophy estimation for Alzheimer Alzheimer's Disease Neuroimaging Initiative (ADNI)  200 NORMAL 3 years  400 MCI 3 years  200 AD 2 years  Visits every 6 month  57 sites Data collected  Clinical, blood, LP  Cognitive Tests  Anatomical images:1.5T MRI (25% 3T)  Functional images: FDG-PET (50%), PiB-PET (approx 100) X. Pennec - ESI - Shapes, Feb 10-13 2015 25

Modeling longitudinal atrophy in AD from images One year structural changes for 70 Alzheimer's patients  Median evolution model and significant atrophy (FdR corrected) Contraction Expansion [Lorenzi et al, in Proc. of IPMI 2011] X. Pennec - ESI - Shapes, Feb 10-13 2015 26

Modeling longitudinal atrophy in AD from images One year structural changes for 70 Alzheimer's patients  Median evolution model and significant atrophy (FdR corrected) Contraction Expansion [Lorenzi et al, in Proc. of IPMI 2011] X. Pennec - ESI - Shapes, Feb 10-13 2015 27

Modeling longitudinal atrophy in AD from images One year structural changes for 70 Alzheimer's patients  Median evolution model and significant atrophy (FdR corrected) Contraction Expansion [Lorenzi et al, in Proc. of IPMI 2011] X. Pennec - ESI - Shapes, Feb 10-13 2015 28

Modeling longitudinal atrophy in AD from images One year structural changes for 70 Alzheimer's patients  Median evolution model and significant atrophy (FdR corrected) Contraction Expansion [Lorenzi et al, in Proc. of IPMI 2011] X. Pennec - ESI - Shapes, Feb 10-13 2015 29

Longitudinal model for AD Modeled changes from 70 AD subjects (ADNI data) Estimated from 1 year changes – Extrapolation to 15 years year Extrapolated Observed Extrapolated X. Pennec - ESI - Shapes, Feb 10-13 2015 31

Modeling longitudinal atrophy in AD from images X. Pennec - ESI - Shapes, Feb 10-13 2015 32

Study of prodromal Alzheimer’s disease  98 healthy subjects , 5 time points (0 to 36 months).  41 subjects A b 42 positive (“at risk” for Alzheimer’s)  Q: Different morphological evolution for A b + vs A b -? Average SVF for normal evolution (A b -) [Lorenzi, Ayache, Frisoni, Pennec, in Proc. of MICCAI 2011] X. Pennec - ESI - Shapes, Feb 10-13 2015 33

Detail: comparison between average evolutions (SVF) A b 42- A b 42+ A b 42+ A b 42+ A b 42- A b 42- X. Pennec - ESI - Shapes, Feb 10-13 2015 34

Time: years A b 42- A b 42+ A b 42- A b 42+ X. Pennec - ESI - Shapes, Feb 10-13 2015 35

Study of prodromal Alzheimer’s disease Linear regression of the SVF over time: interpolation + prediction Multivariate group-wise comparison of the transported SVFs shows statistically significant differences ~  T ( t ) Exp ( v ( t )) * T (nothing significant on log(det) ) 0 [Lorenzi, Ayache, Frisoni, Pennec, in Proc. of MICCAI 2011] X. Pennec - ESI - Shapes, Feb 10-13 2015 36

Statistical Computing on Manifolds for Computational Anatomy Simple Statistics on Riemannian Manifolds Manifold-Valued Image Processing Metric and Affine Geometric Settings for Lie Groups Analysis of Longitudinal Deformations  Measuring Alzheimer’s disease (AD) evolution  Parallel transport of longitudinal trajectories  From velocity fields to AD atrophy models X. Pennec - ESI - Shapes, Feb 10-13 2015 37