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

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

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Xavier Pennec Asclepios team, INRIA Sophia- Antipolis Mediterrane, France Work presented here is taken from the PhD of Marco Lorenzi Statistical Computing on Manifolds for Computational Anatomy 4: Analysis of longitudinal deformations

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Statistical Computing

  • n Manifolds for

Computational Anatomy 4: Analysis of longitudinal deformations

Infinite-dimensional Riemannian Geometry with applications to image matching and shape analysis

Xavier Pennec

Asclepios team, INRIA Sophia- Antipolis – Mediterranée, France

Work presented here is taken from the PhD of Marco Lorenzi

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  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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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

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  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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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

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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

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Longitudinal structural damage in AD

baseline 2 years follow-up

Ventricle’s expansion Hippocampal atrophy Widespread cortical thinning

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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Measuring Temporal Evolution with deformations

Geometry changes (Deformation-based morphometry)

Measure the physical or apparent deformation found by deformable registration

Time i Time i+1

Quantification of apparent deformations

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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Atrophy estimation for Alzheimer’s Disease

Established markers of anatomical changes

Global: BSI / KNBSI (N. Fox, UCL)

Intensity flux through brain surface

SIENA (S.M. Smith, Oxford)

percentage brain volume change

Local: TBM (Paul Thompson, UCLA)

Local volume change: Jacobian (determinant of spatial derivatives matrix)

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  • X. Pennec - ESI - Shapes, Feb 10-13 2015
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=

  • Integrate Jac(f) (~ TBI)  Volume change
  • Integrate log(Jac(f))  Flux-like (~ BSI)
  • Calibrate to obtain “equivalent ”volume changes

Atrophy estimation from SVFs

  • X. Pennec - ESI - Shapes, Feb 10-13 2015
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Groupwise analysis: deformation-based morphometry

 Register subjects and controls to atlas  Spatial normalization of Jacobian maps  Statistical discrimination between groups

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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Population Atlas

)) ( ( det x  

[ N. Lepore et al, MICCAI’06]

Group 1 Group 2

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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

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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PhD Marco Lorenzi - Collaboration With G. Frisoni (IRCCS FateBenefratelli, Brescia)

Patient A Patient B

? ?

Template

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  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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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

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j

M

id v

j

SVF setting

  • v stationary velocity field
  • Lie group Exp(v) non-metric

geodesic wrt Cartan connections LDDMM setting

  • v time-varying velocity field
  • Riemannian expid(v) metric

geodesic wrt Levi-Civita connection

  • Defined by intial momentum

Transporting trajectories: Parallel transport of initial tangent vectors

  • [Lorenzi et al, IJCV 2012]

Parallel transport of deformation trajectories

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

LDDMM: parallel transport along geodesics using Jacobi fields [Younes et al. 2008]

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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  Iterate along the curve

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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P0 P’0 P1 A P2 P’1 A’     C P0 P’0 PN A P’N P(A)

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Schild’s Ladder Intuitive application to images

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P0 P’0 T0 A T’0 PSL(A) time Inter-subject registration

[Lorenzi et al, IPMI 2011]

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

[Lorenzi, Ayache, Pennec: Schild's Ladder for the parallel transport of deformations in time series of images, IPMI 2011 ]

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Pole Ladder: an optimized Schild’s ladder along geodesics

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P0 P’0 T0 A T’0 P(A) A’ A’ C geodesic Number of registrations minimized

  • X. Pennec - ESI - Shapes, Feb 10-13 2015
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Efficient Pole Ladder with SVFs

Numerical scheme

 Direct computation:  Using the BCH:

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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[Lorenzi, Ayache, Pennec: Schild's Ladder for the parallel transport of deformations in time series of images, IPMI 2011 ]

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Left, Right and Sym. Parallel Transport along SVFs

Numerical stability of Jacobian computation

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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Parallel Transport along SVFs

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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Analysis of longitudinal datasets Multilevel framework

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Single-subject, two time points Single-subject, multiple time points Multiple subjects, multiple time points

Log-Demons (LCC criteria) 4D registration of time series within the Log-Demons registration. Schild’s Ladder

[Lorenzi et al, in Proc. of MICCAI 2011]

  • X. Pennec - ESI - Shapes, Feb 10-13 2015
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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)

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

One year structural changes for 70 Alzheimer's patients

 Median evolution model and significant atrophy (FdR corrected)

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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Contraction Expansion

[Lorenzi et al, in Proc.

  • f IPMI 2011]
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Modeling longitudinal atrophy in AD from images

One year structural changes for 70 Alzheimer's patients

 Median evolution model and significant atrophy (FdR corrected)

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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[Lorenzi et al, in Proc.

  • f IPMI 2011]

Contraction Expansion

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Modeling longitudinal atrophy in AD from images

One year structural changes for 70 Alzheimer's patients

 Median evolution model and significant atrophy (FdR corrected)

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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Contraction Expansion

[Lorenzi et al, in Proc.

  • f IPMI 2011]
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Modeling longitudinal atrophy in AD from images

One year structural changes for 70 Alzheimer's patients

 Median evolution model and significant atrophy (FdR corrected)

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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Contraction Expansion

[Lorenzi et al, in Proc.

  • f IPMI 2011]
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Longitudinal model for AD

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Modeled changes from 70 AD subjects (ADNI data) Estimated from 1 year changes – Extrapolation to 15 years

Observed Extrapolated Extrapolated year

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

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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Study of prodromal Alzheimer’s disease

 98 healthy subjects, 5 time points (0 to 36 months).  41 subjects Ab42 positive (“at risk” for Alzheimer’s)  Q: Different morphological evolution for Ab+ vs Ab-?

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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Average SVF for normal evolution (Ab-) [Lorenzi, Ayache, Frisoni, Pennec, in Proc. of MICCAI 2011]

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Detail: comparison between average evolutions (SVF)

Ab42- Ab42- Ab42- Ab42+ Ab42+ Ab42+

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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Ab42- Ab42+ Ab42- Ab42+ Time: years

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  • X. Pennec - ESI - Shapes, Feb 10-13 2015
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Study of prodromal Alzheimer’s disease

Linear regression of the SVF over time: interpolation + prediction

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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* )) ( ~ ( ) ( T t v Exp t T 

Multivariate group-wise comparison

  • f the transported SVFs shows

statistically significant differences (nothing significant on log(det) )

[Lorenzi, Ayache, Frisoni, Pennec, in Proc. of MICCAI 2011]

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  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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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

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Non-rigid registration for longitudinal analysis

Alzheimer’s atrophy trajectory Baseline MRI Follow-up MRI

j=exp(v)

Atrophy flow encoded by the dense stationary velocity field

[Lorenzi et al, MICCAI 2012]

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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Morphological analysis of SVF

Vorticity Structural readjustments Volume changes Atrophy!! Helmholtz decomposition

[Lorenzi et al, MICCAI 2012]

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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Pressure Defines sources and sinks

  • f the atrophy process

Divergence Defines flux across expanding/contracting regions

Divergence Theorem

Morphological analysis of SVF

Discovery Quantification

[Lorenzi et al, MICCAI 2012]

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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Probabilistic definition of the atrophy topography

Nice

E E C C

  • Step1. Finding local maxima and minima for the pressure field (sources,sinks)
  • Step2. Finding surrounding areas of maximal outwards/inwards flux (Expansion and Contraction)

P(Critical area) ≈ Proximity to critical point + Surrounding flux

[Lorenzi et al, MICCAI 2012]

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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From group-wise… …to subject specific

Group-wise flux analysis in Alzheimer’s Quantification

[Lorenzi et al, MICCAI 2012]

From ~106 voxels to 15 regions

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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From discovery to quantification

AD vs controls MCI vs controls Controls Ab42+ vs controls Ab42-

Sample size analysis

[Fox 2000]

p<0.05

Regional flux (all regions) Hippocampal atrophy

[Leung 2010]

(Different ADNI subset)

AD vs controls 164 [106,209] 121 [77, 206] MCI vs controls 277 [166,555] 545 [296, 1331]

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80% power

sample size ∝ sd/(mean1-mean2)

[Lorenzi et al, MICCAI 2012]

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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Top-ranked on Hippocampal atrophy measures

Hippocampal atrophy measures

46 patients, 23 controls, blinded diagnosis 0,2,6,12,26,38 and 52 weeks scans, only baseline information Test on intra-subject pairwise atrophy rates Effect size on left hippocampus Among competitors: Freesurfer (Harvard, USA) Montreal Neurological Institute, Canada Mayo Clinic, USA University College of London, UK University of Pennsylvania, USA

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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Age related Disease related

Morphological changes

Modeling the differential brain atrophy Longitudinal model to separately analyze aging and disease

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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Subject-to-Template Registration (LCC-Demons) Template Observed anatomy Closest point

Disentangling healthy aging from AD

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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  • MFCA Workshop 2011
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Virtual age: Specific changes

Vsub_to_Template Vage Vspecific

tage Vsub_to_Template = tageVage + Vspecific

Projection into the healthy aging trajectory

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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Analysis of the “virtual age” factor

increasing “virtual aging” according to the AD hypothetical clinical evolution

Vsub_to_Template = tageVaging + Vspecific Vaging

Projection into the healthy aging trajectory

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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Analysis of the specific component Vsub_to_Template = tageVageing + Vspecific

Projection into the healthy aging trajectory

  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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High discriminative power between the pathological stages

Spatial information consistent with the biology

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  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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Conclusion

Algorithms for SVFs

 Log-demons: Open-source ITK implementation http://hdl.handle.net/10380/3060  Tensor (DTI) Log-demons: https://gforge.inria.fr/projects/ttk  LCC time-consistent log-demons for AD available soon  ITK class for SVF diffeos currently under development

Schilds Ladder for parallel transport

 Effective instrument for the transport of deformation trajectories  Key component for multivariate analysis and modeling of longitudinal data  Stability and sensitivity

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  • X. Pennec - ESI - Shapes, Feb 10-13 2015

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Publications: http://www.inria.fr/sophia/asclepios/biblio Software: http://www.inria.fr/sophia/asclepios/software/MedINRIA.

Thank You!