personalizing physiological models with ......physiological modelling 6/25/2011 fields - mitacs...
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PERSONALIZING PHYSIOLOGICAL PERSONALIZING PHYSIOLOGICAL
MODELS WITH MEDICAL DATAMODELS WITH MEDICAL DATA
Ender Konukoglu
http://research.microsoft.com/en-us/people/enderk/
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6/25/2011 Fields - MITACS
Physiological Modelling
Models Personalization Simulations for Treatment
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Anatomical ModellingAnatomical Modelling
6/25/2011 Fields - MITACS
Parameterizations-Surfaces-Volumes-Fibre bundles
Geometrical Understanding
Personalization-Segmentation-Registration-Tractography
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Physiological ModellingPhysiological Modelling
6/25/2011 Fields - MITACS
Functional Understanding
Parameterizations-Mathematical Models-Computational Models -Simulations
Personalization-Model Fitting-Parameter Estimation
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Parameterization Parameterization -- PersonalizationPersonalization
6/25/2011 Fields - MITACS
( ) ( ) ( ) 0,1 =∇⋅−+∇⋅∇=∂∂
Ω∂uuuut
uDηD ρ
Behaviour of Tumour Cells
Electrical Conduction
Personalization:
Patient Specific Parameters
Patient Specific Geometry
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Modelling for PatientModelling for Patient--Specific TreatmentSpecific Treatment
6/25/2011 Fields - MITACS
Irradiation Margins taking into Irradiation Margins taking into
account growth dynamicsaccount growth dynamics
Simulating invasive proceduresSimulating invasive procedures
Unkelbach et al. AAIP 2011
Pernod et al. Comp. Graph. 2011
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6/25/2011 Fields - MITACS
Physiological Modelling
Models Personalization Simulations for Treatment
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Personalization
OutlineOutline
6/25/2011 Fields - MITACS
Mathematical
Model
Clinical Data
Growth of Glioma
-Spatio-temporal model,
-Longitudinal images,
-Deterministic method
Cardiac Electrophysiology
-Spatial Model,
-Cardiac Mappings,
-Stochastic method
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GliomaGlioma
6/25/2011 Fields - MITACS
Low Grade Grade IV - GBM
•Enhancement in T2w and in FLAIR
•Hypo-intense in T1w
•No special enhancement in T1w-gad
•Enhancement in T1w-gad
•Enhancement in T2w showing 2 regions
•Hypo-intense T1w
T1w T1w post-gad
T2w
T1w T1w post-gad T2w
FLAIR
Pshychology: An Introduction, C.
G. Morris, A.A. Maisto, 2001
Human Brain Glioma Cells
[www.microscopyu.com]
- Neoplasms of glial cells
- Commonly in cerebral hemisphere- Large variation in characteristics
- Proliferation- Invasion through infiltration
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Growth Model: ReactionGrowth Model: Reaction--DiffusionDiffusion
6/25/2011 Fields - MITACS
( ) ( ) ( ) 0,1 =∇⋅−+∇⋅∇=∂∂
Ω∂uuuut
uDηD ρ
=mattergray ,
matter white,
3I
DD
water
g
w
d
dα
Anisotropic Diffusion
Cell diffusion
Time
Number of cells
Logistic Population
Growth Law No-flux boundary
condition
fiber
boundary the tonormal:
brain theofboundary :
rateion proliferat:
tensordiffusion :
time:
density celltumor :
η
D
Ω∂ρ
t
u Clatz [IEEE TMI 2005]
Jbabdi [MRM 2005]
Hogea [MICCAI 2006, 2007]
Murray [Mathematical Biology 2002]
Prastawa-Gerig [MICCAI 2005, MedIA 2008]
Stein [J Biophys. 2007]
Swanson [Br. J. Cancer 2002, 2008]
Tracqui [Cell Proliferation 1995]
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6/25/2011 Fields - MITACS
Clatz et al. TMI 2005
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6/25/2011 Fields - MITACS
March 2002
March 2002 +
initial contour
September 2002
September 2002
+ simulation contours
Different SlicesManual AdjustmentsManual Adjustments
Clatz et al. TMI 2005
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Data: Longitudinal StudiesData: Longitudinal Studies
6/25/2011 Fields - MITACS
Evolution of a grade II glioma
Evolution of a grade IV glioma
Time 1 Time 3 Time 5
Time 1 Time 2 Time 3
21 days 46 days
Patient DTI120 days 270 days
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Limited ObservationsLimited Observations
6/25/2011 Fields - MITACS
Hypothetical tumor cell distribution
Tumor cell density
space
T2w T1w post-gad
Unknown
Thresholds
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Evolution of the Delineation Evolution of the Delineation notnot the Tumour the Tumour
CellsCells
6/25/2011 Fields - MITACS
tim
e
T2
(+21 days)
T3
(+46 days)
Time T1
space
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Model Reduction for Parameter EstimationModel Reduction for Parameter Estimation
6/25/2011 Fields - MITACS
Evolution of Tumour Cell Densities
( ) ( )uuut
u −+∇⋅∇=∂∂
1ρD
Evolution of Tumour Delineation
Travelling wave solutions of
reaction-diffusion equations
( )vtutu −⋅= nxx ),(
Travelling time formulation for
tumor evolution
( )TT
TTTe
T
Teff
eff
∇∇∇⋅∇==∇∇
−−− −
D
DD
',1'13.0
2
34 3.0/ κρρ
ρ ρκ
Konukoglu et al. TMI 2010
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6/25/2011 Fields - MITACS
Growth of the delineation with RD dynamics
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Parameter EstimationParameter Estimation
6/25/2011 Fields - MITACS
( )∑−=
ΓΓ=1,..,1
2ˆ,
Ni
iidistCMinimization Costρα
,mattergray ,
matter white,
3
=I
DD
water
g
w
d
d
tim
e
T2
(+21 days)
T3
(T2 + 46 days)
Time T1
space
Powell, Mathematical Programming, vol. 92, no. 3, 2002
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Personalization leading to Prediction: Grade Personalization leading to Prediction: Grade
IVIV
6/25/2011 Fields - MITACS
Estimated from Time 1 & Time 2 : ρ dw dg
0.05 / day 0.66 mm2/day 0.0013 mm2/ day
tim
e
Time 2
Time 3 (T2 + 46 days)
space
High grade glioma (glioblastoma multiforme)
MRI T1 Gd, 0.5*0.5*6.5mm3 time points
MR DTI : 2.5mm (time 2)
space
tim
e
Konukoglu et al. TMI 2010
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Personalization leading to Prediction: Grade Personalization leading to Prediction: Grade
IIII--IIIIII
6/25/2011 Fields - MITACS
Estimated from Time 1 to Time 4 : ρ dw dg
0.008 / day 0.20 mm2/day 0.0007 mm2/ day
tim
e
Time 4
Time 5
(T4+180)
space
Low grade glioma (grade 2 Astrocytoma)
MRI T2 Flair 0.5*0.5*6.5mm5 time points
MR DTI : 2.5mm (time 1)
Konukoglu et al. TMI 2010
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A small step back to digestA small step back to digest
• Patient-specific parameters for growth quantification
• Personalized model - predictions on the growth patterns
• Model reduction
• Regularization – Constraint Optimization [Relan et al. IEEE TBE 2011,
Delingette IEEE TBE 2011]
• Restrict the search space
• Enlarge the model via incorporating observation model [Hogea et al. HJ
Theo. Biol. 2011]
• Loss of information vs. uncertainty on the inverse problem
• Influence of uncertain data on the parameters
• Influence of uncertain parameters on the simulations
• Probabilistic treatment [Konukoglu et al. PBMB 2011, Menze et al. IPMI 2011]
6/25/2011 Fields - MITACS 21
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Parameter
Estimation
OutlineOutline
6/25/2011 Fields - MITACS
Mathematical
Model
Clinical Data
Growth of Glioma
-Spatio-temporal model,
-Longitudinal images,
-Deterministic method
Cardiac Electrophysiology
-Spatial Model,
-Cardiac Mappings,
-Stochastic method
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Cardiac ElectrophysiologyCardiac Electrophysiology
6/25/2011 Fields - MITACS
Images taken from Durrer et al., 1970 and Malmivuo and Plonsey, 1995
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Model: EikonalModel: Eikonal--DiffusionDiffusion
• Model of depolarization times
• Fast Simulations
• Purkinje network is
approximated
• Fibres from an atlas
6/25/2011 Fields - MITACS
T: depolarization times in the cardiac tissue
c0: dimensionless constant
τ: cell membrane time constant
D(x): conductivity
M(x): local fibre orientation
ΩE: onset location
Sermesant et al. MedIA 2003, Sermesant et al. FIMH 2003, Konukoglu et al. IPMI 2007, Chipchapatnam et al. TMI 2007
Relan et al. IEEE TBE 2011
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Data: Catheter Mappings and dynamic MRIData: Catheter Mappings and dynamic MRI
6/25/2011 Fields - MITACS
Geometry from MRI:
-Healthy muscle
-Scar regions
-Complex geometry
Depolarization times from Catheter
-Points on the boundary
-Sparse Data
-Noisy Data
Courtesy of Michel Haïssaguerre and Bordeaux University Hospital
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Parameters: Functional ConductivityParameters: Functional Conductivity
6/25/2011 Fields - MITACS
Anatomically defined regionsConductivity:
-Spatially varying conductivity
in the endocardium surface
-Global conductivity for the
healthy muscle
-Conductivity for the scar and
the peri-scar region
RBF Approximation
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Parameters: Onset LocationParameters: Onset Location
6/25/2011 Fields - MITACS
- Onset location on the
septum- Simplification for the
Purkinje network along with
the fast conductivity- 2D parameterization of the
surface patch
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6/25/2011 Fields - MITACS
Probabilistic FormulationProbabilistic Formulation
• Parameter as a random variables
• PDE as conditional dependencies
Joint Distribution of observations and parameters
Assuming independence of observations
Observation
Model
Parameter
priors
Physiological
Model
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Posterior DistributionPosterior Distribution
6/25/2011 Fields - MITACS
Using the uniqueness of the forward problem
Konukoglu et al. PBMB 2011
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6/25/2011 Fields - MITACS
Inferred Posterior Distribution: Toy ExampleInferred Posterior Distribution: Toy Example
Endocardial conductivity
Mu
scle
Co
nd
uctivity
Posterior Distribution
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Posterior DistributionPosterior Distribution
6/25/2011 Fields - MITACS
Using the uniqueness of the forward problem
- Analytical solution only if the model is analytically solvable
- Sampling based methods: MCMC
- Collocation based methods: Sparse grid reconstructions
- Computational bottleneck – # of sample simulations for high
dimensional parameters
- 20 dimensional problem ~ 105 simulations ~ 20 seconds /
simulation ~ 23 days…
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Spectral Representations for Random Spectral Representations for Random
VariablesVariables
6/25/2011 Fields - MITACS
Polynomial Chaos Expansion for random variables
[Ghanem 2006, Xiu 2009]
Gaussian – HermiteUniform – Legendre…
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Spectral Representation of the PDESpectral Representation of the PDE
6/25/2011 Fields - MITACS
Randomness in model solution can be represented using the same spectral basis
[Ghanem 2006, Xiu and Karniadakis 2002, Xiu 2009,…]
Linear superposition instead of solving the PDE numerically
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Expansions for ED parametersExpansions for ED parameters
6/25/2011 Fields - MITACS
Uninformative prior for the vector
Spectral basis:
Same construction for the onset location
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Expansion for the Depolarization TimesExpansion for the Depolarization Times
6/25/2011 Fields - MITACS
Computation of the basis for
• Galerkin projections into the PDE [Marzouk 2007, 2009]
– Nonlinearities become a problem
– Computationally expensive for high dimensional problems, i.e. high M
• Numerical integration / Stochastic Collocation [Marzouk 2009]
– Curse of dimensionality
– Computationally expensive for high dimensional problems, O(105)
samples for a problem of 15 dimensions. [Ma 2009]
• Solution: Sparse reconstructions…
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Computational
Bottleneck
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Sparse link between parameters and Sparse link between parameters and
solutionssolutions
6/25/2011 Fields - MITACS
• For a given point only a small number of parameters influence the result of the ED model.
• We expect a sparse representation in the spectral domain:
• Only a small number of basis functions
• They can be recovered by a small number of random projections
• And they can be computed by the following problem
[Doostan 2009] studied 1D linear diffusion equation
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Sparse Reconstruction for the ED ModelSparse Reconstruction for the ED Model
6/25/2011 Fields - MITACS
a b c
• 18 parameters for conductivity
• 2 parameters for onset location
• Cartesian grid in 3D
• Size: 15x15x10 cm3,
• Resolution: 1mm3
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Patient Data: Observation ModelPatient Data: Observation Model
6/25/2011 Fields - MITACS
Sources of Uncertainty:
-Acquisition via catheter: motion artefacts
-Projection of observation points onto the patient mesh
-Depolarization times are estimated from potential measurements
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PersonalizationPersonalization
6/25/2011 Fields - MITACS
Endocardial Observation Points Epicardial Observation Points
.
Range for all possible personalized
simulations
Blue region ± one standard deviation of ǫ(x)
Measured depolarization times
Quantifying Uncertainty
on estimated parameters
Model FitModel Fit
Konukoglu et al. PBMB 2011
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Personalization and Endocardial PredictionPersonalization and Endocardial Prediction
6/25/2011 Fields - MITACS
Endocardial Observation Points Epicardial Observation Points
Model FitPrediction
.
Range for all possible personalized
simulations
Blue region ± one standard deviation of ǫ(x)
Measured depolarization times
Quantifying Uncertainty
on the model predictions
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Personalization and Epicardial PredictionPersonalization and Epicardial Prediction
6/25/2011 Fields - MITACS
Endocardial Observation Points Epicardial Observation Points
Model Fit Prediction
.
Range for all possible personalized
simulations
Blue region ± one standard deviation of ǫ(x)
Measured depolarization times
Not all observations are
the same
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ChallengesChallenges
• Detailed yet tractable models
• Fusion of information from more sources
• Novel methods for taking into account high dimensional uncertainty
• Multi-scale models
• Integrating more information: micro arrays, pathology results, history
• Integrating different scales of computation
• Numerical challenges
• Inference of micro-scale information from macro-scale
data.
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ThanksThanks
• Nicholas Ayache
• Maxime Sermesant
• Olivier Clatz
• Xavier Pennec
• Bjoern H. Menze
• Hervé Delingette
• Eric Pernod
• Michel Haïssaguerre
6/25/2011 Fields - MITACS
• Andrew Blake
• Antonio Criminisi
• Kilian M. Pohl
• Ben Glocker
• Emmanuel Mandonnet
• Pierre-Yves Bondiau
• Erin Stretton
• Phani Chipchapatnam
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• Health-e-Child European project
• Compu-Tumor initiative
• Microsoft-Amalga product team
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ThanksThanks
6/25/2011 Fields - MITACS 44
• Health-e-Child European project
• Compu-Tumor initiative
• Microsoft-Amalga product team
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Personalization under UncertaintyPersonalization under Uncertainty
6/25/2011 Fields - MITACS
Homogeneous
Uncertainty
Uncertainty
based on
Projection
Distances
[PBMB under revision]
• Functional Conductivity
• Onset Location
• Efficient Inference using Spectral Methods
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