5. Biophysical Modeling...3. Relaxation Early Isovolumic 4. Filling a) Early, rapid b) Late,...

1

Medical Imaging : Biophysical Modeling :

application to cardiac modeling,

Hervé Delingette

Epione Team

[email protected]

1Hervé Delingette

5. Biophysical Modeling : application to cardiac modeling

• 5.1 Introduction to Biophysical Modeling

• 5.2 Cardiac Anatomy

• 5.3 Electro-Mechanical Modeling of the Heart

• 5.4 Principles of Biophysical Model Personalization

• 5.5 Variational Data Assimilation

• 5.6 Sequential Data Assimilation c

Hervé Delingette 2

Hervé Delingette 3

Modeling the human body : Physiome Project

A multiscale and multiphysics problem

Peter Hunter (Auckland & Oxford Universities)

2

Objectives of Physiological Models

• Descriptive / Phenomenological Models :• Quantification

• Teaching

• Training

• Discriminative Models• Patient Selection

• Diagnosis

• Clinical Trial

• Predictive Models• Therapy Planning

Increasing

Accuracy /

Complexity

Example :Cardiac Imaging & Modeling

• Main Cardiac Images :

• Echography: 2D and 3D+t. Can be real-time. Invasive vs. Non invasive. Low-Cost. Low-Quality.

• CT: DSR (historical), Spiral CT can be fast. X-Rays, limited orthogonal resolution.

• Angiocardiography : 2D projective, 3D from stereo and/or motion. Afternoon session

• SPECT: perfusion + motion; stress-rest (2M exams per year in USA). Gated imagery.

• Cine MRI and Tagged MRI : displacement field (sparse). Gated imagery. Non invasive. Expensive.

Hervé Delingette 5

Hervé Delingette 6

Towards a Personalized Virtual Physiological Human

diagnosis

personalization

evolution

simulation

planning

geometrystatistics

physics…

physiology

MedicalImages

&Signalsin

viv

o ComputationalModels of

Human Organs& Pathologies

in silico

prevention

therapy

• Virtual Physiological Human (VPH) European Program (2008-2012)• The physiome project (P. Hunter, D. Noble et al.)• Computational Models for the Human Body, N. A. (Editor), Elsevier, July 2004.

mul

tisca

le

3

Hervé Delingette 7

Objectives of 4-D Cardiac Image Analysis

• Diagnosis• quantify ventricular

function• detect ischemic/infarcted

regions• Detect abnormal behavior

• Therapy• plan, simulate, control and

evaluate

4D Cardiac Models

• Geometry Based :• Shape Information

• Physically Based :• Shape & Mechanical Information

• Physiology Based :• Shape & Mechanical & Physiology Information

Hervé Delingette 8

Geometric Cardiac Models

Global Volume

2D/3D/4D Image

SegmentationIRM-CT-US

time

volu

me

Deformable Models [Montagnat-Delingette-IVC, 2001]

4

4D Deformable modelfor Cardiac Motion Tracking

Pathological case

Electromechanical Cardiac Model

Global Volume

2D/3D/4D Image

SegmentationIRM-CT-US

2D/3D/4D Image

Tracking

Local Motion Analysis

IRM-CT-US

Contractility & Therapy Planning

Electro-mechanical

ModelIRM-CT-US + ECG-3D Maps

mechanics

blood flow electro-physiology

perfusion & metabolism

anatomy

Cardiac Modeling :A Multi-Physics Problem

5








Hervé Delingette 13


Cardiac Anatomy

Base

Apex

RVLV

Epicardium

Endocardium

Septum

RALA

AortaPulmonaryartery

SVC

Pulmonic

Tricuspidvalves

Mitral

Cardiac Microstructure [LeGrice,1995]

Myocardial fibers

• Laminar sheets

• Play an important role in cardiac modeling (Electrophysiology, Mechanics)

6

Fiber Directions (Canine Data)From high resolution Diffusion Tensor MRI

E.W. Hsu and C.S. Henriquez, Myocardial fiber orientation mapping using reducedencoding diffusion tensor imaging, Journal of Cardiovascular Magnetic Resonance, 2001.


Average structure

Geometry & Statistics

• Heart Database (E. McVeigh, NIH)

DTI Image Statistical Analysis

J.M. Peyrat, M. Sermesant, X. Pennec, H. Delingette, C. Xu, E. McVeigh, N. Ayache A Computational Framework for the Statistical Analysis of Cardiac Diffusion Tensors: Application to a Small Database of Canine Hearts. IEEE Transactions on Medical Imaging, 26(11):1500-1514, November 2007


Cardiac Anatomical Divisions• Normalised by the American Heart Association

(AHA), 17 segments+ right ventricle

• Important for models• Visualisation of results

• Parameter estimation

7










Heart Electrical Activity

Atrialdepolarisation

Ventriculardepolarisation

Ventricularrepolarisation

Cardiac Electrophysiology


8


Physiology of the heart

left bundle branchrightbundlebranch

sinusnode

AV-node

Purkinje fibers

left ventricle

right ventricle

left atrium

right atrium

Conduction System


Multiple ion channels exist in the cell membrane

K loss


Cardiac Action Potential

Vm

Na+ entry

Ca2+ entry

K+ lossK+ loss

Stimulus

9


Rise in [Ca2+]i → mechanical contraction

Sarcoplasmic Reticulum

Contractile Elements

Measuring Cardiac Electrophysiology

• Various methods:• From very invasive to non-invasive

• Measuring extra-cellular potential or action potential

• Measuring on the endocardium, epicardium or thorax level


Precise sequence of electrical activation →well-coordinated & efficient contraction

An electrocardiogram (ECG) is used to measure the electrical activity of the heart and can detect “arrhythmias”(conduction abnormalities)

3. Right and left ventricles recover

1. Right and left atria activate

2. Right and left ventricles activate

Einthoven (1912)

Sinoatrial node

Atrioventricular

node

Right atriumLeft atrium

Right ventricle

Left ventricle

10

Electrophysiology Data (2)

• Body Surface Mapping (ECGI)

Patient Data Sparse data

Not invasive Only in the Thorax not in the myocardium

Data Source KIT : Pr O. DoesselP Chinchapatnam, K Rhode, A. King, G. Gao, Y. Ma, T. Schaeffter, D. Hawkes, R. Razavi, D. Hill, S. Arridge, and M. Sermesant. Anisotropic Wave Propagation and Apparent Conductivity Estimation in a Fast Electrophysiological Model: Application to XMR Interventional Imaging. In Nicholas Ayache, Sébastien Ourselin, and Anthony Maeder, editors, Proc. Medical Image Computing and Computer Assisted Intervention (MICCAI'07), volume 4791 of LNCS, Brisbane, Australia, pages 575-583, October 2007


Electrophysiological models

1. Ionic modelsHodgkin-Huxley, Luo-Rudy, Noble… TenTusscher et al. 2004 (17 state variables)

2. Phenomenological models FitzHugh-Nagumo, Aliev-Panfilov, Mitchell-Schaeffer,

3. Eikonal EquationKeener, Colli-Franzone

Cellular automata


Biophysical model: Beeler-ReuterIonic currents

• 4 ionic membrane currents plus a stimulus current are included

• Currents are functions of the independent variables of the ODE set:• 6 gating variables• Calcium concentration,

[Ca]i• Membrane potential, Vm Iion = f (Vm, [Ca]i, x1, m, h, j, d, f)

1KICas II NaI

1xI

Fast inward Na+

current

Slow inward Ca2+

current

Time & voltage dep. outward K+

current

Time indep. outward K+ current

stimsNaxKm

m IIIIICdt

dV

11

1

Beeler GW, Reuter H (1977) J Physiol 268(1): 177-210

11


a) Ionic modelsHodgkin-Huxley, Luo-Rudy, Noble, …

b) Phenomenological models FitzHugh-Nagumo, Aliev-Panfilov, Mitchell-Schaeffer, …

c) Eikonal EquationKeener, Colli-Franzone, …

Cellular automata

PDE

FMA

More compact (few variables)

Not Physiology-based

Easier numerically

Mitchell-Schaeffer• Simplified from Fenton- Karma

u action potentialz gating variableParameters :τin time constant for inward sodium currentτout time constant for outward potassium currentτopen time constant for z (gate open)τclose time constant for z (gate close)D Diffusion Tensor (Fibre Orientations)

•Decoupling of D and APD•APD as a function of diastolic interval (Restitution curve)

DIn

APDn+1

APDn+1

DIn

gate

gate

2

zz if

zz if1

1

close

open

stimoutin

z

z

t

z

tJuuzu

uDdivt

u

ElectroPhysiology Simulation

Color : Action Potential uC. C. Mitchell and D. G. Schaeffer, A two current model for the dynamics of cardiac membrane B. of Mathematical Biology 2003

Depolarisation Time Isochrones

Pacing Location

12


a) Ionic modelsHodgkin-Huxley, Luo-Rudy, Noble

b) Phenomenological models FitzHugh-Nagumo, Aliev-Panfilov, Mitchell-Schaeffer,

c) Eikonal EquationKeener, Colli-Franzone

Cellular automata

PDE

FMA

Only models time of flight not action potential

Not Physiology-based

Fast and Easier to identify parameters

Eikonal Formulation

• Hypothesize a propagating wave

• Only interest in the wave speed not its shape

• Unknown :• T = time at which the wave reaches a given

point

• A wave can be characterized by a function of T :• Example : isotropic propagation with speed c

Hervé Delingette 351Tc

Eikonal Modeling

• Simulation propagation of a single traveling wave (depolarization wave)

Eikonal-Diffusion[Colli-Franzone et al.]

1 TDdivkTDTc t

T: Depolarisation time; c0, k, D: speed parameters

1

T

TdivTkTDTc tEikonal-Curvature

[Keener et al.]

• Fast Solution based on (Anisotropic) Fast Marching Method

13

Extension for electrophysiology

• Fast Marching Method (FMM) with correction term for curvature effect [Sermesant, MICCAI’05]

• Anisotropic Fast Marching Method with recursive update [Konukoglu, IPMI’09]

• Multi-front Anisotropic FMM [Sermesant, FIMH’07] :• Put the FMM in a Time-stepping scheme

• Approximate Action Potential with discrete states

Example:six simultaneous

fronts

Pseudo-potential:Blue: excitable

Red: depolarisedYellow: refractory

Simulation of Pathological Cases

Ectopic Focus Pseudo-potentialBlue: excitable

Red: depolarisedYellow: refractory

M. Sermesant, E. Konukoglu, H. Delingette, Y. Coudiere, P. Chinchaptanam, K.S. Rhode, R. Ra zavi, and N. Ayache. An anisotropic multi-front fast marching method for real-time simulation of cardiac electrophysiology. In Proceedings of Functional Imaging and Modeling of the Heart 2007 (FIMH'07), volume 4466 of LNCS, pages 160-169, 7-9 June 2007


Overview

• Introduction on Biophysical Modeling

• Cardiac Anatomy

• Electro-Mechanical Modeling of the Heart• Electrical Modeling

• Mechanical Modeling

• Biophysical Model Personalization • Principles of personalization

• Variational Data Assimilation

• Sequential Data Assimilation

14


The Cardiac Cycle

Systole:1. Isovolumic

contraction2. Ejection

Diastole:3. Relaxation

EarlyIsovolumic

4. Fillinga) Early, rapidb) Late, diastasis

1

2

3

4a4b

1

2

3

4b4a


Pressureand Volume

2

0

4

6

8

10

12

14

16

MVC

AVO

AVC

MVO

30

60

90

120

150

Time (msec)

700500400300200100 6000

Aorta

Left ventricle

Vo

lum

e (m

l)P

ress

ure

(kP

a)

1 2 3 4a4b


The Pressure-Volume Diagram

2001501005000

4

8

12

16

20

Pre

ssu

re (

kPa

)

AVO

MVO MVC

AVC

Volume (ml)

Isov

olum

ic

cont

ract

ion

Ejection

Isov

olum

ic

rela

xatio

n

Filling

Strokevolume

(SV)

SV=EDV-ESV

Ejection FractionEF=SV/EDV

End-diastole (ED)

End-systole (ES)

15


The Pressure-Volume Diagram

2001501005000

4

8

12

16

20

Pre

ssu

re (

kPa

)

AVO

MVO MVC

AVC

Stroke(external)

work

Volume (ml)

Isov

olum

ic

cont

ract

ion

Ejection

Isov

olum

ic

rela

xatio

n

Filling

EW P(t )d V

EDV

ESV


Electro-mechanical Model

Kc stiffnessu action potentialc strainc stress

J. Bestel, F. Clément, and M. Sorine. A Biomechanical Model of Muscle Contraction MICCAI 2001.

Inspired by Hill-Maxwell rheological modelModel of Bestel-Clément-Sorinenano

micro

méso

macro

ATP

sarcomeres

fibers

organ

active non-linear viscoelastic anisotropic incompressible material.

ES and Ep: elastic material laws,

Ec contractile electrically-activated element.

J. Sainte-Marie, D. Chapelle, R. Cimrman and M. Sorine. Modeling and estimation of the cardiacelectromechanical activity. Computers & Structures, 84:1743-1759, 2006


Electro-Mechanical Simulation

• Action potential u controls contractile element:

u > 0 : Contraction u 0 : Relaxation

• u also modifies stiffness k of

the material.

Ayache-Chapelle-Clément-Coudière-Delingette- Sermesant-Sorine (FIMH’01)

M. Sermesant, H. Delingette, N. Ayache. An Electromechanical Modelof the Heart for Image Analysis and Simulation.IEEE Transactions on Medical Imaging. 2006 May;25(5):612-25.

action potential u

16


action potential u

Electro-Mechanical Simulation• 4 Physiological Phases:

• Filling• Isovolumetric Contraction• Ejection• Isovolumetric Relaxation

• 2 Volumetric Conditions:

• Pressure Field on endocardium

• Isovolumetric Constraint of blood pools

Sermesant et al.


Physiological Parameters

Color: Action

potential

D. Chapelle, P Moireau, M. Sermesant, M. Fernandez, H. Delingette: The Digital Heart – INRIA DVD









17


Personalized Cardiac Models

Clinical Data• Electrophysiology• Image Motion

in silico Heart Model• Simulated Electrophysiology• Simulated Motion

Feedback

Patient Parameters• Electrical• mechanical

Compare simulation & measurements to learn model parameters

• Moreau-Villeger, Delingette, Sermesant, Mc Veigh, N.A. et al., IEEE Trans. on BioEng. 2006

•Sermesant, Peyrat, Chinchapatnam, Billet, Mansi, Rhode, Delingette, Razavi, Ayache, Toward Patient-Specific Myocardial Models of the Heart, Heart Failure Clinics, July 2008.

Why is the simulation different from the observation ?

Source of Errors

• Errors from the observation :• Noise & Artefacts

• Errors from the computational model :• Computational domain (mesh)

• Errors in the “parameters” : IC, BC

• Errors in the implementation (bug)

• Errors in the discretization (grid size)

• Errors in the Model (False Hypothesis)

Acquisition &Signal ProcessingIssue

VerificationIssue

ModelingIssue

PersonalizationIssues

18

Objectives of Model Personalization

• Model Validation (knowledge Building):• Model can represent observations ?

• Yes -> can be tested for model prediction

• No -> model should be modified

• Model Prediction

• Parameter analysis :• Parameter can be used for diagnosis

Parameter Estimation Issues

• Observability of the parameters

• Dimensionality of the parameters vs Dimension of the observations

• Optimization Techniques

Parameter Observability

• Not all parameters can be estimated from observations

54

dx

Cannot estimate spring stiffness kfrom dx!!

dx

Fk

?k

19


• Can estimate combination of parameters from observation

55

dx

Only estimate spring stiffness k1+k2

from dx and F!!

k1

k2

F


• Can estimate combination of parameters from observation

56

dx2

Can estimate the ratio of spring stiffness k1/k2

from displacements !!

k1 k2k1k2

dx1

Biophysical Model as a tool

• Models should be designed to answer a given question

“Essentially, all models are wrong, but some are useful.” George E. P. Box

• Avoid overfitting of parameters :

• Adapt model complexity to the complexity of the observations

• Follow Lex Parsimonia (Ockam’s razor) : among all suitable

models, select the most simple one

« The ideal model will be as simple as possible and as complex as

necessary for the particular question raised. »

Garny, Noble, Kohl, Dimensionality in cardiac modelling, Progress in Biophysics and Molecular Biology, Volume 87, Issue1 January 2005, Pages 47-66 Biophysics of Excitable Tissues

20







• 5.6 Sequential Data Assimilation


Basic Model Hypothesis

• State of a system is described as • Example :

• positions and velocities in the myocardium

• (Extracellular) Potential in the heart

• The temporal evolution of state is modeledas a dynamical system :

• , , where is a set of parameters

• Example : Mechanical 2nd law of Newton


Basic Model Hypothesis

• System State is measured with sensors (withnoise) modeled as :


Measuring Devices Estimator

MeasurementError Sources

System State X (desired but not known)

External Controls

Observed Measurements Y

Optimal Estimate of System State

SystemError Sources

System

Black Box

21

Objectives of Data Assimilation

1. Estimate the State from :• Noisy measurements

• Approximate evolution model : , , ,

0

2. Estimate various parameters :• Noise of sensors

• Model errors

• Model parameters

• Initial and boundary values Hervé Delingette 61

Data Assimilation

• Trade-off between trusting the model and the trusting the measurements


63

Offline algorithm :

• Use all measurements at once

• Perform forward and backwardsimulations

Sequential vs Variational Data Assimilation

Sequential DA Variational DA

++

+

+ ++

+X0

++

+

+ ++

+X0

Online algorithm :

• update state and error estimations fromnew measurements

• Combines predictions and corrections

ttTrajectory without assimilationTrajectory with Data assimilation

Prévision

Etat analysé

22

Variational Data Assimilation

• State depends on initial value and

parameters : , ,

• Assumptions :• Gaussian Likelihood for the observation :

• ; Σ where :– H(X) is a known model of sensor and Σ is a covariance matrix

of the measurement

• Initial Value is known a priori : ; Σ where :

– is the prior value of and Σ is the covariance on that value estimate

• Parameters are known a priori : ; ΣHervé Delingette 64


• 4D-Var minimize this functional :

• ,

• Discrete version : , ,• Add constraint : , , 1 ,

, , 1 , , 2 , , , 1 , , 2 , , 3 ….



• Difficulty : compute , which requires

to compute /• Use Adjoint Method to compute gradient

Provide gradients of the function to optimization algorithm (e.g. L-BFGS)

Computation time independent of the number of parameters

Perform Sensitivity Analysis

Need to implement adjoint model (may use automatic differentiation)

Costly in memory


23

Adjoint Method

• Sketch of the method :• Consider the Lagrange multiplier associated with

constraint . . , , 1 , . . ,• It follows a recursive relation :

Hervé Delingette







• 5.6 Sequential Data Assimilation


Sequential Data Assimilation

• Kalman Filtering (aka Linear-Gaussian State Model)

• In probabilistic term, similar than HiddenMarkov Models with continuous latent variables


xt-1 xt xT

y1 yt-1 yt yT

x1

Observed quantities

Hidden State

24

Notations

• Available quantities :

• Initial State

• Observations : …

• System (motion) Model : |

• Measurement (observation) Model |


Hypothesis of KF

• Gaussian State Model :

| ; Γ where• A is a state transition matrix (size nxn)

• Γ is covariance of state transition noise

• Gaussian Observation Model :

| ; Σ where• C is the measurement matrix (size mxn)

• Σ is covariance of measurement noise


,0

,0

1

Nv

Nw

vxCy

wxAx

t

t

tttt

tttt

72

Bayesian Estimation

xt-1 xt xT

y1 yt-1 yt yT

x1

t

ttttt PPPP

21111 xyxxxyx tt ,...,,P yyyx 21

Inference task : Compute the probability that the system is at state z at time t given all observations up to time t

Bayesian estimation: Attempt to construct the posterior distribution of the state given all measurements.

Source : Kalman/Particle Filters Tutorial, Haris Baltzakis

25

73

Bayes Filter

Two steps: Prediction Step - Update step

Advantages over batch processing Online computation - Faster - Less memory - Easy adaptation

Example: two states: A,B

Recursive Bayes Filter

Z

ttttttt

tt dzyzxPzxxPxyPc

yxP 1:111:1 |1

|

1:1111:111:1 ||1

| ttttttttttt

tt yBxPBxBxPyAxPAxBxPBxyPc

yBxP

1:1111:111:1 ||1

| ttttttttttt

tt yBxPBxAxPyAxPAxAxPAxyPc

yAxP

74

Kalman Filters - Update

Ttt

tt

AAPP

xAx

1

1ˆˆ

1)( Tt

Ttt CCPCPK

,0

,0

1

Nv

Nw

vxCy

wxAx

t

t

tttt

ttttPredict

Compute Gain

Compute Innovation

ttt xCyJ ˆˆ

Update

ktt

tttt

PCKIP

Jxx

)(

ˆˆ

Source : Kalman/Particle Filters Tutorial, Haris Baltzakis

Posterior | : is a Gaussian ; where meanand covariance can be computed iteratively

75

Kalman Filter - Example

,0

,0

1

Nv

Nw

vDxCy

wBxAx

t

t

tttt

tttt]1[tA

tt uB

]1[tC

]1[tD

,0

,0

1

Nv

Nw

vxdy

wuxx

t

t

ttt

tttt

26

76


Ttt

tt

AAPP

BxAx

1

1ˆˆPredict

77


Ttt

tt

AAPP

BxAx

1

1ˆˆPredict

78


Ttt

tt

AAPP

BxAx

1

1ˆˆPredict

Compute Innovation

ttt xCyJ ˆˆ

1)( Tt

Ttt CCPCPK

Compute Gain

27

79

Kalman Filter – Example

Ttt

tt

AAPP

BxAx

1

1ˆˆ

1)( Tt

Ttt CCPCPK

Predict

Compute Gain

Compute Innovation

ttt xCyJ ˆˆ

Update

ktt

tttt

PCKIP

Jxx

)(

ˆˆ

80

Non-Linear Case

Kalman Filter assumes that system and measurement processes are linear

Extended Kalman Filter -> linearized Case

,0

,0

1

Nv

Nw

vxCy

wxAx

t

t

tttt

tttt

,0

,0

)(

)( 1

Nv

Nw

vxgy

wxfx

t

t

ttt

ttt

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5. Biophysical Modeling...3. Relaxation Early Isovolumic 4. Filling a) Early, rapid b) Late,...

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