Modeling of myocardial - PowerPoint Presentation

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Modeling of myocardial mechanical function: from tissue model to organ model

Vladimir Zverev, Sergey Pravdin, Leonid Katsnelson, Olga Solovyova

Ural Federal University,

Institute of Immunology and Physiology,

Ekaterinburg

Schematic representation of heart and MRI of the heart

BIOMECHANICS IN THE HEART: AN INTRODUCTION, COMMAS SUMMER SCHOOL Universitat Stuttgart, DE, 2013

Abbreviations: L.A. left atrium, L.V. left ventricle, R.A. right atrium, R.V. right ventricle

Pressures and volumes during the cycle

2

Right atrium

Left atrium

Right ventricle

Heart cycle

Left ventricle

Reference configuration

Current configuration

displacement

3

is right Cauchy tensor

, where

Notations for kinematic

4

P is first

Piola

-Kirchhoff stress tensor(relates forces in the current configuration with areas in the reference )

BIOMECHANICS

IN THE HEART: AN INTRODUCTION

,

COMMAS SUMMER SCHOOL

Universitat

Stuttgart

, DE

, 2013

Mechanical equation

Holzapfel

, Ogden// Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 2009,

Adapted

from

Dokos

et

al

. (2002)

shear stress (

kPa

)

amount of shear

(

adapted

from Dokos et al. ,

2002)

Le

Grice

et

al

. (1995)

5

is transmembrane potential

is

diffusion tensor

is

vector of other state

variables (

T

is element

of

S

)

Equations. Active forces

Phenomenological model

of the electrical activity

Detailed

model

of the

electrical-mechanical

activity

approximately 30 equations

Katsnelson

L.B.

et al //

Russ. J.

Numer

. Anal

.

Math. Model. 2014.

V

.

29. № 5.

p

.

1–26.

Aliev

R.R.,

Panfilov

A. V

. //

Chaos, Solitons and Fractals.

1996

.

v.

7.

p.

293–301.

3

equations

,

,

,

,

,

are parameters of model

initial fiber direction

6Numerical approximation:

key pointsNewton method for solving nonlinear equationsLagrange extrapolation

in order to find initial point for newton iterationWe decrease the time step

when iteration process does not converge

Finite element

method (

FEniCS

framework)

Operator splitting

scheme

M. S.

Alnæs

, J.

Blechta

, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E.

Rognes

and

G. N. Wells (2015). 

The

FEniCS

Project Version 1.5

, Archive

of

Numerical

Software, 3(100)

7

Electrical function in 1D Domain

We also emulated the 1D case on

bar-shaped

geometry using an adaptive mesh.

Red color correspond the hugest value of v (transmembrane potential)

Arrival time is first moment

when time when

position [dimensionless]

Study of scalability

of the LV simulation in 3d (electrical function only)

Detailed model

of the electrical activity

Katsnelson L.B. et al // Russ. J.

Numer

. Anal

.

Math. Model. 2014.

v

.

29. № 5. p. 1–26.Sozykin A. et al//Mathematical and

Information Technologies, MIT-2016 7178 points, 26156 tetrahedrons

8

9Simulation of the electrical and mechanical function in 3D model

fibers are parallel to the longest side

10

Model a wedge preparation of the LV free wallwith the characteristic fiber rotation

fiber orientation

I

. J.

LeGrice

et al //

Am

. J. Physiol.

, vol. 269, no. 2 Pt 2, pp. H571–H582, 1995.

11

Simulation of the LV diastolic filling

My deeply acknowledge

to colleges

Sergey Pravdin,

Alexandr Kursanov,

Natalya Vikulova, Anastasya

Khohlova

Pavel

Konovalov

, Tatyana

Chumarnya

,

and prof. Leonid Katsnelson, prof

. Olga SolovyovaThe work is supported by Russian

Science Foundation

Thank you for your interest