Second African Conference on Computational Mechanics - An International Conference - AfriCOMP11

January 5 - 8, 2011, Cape Town, South Africa

A.G. Malan, P. Nithiarasu and B.D. Reddy (Eds.)

HIGH PERFORMANCE COMPUTATIONALELECTROMECHANICAL MODEL OF THE HEART

Mariano V azquez⋆, Pierre Lafortune⋆, Ruth Ar ıs⋆, Guillaume Houzeaux⋆, and AntoineJerusalem⋆⋆

⋆Barcelona Supercomputing Center BSC-CNS, Campus Nord UPC,Spain,mariano.vazquez@bsc.es

⋆⋆IMDEA Materials Institute, Madrid, Spain, antoine.jerusalem@imdea.org

SUMMARY

In this paper, we present a high performance computational electromechanical model of the heartthat includes coupling between electrical activation and mechanical deformation, capable of run-ning efficiently in up to thousands of processors. The electrical activation propagation is modelledby different strategies, such as FitzHugh-Nagumo’s or Fenton-Karma’s, including fiber orienta-tion with a special treatment for regions with large gradients in the fiber fields. The mechanicaldeformation is treated using anisotropic hyper-elastic materials in a total Lagrangian formulation.We tested several material models, such as models based on biaxial tests on excised myocardiumor orthotropic formulations. Coupling is treated using theCross-Bridges model of Peterson. Thescheme is implemented in Alya, which run simulations in parallel with almost linear scalabilityproperties in large-scale computers. The computational model is assessed through several tests,including those to evaluate its parallel performance.

Key Words: Computational Electrophysiology, Computational Solid Biomechanics, Cardiac Me-chanics, Parallelization

1 INTRODUCTION

This paper continues previous works in Cardiac Computational Mechanics [1,2,3,4,5,6] where weprogressively go from Electrophysiology models in simple geometries up to coupled Electrome-chanical models for subject-specific geometries running inlarge scale parallel computers. Dueto our background as computational mechanics researchers,it is not our intention to make directcontributions in Physiology or Medical research but to develop tools to be used decisively by thosewho can do those contributions. Of course, in the process of development, Physiologists and Med-ical doctors play a central role because they pose the problems to be solved. However, the finalobjective is to develop flexible simulation tools for complex biomechanical problems capable ofrunning efficiently in supercomputers and made them available to researchers in the medical field.This concept is thoroughly defined and explored in [7].

2 THE COMPUTATIONAL MODEL

The computational model presented simulates two differentPhysical problems and a couplingstrategy between them.

2.1 Electrophysiology

The basic form of the electrical activation potentialsφα propagation equation is1:

∂φα

∂t=

∂

∂xi

(

Dij∂φα

∂xj

)

+ L(φα). (1)

Latin subindices counts the space dimension of the problem.Greek subindices counts the numberof activation potentials involved, beingα = 1 for monodomain models andα = 1, 2 for bido-main ones, extracellular and intracellular. The equation’s right hand side represents the transientmacroscopical model, based in the well knowncontinuum cable equation. The diffusion term isgoverned by the diffusion tensorDij . The equation is set in a fixed reference frame andDij mustdescribe the cable (i.e. cardiac tissue fibers) orientationin the fixed reference frame. Then,Dij

can be written asDij = C−1

ik Dloc

lk Clj, (2)

whereClk is the base change matrix from the local fiber-aligned reference frame(ai, c1i , c

2i ) (i.e.

one axial vector and two crosswise ones) to the global reference frame.Dloc

lk is the local diagonaldiffusion matrix, whose diagonal components are the axial and crosswise fiber diffusions.L(φα)is a reaction non-linear local term. We present results for FitzHugh-Nagumo and Fenton-Karmamodels.

2.2 Mechanical Deformation

The local form of the linear momentum balance, to be solved using a Lagrangian (solid mechanics)FEM formulation, is written as

ρo∂2ui

∂t2=

∂PiJ

∂XJ

+ ρoBi, (3)

which in its weak form is

∫

Ωo

Φρo∂2ui

∂t2= −

∫

Ωo

∂Φ

∂xJ

PiJ +

∫

∂Ωo

NJPiJΦ +

∫

Ωo

ΦρoBi (4)

for each space dimensioni. PiJ , the nominal stress or first Piola-Kirchoff stress tensor, dependson the second Piola-Kirchoff stress tensorSIJ . Depending on the model, it includes active andpassive stresses and a volumetric correction. We present results for a transversally isotropic law,based on Lin-Yin [8] and an orthotropic law based on Holzapfel and Ogden [9].

2.3 Coupling

The excitation-contraction (E-C) coupling use by Watanabeet al. [10] is based on the four-statemodel of Peterson. The excitation model dictates, for a specific location, the instant the concen-tration ofCa2+ ion concentration starts changing. The active force is proportional to the concen-tration of attached cross-bridges, created by the binding of Ca2+ with the protein TnC. This isobtained by solving a set of four ordinary differential equations at each Gauss point.

1Einstein convention on repeated indexes is used.

Figure 1: Activation potentialφ evolution in the two ventricles (left) on a 12M tetrahedra meshand scalability plot up to 1000 CPUs.

2.4 Parallelization

The simulation models presented here are implemented in theAlya System, conceived for simu-lating complex computational mechanics problems in parallel. Alya is designed from scratch totake profit of parallel architectures with two premises: programming flexibility and parallel effi-ciency. In order to have an efficient algorithm to run on thousands of processors, some importantaspects of the parallelization must be carefully treated: mesh partitioning, node numbering andcommunication scheduling. These issues will be discussed at the conference presentation.

3 CONCLUSIONS

We present a high performance computational electromechanical model of the heart that includescoupling between electrical activation and mechanical deformation, capable of running efficientlyin up to thousands of processors meshes of millions of elements. The computational model isassessed through several numerical tests for different electrophysiology, mechanical and couplingmodels.

REFERENCES

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