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Multiscale Numerical Simulation of Heart

Electrophysiology

Andres Mena*, Jose A. Bea †
†
*CIBER, Zaragoza, Spain Aragon Institute of Engineering Research, University of Zaragoza, Zaragoza, Spain

7.1 CARDIAC ELECTROPHYSIOLOGY: INTRODUCTION

In the recent decades, mathematical modeling and computer simulations have become a useful tool for tackling
problems in science and engineering. In this regard, modeling the electric activity of the heart, under physiological
and pathological conditions, has attracted the attention of researchers [1] because ventricular tachycardia and fibril-
lation are among the major causes of sudden death [2]. Because direct measurements are many times limited to only
surface signals, multiscale numerical simulations where the electrical activity at the surface as well as in the myocar-
dium can be related to the underlying electrochemical behavior of the cell, help to gain further insights into the
problem.
The electric activity of the heart is usually studied using the well-known bidomain model [3, 4]. It consists of an
elliptic partial differential equations and a parabolic partial differential equation coupled to a system of stiff nonlinear
ordinary differential equations (ODEs) describing the ionic current through the cellular membrane. This model can be
simplified to the so-called anisotropic monodomain equation [3], a parabolic reaction-diffusion equation describing
the propagation of the transmembrane potential coupled to a system of ODEs describing the cellular ionic model.
The monodomain model represents a much less computationally expensive model for the electric activity of the heart,
and has been extensively used [5–8].
The high computational cost of the bidomain and monodomain models is due to the stiffness of the system of ODE
describing the transmembrane ionic current, which introduces different space and time scales. The depolarization
front is localized in a thin layer of less than a millimeter. Therefore, this requires discretizations of the order of tenths
of millimeters in order to accurately resolve the depolarization front, implying models with millions of degrees of free-
dom to simulate the heart. The time scale is another fundamental issue in cardiac simulations. The time constants
involved in the kinetics of cellular models range from 0.1 to 600 ms, requiring in some phases of the process the
use of time steps of the order of a hundredth of a millisecond. Hence, solving a single heartbeat requires thousands
of time steps.
A number of alternatives have been proposed to solve this problem. In this particular, the multilength scale nature
of the problem has inspired the development of adaptive techniques, where the mesh is allowed to change with time
coupled with adaptive time integration schemes, to improve the computational performance [9–11]. However,
dynamic loading for these adaptive schemes is still cumbersome, limiting their application in massively parallel archi-
tectures. Recent efforts [12–15] suggest the use of multilevel meshes, fixed in time, along with adaptive time schemes
that take advantage of the different kinetics of the ionic currents. This allows reductions of up to two orders of mag-
nitude in CPU time with respect to traditional explicit algorithms. However, these techniques require a fine mesh
(lower level mesh) for solving the partial differential equations (responsible for the propagation of the
depolarization front).
Despite the efforts at designing more efficient schemes, the solution of the electrophysiology problem requires
the use of algorithms with higher levels of parallelism in multicore platforms. In this regard, the next generation of

Advances in Biomechanics and Tissue Regeneration 115 © 2019 Elsevier Inc. All rights reserved.
https://doi.org/10.1016/B978-0-12-816390-0.00007-8