On the Computational Complexity of the Bidomain and the Monodomain Models of Electrophysiology

Abstract

The bidomain model, coupled with accurate models of cell membrane kinetics, is generally believed to provide a reasonable basis for numerical simulations of cardiac electrophysiology. Because of changes occurring in very short time intervals and over small spatial domains, discretized versions of these models must be solved on fine computational grids, and small time-steps must be applied. This leads to huge computational challenges that have been addressed by several authors. One popular way of reducing the CPU demands is to approximate the bidomain model by the monodomain model, and thus reducing a two by two set of partial differential equations to one scalar partial differential equation; both of which are coupled to a set of ordinary differential equations modeling the cell membrane kinetics. A reduction in CPU time of two orders of magnitude has been reported. It is the purpose of the present paper to provide arguments that such a reduction is not present when order-optimal numerical methods are applied. Theoretical considerations and numerical experiments indicate that the reduction factor of the CPU requirements from bidomain to monodomain computations, using order-optimal methods, typically is about 10 for simple cell models and less than two for more complex cell models.

Modeling Defibrillation

“…Typically, monolithic meshes consisting of one element type only were used, but exception exist Rocha et al, 2011) where hybrid meshes consisting of tetrahedra, hexahedra, pyramids and prisms were used. Further, most FEM studies relied on the Galerkin FEM where linear test functions with tetrahedral elements (Franzone et al, 2006;Sundnes et al, 2006;Vigmond et al, 2002), isoparametric trilinear test functions with hexahedral elements (Munteanu et al, 2009) or cubic-hermite hexahedral elements (Rogers & McCulloch, 1994;Saucerman et al, 2004) were used. Independently of the spatial discretization technique, the choice of space step, h, is of major importance.…”

Simulation of cardiac electrophysiology on next-generation high-performance computers

“…However, the computational benefits of using the monodomain equations are much less significant for the more realistic cell models that are likely to be used in the future than they are for simpler ones. Sundnes et al (2004) showed that, when using order optimal methods, the theoretical reduction in computational requirements of the monodomain equations when compared with the bidomain is only two (compared with approximately ten for simpler models).…”

C haste : incorporating a novel multi-scale spatial and temporal algorithm into a large-scale open source library

“…When using this technique, or a related technique, the time step used is usually approximately 0.1 ms, i.e. an order of magnitude larger than that used in conjunction with the explicit method (Sundnes et al 2001(Sundnes et al , 2005(Sundnes et al , 2006Trew et al 2005;Whiteley 2006Whiteley , 2007. We note that the only additional complexity introduced when using a semi-implicit technique instead of an explicit technique is that a larger linear system must be solved on each time step.…”