A Model of Electromechanical Coupling in the Small Intestine - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 184

advances in computational power since, the discrete approach is still challenging
to apply to intestinal slow wave propagation problems that need to be studied over
seconds, if not minutes. In another approach, the basic unit of an electrical con-
tinuum is treated in a spatially averaged sense, i.e., the conductive medium is
modeled as continuous rather than consisting of discrete cells [ 45 ]. The spatially-
averaged treatment of the conductive medium reduces the size of the model and
thereby reduces the computation time required to solve the model.
The bidomain model is one of the main continuum electrical models employed
to simulate intestinal slow wave propagation. The principle of the bidomain
equations is to model the electrical current flow between two inter-penetrating
domains: (i) the intracellular domain (usually denoted by subscript i); and (ii) the
extracellular domain (usually denoted by subscript e). The two domains are
divided by a continuum cell membrane. The current fluxes across the cell mem-
brane are described by an I ion term denoting the ionic current term in Eq. ( 3 ). This
critical step links the bidomain equations to the cell model. More specifically, the
bidomain formulation involves two equations,

rð r i r V m Þ ¼ r ðð r i þ r e Þr / e Þ ; ð 8 Þ

rð r i r / e Þ¼rð r i r V m Þ ;

C m o V m A m

ot þ I ion

ð 9 Þ

where the r terms denote tissue conductivity tensors, with subscript i denoting the
intracellular domain and subscript e denoting the extracellular domain. The I ion
denotes the current flow through the cell membrane, as described in the cell models.
Equation ( 8 ) describes the relationship between V m and / e . Equation (9) is a reac-
tion-diffusion equation in terms of the V m , where the sum of ion conductances from
cell models provides the non-linear reaction term [ 45 ]. The bidomain model pre-
sented here is a voltage dependent system, in line with the standard of previous
simulation studies of both cardiac and gastrointestinal electrical activity [ 45 ].
In the numerical solution process, the bidomain equations can be approximated
using a weighted residual approach, similar to that used for the finite deformation
approximations (described in the following section). The bidomain equations
approximated over a physical solution domain can therefore be expressed as,
Z

½r ð r i r V m Þ wdX ¼ 0 ; ð 10 Þ

X

Z wdX ¼ 0 ;

C m o V m rð r i r V m Þþrð r i r / e Þ A m

ot þ I ion

ð 11 Þ

X

where w is the weighting factor, specified using the interpolating basis function.
The basis functions are also used to interpolate parameter variables over the local
coordinate of the geometric elements. The time derivative o V m
ot

can be approximated

using a difference method, for example,