A DOMAIN EMBEDDING STRATEGY FOR SOLVING THE BIDOMAIN EQUATIONS ON COMPLICATED GEOMETRIES

G. T. Lines, J. Sundnes, A. Tveito
Simula Research Laboratory,
P.O. Box 134 Lysaker, N-1325, Norway

Abstract: Simulations with the bidomain equations are often solved on simple geometries due to the complexity introduced by using more realistic boundaries. In this work we propose a method where all the calculations are carried out on squares and boxes even when the tissue boundaries are complex. This is achieved by domain embedding and proper extension of the conductivity tensors in the model.We show in this work how realistic geometries can be implemented for a simulator working on lattice grids only.

INTRODUCTION

In many applications of the bidomain model it is important to use realistic geometries, re-entry simulations being one example. The geometry is then often represented by an unstructured grid, which makes the implementation of the simulator hard compared to the case where the equations are defined over a structured grid. Although there exists tools for dealing with these complications, the fact remains that a lot of work is still carried out on idealized geometries.  
METHOD
The coupled bidomain[1] and forward problem can be formulated as:

H: ¶s ¶t = F(t,s,v;x) H:  C c ¶v ¶t + cIion(v,s) = Ñ·(Mi Ñ(v+ue)) H: Ñ·((Mi+Me) Ñue) = - Ñ·(Mi Ñv) ¶H: nH ·(Mi Ñ(v+ue)) = 0 ¶H: ue = uT ¶H: nH·Me Ñue = - nT·MT ÑuT T: Ñ·(MT ÑuT) = 0 ¶T: nT ·(MT ÑuT) = 0

(1)

where H is the myocardial region and T is the rest of the computational domain. The physical parameters C and c are membrane capacitance and membrane surface area per volume tissue. The matrices M_i,M_e and M_T are conductivity tensors for intra- and extra-cellular space in H and for extra cardiac tissue in T. The primary unknowns are the transmembrane potential v, the extra cellular potential u_e, and the extra cardiac potential u_T. To compute the ion-current I_ion a set of states s must be known, these are determined from an ODE system[2] with right hand side F. Our aim is to construct a method were a simple grid can be used to solve (1) even if H and T are complex. We first note that (1) can be reduced to a simpler system[3]. In particular the boundary conditions for u_T and u_e make it possible to combine the elliptic equations on H and T. To this end we define W = TÈH and

u(x)= ì í î ue(x), if x Î H uT(x), if x Î T

The equation for u becomes

Ñ·(M Ñu) = f, x Î W

where

f(x) = ì í î -Ñ·(MiÑv) if x Î H 0 if x Î T

(2)

and

M(x) = ì ï í ï î Mi(x)+Me(x), if x Î H MT(x), if x Î T

(3)

The next simplification is to expand the domains H and T into two circumscribing boxes H^¢ and T^¢, see Figure 1. The parabolic part of the problem is only defined over the myocardial domain H. The boundary condition states that the intra cellular current does not leave the domain. This is implemented by setting M_i=0 on H^¢-H. A similar approach is used to implement the no-flow condition on ¶W, i.e. M_T=0 on T^¢-T. There will be overlap between T and H^¢, M_T will in this area be defined according to its extra cardiac location. As a third and final simplification we propose an operator splitting method to solve the reduced system. The ODEs and the two PDEs are thus not solved simultaneously, but in three separate steps.

Figure 1: The original domains H and T embedded in boxes H^¢ and T^¢ .

 
IMPLEMENTATION

Constructing appropriate grids for the the problem is now fairly simple since the domains are only rectangles or boxes. If the finite difference scheme will be used a lattice grid is suitable. In the test cases presented here we have used the finite element method with triangular elements i 2D. Since the solution from one domain is needed to compute the solution in the other domain a mapping is needed between the domains. This is particularly easy to construct if the inner grid is a subset of the outer grid, i.e. with common nodes and elements. Such an arrangement is shown in Figure 2.

Figure 2: Cross section of the torso. The drawn boundaries are from outer to inner: ¶W ^¢, ¶ W, ¶H ^¢ and ¶H (both epi- and endocardium). The coarsest grid is shown with dotted lines. Note that the grid over H^¢ is a subset of the grid over W^¢.