Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol166.Issue1.Jul2000:

Mathematical Biosciences 166 (2000) 85±100
www.elsevier.com/locate/mbs

A numerically ecient model for simulation of de®brillation in
an active bidomain sheet of myocardium
Kirill Skouibine a, Natalia Trayanova b,*, Peter Moore c
a

b

Department of Mathematics, Duke University, USA
Department of Biomedical Engineering, Tulane University, New Orleans, LA 70118, USA
c
Department of Mathematics, Tulane University, New Orleans, LA 70118, USA

Received 18 November 1999; received in revised form 13 April 2000; accepted 1 May 2000

Abstract
Presented here is an ecient algorithm for solving the bidomain equations describing myocardial tissue
with active membrane kinetics. An analysis of the accuracy shows advantages of this numerical technique
over other simple and therefore popular approaches. The modular structure of the algorithm provides the

critical ¯exibility needed in simulation studies: ®ber orientation and membrane kinetics can be easily
modi®ed. The computational tool described here is designed speci®cally to simulate cardiac de®brillation,
i.e., to allow modeling of strong electric shocks applied to the myocardium extracellularly. Accordingly, the
algorithm presented also incorporates modi®cations of the membrane model to handle the high transmembrane voltages created in the immediate vicinity of the de®brillation electrodes. Ó 2000 Elsevier
Science Inc. All rights reserved.
Keywords: Transmembrane potential; De®brillation; Bidomain model; Active membrane kinetics

1. Introduction
The bidomain representation of cardiac tissue has been widely accepted and is now often used
in modeling studies [1]. It is of particular interest for the simulation of de®brillation, since it allows
modeling of extracellular shocks. The computational expense of solving bidomain equations has
previously limited de®brillation simulations mainly to the case of passive tissue [2±5]. Active bidomain model implementations for de®brillation are relatively few [6±9]. The forward Euler rule
is a popular choice for time-stepping in most of them (except for [9]). Additional time savings are

*

Corresponding author. Tel.: +1-504 862 8934; fax: +1-504 862 8779.
E-mail address: nataliat@tulane.edu (N. Trayanova).

0025-5564/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved.

PII: S 0 0 2 5 - 5 5 6 4 ( 0 0 ) 0 0 0 1 9 - 5

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K. Skouibine et al. / Mathematical Biosciences 166 (2000) 85±100

achieved by solving the full bidomain equations only along the ®ber and later coupling the ®bers
[8]. Another problem that modeling research in de®brillation faces is the need for modi®cation of
the membrane equations outside the normal signal range of the transmembrane potential in order
to accommodate the e€ect of strong electric ®elds. This is true for all popular ionic models such as,
for instance, Beeler±Reuter [10], BRDR [11], Luo±Rudy phase I [12] and to some degree, the
Luo±Rudy phase II model [13].
The goal of the present study is to o€er the electrophysiology community an ecient way of
solving the full bidomain system of equations. The equations include, without loss of generality,
the BRDR ionic membrane representation adjusted here to accommodate strong electric ®elds.
The major advantage of this method is the time stepping predictor±corrector technique that allows higher temporal accuracy and better stability as compared to the forward Euler method that
is widely used in computational electrophysiology. Since the implementation of the predictor±
corrector scheme is almost as easy as that of the forward Euler method, we provide detailed
comparisons between the two so as to encourage the use of the predictor±corrector method rather
than forward Euler even when the simplicity of the model is the major consideration. It is important to note that the predictor±corrector time stepping is independent of the rest of the model

presented here.

2. Model equations
We model a two-dimensional slice of cardiac tissue using the bidomain representation [1]. The
intracellular, Ui (mV), and extracellular, Ue , potentials, as well as the transmembrane potential,
Vm ˆ Ui ÿ Ue , are de®ned everywhere in the cardiac domain X. The following coupled reaction±
di€usion equations constitute the bidomain model:
r
…^
ri rUi † ˆ im ;

…1†

r
…^
re rUe † ˆ ÿim ÿ i0 in X;


oVm
‡ Iion …Vm ; t† ‡ G…Vm ; t†Vm ;
i m ˆ b Cm
ot

…2†
…3†

where r^i (mS/cm) and r^e are conductivity tensors in the corresponding domains, im (lA=cm3 ) is
the volume density of the transmembrane current, b (cmÿ1 ) is the surface-to-volume ratio (i.e., the
ratio of total membrane area to total tissue volume), Cm (lF=cm2 ) is the speci®c membrane ca3
pacitance, and i0 (lA=cm ) is the volume density of the extracellular (shock) current.
By eliminating Ui , we obtain the following system:
r
…^
ri rVm † ‡ r
……^
re ‡ r^i †rUe † ˆ ÿi0 ;

…4†

r
…^
re rUe † ˆ ÿim ÿ i0

…5†

in X;



oVm
i m ˆ b Cm
‡ Iion …Vm † ‡ G…Vm ; t†Vm ;
ot

…6†

K. Skouibine et al. / Mathematical Biosciences 166 (2000) 85±100

87

d‰CaŠi
ˆ fCa …Vm ; y; ‰CaŠi †;
dt

…7†

dG
ˆ fG …Vm †;
dt

…8†

dyk
ˆ fyk …Vm ; yk †;
dt

k ˆ 1; . . . ; 5; in X;

…9†

where X ˆ ‰0; aŠ  ‰0; bŠ is a rectangle. Functions yk represent the gating variables x1 ; m; h; d; and f ;
‰CaŠi is the intracellular calcium concentration, and G is the electroporation function. The
boundary conditions re¯ect the fact that the tissue is surrounded by an insulator, except where
stimulated
~

n
…^
ri r…Ue ‡ Vm †† ˆ 0;

~
n
…^
re rUe † ˆ 0

~
n
…^
ri r…Ue ‡ Vm †† ˆ 0;

Ue ˆ Vstim

on oX1 ;

on oX2 ;

…10†
…11†

where oX2 is the part of the boundary where the extracellular stimulus is applied and oX1 represents the rest of the boundary.
Here the ionic current (the term Iion …lA=cm2 † in (6)) is represented by the Drouhard±Roberge
modi®cation [11] of Beeler±Reuter kinetics [10] (BRDR model). Action potential duration (APD)
in a ®brillating ventricle is considerably shorter than a normal action potential. To account for
this in our model, we decrease the value of the time constants of the slow inward current by a
factor k ˆ 8. This modi®cation follows the procedure suggested in [35]. This results in a solitary
APD of approximately 100 ms.
We alter the original BRDR model in order to accommodate strong electric ®elds. Since the
behavior of the ionic currents under strong electric ®elds remains unknown, our modi®cations
amount only to changes that alleviate the inherent numerical instability of the original BRDR
model at the range of external stimuli used for de®brialltion. Speci®cally, the equations for the
rate coecients am;h;x1 ; bm;h;x1 are extended outside the normal range of Vm ; so is the original differential equation for the intracellular calcium concentration ‰CaŠi . We ensure that: (1) the sodium
activation gates remain closed …m ˆ 0† for Vm < ÿ85 mV and open …m ˆ 1† for Vm > 100 mV; (2)
the sodium inactivation gates are open …h ˆ 1† for Vm < ÿ90 mV; (3) the outward recti®er current
activation gate stays open …x1 ˆ 1† for Vm > 400 mV; and (4) ‰CaŠi is kept constant for
Vm > 200 mV. Modi®cations (1)±(3) do not introduce any changes in the physiology of channel
behavior. Modi®cation (4) ensures that the ‰CaŠi concentration does not become negative for
Vm > 200 mV. The latter is a limitation of the BRDR model that is corrected here to the best of
our ability. The summary of rate coecient revisions for the various gates is given in Appendix B.
The modi®cations presented here are natural extensions of the probabilities of channel opening

and closing and do not a€ect the kinetics of the ionic currents. It is important to underscore that
these modi®cations take e€ect only under the de®brillation electrodes where the shock-induced
transmembrane potential is large. The potentials in rest of the tissue are typically within the range
of the original BRDR model. Using the approach, we are able to achieve physiologically
meaningful results while maintaining numerical stability of our transmembrane potential solutions under the de®brillation electrodes.

88

K. Skouibine et al. / Mathematical Biosciences 166 (2000) 85±100

The additional variable membrane conductance G…Vm ; t† accounts for the pore generation in the
membrane during strong electric shocks [14]. G grows at the following rate [15],
2
2
dG
ˆ aeb…Vm ÿVrest † …1 ÿ eÿc…Vm ÿVrest † †; G…0† ˆ G0 :
…12†
dt
Values of a …mS=cm2 ms†, b …1=mV2 †, and c …1=mV2 † are given in Appendix A.
To solve the system of equations (4)±(11), we ®rst replace the spatial di€erential operators in

Eqs. (4) and (5) with di€erence operators de®ned on a ®nite grid. This leads to a system of differential-algebraic equations (DAEs). We then employ a semi-implicit predictor±corrector scheme
that, in combination with the Generalized Minimal Residual Method (GMRES) iterative solver
for large linear algebraic systems, eciently solves our system of DAEs (Section 3). While relatively simple in implementation, this method has a clear advantage over the Euler method traditionally used in theoretical electrophysiology to solve bidomain and monodomain equations.
Comparative studies to this e€ect are presented in Section 4.

3. Predictor±corrector solution scheme
We rewrite (4) and (5) in the following form:
ri †…Vm †i;j ˆ …i0 †i;j
K…^
re ‡ r^i †…Ue †i;j ‡ K…^


dVm
K…^
re †…Ue †i;j ‡ b Cm
‡ Iion ‡ …GVm †
dt

…13†

on Xh ;


ˆ ÿ…i0 †i;j

on Xh ;

…14†

i;j

where K…^
r†…f † is a ®nite-di€erence approximation of the di€usion term r
…^
rrf †.
By projecting the unknowns Vm , Ue , etc., in (4)±(11) that are de®ned everywhere in X onto a
®nite grid
Xh ˆ 
Xh ‡ oXh
ˆ …xi ; yj † j xi ˆ iDx;


a
b
i ˆ 0; . . . ; Nx ; yj ˆ jDy; j ˆ 0; . . . ; Ny ; Dx ˆ ; Dy ˆ
;
Nx
Ny

~e , etc., with the components …Vm † ˆ Vm …xi ; yj †, …Ue † ˆ Ue …xi ; yj †; etc.
we obtain the vectors V~m , U
i;j
i;j
Here oXh consists of the mesh points on the boundary oX. This leads to the following system of
DAEs:
dV~m ~ ~ ~ ~
y; t†;
ˆ FVm …Vm ; Ue ; G;~
dt

…15†

ƒ!
d‰ Ca Ši ~ ~
ƒ!
y; ‰ Ca Ši †;
ˆ FCa …Vm ;~
dt

…16†

~
dG
ˆ~
FG …V~m †;
dt

…17†

K. Skouibine et al. / Mathematical Biosciences 166 (2000) 85±100

d~
yk ~ ~
ˆ Fyk …Vm ;~
yk †;
dt
~e ˆ ~
AU
FUe …V~m †

k ˆ 1; . . . ; 5; on Xh ;
~h ;
on X

~e †
BV~m ˆ ~
FB …V~m ; U

89

…18†
…19†
…20†

on oXh ;

where the components of the vector function in the right-hand side of (15) are found from (14):

1 
1
K…^
re †…Ue † ‡ …i0 † ÿ
…Iion ‡ …GVm ††i;j on Xh ;
…21†
…FVm †i;j ˆ ÿ
i;j
bCm
Cm
and the functions on the right-hand side of (16)±(18) have the components from (7)±(9)
…FCa †i;j ˆ fCa ……Vm †i;j ; …y†i;j ; …‰CaŠi †i;j †;

…22†

…FG †i;j ˆ fG ……Vm †i;j †;

…23†

…Fyk †i;j ˆ fyk ……Vm †i;j ; …yk †i;j † on Xh :

…24†

Thus, systems (15)±(20) consists of 8  Nx  Ny ODEs (15)±(18) that are coupled with two algebraic systems. The sparse linear algebraic system (19), represented by the …Nx ‡ 1†…Ny ‡ 1†  …Nx ‡
1†…Ny ‡ 1† matrix A, is obtained from (13) and the ®nite di€erence approximation of the boundary
conditions (10) and (11) (equations involving Ue only). System (20) results from the ®nite difference approximation of the boundary conditions (10) and (11) using ®ve-point di€erence second-order formulae (equations involving Vm ). The 2…Nx ‡ Ny †  2…Nx ‡ Ny † matrix B of (20) is
tridiagonal: the values of Vm at the internal nodes are considered known and the boundary nodes
are updated. At each boundary node the di€erence equation has only three unknowns.
The predictor±corrector scheme for the solution of (15)±(20) involves four sub-steps for each
vn ˆ ~
v…tn † and ~
v denotes the predicted value of ~
v. The steps are
full time step tn‡1 ˆ tn ‡ Dt. Here ~
as follows (we use (15) as an example, the rest of the ODEs are solved using identical steps):
(P) The predictor step is the explicit two-step Adams±Bashforth rule


3 ~n
1 ~nÿ1
n

~
~
on Xh ;
F ÿ F
…25†
Vm ˆ Vm ‡ Dt
2 Vm 2 Vm
where


n n
~n ;~
~n ; G
~
FVnm ˆ ~
y
;
t
FVm V~mn ; U
e

on Xh :

…26†

(E) The evaluation step makes use of the predicted value V~m in order to update the right-hand
~e
side functions of (15)±(18). First, we solve the algebraic system (19) for U
~ ˆ ~
AU
FUe …V~m †
e

~h
on X

…27†

employing an iterative technique, GMRES [16], a variation of the conjugate gradient method that
proves ecient in handling large sparse non-symmetric systems. The eciency of GMRES is
increased by preconditioning. We precondition with the diagonal of A. To obtain the values of V~m
on the boundary of Xh , we solve the tridiagonal system

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K. Skouibine et al. / Mathematical Biosciences 166 (2000) 85±100



~
FB V~m ; U
BV~m ˆ ~
e

…28†

on oXh :

We then obtain the new values of the function


 
~ ;~
~ ; G
~
FVm ˆ ~
FVm V~m ; U
y
;
t
on Xh
e

…29†

and proceed to the corrector step.
(C) The corrector step is the implicit two-step Adams±Moulton rule with ~
FVn‡1
approximated by
m

~
FVm , the result of the predictor step

Dt  ~
…30†
FVnÿ1
5FVm ‡ 8~
V~mn‡1 ˆ V~mn ‡
on Xh ;
FVnm ÿ ~
m
12
(E) Finally, we re-evaluate the right-hand side function, using the corrected value V~mn‡1 . Again,
~n‡1
we ®rst solve (19) to get U
e


~h ;
~n‡1 ˆ ~
FUe V~mn‡1
…31†
AU
on X
e
and (19) to obtain V~mn‡1 on the grid boundary


~n‡1
FB V~n‡1 ; U
BV~n‡1 ˆ ~
on oXh
m

m

e

and then determine


n‡1 ~n‡1 ~n‡1 n‡1 n‡1
~
~
~
V
;
G
;~
y
t
;
U
ˆ
F
FVn‡1
Vm
m
e
m

on Xh

…32†

…33†

for the next time step.
Our multi-step method requires two initial vectors V~m0 and V~m1 to begin the iteration described
above. The initial condition to our problem provides V~m0 (see Appendix A). We use the forward
2
Euler method with the time step …Dt† to generate V~m1 . This choice of the time step guarantees that
the accuracy of PECE is preserved.
When Neumann boundary conditions for the extracellular potential are applied at all tissue
borders (i.e., there are no border electrodes), Ue is determined only up to a constant, whereas the
di€erence between Ui and Ue , the transmembrane potential, is determined uniquely. To avoid the
computational problems that arise (the algebraic system constituting (13) will have in®nitely many
solutions) we have to select a point of reference that will single out one Ue from the family of
solutions. We choose to ®x Ue to be zero in the middle of the tissue. This is done by substituting
the equation for Ue at that node in (13) with …Ue †imiddle; jmiddle ˆ 0:
4. Convergence rate estimates and accuracy comparison
The choice of the presented numerical technique was determined in accordance with the following considerations. Cardiac action potential is characterized by a sharp increase during upstroke (e.g., 500 mV/ms in the BRDR model). The equations of the system describing the ionic

K. Skouibine et al. / Mathematical Biosciences 166 (2000) 85±100

91

current Iion belong to the category of sti€ di€erential equations that ideally require treatment with
fully implicit numerical methods. This holds true for any realistic ionic model that one might
consider in place of BRDR, including Luo±Rudy I and II, DiFrancesco±Noble and others. Implicit methods for DAE [17] require the solution of a system of algebraic equations, similar to (19)
for all unknowns (in our case there are 10 of them) at all nodes of the discrete spatial mesh. This
would involve matrices an order bigger than A from our predictor±corrector stepping. We
compromise by using a semi-implicit method that provides a higher order of accuracy than that of
commonly used explicit solvers, i.e., forward Euler method, while avoiding the computational
expense of fully implicit methods. A two-step Adams±Bashforth predictor with a two-step Adams±Moulton corrector is a classical method with good stability properties [18]. Still, as computer
power increases, the fully implicit option will look more and more attractive.
Contributions to the numerical error in our model include round-o€, the spatial and temporal
truncation errors, and the residual error in (19) from the matrix solver GMRES. They are considered below.
4.1. GMRES accuracy analysis
To solve an n  n linear system
A~
u ˆ~
b

…34†

for ~
u (the extracellular potential function in our case) GMRES, a projection method, constructs
v1 ; . . . ;~
vm Š of the Krylov subspace
an l2 -orthonormal basis Vm ˆ ‰~
Km ˆ spanf~
r0 ; A~
r0 ; . . . ; Amÿ1~
r0 g;

…35†

b ÿ A~
u0 is the residual, and ~
u0 is the initial guess. To do this, the method uses a
where ~
r0 ˆ ~
procedure called Arnoldi's method. The solution approximation on the mth step of the iteration
procedure is obtained by ®nding the unique minimizer ~
um of k~
b ÿ A~
uk2 , such that ~
um 2 ~
u0 ‡ Km .
The minimization, as implemented in this algorithm (see [16] for details), is equivalent to solving
an m  m upper-triangular system. By construction, the algorithm converges in at most n steps (in
exact arithmetic). Since the method becomes increasingly expensive as m grows (the number of
multiplications is O…m2 np†, where p is the average number of unknowns per row), we employ a
banded version of GMRES. Here, when constructing the basis Vm , we require vj to be orthogonal
only to the previous l vectors vjÿl ; . . . ; vjÿ1 , where l is a small ®xed number that depends only on
the dimension of A and is chosen experimentally (this is similar to an earlier method called incomplete orthogonalization) [19]). This allows us to reduce storage requirements and to bring the
number of multiplications to O…mnp† for m steps. Although a theory that would guarantee
convergence of this banded iterative approach has not yet been developed, our simulation studies
show that the method is rather robust. As suggested in [16], we further accelerate the algorithm by
ˆ~
um and ~
r0new ˆ ~
b ÿ A~
um . The optimal value of m is
restarting it every m steps by setting ~
unew
0
determined experimentally. It remains ®xed throughout the simulation and takes values anywhere
from 5 to 35, depending on the size of the matrix A. Finally, we use diagonal preconditioning in
all our simulations. The algorithm stops when the norm of the residual becomes less than a ®xed
tolerance TOL.

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K. Skouibine et al. / Mathematical Biosciences 166 (2000) 85±100

Table 1
Relative di€erence between the solutions ~
u1 and ~
u2 obtained with tolerances TOL ˆ 10ÿ5 and TOL ˆ 10ÿ7 , respectively
(here ~
u1 and ~
u2 are the vectors that include all the unknowns in (15)±(20))
Total time T

Matrix size n

Average number of el-ts per row u

k~
u1 ÿ ~
u2 k2
k~
u2 k2

k~
u1 ÿ ~
u2 k1
j~
u2 k1

k~
u1 ÿ ~
u2 k1
j~
u2 k1

1. Straight ®bers. Transmembrane stimulation of tissue in diastole (1 ms S1)
10 ms
1089
5
0.000057

0.000057

0.000062

2. Curved ®bers. Transmembrane stimulation of tissue in diastole (1 ms S1)
5 ms
4961
9
0.000092

0.000105

0.000059

3. Curved ®bers. Extracellular stimulation of tissue in diastole (3 ms pulse)
5 ms
4961
9
0.000036

0.000039

0.000039

k~
rk2 ˆ k~
b ÿ A~
uk2 ˆ e < TOL:

…36†

This is equivalent to the following requirement for the relative error
k~
u ÿ~
uexact k2
j2 …A†
6e
;
k~
uexact k2
k~
bk2

…37†

where j2 …A† is the condition number of A in l2 -norm. We verify experimentally that this stopping
criterion provides a satisfactory error bound for our numerical solutions. The results are summarized in Table 1. The relative di€erence between the solutions, obtained with TOL ˆ 10ÿ5 and
TOL ˆ 10ÿ7 in (36) in the three di€erent simulation scenarios allows us to conclude that the error
introduced by the matrix solver is at least an order less that the truncation errors that result from
the predictor±corrector scheme (cf. Table 2). The matrix A is the same in the second and third
simulations. A smaller error in the latter simulation is due to the larger k~
bk2 , resulting from the
high values of the shock current introduced into the right-hand side of (19).
4.2. Temporal accuracy and convergence comparison
We now examine the issue of temporal accuracy and convergence of the method and compare it
to the performance of the forward Euler method. The experimental results are summarized in
Table 2. The relative errors of the numerical solutions, obtained using our predictor±corrector
scheme and the forward Euler method are estimated as follows. In each simulation scenario we
use the time steps Dt ˆ 5 ls; Dt ˆ 2:5 ls, and Dt ˆ 0:5 ls. The result that is generated using the
smallest time step is then assumed to be the exact solution ~
u…t†. The solutions, obtained with the
coarser time step are then compared to ~
u…t†. We summarize the relative errors in the ®rst part of
our table. The error of PECE method is 5±10 times smaller than that of the forward Euler method
(compare the relative error columns for the same Dt).
To estimate the convergence rate of each method we divide the error of the solution with
Dt ˆ 2:5 ls by that of the solution with Dt ˆ 5 ls. In the case of linear convergence, this ratio is
expected to be near 0.5: when the time step is halved, the error is also halved. Being a linear
method, forward Euler demonstrates just that. The ratios in all four experiments are near 0.5.

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K. Skouibine et al. / Mathematical Biosciences 166 (2000) 85±100
Table 2
Relative error and convergence ratio comparison for PECE vs forward Euler time stepping
Norms

Relative error

Convergence ratio

Forward Euler
Dt ˆ 0:005

PECE
Dt ˆ 0:0025

Dt ˆ 0:005

1. 3 ms transmembrane (S1) stimulation of tissue in diastole
0.02982
0.01549
0.00487
k~
uDt ÿ ~
u…t†k2
k~
u…t†k2

Forward Euler

PECE

0.00183

0.52

0.38

Dt ˆ 0:0025

0.02346

0.01223

0.00407

0.00153

0.52

0.38

0.04593

0.02384

0.00702

0.00264

0.52

0.38

2. 5 ms extracellular stimulation of tissue in diastole (no electroporation)
0.02038
0.01027
0.00322
0.00143
k~
uDt ÿ ~
u…t†k2
k~
u…t†k2

0.50

0.45

k~
uDt ÿ ~
u…t†k1
k~
u…t†k1
k~
uDt ÿ ~
u…t†k1
k~
u…t†k1

k~
uDt ÿ ~
u…t†k1
k~
u…t†k1
k~
uDt ÿ ~
u…t†k1
k~
u…t†k1

0.01490

0.00751

0.00261

0.00117

0.50

0.45

0.04641

0.02345

0.00746

0.00319

0.51

0.43

0.48

0.43

3. 5 ms extracellular stimulation of tissue in diastole (electroporation included)
0.01296
0.01063
0.00240
0.00107
k~
uDt ÿ ~
u…t†k2
k~
u…t†k2
k~
uDt ÿ ~
u…t†k1
k~
u…t†k1
k~
uDt ÿ ~
u…t†k1
k~
u…t†k1

0.01577

0.00759

0.00178

0.00077

0.48

0.43

0.02960

0.01432

0.00389

0.00173

0.48

0.44

PECE exhibits supralinear convergence performance in all the four experiments. While the
computational work required to do one PECE time step is double that of the Euler method, a gain
of order in accuracy more than justi®es the expense.
The time step in our method remains ®xed. The implementation of adaptive time stepping,
based on the a-posteriori error control (as in [20], for example) did not give a clear advantage over
using a ®xed time step. The inherent sti€ness of the problem results in unrealistic error estimates,
that are produced by comparing the predicted and corrected values of the solution on each given
step. Again, the implicit methods for systems of DAEs [17] can ®x this problem and will allow
adaptivity.

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K. Skouibine et al. / Mathematical Biosciences 166 (2000) 85±100

4.3. Stability considerations
While the accuracy requirement keeps the time steps very small in the ®rst-order methods, it is
the stability condition that restricts the step size in the more accurate higher-order explicit
methods. As expected, our predictor±corrector scheme behaves similarly to an explicit method
from the standpoint of stability.To simplify the stability analysis of (4)±(11) we assume that the
®bers are straight and the anisotropy ratios of the conductivities are equal
ret rel
ˆ :
rit ril

…38†

In this case the reaction±di€usion equations (4) and (5) are reduced to a single parabolic equation
for Vm


o2 Vm
o2 Vm
oVm
‡ Iion …Vm † ‡ G…Vm ; t†Vm ;
…39†
vx 2 ‡ vy 2 ˆ b Cm
ox
oy
ot
where vx ˆ ril rel =ril ‡ rel and vy ˆ rit ret =rit ‡ ret . Fourier stability analysis of an explicit method for
the corresponding homogeneous equation produces the following necessary condition for the time
step (cf. [21]):
Dt
1
6
2
bCm h
2

…40†

assuming Dx ˆ Dy ˆ h. Using the values from Appendix A we found this stability bound to be in
the range 0.06±0.15 ms, depending on what conductivity value we adjust to obtain (38). Unless the
spatial resolution is increased, this bound exceeds the time step restriction Dt 6 0:03 ms on the
scheme that solves the inhomogeneous equation (39). This restriction is imposed by the sti€ness of
systems (6)±(9) that describes the ionic current Iion …Vm †. Our time step is therefore chosen to satisfy
this experimentally obtained bound. The stability region of the scheme becomes smaller in the
presence of high gradients of the shock current i0 . In fact, the original BRDR ionic model does
not allow (and was not devised for) simulation of high-strength shocks. This part of the stability
problem is solved by modifying the gating variable equations as described in the previous section.
The time step is automatically lowered during the shock simulation.
4.4. Spatial convergence estimate
Finally, we test spatial convergence of our method. To minimize the temporal error contribution, we use the passive membrane model in this series of tests: Iion …Vm † ˆ Vm =Rm ; where
Rm ˆ 2k X cm2 is the speci®c membrane resistance of the passive tissue. Starting with the grid
with h ˆ 0:0083 cm (12 grid points per mm), we double the resolution by uniformly re®ning the
grid with each new simulation. We then compare the results obtained using these progressively
u24 ,
®ner grids. We denote the solutions on these grids by the number of points per millimeter: ~
u12 ;~
and ~
u48 . To satisfy the stability requirement, we reduce the time step by a factor of four each time
the spatial resolution is doubled.
u24 ; as well as the
The relative di€erences between each consecutive pair of the solutions ~
u12 ;~
ratios of the relative di€erences are summarized in Table 3. Since a very close approximation of

95

K. Skouibine et al. / Mathematical Biosciences 166 (2000) 85±100

Table 3
Relative di€erence between the solutions ~
u12 ;~
u24 and ~
u48 and the convergence ratio estimate (Here the solution vectors
include only Vm and Ue )
Curved ®bers. Extracellular stimulation of tissue
k~
u24 ÿ ~
u48 k2
k~
u48 k2

k~
u12 ÿ ~
u24 k2
k~
u48 k2

k~
u24 ÿ ~
u48 k2
k~
u12 ÿ ~
u24 k2

0.00986

0.06255

0.158

k

2.67

the exact solution ~
u1 in this case cannot be computed within reasonable time, we obtain the
convergence estimates based on the available data following the idea of Richardson extrapolation.
Assuming h to be the value above, we can represent the error for each grid solution as
u1 k2 ˆ Chk ‡ O…hk‡1 †;
k~
u12 ÿ ~

…41†

u1 k2 ˆ C…h=2†k ‡ O……h=2†k‡1 †;
k~
u24 ÿ ~

…42†

k~
u48 ÿ ~
u1 k2 ˆ C…h=4†k ‡ O……h=4†k‡1 †;

…43†

where k is the order of our method and is to be estimated. Then
k~
u24 ÿ ~
u48 k2
C……h=2†k ÿ …h=4†k † ‡ O……h=2†k‡1 †
ˆ
k
k~
u12 ÿ ~
u24 k2
C…hk ÿ …h=2† † ‡ O…hk‡1 †

…44†

k

…h=2† …C…1 ÿ 2ÿk † ‡ O…h††
ˆ
Chk …1 ÿ 2ÿk ‡ O…h††
1 ÿ 2ÿk
O…h†
‡ k ;
ˆ 2ÿk
2
1 ÿ 2ÿk ‡ O…h†

…45†
…46†

and as h ! 0 this ratio approaches 2ÿk . The term in the left-hand side of (44) (Table 3, column 3)
u24 and ~
u48 (Table 3,
is equal to 0.158, the ratio of the relative di€erences in the solutions ~
u12 , ~
columns 1 and 2). By solving 2ÿk ˆ 0:158 we get the estimate for the order of our method, k, to be
2.67. It is a little optimistic due to the non-zero term O…h† in the denominator of (46), but still it
clearly shows the superlinear convergence.
The present analysis validates our technique as superior to the popular forward Euler time
stepping and, at the same time, comparable to it in terms of simplicity in implementation and
practical use.

5. Discussion
Studies of cardiac de®brillation were given a boost by the advent of optical recordings of
transmembrane potentials [22±26]. Optical measurements are not a€ected by the high-intensity
electric ®eld of the shock thus allowing examination of transmembrane potential distributions
during and immediately after the shock. However, simulation studies of de®brillation are
currently being hampered by (1) the computational expense associated with simulating hundreds

96

K. Skouibine et al. / Mathematical Biosciences 166 (2000) 85±100

of milliseconds of electrical activity in the myocardium, and (2) the inappropriateness of most of
the currently available ionic models to handle large transmembrane potential changes. Indeed, the
ionic models are based on voltage clamp data ®tted predominantly over the range of a normal
action potential, thus they `blow up' for large values of membrane hyper- and depolarization. For
instance, in several de®brillation studies [27±29] the occurrence of this problem at the points (or
cells) where current was delivered has forced researches to examine only low-intensity de®brillation shocks (up to 3±5 times diastolic threshold). The Luo±Rudy phase II model [13] appears to
be best suited for de®brillation studies since it allows for transmembrane potential excursions over
500 mV; however it is associated with considerable additional computational expense in monitoring the shock-induced transmembrane potential patterns over time.
In this article we o€er a numerical recipe for eciently conducting simulation studies of de®brillation shocks and post-shock electrical activity in the myocardium. The recipe incorporates the
time stepping predictor±corrector technique as well as a low-expense modi®ed BRDR model that
accommodates large membrane depolarization or hyperpolarization.
The predictor±corrector scheme is shown to have higher temporal accuracy and better stability
than the forward Euler method while maintaining simplicity and low storage requirements. Thus,
we o€er a numerical technique that holds a middle ground between the explicit Euler and a fully
implicit method. It appears most suitable for bidomain simulation studies in view of the level of
storage and speed of the current computational resources. The technique allows us to examine
wavefront propagation in the myocardium over considerable time intervals with minimal error
accumulation.
The modi®cation of the BRDR model provides methodology to handle the high transmembrane
voltages created in immediate vicinity of the de®brillation electrodes. This methodology can be
successfully used with other ionic models. By extending the range of validity of the rate-constant
equations in the currently available membrane models and by including the di€erential equation
representing membrane electroporation under strong electric ®elds, we o€er a solution to a
problem that has impeded modeling research in de®brillation for many years.
The numerical scheme presented here has been already used successfully in several de®brillation
studies of ours conducted recently. These include stimulation of tissue in diastole via the virtual
electrode mechanism [30], examination of spiral wave termination and reorganization in myocardial slices subjected to de®brillation shocks delivered via small-size electrodes [31], study of the
impact of electroporation in anode/cathode break excitations [15], and examination of the role of
curvature-induced virtual electrodes in extending refractoriness of the tissue [32] and terminating
reentry [33]. These studies only referred to the model without providing necessary details that
would enable other researchers in the ®eld of de®brillation to take advantage of its numerical
eciency. This study does exactly that: it provides a thorough description of our de®brillation
model. It is our hope that other researchers will take advantage of its capabilities.

Acknowledgements
This research is supported by contract LEQSF (1998-01)-RD-A-30 from the Louisiana Board
of Regents through the Board of Regents Support Fund, by NIH Award HL63195, by NSF GIG
Award DMF-9709754, and by NSF GOALI Award BES-9809132.

97

K. Skouibine et al. / Mathematical Biosciences 166 (2000) 85±100

Appendix A. Parameters used in simulations
Material constants and electrical parameters [34,35]
Membrane capacitance per unit area
Cm
Surface-to-volume ratio
b
Intracellular conductivity across the ®ber
rix
Intracellular conductivity along the ®ber
riy
Extracellular conductivity across the ®ber
rex
Extracellular conductivity along the ®ber
rey
Transmembrane stimulus (S1±S2) current
Istim
Slow inward current coecient
k
Electroporation parameters [14]
Rate constant of electroporation
a
Rate constant of electroporation
b
Rate constant of electroporation
c
Discretization parameters
Tissue size
LL
Grid cell size in one direction
Dx
Time step (variable)
Dt

2

1.0
3000
0.375
3.750
2.140
3.750
50
8

lF=cm
1/cm
mS/cm
mS/cm
mS/cm
mS/cm
lA=cm2

2:5  10ÿ3
2:5  10ÿ5
1:0  10ÿ9

mS=cm2 ms
1=mV2
1=mV2

14  5±20  20
0.0125±0.02
0.005±0.02

mm2
cm
ms

Appendix B. Modi®ed BRDR ionic membrane model
· Ionic current densities
iNa ˆ fNa …Vm ; y† ˆ GNa m3 h…Vm ÿ ENa †;
iK1 ˆ fK1 …Vm ; y†


4…exp…0:04…Vm ‡ 85†† ÿ 1†
Vm ‡ 23
ˆ 0:35
‡ 0:2
;
exp…0:08…Vm ‡ 53†† ‡ exp…0:04…Vm ‡ 53††
1 ÿ exp…ÿ0:04…Vm ‡ 23††
ix1 ˆ fx1 …Vm ; y† ˆ

0:8x1 …exp…0:04…Vm ‡ 77†† ÿ 1†
;
exp…0:04…Vm ‡ 35††

is ˆ fs …Vm ; y; ‰CaŠi † ˆ Gs df …Vm ÿ Es †;

Es ˆ 82:3 ÿ 13:0287 ln‰CaŠi ;

· Gating variables
dyk
ˆ fyk …Vm † ˆ ak …1 ÿ yk † ÿ bk yk ;
dt

ak ˆ ak …Vm †; bk ˆ bk …Vm †; k ˆ 1; . . . ; 5;

where yk are the variables x1 ; m; h; d and f.

98

K. Skouibine et al. / Mathematical Biosciences 166 (2000) 85±100

· Sodium current activation gate rates
8
Vm ‡ 42:65
>
>
;
Vm < 100 mV;
0:9
>
>
1 ÿ exp…ÿ0:22Vm ÿ 9:3830†
<
am ˆ
>
>
>
exp…0:0486479 Vm ÿ 4:8647916†
>
: 890:9437890
; Vm > 100 mV:
1 ‡ 5:93962526 exp…0:0486479 Vm ÿ 4:8647916†
8
Vm > ÿ85 mV;
< 1:437 exp…ÿ0:085 Vm ÿ 3:37875†;
100
bm ˆ
; Vm < ÿ85 mV:
:
1 ‡ 0:4864082 exp…0:2597504 Vm ‡ 22:0787804†
· Sodium current inactivation gate rates

0:1 exp…ÿ0:193Vm ÿ 15:37245†; Vm > ÿ90 mV;
ah ˆ
ÿ12:0662845 ÿ 0:1422598Vm ;
Vm < ÿ90 mV:
1:7
:
1 ‡ exp…ÿ0:095Vm ÿ 1:9475†
· Outward recti®er current activation gate rates
8
exp…0:083 Vm ‡ 4:150†
>
>
0:0005
;
>
>
exp…0:057
Vm ‡ 2:850† ‡ 1
<
ax1 ˆ
>
>
>
exp…0:0654679 Vm ÿ 26:1871448†
>
: 151:7994692
1 ‡ 1:5179947 exp…0:0654679 Vm ÿ 26:1871448†
bh ˆ

Vm < 400 mV;

Vm > 400 mV:

exp…ÿ0:06Vm ÿ 1:20†
:
exp…ÿ0:04Vm ÿ 0:80† ‡ 1
· Slow inward calcium activation gate rates
bx1 ˆ 0:0013

ad ˆ 0:095

exp…ÿ0:01…Vm ÿ 5††
k;
exp…ÿ0:072…Vm ÿ 5†† ‡ 1

exp…ÿ0:017…Vm ‡ 44††
k:
exp…ÿ0:05…Vm ‡ 44†† ‡ 1
· Slow inward calcium inactivation gate rates
exp…ÿ0:008…Vm ‡ 28††
k;
af ˆ 0:012
exp…ÿ0:15…Vm ‡ 28†† ‡ 1
bd ˆ 0:07

exp…ÿ0:06…Vm ‡ 30††
k:
exp…ÿ0:2…Vm ‡ 30†† ‡ 1
· Calcium concentration inside the cell

d‰CaŠi
ÿ10ÿ7
is ‡ 0:07…10ÿ7 ÿ ‰CaŠi †;
ˆ fCa …Vm ; y; ‰CaŠi † ˆ
0;
dt
bf ˆ 0:0065

Vm < 200 mV;
Vm > 200 mV:

K. Skouibine et al. / Mathematical Biosciences 166 (2000) 85±100

99

· Initial conditions and constants
Vm0
G0
x10
m0
h0
d0
f0
‰CaŠi0
GNa
ENa
Gs

)84.35
0
0.0241
0.01126
0.9871
0.0030
1.0
3  10ÿ 7
15.0
40.0
0.09

mV
mS=cm2

M
mS=cm2
mV
mS=cm2

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