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

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

Department of Mathematics, Duke University, USA b

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 trans-membrane 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 bi-domain 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.}

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 al-lows 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 im-portant 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,

VmˆUiÿUe, are de®ned everywhere in the cardiac domain X. The following coupled reaction±

di€usion equations constitute the bidomain model:

r …r^irUi† ˆim; …1†

r …r^erUe† ˆ ÿimÿi0 inX; …2†

imˆb Cm

o_Vm o_t

‡Iion…Vm;t† ‡G…Vm;t†Vm

; …3†

wherer^i (mS/cm) andr^e are conductivity tensors in the corresponding domains,im (lA=cm

3

) 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=cm

2

) is the speci®c membrane ca-pacitance, and i0 (lA=cm

3

) is the volume density of the extracellular (shock) current. By eliminatingUi, we obtain the following system:

r …r^irVm† ‡ r ……r^e‡r^i†rUe† ˆ ÿi0; …4†

r …r^erUe† ˆ ÿimÿi0 inX; …5†

imˆb Cm

o_Vm o_t

‡Iion…Vm† ‡G…Vm;t†Vm

d‰CaŠ_i

dt ˆfCa…Vm;y;‰CaŠi†; …7†

dG

dt ˆfG…Vm†; …8†

dyk

dt ˆfyk…Vm;yk†; k ˆ1;. . .;5; inX; …9†

whereXˆ ‰0;aŠ ‰0;bŠis a rectangle. Functionsykrepresent the gating variablesx1;m;h;d;andf;

‰CaŠ_i 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 …r^ir…Ue‡Vm†† ˆ0; ~n …r^erUe† ˆ0 onoX1; …10†

~n …r^ir…Ue‡Vm†† ˆ0; UeˆVstim on oX2; …11†

whereo_{X2 is the part of the boundary where the extracellular stimulus is applied and} o_X1

rep-resents the rest of the boundary.

Here the ionic current (the termIion …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 factorkˆ8. 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cientsam;h;x1;bm;h;x1 are extended outside the normal range of Vm; so is the original dif-ferential equation for the intracellular calcium concentration‰CaŠ_i. We ensure that: (1) the sodium activation gates remain closed…mˆ0†forVm<ÿ85 mV and open…mˆ1†forVm>100 mV; (2)

the sodium inactivation gates are open…hˆ1†forVm<ÿ90 mV; (3) the outward recti®er current

activation gate stays open …x1ˆ1† 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 ‰CaŠ_i 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

The additional variable membrane conductanceG…Vm;t†accounts for the pore generation in the membrane during strong electric shocks [14]. Ggrows at the following rate [15],

dG dt ˆae

b…VmÿVrest†2_1ÿeÿc…VmÿVrest†2_{†; G…0† ˆG}

0: …12†

Values of a …mS=cm2 ms†, b …1=mV2_{†, andc …1=mV}2_{† 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 dif-ferential-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 rela-tively simple in implementation, this method has a clear advantage over the Euler method tra-ditionally 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:

K…r^e‡r^i†…Ue†_i;j‡K…r^i†…Vm†_i;j ˆ …i0†_i;j on Xh; …13†

K…r^e†…Ue†i;j‡b Cm

dVm

dt

‡Iion‡ …GVm†

i;j

ˆ ÿ…i0†i;j on Xh; …14†

whereK…r^†…f†is a ®nite-di€erence approximation of the di€usion term r …r^rf†.

By projecting the unknownsVm, Ue, etc., in (4)±(11) that are de®ned everywhere in X onto a

®nite grid

XhˆXh‡oXh

ˆ …xi;yj† jxi

ˆiDx; iˆ0;. . .;Nx; yj ˆjDy; jˆ0;. . .;Ny; Dxˆ

a Nx

; Dyˆ b

Ny

;

we obtain the vectors~Vm,~Ue, etc., with the components…Vm†i;j ˆVm…xi;yj†, …Ue†_i;jˆUe…xi;yj†;etc.

Here o_{Xh consists of the mesh points on the boundary}o_{X. This leads to the following system of}

DAEs:

d~Vm

dt ˆ~FVm…~Vm; ~Ue; ~G;~y;t†; …15†

d‰ƒCa!Š_i

dt ˆ~FCa…~Vm;~y;‰Ca

ƒ!_Š

i†; …16†

d~G

d~yk

dt ˆ~Fyk…~Vm;~yk†; kˆ1;. . .;5; on Xh; …18†

A~Ueˆ~FUe…~Vm† onX~h; …19†

B~Vmˆ~FB…~Vm; ~Ue† on oXh; …20†

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

FVm

and the functions on the right-hand side of (16)±(18) have the components from (7)±(9)

…FCa†_i;jˆfCa……Vm†i;j;…y†i;j;…‰CaŠi†i;j†; …22†

…FG†i;jˆfG……Vm†i;j†; …23†

…Fyk†i;j ˆfyk……Vm†i;j;…yk†i;j† on Xh: …24†

Thus, systems (15)±(20) consists of 8NxNy ODEs (15)±(18) that are coupled with two

alge-braic systems. The sparse linear algealge-braic system (19), represented by the…Nx‡1†…Ny‡1† …Nx‡

1†…Ny‡1†matrixA, 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

dif-ference approximation of the boundary conditions (10) and (11) using ®ve-point di€erence sec-ond-order formulae (equations involving Vm). The 2…Nx‡Ny† 2…Nx‡Ny† matrix B of (20) is

tridiagonal: the values ofVm 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 full time steptn‡1 _ˆtn_{‡Dt. Here~v}n_ˆ~v…tn_{†and~v denotes the predicted value of~v. The steps are} 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

~

(E) The evaluation step makes use of the predicted value~V

m in order to update the right-hand

side functions of (15)±(18). First, we solve the algebraic system (19) for~Ue

A~U_e ˆ~FUe…~V

m† on~Xh …27†

B~V_m ˆ~FB ~Vm; ~U

e

on o_{Xh: …28†}

We then obtain the new values of the function

~

and proceed to the corrector step.

(C) The corrector step is the implicit two-step Adams±Moulton rule with~Fn‡1

Vm approximated by

(E) Finally, we re-evaluate the right-hand side function, using the corrected value~Vn‡1

m . Again,

Our multi-step method requires two initial vectors~V0

m and ~Vm1 to begin the iteration described

above. The initial condition to our problem provides ~V0

m (see Appendix A). We use the forward

Euler method with the time step…Dt†2to generateV~_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 betweenUiandUe, 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 ®xUeto be zero in the middle of the tissue. This is done by substituting

the equation for Ueat that node in (13) with …Ue†_{imiddle;jmiddle} ˆ0:

4. Convergence rate estimates and accuracy comparison

currentIionbelong 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. Im-plicit methods for DAE [17] require the solution of a system of algebraic equations, similar to (19) forallunknowns (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 Ad-ams±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 con-sidered below.

4.1. GMRES accuracy analysis

To solve an nn linear system

A~uˆ~b …34†

for~u(the extracellular potential function in our case) GMRES, a projection method, constructs an l2-orthonormal basisVmˆ ‰~v1;. . .;~vmŠ of the Krylov subspace

K_{mˆspanf~r0;A~r0;. . .;A}mÿ1_~r

0g; …35†

where~r0 ˆ~bÿA~u0 is the residual, and~u0 is the initial guess. To do this, the method uses a

procedure called Arnoldi's method. The solution approximation on themth step of the iteration procedure is obtained by ®nding the unique minimizer~umofk~bÿA~uk2, such that~um2~u0‡Km.

The minimization, as implemented in this algorithm (see [16] for details), is equivalent to solving anmmupper-triangular system. By construction, the algorithm converges in at mostnsteps (in exact arithmetic). Since the method becomes increasingly expensive as m grows (the number of multiplications is O…m2np†, where p is the average number of unknowns per row), we employ a banded version of GMRES. Here, when constructing the basisVm_{, we requirevjto be orthogonal} only to the previousl vectorsvjÿl;. . .;vjÿ1, wherel 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 in-complete 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 restarting it every m steps by setting~unew

0 ˆ~um and~rnew0 ˆ~bÿA~um. The optimal value of m is

k~rk₂ ˆ k~bÿA~uk₂ˆe<TOL: …36†

This is equivalent to the following requirement for the relative error

k~uÿ~uexactk2

k~uexactk2

6ej2…A†

k~bk₂ ; …37†

wherej2…A†is the condition number ofAinl2-norm. We verify experimentally that this stopping

criterion provides a satisfactory error bound for our numerical solutions. The results are sum-marized in Table 1. The relative di€erence between the solutions, obtained with TOLˆ10ÿ5 _and

TOLˆ10ÿ7 _{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 largerk~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 stepsDtˆ5 ls;Dtˆ2:5 ls, andDtˆ0:5 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 sameDt).

To estimate the convergence rate of each method we divide the error of the solution with

Dtˆ2:5ls by that of the solution with Dtˆ5 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. Table 1

Relative di€erence between the solutions~u1and~u2obtained with tolerances TOLˆ10ÿ5and TOLˆ10ÿ7, respectively (here~u1 and~u2are the vectors that include all the unknowns in (15)±(20))

Total timeT Matrix sizen Average number of el-ts per rowu k~u1ÿ~u2k2 k~u2k2

k~u1ÿ~u2k1 j~u2k1

k~u1ÿ~u2k1

j~u2k1

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)

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.

Table 2

Relative error and convergence ratio comparison for PECE vs forward Euler time stepping

Norms Relative error Convergence ratio

Forward Euler PECE Forward Euler PECE

Dtˆ0:005 Dtˆ0:0025 Dtˆ0:005 Dtˆ0:0025

1. 3 ms transmembrane (S1) stimulation of tissue in diastole

k~uDtÿ~u…t†k2 k~u…t†k2

0.02982 0.01549 0.00487 0.00183 0.52 0.38

k~uDtÿ~u…t†k1 k~u…t†k1

0.02346 0.01223 0.00407 0.00153 0.52 0.38

k~uDtÿ~u…t†k1

k~u…t†k₁

0.04593 0.02384 0.00702 0.00264 0.52 0.38

2. 5 ms extracellular stimulation of tissue in diastole (no electroporation)

k~uDtÿ~u…t†k2 k~u…t†k2

0.02038 0.01027 0.00322 0.00143 0.50 0.45

k~uDtÿ~u…t†k1 k~u…t†k₁

0.01490 0.00751 0.00261 0.00117 0.50 0.45

k~uDtÿ~u…t†k1

k~u…t†k₁

0.04641 0.02345 0.00746 0.00319 0.51 0.43

3. 5 ms extracellular stimulation of tissue in diastole (electroporation included)

k~uDtÿ~u…t†k2 k~u…t†k₂

0.01296 0.01063 0.00240 0.00107 0.48 0.43

k~uDtÿ~u…t†k1 k~u…t†k1

0.01577 0.00759 0.00178 0.00077 0.48 0.43

k~uDtÿ~u…t†k1

k~u…t†k₁

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

re

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

t. Fourier stability analysis of an explicit method for

the corresponding homogeneous equation produces the following necessary condition for the time step (cf. [21]):

Dt

bCmh2

61

2 …40†

assumingDxˆDyˆh. 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 Dt60:03 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 currentIion…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 contri-bution, we use the passive membrane model in this series of tests: Iion…Vm† ˆVm=Rm; where

Rmˆ2k X cm2 is the speci®c membrane resistance of the passive tissue. Starting with the grid

withhˆ0:0083 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 ®ner grids. We denote the solutions on these grids by the number of points per millimeter:~u12;~u24,

and~u48. To satisfy the stability requirement, we reduce the time step by a factor of four each time

the spatial resolution is doubled.

The relative di€erences between each consecutive pair of the solutions~u12;~u24; as well as the

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. Assumingh to be the value above, we can represent the error for each grid solution as

k~u12ÿ~u1k2 ˆChk‡O…hk‡ 1

†; …41†

k~u24ÿ~u1k2 ˆC…h=2†

k

‡O……h=2†k‡1†; …42†

k~u48ÿ~u1k2 ˆC…h=4†

k

‡O……h=4†k‡1†; …43†

wherekis the order of our method and is to be estimated. Then

k~u24ÿ~u48k2

k~u12ÿ~u24k2

ˆ C……h=2† k

ÿ …h=4†k† ‡O……h=2†k‡1†

C…hk_{ÿ …h=2†}k_{† ‡O…h}k‡1_† …44†

ˆ …h=2† k

…C…1ÿ2ÿk_{† ‡O…h††}

Chk_1ÿ2ÿk_‡O…h†† …45†

ˆ2ÿk 1ÿ2ÿk

1ÿ2ÿk_‡O…h†‡

O…h†

2k ; …46†

and ash!0 this ratio approaches 2ÿk_{. The term in the left-hand side of (44) (Table 3, column 3)}

is equal to 0.158, the ratio of the relative di€erences in the solutions~u12 ,~u24 and~u48 (Table 3,

columns 1 and 2). By solving 2ÿk _{ˆ0:158 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 Table 3

Relative di€erence between the solutions~u12;~u24and~u48and the convergence ratio estimate (Here the solution vectors include onlyVm andUe)

Curved ®bers. Extracellular stimulation of tissue

k~u24ÿ~u48k2 k~u48k2

k~u12ÿ~u24k2 k~u48k2

k~u24ÿ~u48k2 k~u12ÿ~u24k2

k

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®bril-lation 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 moni-toring 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 schemeis 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 modelprovides 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 myo-cardial 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

Appendix A. Parameters used in simulations

Appendix B. Modi®ed BRDR ionic membrane model

· _{Ionic current densities}

iNaˆfNa…Vm;y† ˆGNam3h…VmÿENa†;

iK1 ˆfK1…Vm;y†

ˆ0:35 4…exp…0:04…Vm‡85†† ÿ1†

exp…0:08…Vm‡53†† ‡exp…0:04…Vm‡53††

‡0:2 Vm‡23

1ÿexp…ÿ0:04…Vm‡23††

;

ix1 ˆfx1…Vm;y† ˆ

0:8x1…exp…0:04…Vm‡77†† ÿ1† exp…0:04…Vm‡35††

;

is ˆfs…Vm;y;‰CaŠi† ˆGsdf…VmÿEs†; Esˆ82:3ÿ13:0287 ln‰CaŠi;

· _{Gating variables}

dyk

dt ˆfyk…Vm† ˆak…1ÿyk† ÿbkyk; akˆak…Vm†; bkˆbk…Vm†; k ˆ1;. . .;5;

whereyk are the variablesx1;m;h;d and f.

Material constants and electrical parameters[34,35]

Membrane capacitance per unit area Cm 1.0 lF=cm

2

Surface-to-volume ratio b 3000 1/cm

Intracellular conductivity across the ®ber rix 0.375 mS/cm

Intracellular conductivity along the ®ber riy 3.750 mS/cm

Extracellular conductivity across the ®ber rex 2.140 mS/cm

Extracellular conductivity along the ®ber rey 3.750 mS/cm

Transmembrane stimulus (S1±S2) current Istim 50 lA=cm 2

Slow inward current coecient k 8

Electroporation parameters[14]

Rate constant of electroporation a 2:510ÿ3 _mS=cm2

ms Rate constant of electroporation b 2:510ÿ5 _1=mV2

Rate constant of electroporation c 1:010ÿ9 _1=mV2

Discretization parameters

Tissue size LL 145_±2020 mm2

Grid cell size in one direction Dx 0.0125±0.02 cm

· _{Sodium current activation gate rates}

· _{Sodium current inactivation gate rates}

ah ˆ

· _{Outward recti®er current activation gate rates}

ax1 ˆ

· _{Slow inward calcium activation gate rates}

ad ˆ0:095

· _{Slow inward calcium inactivation gate rates}

af ˆ0:012

· _{Calcium concentration inside the cell}

· _{Initial conditions and constants}

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