Ansys Pacing Threshold Model

Arxiv:0805.4444V2 [Physics.Bio-Ph] 14 Aug 2008

Action potential restitution and hysteresis in a reaction-diffusion system with pacing rate dependent excitation threshold

Mediwave Star Technology, Inc., Greensboro, Nc

C. P. Danford, V. Varadarajan, A. J. Starobin, and V. N. Polotski

Abstract

We have demonstrated that rate dependent restitution and action potential duration-refractory period hysteresis can be reproduced in a one-dimensional two-variable Chernyak-Starobin-Cohen reaction-diffusion medium with variable excitation threshold. We show that restitution and hys- teresis depend on the relationship between pacing period and steady state excitation threshold and also on the rate of excitation threshold adaptation after an abrupt change in pacing period. It was also observed that the onset of action potential duration alternans is determined by the minimal stable wavefront speed, which could be approximated by the analytical critical speed of a stable solitary pulse. This approximation was suitably accurate regardless of the adaptation constant of excitation threshold, its dependence on pacing interval, or magnitude of the slopes of restitution curves.

Introduction

Repetitive pacing of biological reaction diffusion media by over-threshold stimuli gives rise to periodically propagating excitation waves. In cardiac tissue the duration of excitation referred to as action potential duration, TAP, as well as the speeds of excitation wavefront and waveback, depend on previous stimulation periods and refractory (diastolic) intervals, TDI [1, 2, 3]. The analysis of such dependences known as restitution curves has been established as an effective method for evaluation of normal functioning of the heart [4, 5, 6, 7, 8, 9].

Pioneering experimental studies [10, 11, 12] established two major pacing sequences such as steady state (dynamic) and S1-S2 pacing protocols, which led to two standard TAP resti- tution dependences. It has been shown that the protocol dependent rates of TAP adaptation were different for stepwise S1-S2 perturbations of cycle length acceleration and deceleration [12, 13]. This phenomenon was later experimentally introduced as action potential dura- tion cycle length hysteresis [14, 15] and has been recently associated with cardiac ischemia, coronary flow reduction [16, 17], and cardiac memory [18, 19].

A stepwise change in pacing rate following a long series of conditioning S1 stimuli may result in prolonged adaptation of action potential duration to its new steady-state value. The process of such adaptation can extend well beyond the TAP response to the first S2 test stimulus [11, 12] and give rise to the additional constant BCL (basic cycle length) transient restitution component attributed to cardiac memory [2, 20, 21, 22]. Relationship between this phenomenon and stability of pulse propagation as well as with rate dependent TAP-TDI hysteresis has been recently investigated in experimental [17, 20, 22, 23] and theoretical [24, 25, 26, 27, 28] studies.

It has been found that in the presence of restitution transients cardiac dynamics is more complex than predicted by Nolasco and Dahlen restitution criterion . Specifically, it was demonstrated that dynamic restitution curve slopes > 1 may not automatically indicate loss of excitation wave stability and subsequent appearance of alternans [20, 24, 25, 26, 27, 30].

Computational experiments with Fenton-Karma and Mitchell-Schaeffer ionic models readily identified that such effects can be quantified by fitting TAP and TDI for specific restitution curves using time dependent gate variables [24, 25, 31, 32, 33].

In this paper, we analyze excitation wave propagation in a one-dimensional cable based on the approach, which follows from direct experimental observations of the dependence of

2

cardiac muscle resting potential on frequency of external pacing [34, 35, 36]. We implement a two-variable exactly solvable Chernyak-Starobin-Cohen (CSC) reaction-diffusion model [37, 38] and modify it accordingly to incorporate pacing rate driven adjustments of resting potential as a rate dependent excitation threshold. We use an exponential-like evolution of excitation threshold, Vr, that takes place over the course of multiple heart beats following stepwise changes in pacing rate . We demonstrate that adaptation of Vr significantly affects the stability of pulse propagation and gives rise to rate dependent TAP-TDI hysteresis.

In particular, we find that under given medium parameters, regardless of the slopes of restitution curves, the appearance of alternans is determined by proximity of the wavefront speed to the minimal speed of a stationary solitary pulse determined analytically in .

Methods

Basic equations that describe a class of exactly solvable models for excitable media have been defined in [37, 38]. Here we introduce a modification of this analytical model by adjusting the excitation threshold, Vr, in response to changes in frequency of external pacing.

(3)

u(x, t) and v(x, t) are a membrane potential and slow recovery current, respectively. λ, ǫ, ζ, and τ are the model parameters, where τ −1 < ǫ. The scaling of the system is described in the Appendix.

The pacing function, P(x, t), is defined as a product of two functions, X(x) and T(t). Each is composed of the Heaviside step function, Θ(x), as follows: X(x) = A[Θ(x −δ1) − Θ(x−δ2)] and T(t) = Θ(tk)−Θ(tk +Ts), where A and δ2−δ1 are the amplitude and width of the pulse, respectively, Ts is the pulse duration, and tk = tN(m−1) +[k −N(m−1)]Tm are the instants of time when stimuli are delivered. We use this to construct a pacing protocol where

3

the pacing period, Tm, is stepwise constant (Tm > Ts). N represents the number of stimuli at each pacing interval plateau, the index m denotes each pacing plateau, m = 1, . . . , M, and k is an integer in the range k = 1, . . . , NM. The number of stimuli N is the same for all plateaus. Overall during the course of the protocol, Tm progressively decreases to a minimum and then increases to its starting value.

The right-hand term B in Eq. 3 responds to the stepwise evolution of pacing period, Tm. Stepwise changes in B result in smooth exponential transition of the excitation threshold from one steady-state plateau to another. A steady-state value of excitation threshold at

≡B±

m, was chosen to be linearly dependent on the corresponding pacing interval, Tm [34, 35].

(4)

Here α± and β± are positive parameters that determine the amplitude of change of B±

M

between two consecutive pacing plateaus for the increasing and decreasing rate, respectively.

Numerical Simulations

The system of Eqs. 1-3 was solved numerically on a short cable of 250 grid points with spatial and temporal grid intervals of ∆x = 0.13 and ∆t = 7.2 × 10−4, respectively. The length of the cable was approximately equal to the width of the pulse to reflect the relative dimensions of the heart and a propagating cardiac pulse at moderate heart rates. Periodic wavetrains were produced by stimulating the cable with a square wave at the left end using the function P(x, t) = X(x)T(t), defined above, where A = 10, δ1 = 2∆x, δ2 = 15∆x, and Ts = 103∆t. The model parameters λ, ǫ, and ζ were equal to 0.4, 0.1, and 1.2, re- spectively, for all simulations.

Numerical solutions were computed using a second-order explicit-difference scheme . A typical solution is depicted in Fig. 1, showing the propaga- tion of a single pulse at three instants of time.

In order to quantify the dynamics of the system (Eqs. 1-3), we computed the action potential duration, TAP, the diastolic interval, TDI, and the wavefront velocity, c.

The

action potential duration was defined as the interval of time when u > v at a specified node, x0. Accordingly, the refractory period, TDI, was defined as the interval of time when u < v. The speed of the wavefront, c, was calculated based on the time it took for a point of constant phase on the wavefront (u = 0.5) to travel a span of 10 grid points centered around

Uv

FIG. 1: Spatio-temporal dynamics of u,v variables. Three snapshots of u and v are shown for a excitation threshold Vr = 0.19. The first snapshot occurs at the instant of the pacing stimulus and shows the formation of the wavefront. The next two snapshots are taken after periods of 7.2 (20% of the pacing period) and 14.4 and show formation and propagation of the pulse.

x0. In order to analyze a developed pulse, we measured intervals and speeds at x0 = 20 (except Sec. III D) where the speed of the wavefront had reached a constant steady-state value.

Restitution For Constant Excitation Threshold

The system of equations 1-3 was initially studied with Eq. 3 replaced by its asymptotic

Form ∂Vr

∂t = 0, which is equivalent to τ = ∞. The cable was stimulated periodically for forty consecutive pacing periods over the range Tm = 70 to 25 with decrements of 1.5. At the end of each plateau, TAP and TDI had reached steady-state values that were used to compose the steady-state restitution curve. Two values of Tm, at 30 and 27, were used to obtain conventional S1-S2 restitution curves[10, 11]. The pulse, T n+1

Ap , Resulting From The

test stimulus, S2, following the conditioning sequence and the last diastolic interval, T n

T

FIG. 2: Steady-state restitution curves are shown with solid lines, and S1-S2 restitution curves computed for S1 BCLs of 27 and 30 are shown with dashed lines. Critical TAP and TDI values at which pulse durations begin to oscillate are at the ends of the curves where T crit

= 20.8, T Crit

AP = 5.5 for Vr = 0.19 and Vr = 0.215, respectively. Open markers in the upper left insert indicate corresponding critical speeds plotted with the analytical dispersion curve for a solitary pulse . The mid and lower inserts demonstrate oscillation of pulse duration at and below the critical speed.

from the conditioning S1 plateau composed the S1-S2 restitution curve. Steady state and S1-S2 restitution curves computed for a pair of constant excitation thresholds and a pair of S1 basic cycle lengths are shown in Fig. 2. We observed that the maximal difference between steady state and S1-S2 restitution curves comprised less than 15% of the corresponding steady state value of TAP, which implied that transient responses to any premature stimulus were on average limited to just a single non-stationary pulse.

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The ends of both steady state restitution curves shown in Fig. 2 indicate the critical points below which no stable propagation and no 1 : 1 responses were observed. We found that when the pacing interval reached the value of Tm = 26.3 at Vr = 0.215, the resulting action potentials oscillated in duration as shown in the mid level insert panel. Further reduction of the pacing interval from Tm = 26.3 to Tm = 25.9 induced an even more complex 3 : 2 TAP response pattern depicted in the lower level insert panel.

The upper level insert shows the dispersion curve computed analytically for a steady state solitary pulse . The critical speed and duration of a stable solitary pulse for the model parameters, λ, ζ, and ǫ, described above is ccrit = 0.48 and T crit

Ap = 5.4. As Shown In The

insert, the difference between the numerical and analytical critical speeds is relatively small and amounts only to 10% of the wavefront speed determined from our numerical model at Vr = 0.215. This suggests a criterion for determining the critical speed below which pulse durations start to oscillate. On the contrary, the conventional Nolasco-Dahlen critical slope stability criterion is unsuitable as the maximum slope of the dynamic restitution curve at this Vr is 30% greater than one .

Restitution And Hysteresis For Rate-Dependent Vr

Unlike the previous section, the evolution of Vr according to Eq. 3 resulted in a set of prolonged transients initiated by abrupt changes in stimulation rate. These TAP transients, which constituted the constant BCL restitution (negative slope, Fig. 3A), had a duration of 5-50 stimulation periods depending on the adaptation constant, τ (Fig. 3D). The phase of constant BCL adaptation followed the immediate S1-S2 responses, which were on the contrary positive and aligned with the steady state restitution curves for each steady state excitation threshold (gray lines, Fig. 3A).

Different steady state values of excitation threshold during progressively increasing and decreasing stimulation rates gave rise to TAP-TDI hysteresis.

The Cable Was Stimulated

periodically for fifty consecutive periods at a series of S1 conditioning plateaus. The series consisted of five plateaus with decreasing Tm followed by the same number of plateaus with increasing Tm. The excitation threshold evolved according to Eq. 3, and its steady state value at each plateau was related to the pacing period by Eq. 4.

Using this protocol we demonstrated that higher values of TAP-TDI hysteresis corre-

D

FIG. 3: Restitution relations and TAP -TDI hysteresis for a stepwise stimulation protocol with rate dependent excitation threshold. (A) A series of five stepwise changes of pacing rate are shown with individual TAP and TDI as black circles. Open circles mark steady-state TAP and TDI intervals. Portions of constant excitation threshold, V m

M, And Steady State Restitution Curves

are shown in gray. (B) TAP -TDI hysteresis produced by increasing differences between B+

B−

m. β+ = 6 × 10−3 and α+ = 0.37 for the top curve (solid line, squares), β−= 4 × 10−3 and α−= 0.31 (dotted line, solid circles), β−= 2 × 10−3 and α−= 0.25 (dashed line, triangles), and β−= 0.1 × 10−3 and α−= 0.19 for the lowest curve (dash-dotted line, circles). (C) TAP-TDI hysteresis for different values of the adaptation constant, τ (τ = 216, dotted line; τ = 32, solid line), when β+ = β−and α+ = α−. (D) Adaptation of action potential duration after an abrupt decrease in stimulation interval for two different rate constants (β+ = 6 × 10−3 and α+ = 0.37).

The measured transient intervals in Panel C are highlighted with corresponding markers.

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