Electrical Instability Due To Regional Increase In Extracellular Potassium Ion Concentration
Sunil M Kandel* and Bradley J Roth
Department of Physics, Oakland University, Rochester, Michigan, USA
Introduction: Ventricular tachycardia and ventricular fibrillation are the two most dangerous arrhythmias. Both are related to reentrant electrical activity in the ventricles. Many studies of arrhythmias consider a homogeneous sheet of cardiac tissue. Since normal ventricular myocardium is inhomogeneous and inhomogeneities play an important role in the induction of reentry, we investigate the effect of a localized inhomogeniety developed at the border between normal and ischemic region.
Methods: We used the bidomain model to represent the electrical properties of cardiac tissue and a modified version of the dynamic Luo-Rudy (LRd) model to represent the active properties of the membrane. To investigate the effect of a localized inhomogeneity, the extracellular potassium [K]e concentration is raised to 10 mM from normal [K]e (4 mM) on the right half of the tissue.
Results and Discussion: A train of cathodal stimuli are applied from the lower left corner of the tissue with different basic cycle lengths (BCL). At certain BCL, the spatial heterogeneity created with regional elevation of [K]e can lead to action potential instability (alternans) in the normal and border regions, and 2:1 conduction block in the ischemic region. We observed the reentry when local heterogeneity in [K]e is changed from 10 to 12 mM on the right half of the virtual ventricular myocardium sheet.
Conclusion: Electrical alternans occur during high heart rates and are observed in patients suffering from ventricular tachycardia. It is an early indication of left ventricular systolic impairment. This study will help to evaluate alternans as a predictor and guide for antiarrhythmic therapy. Journal of Nature and Science, 1(8):e160, 2015.
Arrhythmias | alternans | bidomain model | Luo-Rudy Model
Ventricular tachycardia (VT) and ventricular fibrillation (VF) are the two most dangerous arrhythmias. During VT, the ventricles of the heart contract very fast, and during VF the ventricles contract in an asynchronous fashion. Both of these arrhythmias are related to electrical instability in the heart. During arrhythmias, beat-to-beat alternation in the shape of electrocardiogram has been noticed for several decades, but the cause and effect relationship between alternans and arrhythmia has not been confirmed.
A regional increase of extracellular potassium concentration [K]e is prominently responsible for arrhythmias during the early stage of acute ischemia (called phase 1a arrhythmias). During this phase, which lasts for 5 -10 min, [K]e increases from 4.5 to 14 mM. The accumulation of [K]e during this phase changes the transmembrane potential, which affects tissue excitability and wave propagation, and ultimately results in wave fragmentation and reentry. But the role of a regional increase in [K]e (heterogeneity) has not been studied extensively. Sidorov et al.1 created a local [K]e heterogeneity by perfusing solution containing different concentrations of [K]e in a Langendroff-perfused rabbit heart and investigated its role in fast pacing response and arrhythmia induction. They found that the spatial heterogeneity created by changing [K]e causes action potential alternans (beat to beat alternation in the shape of the action potential), 2:1 conduction block, and reentry.
Most of the studies related to electrical instability (alternans) consider a homogeneous sheet of cardiac tissue 2-8. In such tissue, all the cells that make up the tissue are identical and are connected to one another in a uniform way. But it has been known that normal ventricular myocardium is inhomogeneous and inhomogeneities play an important role in the electrical instability in the heart. We consider a 3D cylindrical model to represent ventricular myocardium to investigate the effect of a localized inhomogeneity at the border between normal and ischemic tissue.
We used the bidomain model9 to represent the electrical properties of cardiac tissue and the dynamic Luo-Rudy (LRd) model10 to represent the active properties of the membrane. To investigate the effect of a localized inhomogeneity, we consider an ischemic region in a volume of ventricular myocardium. The extracellular potassium concentration, [K]e, is raised on the right to 10 mM compared to the normal 4 mM on the left. Figure 1 depicts the spatial distribution of the resting transmembrane potential.
The tissue is stimulated through a 1 mm long, 0.4 mm diameter cylindrical electrode, placed in the center of a 20 mm long and 8 mm diameter cylinder of cardiac tissue. Because of the symmetry, our calculations involve only a quarter of the tissue (z > 0 and ρ > 0). Our implementation of the bidomain model is the same as we described in our previous paper11. We used unequal anisotropy ratios for this simulation.
The time step is 10 or 5 μs depending upon the calculations, and the space steps along z and ρ axis are 0.1 and 0.04 mm respectively. The number of grid points used in each direction is 101.
During ischemia, the basic cycle length (pacing interval) gradually decreases. Hence, we apply stimuli at different pacing intervals gradually decreasing from 200 to 120 ms. The stimulus has a duration of 4 ms and its strength is adjusted to be three to four times the diastolic threshold of excitation as done by Sidorov et al.1.
The resting potential is depolarized to about -57 mV on the right (in the ischemic area) as a consequence of [K]e being set to 10 mM. We run the simulation for 100 s to allow electrotonic interactions to occur and to set the steady state space-clamped values before applying the stimulus. Hence we see the spatially extended boarder zone due to spatial variation of voltage and other variables in the border zone. The border zone is a region defined by the large gradient in cellular electrophysiology and structural architecture. In our case it extends about 1 mm on each side of the discontinuity (Figure 1).
The electrode in the lower left corner of the tissue is first stimulated with a cathodal stimulus (S1) of strength 0.25 mA at time 100 ms for 4 ms pulse duration. This stimulus excites an action potential that propagates outward from the electrode. Figure 2 shows a plot of the transmembrane potential versus time at three locations shown in Figure 1. The blue, red and green curves represent the transmembrane potential near the left edge of the tissue sheet in the region of normal [K]e (z= 1 mm, ρ = 0), at the center (just on the boundary between normal and ischemic tissue) (z = 5 mm, ρ= 0), and near the right edge in the region of elevated [K]e (z = 9 mm, ρ = 0), respectively.
Figure 1. The spatial distribution of resting transmembrane potential. The tissue has [K]e =4 mM (normal) on the left, and [K]e=10 mM (ischemic) on the right. The three black dots at ρ = 0 and z = 1, 5 and 9 mm show the position where transmembrane potential is calculated in Figure 2.
Figure 2. Transmembrane potential (Vm) versus time at three different locations shown in Figure 1. The blue curve is at z = 1 mm in normal tissue, the red curve is at z = 5 mm in the border zone, and the green curve is at z = 9 mm in the ischemic tissue. The stimulus of -0.6 mA is applied for 4 ms duration at BCL of 210 ms.
A train of cathodal stimuli is applied at the same location with a variable basic cycle length (BCL). The BCL is initially 220 ms and is reduced stepwise to 170, 140, and then held constant at a lower value for 50 pacing intervals. A wave front is propagated successfully and slowly through the ischemic area for all beats when the BCL is sufficiently large (≥ 123 ms) (Figure 3(a)). As the BCL is decreased to 122 ms (or 121 ms), we see alternans on the left and at the boundary region between normal and ischemic tissue (blue and red curves of Figure 3(b, c)) and 2:1 conduction block in the ischemic region (green curve of Figure 3(b, c)). When the BCL is decreased to 120 ms, we see 2:1 conduction block in all regions. For a few cycles (t ≤ 4000 ms) in the border zone, we see Wenckebach-like rhythms that transform to alternans in later cycles.
Figure 4(a) shows the variation of Vm and calcium cycling with time. It confirms that Vm and Cai are bi-directionally coupled. Changes in Vm alter Cai cycling, and Cai cycling influences other ion channels and the AP morphology. Hence, Cai dynamics play a major role in electrical activities such as wave propagation. Figure 4(c) shows the L-type calcium current as a function of time. It has been known that membrane excitability is reduced at the border zone and in ischemic cardiac tissue. Hence, in these regions the depolarization due to sodium current is reduced and conduction and wave propagation are sustained by the calcium current (Fig. 4(b)). The red curve of Figure 4(d) shows the electrotonic current flowing through the ischemic border and is responsible for ectopic activity and reentry induction.
Figure 3. Action potential propagation in different regions. Figure (a) shows the regular action potential propagation in different regions when BCL ≥ 123 ms. Figures (b) and (c) show alternans at the border and 2:1 conduction block in the ischemic region when the BCL is reduced to 122 or 121 ms.
Figure 4. The dynamics of different parameters at different regions when BCL is 121 ms. Figures (a), (b), (c) and (d) show the dynamics of transmembrane potential (Vm) (represented by the solid curve) and intracellular calcium concentration (Cai) (represented by the dashed curve), sodium current (INa), calcium current (ICa), and electrotonic current (Ielec) as function of time respectively. All the currents have the same unit of μA/μF. Blue, red and green curves are calculated at three different locations (ie. normal, border and ischemic regions) shown in Figure 1.
Figure 5. Regional increase of [K]e and action potential propagation. Figure A: An image of the whole heart with superimposed isochrones (black lines). The white curve represents the border between the normal and high potassium (ischemic) regions, black arrows show the direction of wave propagation, and the numbers 1 to 5 represents the locations where action potentials are measured in Figure B. A normal action potential is altered throughout the border zone and can lead to conduction block during fast pacing at the border of the K+ heterogeneity. Reproduced with permission from Sidorov et al. (2011).
Figure 3 shows the effect of [K]e on electrical stability. At certain BCLs, the spatial heterogeneity created with regional elevation of [K]e can lead to action potential instability (alternans) and 2:1 conduction block. Sidorov et al.1 created a local [K]e heterogeneity by perfusing left marginal vein of a Langedoroff-perfused rabbit heart with solutions containing 4, 6, 8, and 10 mM of potassium and investigated its role in fast pacing response and arrhythmia induction. They found local [K]e heterogeneity causes action potential alternans, 2:1 conduction block and wave breaks. More specifically, they detected normally propagating wave fronts in the normal [K]e region, alternans in the border zone, and 2:1 conduction block in the high [K]e (ischemic) region (Figure 5). Our results are consistent with their experimental findings.
Arce et al.12 examined propagation in 3 cm long fibers with a centrally located 1-cm long segment exposed to higher concentration of [K]e. They increased [K]e and found the transition in the action potential propagation from a normal rhythm (1:1) to alternans (2:2) and to conduction block (2:1). The pacing cycle length they used was 400 ms. They used a one-dimensional cable model with the 1991 version of the Luo-Rudy model13 to represent a fiber and active properties of cell membrane respectively. The original version of the Luo-Rudy model contains a preliminary model for calcium cycling. Our work is the expansion of their 1-dimensional analysis to 3-dimensions, and considers the dynamics of calcium currents.
Livshitz and Rudy (2007)14 observed alternans in simulations using fast pacing. To determine if such purely membrane-based alternans play a role in our simulation, we removed the heterogeneity by making the entire tissue have a normal (4 mM) extracellular potassium concentration. In this case, we never observed alternans. Libshitz and Rudy (2007) observed alternans over a wide window of BCLs (150-250 ms), whereas we observed BCLs over a narrow window at more rapid BCLs (121-122). Therefore, we believe these alternans arise from a different mechanism than those observed by Livshitz and Rudy.
The occurrence of alternans depends upon the pacing interval and occurs in a narrow window of intervals. Sidorov et al.1 confirmed that the stability of action potential depends on the pacing interval and the pacing rate.
Our model has several limitations. We used a sharp transition between the normal and ischemic zones, but in fact there is a graded border zone. The transmembrane potential varies over a distance of a few length constants (about two millimeters). As long as the transition between high and low potassium occurs over a distance small compared to the length constant, our model should be appropriate. For a complete ischemic model, the ATP-activated K+ current (IK(ATP)), transient outward current (Ito), acidosis and hypoxic current should be incorporated into the model15. Fiber axis rotation is not considered. In our simulation, fibers are oriented along z-direction. The dynamic Luo-Rudy model is based on the guinea-pig ventricle10. Most of the experiments (including Sidorov et al.,1) on ischemia have been performed using larger animals including rabbits, dogs, sheeps, and pigs. So the results might be inconsistent.
We created 3D inhomogeneous sheet of ventricular myocardium by raising the extracellular potassium ion concentration from 4 mM (normal) to 10 mM (ischemia) regionally to investigate the effect of a localized inhomogeneity at the border between normal and ischemic tissue. At certain BCLs, the spatial heterogeneity created with regional elevation of [K]e led to action potential instability (alternans) and 2:1 conduction block. Electrical alternans occur during high heart rates and are observed in patients suffering from ventricular tachycardia. It is an early indication of left ventricular systolic impairment. This study will help to evaluate alternans as a predictor and guide for antiarrhythmic therapy.
This research was supported by NIH grant R01HL118392.
1. Sidorov VY, Uzelac I, and Wikswo JP Jr (2011). Regional increase of extracellular potassium leads to electrical instability and reentry occurrence through the spatial heterogeneity of APD restitution. Am J Physiol Heart Circ Physiol, 301: H209–H220.
2. Winfree AT (1989). Electrical instability in cardiac muscle: Phase singularities and rotors. J. Theor. Biol., 138:353–405.
3. Winfree AT (1991). Varieties of spiral wave behavior: An experimentalist’s approach to the theory of excitable media. Chaos, 1:303–334.
4. Thakor NV, and Fishler MG (1996). Initiation and termination of spiral waves in a two-dimensional bidomain model of cardiac tissue. Computers in Cardiology IEEE Computer Society, Silver Springs, 229–232.
5. Courtemanche M, and Winfree AT (1991). Re-entrant rotating waves in a Beeler–Reuter based model of two-dimensional cardiac electrical activity. Int. J. Bifurcation Chaos Appl. Sci. Eng., 1: 431–444.
6. Efimov IR, Krinsky VI, and Jalife J (1995). Dynamics of rotating vortices in the Beeler–Reuter model of cardiac tissue. Chaos Solitons Fractals, 5:513–526.
7. Fenton F, and Karma A (1998). Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: Filament instability and fibrillation. Chaos, 8: 20 –47.
8. Trayanova N, Skouibine K, and Aguel F (1998). The role of cardiac tissue structure in defibrillation. Chaos, 8: 221–233.
9. Henriquez CS (1993). Simulating the electrical behavior of cardiac tissue using the bidomain model. Crit. Rev. Biomed. Eng., 21: 1–77.
10. Luo CH, and Rudy Y (1994). A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes. Circ Res, 74: 1071 – 1096.
11. Kandel SM, and Roth BJ (2014). Intracellular calcium and the mechanism of the dip in the anodal strength-interval curve in cardiac tissue. Circulation J., 78:1127-1136.
12. Arce H, Lopez A, and Guevara MR (2002). Triggered alternans in an ionic model of ischemic cardiac ventricular muscle. Chaos, 12: 807–818.
13. Luo CH, and Rudy Y (1991). A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction. Circ Res, 68: 1501 – 1526.
14. Livshitz LM, and Rudy Y (2007). Regulation of Ca2 and electrical alternans in cardiac myocytes: role of CAMKII and repolarizing currents. Am J Physiol Heart Circ Physiol 292: H2854 – H2866.
15. Shaw RM, and Rudy Y (1997). Electrophysiologic effects of acute myocardial ischemia: a theoretical study of altered cell excitability and action potential duration. Cardiovascular Research, 35(2):256–272.
Conflict of interest: No conflicts declared.
*Corresponding Author. Sunil M Kandel. Department of Molecular and Integrative Physiology, University of Michigan, 2800 Plymouth Road
NCRC 10-A122, Ann Arbor, MI 48105, USA. Phone: 248-917-5880. Email: firstname.lastname@example.org, email@example.com
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