The role of the hyperpolarization-activated inward current f in arrhythmogenesis: a computer model study

Transcription

1 1 The role of the hyperpolarization-activated inward current f in arrhythmogenesis: a computer model study Nico H.L. Kuijpers, Rikkert H. Keldermann, Huub M.M. ten Eikelder, Theo Arts, and Peter A.J. Hilbers Abstract Atrial fibrillation is the most common cardiac arrhythmia. Structural cardiac defects such as fibrosis and gap junction remodeling lead to a reduced cellular electrical coupling and are known to promote atrial fibrillation. It has been observed that the expression of the hyperpolarization-activated current f is increased under pathological conditions. Recent experimental data indicate a possible contribution of f to arrhythmogenesis. In this study, the role of f in action potential propagation in normal and in pathological tissue is investigated by means of computer simulations. The effect of diffuse fibrosis and gap junction remodeling is simulated by reducing cellular coupling non-uniformly. As expected, the conduction velocity decreases when cellular coupling is reduced. In the presence of f the conduction velocity increases both in normal and in pathological tissue. In our simulations, ectopic activity is present in regions with high expression of f and is facilitated by cellular uncoupling. We conclude that an increased f may facilitate propagation of the action potential. Hence, f may prevent conduction slowing and block. Overexpression of f may lead to ectopic activity, especially when cellular coupling is reduced under pathological conditions. Index Terms Atrial Fibrillation, Ectopic Activity, Bidomain Model, Hyperpolarization-activated Inward Current, Fibrosis, Gap Junction Remodeling I. INTRODUCTION ATRIAL FIBRILLATION (AF) is the most common cardiac arrhythmia [1]. The prevalence of AF increases with age from.5 percent of people under the age of 6 to almost 1 percent of people over the age of 8 [2]. Related to the occurrence of AF is structural remodeling of the tissue, including increase of cell size, erstitial fibrosis and gap junction remodeling [3]. These effects lead to the loss of electrical coupling between cardiac cells and are known to promote recurrence of AF episodes [4], [5]. Changes in gap junction distribution may be involved in the initiation and persistence of atrial fibrillation [6], [7], [8], [9]. Besides structural remodeling, electrophysiological remodeling is related to AF, including shorter atrial effective refractory periods (AERP), greater dispersion of atrial refractoriness, and atrial conduction delay [1]. However, even if N.H.L. Kuijpers, H.M.M. ten Eikelder, T. Arts and P.A.J. Hilbers are with the Department of Biomedical Engineering, Technische Universiteit Eindhoven, P.O. Box 513, 56 MB, Eindhoven, The Netherlands. R.H. Keldermann is with the Department of Theoretical Biology, Universiteit Utrecht, The Netherlands. T. Arts is also with the Department of Biophysics, Universiteit Maastricht, The Netherlands. P.A.J. Hilbers is also with the Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, The Netherlands. AF initiation and AF perpetuation are facilitated by electrophysiological remodeling, the initiation of AF still requires a trigger [11]. Such a trigger may come from ectopic foci. Especially in the case of paroxysmal AF, episodes of atrial fibrillation are often triggered by ectopic foci that are located in the pulmonary veins [12]. The mechanism of focal activity leading to AF is, however, still unknown [12]. Two possible mechanisms are micro-reentry within the pulmonary veins [13] and spontaneous depolarization of cells located in the pulmonary veins [14]. The ionic current responsible for pacemaker activity of sinoatrial node cells and Purkinje fibers is the hyperpolarizationactivated inward current f [15]. f channels are believed to be complexes of hyperpolarization-activated cyclic nucleotidegated (HCN) channels [16]. f channels in atrial myocardium are most likely heteromeric complexes composed of HCN4 and/or HCN2 [17], [18]. f conducts both K and Na, with about a 3:1 preference for K [15]. It has some unusual features and is thereby known as the funny current [15], [16]. The first unusual feature is that f is activated by hyperpolarization with a threshold of approximately to mv in the sinus node [16] and about mv in other myocardial cells [19]. The second unusual feature is that the fully activated current/voltage relation reverses near to mv in physiological solutions as a consequence of the channel s mixed permeability to Na and K [16]. The activation by hyperpolarization and permeability to Na and K are important properties with respect to the role of f in diastolic depolarization and spontaneous activity [16]. As opposed to earlier results [2], [21], [22], [23], it was recently observed by Michels et al. [18] that half-maximum activation of single-channel f is within the diastolic range of human atrial myocardium. These observations support a possible contribution of HCN-gated channels and f to arrhythmogenesis under pathological conditions [18]. f has been identified in cardiac tissue that is normally not capable of pacemaking, including human left ventricular myocytes [2], [21] and human atrial myocytes [19], [22], [24]. Interestingly, the expression of f in left ventricular myocytes of the rat increases with age [25]. Also in ventricular myocytes of hypertrophied and failing hearts of the rat, the expression of f is increased and leads to diastolic depolarization in isolated myocytes [25], [26], [27]. Sartiani et al. [28] observed two action potentials of different HL-1 cells derived from the atria of a transgenic mouse. One of the action potentials showed spontaneous diastolic depolarization and the other a

2 2 flat diastolic potential. A hyperpolarization-activated inward current was observed in the cell with a spontaneous diastolic depolarization phase, but not in the other cell [28]. Besides, they observed spontaneously beating cells in some regions of HL-1 cell cultures with a frequency varying between 1.3 and 5 Hz [28]. Since the beating stopped in the presence of f-blocker Cs, it is suggested that the spontaneous activity is caused by f [28]. However, whether f may favor spontaneous diastolic depolarization in individual human atrial myocytes remains to be determined [22]. Previous simulation studies to investigate conduction velocity and arrhythmia in structurally remodeled tissue did not include f. Fast and Kléber showed that conduction slowing and block may occur at an abrupt tissue expansion [29]. Shaw and Rudy applied a multicellular monodomain fiber model to investigate conduction slowing and block in relation to reduced brane excitability and decreased gap junction coupling [3], [31]. Street and Plonsey applied a multi-fiber bidomain model to investigate conduction slowing and block in regions of passive, connective tissue representing infarcted regions [32]. Recent large-scale simulation studies to atrial arrhythmia apply detailed models of the ionic brane currents, electrophysiological remodeling, anisotropy and 3D geometry [33], [34], [35], [36], [37], [38], [39]. The aim of the present simulation study is to investigate the influence of f on the conduction velocity and ectopic activity under normal and pathological conditions. We apply a discrete bidomain model with active brane behavior to represent the cardiac tissue. Diffuse fibrosis and gap junction remodeling are modeled by decreasing racellular coupling at random throughout the tissue. The amount of f current and of cellular uncoupling are varied to investigate their respective influence on the conduction velocity and ectopic activity. II. METHODS We have developed a new discrete bidomain model, the Cellular Bidomain Model, which we previously applied to investigate virtual electrode polarization during ernal field stimulation under normal and pathological conditions [4]. Active brane behavior as well as racellular coupling and erstitial currents are described by this model. Similar to the volume averaging approach of the continuous bidomain model, each po in the tissue is assigned both an racellular and an racellular potential [32], [41]. In our model, the bidomain is subdivided in segments and the same state is assumed everywhere within a single segment, which is why we call it a discrete bidomain model, or Cellular Bidomain Model [4]. Although a segment may represent a single myocyte, we usually apply somewhat larger segments such that less computational power is required to simulate sheets of cardiac tissue. The segment sizes may vary as well as the electrical coupling between adjacent segments. Pathology such as fibrosis and gap junction remodeling can be modeled by reducing the racellular coupling between the segments. Furthermore, active brane behavior can be varied throughout the tissue by assigning different brane properties to individual segments. Fig. 1. Graphical representation of a simulation graph. Each node represents a rectangular segment of cardiac tissue. The state of each node is represented by the ernal potential, the ernal potential, and the brane potential. Electrical coupling between the nodes is indicated by the racellular and erstitial conductances denoted by and. The racellular and erstitial current flowing between the nodes are represented by the arrows labeled and. A. Cellular Bidomain Model In the Cellular Bidomain Model, the structure of the cardiac tissue is represented by a graph consisting of nodes and edges, where a node represents a segment and an edge the electrical coupling between the segments. Let represent the set of nodes and the set of edges connecting the nodes. Each node has its own brane model describing the ionic brane currents. The state of each node is defined by the ernal potential and the ernal potential. The brane potential for node is defined as. Edges define the conductance for racellular and racellular currents between two adjacent nodes. We distinguish the ernal and ernal conductance, which are denoted by and, respectively, see Fig. 1. It is assumed that for each edge it holds!#" $ and %!#" $. The ernal and ernal currents flowing from node to node are denoted by &('! and (')!, and are given by Ohm s law: (')! * +! &!#" (1) (')! * +! &!#", (2) The racellular current entering node coming from all adjacent nodes -,.-/ 12, is denoted by & and the racellular current by 3. These currents are defined by "98(: "98(: 5 ') (3) 5 '), (4) According to Kirchhoff s law, current entering a node as racellular current must flow to the erstitial space as transbrane current and leave the node as racellular current. By choosing the transbrane current, denoted by

3 . 3 trans, flowing from the racellular space to the erstitial space, we obtain for node trans, (5) The transbrane current is the sum of capacitive and ionic currents, i.e., trans ion * (6) where represents the brane capacitance of node the brane surface in cm. The ionic in F and brane current of node, denoted by ion, is expressed in A per cm brane surface and depends on the brane potential as well as gating variables and ionic concentrations denoted by the vector 1. In the present study, we apply a modified version of the Human Atrial Action Potential Model of Courtemanche et al. [42] to model the ionic currents. We ended this model with the hyperpolarization-activated current f and adapted the Na kinetics as described below. The total ionic current is given by ion Na K1 to Kur Kr Ks Ca,L p,ca NaK NaCa b,na b,ca f, (7) The ionic and pump currents, including the handling of the racellular Ca concentration ( Ca i) by the sarcoplasmic reticulum (SR), are described in Ref. [42]. The model keeps track of Ca i as well as Na i and K i, while the racellular concentrations Ca o, Na o and K o are constant [42]. When solving the equations of the Cellular Bidomain Model, no-flux boundary conditions are assumed for both the racellular and the erstitial domain. The numerical egration scheme is described in the appendix. Below, we describe how the pacemaker current f is modeled and the inward Na current Na is modified. N, it is described how normal and pathological atrial tissue is modeled. Furthermore, the simulation protocol for the present study is described. B. Hyperpolarization-activated inward current f The f current is expressed in pa/pf and is defined by f f o f (8) where f is the maximum brane conductance for f in ns/pf, o the fraction of channels in the open state, and f the reversal potential. The dynamics of o is defined by o o o o o (9) where o and o are defined as in DiFrancesco [43]: o,!#"%$'&)(, o, $'&)(, (1), (11) Moroni et al. [44] describe human f kinetics based on HCN2 only. Ludwig et al. [17] concluded that pacemaker activity in the human heart is a combined effect of HCN2 and HCN4 channels. Michels et al. [18] observed that single-channel characteristics of f in human atrial myocytes resemble those of HCN4 or HCN2 + HCN4. Since HCN4 kinetics are slower than HCN2 kinetics [17], [44], and the kinetics obtained by DiFrancesco [43] for rabbit sino-atrial nodal f are slower than the HCN2 kinetics obtained by Moroni et al. [44], we decided to model activation parameter o as described by DiFrancesco [43]. Using these parameters, the half activation channels potential is +* mv and only a small fraction of f will be open in diastolic range (above mv) [43]. The reversal potential f is defined as in Moroni et al. [44]: f,%-./21 K o 43 Na5 3 K' Na o K i 43 Na5 3 K' Na i (12) where 3 Na and 3 K are permeabilities to Na and K,, the universal gas constant, - the temperature (31 K [42]), and Faraday s constant. The ratio 43 Na5 3 K is.41 [44]. In the Courtemanche model, the racellular ion concentrations are constant ( K o, mm, Na o mm [42]), while the racellular concentrations are dynamic ( K i * mm and Na i, mm when, mv [42]). We have adapted the Courtemanche model to take the influence of f on the racellular concentrations o account as described by DiFrancesco and Noble [45]. With the parameters defined in the Courtemanche model, the reversal potential f is approximately mv. C. Fast inward Na current Na Na is modeled in the Courtemanche model as in the Luo- Rudy phase-2 model [46], and is given by Na Na7698;:/* < Na, (13) Here, Na is the maximum Na conductance (=, ns/pf) and Na the equilibrium potential for Na [42]. Further, is the fast activation variable, and 8 and : are the fast and slow inactivation variables [42], [46]. To obtain diastolic action potentials, we adapted the Na kinetics by doubling the forward rates m and h of the fast gating variables and 8 (see equations (3), (31) and (34) of the Courtemanche model [42]). With this change, the Na channels open and diastolic action potentials occur with a basic cycle length (BCL) of 1.5 seconds for f, = ns/pf. The action potential morphology and duration remain unchanged when a stimulation current of 2 pa/pf is applied during 2 ms as in Ref. [42]. D. Modeling normal atrial tissue The modifications of the Na kinetics have effect on the maximum upstroke velocity 5 max and thus on the conduction velocity. In our simulations, 5 max increased from 167 V/s to 288 V/s. To obtain conduction velocities similar to those reported by Spach and Boineau (.48 m/s longitudinal and.15 m/s transverse [47]), we have scaled the conductivity parameters as reported by Clerc [41], [48] with a factor.6. The bidomain parameters used in the present study for normal atrial tissue are listed in Table I. To obtain criteria for the size of individual segments, we apply cable theory and consider subthreshold behavior along

4 , 4 TABLE I BIDOMAIN PARAMETERS FOR ATRIAL TISSUE Parameter Definition Value longitudinal ernal conductivity 1.44 ms/cm transverse ernal conductivity.1158 ms/cm longitudinal ernal conductivity 3.75 ms/cm transverse ernal conductivity ms/cm brane capacitance 1. F/cm surface-to-volume ratio 2/cm a fiber as described in Ref. [41]. The application of a stimulus current produces a spatial change in the brane potential along the fiber. For subthreshold behavior, the transbrane current trans can be described by trans, (14), where is the brane resistance in cm and the brane capacitance in F/cm [41]. The steady-state response of along the fiber to a subthreshold current at position is exponential and can be described by / 3 $'&)(1 (15) and, where / is the brane potential on distance from the stimulus site, the brane potential at the stimulus site ( ), and the length constant [41]. Using the bidomain parameters listed in Table I, the length constants in longitudinal and transverse direction, denoted by can be expressed as and,, where, as before, cm, [32], [41]. We estimated (16) (17) is the brane resistance in for the modified Courtemanche model by applying a subthreshold stimulation current of, pa/pf during, ms. was estimated using rest, ion (18) is in between, * and, is in between, cm and, cm. To obtain for rest mv and, ranging from mv to = mv. We found values for between and cm., For these values of, cm, and accurate simulation results, the tissue is modeled as a brickwall with segments of length 2 m and width 8 m. Hence, the segment sizes are approximately one fifth of the length constant in both directions. E. Modeling pathological tissue In the Cellular Bidomain Model a distinction is made between longitudinal coupling (along the fiber direction) and transverse coupling (side-to-side). The effects of gap junction remodeling are simulated by varying the ernal conductivities non-uniformly throughout the tissue. A normal distribution with parameters is applied to determine a factor by which the initial values for the ernal conductivities ( ) are multiplied for each edge. Negative factors and factors larger than one are replaced by zero and one, respectively. The average and standard deviation of the gap junction remodeling factor in longitudinal direction are denoted by L and L, respectively, and the average and standard deviation in transverse direction by T and T, respectively. Fibrosis causes the loss of side-to-side cell coupling which can result in a disturbed wavefront propagation and may isolate groups of myocytes [49]. Since side-to-side coupling is affected, the main effect of fibrosis is a reduction of the transverse conduction velocity [47]. We model the effect of diffuse fibrosis [5] by randomly removing some of the ernal conductivities in transverse direction, i.e., becomes zero for a fraction of the edges in transverse direction. This fraction is called fibrosis fraction and is abbreviated by FF. F. Simulation protocol 1) Single cell simulations: A series of single cell simulations over a period of four seconds was performed with varying expression of f ( f,,,,, and, ns/pf). Initially, the brane state was equal to the resting state of a human atrial cell without f ( mv [42]). No stimulation current was applied in these simulations. 2) Gap junction remodeling and fibrosis: To investigate the effect of gap junction remodeling and fibrosis on the longitudinal conduction velocity L and the transverse conduction velocity T, three series of simulations were performed on a cm, cm sheet of atrial tissue composed of uniform m m segments. An action potential front was initiated by stimulating the left-most segments of each row (first series) or the upper row of segments (second and third series). The segments were stimulated by adding a stimulus current of pa/pf to the ionic brane currents until the brane was depolarized. In the first series, L ranged from, through, T, L,, and, and T. In the second series, L, T ranged from, through, L, and T,, and. In the third series, fibrosis fraction FF ranged from through, *. No gap junction remodeling was applied in the third series. L and T were determined by averaging the velocity of the action potential front along nine parallel tracks of length, cm in longitudinal direction and, cm in transverse direction. 3) Uniform expression of f: To investigate the influence of a uniform expression of f on L in normal, remodeled and fibrotic tissue, five series of simulations were performed on a 2 cm.5 cm sheet of atrial tissue with segment lengths varied randomly between 15 and 25 m and width 8 m. Two types of remodeled tissue were simulated by applying gap junction remodeling ( L T, = and L T,, and L T, and L T ). Two types of fibrotic tissue were simulated by applying both gap junction remodeling and fibrosis (FF, ). The expression of f ranged from, through, ns/pf. The left-most segments of the center fibers

5 5 were stimulated at the start of the simulation. Stimulation was repeated after 8 ms. Four independent measurements of L were obtained by measuring over four consecutive tracks of length, cm along the central fiber. 4) Regional expression of f: To investigate the critical size of an ectopic pacemaker in relation to cellular coupling, a series of simulations was performed on a cm, cm strip of tissue. f was present in the right part of the strip ( f, ns/pf), but not in the left part. The size of the region in which f was present was varied between and of the tissue. Uniform, reduced cellular coupling was simulated ( L T ranged from, through, and L T ). Each simulation lasted seconds. No stimulation current was applied. In case ectopic activity occurred, the basic cycle length (BCL) was determined for each segment by averaging the erval times between consecutive depolarizations. 5) Non-uniform expression of f: To compare ectopic activity in normal and remodeled atrial tissue when the expression of f is non-uniform, two series, A and B, of simulations were performed on a cm, cm sheet of tissue composed of uniform m m segments. In series A, f was uniformly increased from, = ns/pf (BCL 1.5 seconds) on the left to, ns/pf (BCL.75 seconds) on the right. In series B, f was randomly distributed throughout the tissue with average, ns/pf and standard deviation, * ns/pf. f was bound by a minimum of, = ns/pf (BCL 1.5 seconds) and a maximum of, ns/pf (BCL.75 seconds). Besides normal tissue, various types of remodeled tissue were simulated ( L T, and,, and L T,,,,,,,, and, ) as well as totally uncoupled tissue ( L T, and L T, ). The distribution of f was exactly the same for all simulations of a series, thus the simulations of one series only differ in cellular coupling. Each simulation lasted seconds. No stimulation current was applied. The basic cycle length (BCL) was determined for each segment by averaging the erval times between consecutive depolarizations. 6) Large-scale simulations: To investigate the effect of a non-uniform expression of f under normal and pathological conditions, three large-scale simulations were performed on an 8 cm 3 cm sheet of atrial tissue with non-uniform expression of f. The width of the segments was m, while the length was varied using a normal distribution with average m and standard deviation m. The segment lengths were bound by a minimum of m and a maximum of m. f was, ;= ns/pf in the leftmost, cm of tissue and f was varied in the rightmost 1.6 cm of the tissue with average, ns/pf and standard deviation, 9* ns/pf. f was bound by a minimum of, ;= ns/pf (BCL 1.5 seconds) and a maximum of, ns/pf (BCL.75 seconds). The left-most segments of the center fibers were stimulated at the start of the simulation. Stimulation was repeated each 8 ms. To simulate pathological tissue, two simulations were performed on the same 8 cm 3 cm sheet of atrial tissue with the same distribution of f. The fibrosis fraction FF was.35. After the first stimulation, the action potential propagated from left to right such that the entire tissue was depolarized after approximately 25 ms. Just before the second stimulation at simulation time 8 ms, the conductivity properties of the V [mv] I f [pa/pf] g f [ns/pf] I K1 [pa/pf] Fig Simulation time [ms] Membrane potential (mv), hyperpolarization-activated inward current f (pa/pf), time-independent rectifier K current K1 (pa/pf), and conductance f f o (ns/pf) for single cell simulations with different f values: dotted f ns/pf, short dash f ns/pf, long dash f ns/pf, solid f ns/pf. No ernal stimulation current was applied. Initially, the brane state was equal to the resting state of a human atrial cell without f ( mv). tissue were changed to simulate gap junction remodeling with average L T equal to.5 and.3, and standard deviation L T equal to 1 and 2, respectively. The second simulation of pathological tissue lasted 15 seconds of simulation time. A. Single cell simulations III. RESULTS The brane potential, the hyperpolarizationactivated inward current f, the time-independent K current K1, and conductance f f o are presented in Fig. 2 for f,,,,,, and, ns/pf. Spontaneous diastolic action potentials occurred for f, and, ns/pf with basic cycle length (BCL) 126 ms and 76 ms, respectively. f, and, ns/pf resulted in a depolarized steadystate brane potential of =, mv and =, mv, respectively. When comparing the ionic currents for f, ns/pf with f, ns/pf, the inward current f was increased from, to, * pa/pf and the rectifier current K1 was increased by the same magnitude from, to, pa/pf (see Fig. 2). The changes in the remaining currents were marginal (not shown). The change in K1 is expected, because K1 is voltage dependent with a maximum current size of, pa/pf for = mv [42]. Thus, for increasing f, the brane potential rises and K1 increases. For f smaller than, ;= ns/pf, a steady-state brane potential is reached at which the depolarizing f current is compensated by the rectifying K1 current. The fraction of f channels being in the is of the order of a few percent. open state o f5 f B. Gap junction remodeling and fibrosis In Fig. 3 the longitudinal and transverse conduction velocities L and T are presented for varying L and T (top), L and varying T (center) and varying FF (bottom). For

6 T 6 θ L [m/s] θ T [m/s] σ L = σ L =.5 σ L = µ L.1.5 σ T = σ T =.5 σ T = 1 θ L [m/s] µ = 1., σ =., FF =. µ =.75, σ =.25, FF =. µ =.75, σ =.25, FF =.35 µ =.5, σ = 1., FF =. µ =.5, σ = 1., FF = µ T.15.2 θ T [m/s] FF Fig. 3. velocity. Top: longitudinal conduction velocity L for varying L, T T. Center: transverse conduction velocity T for varying T, L The effect of gap junction remodeling and fibrosis on the conduction and and L. Bottom: transverse conduction velocity T for varying fibrosis fraction FF, L T and L T. The solid lines represent L and T predicted by cable theory for the continuous bidomain model. The dashed line in the lower figure represents T for a linear descent. larger values of L and T, the conduction velocities are smaller for large L and T, while larger conduction velocities are obtained for small L and T. This is caused by the requirement that the factors by which is multiplied must lie between and. The average of these factors is thus smaller than L or T for L T $, and larger than L or T for L T,. The solid lines indicate L and T predicted by cable theory. For non-curved action potential fronts, L and T are proportional to the length constants and (see equations (16) and (17)) [32], [41]. The prediction of L using cable theory is accurate for L and L,. Cable theory overestimates for both transverse gap junction remodeling and fibrosis. As indicated by the dashed line, the effect of fibrosis is better estimated assuming a linear descent of T. A reduction of in L due to gap junction remodeling in rat ventricular tissue was reported by Uzzaman et al. [51]. A similar reduction of L in our model is obtained for L, and L,. Spach and Dolber observed transverse conduction velocities related to uncoupling of side-to-side connections as low as, m/s in human cardiac tissue of adults over age 5 [49]. From Fig. 3 it can be observed that such low values for T in our model can be obtained for fibrosis fraction FF between, and,. C. Uniform expression of f The longitudinal conduction velocity L versus f after the second stimulation is presented for five different types of tissue in Fig. 4. The standard deviation is indicated by the error bars. A similar increase in L is obtained for normal, remodeled and is fibrotic tissue. For f above, ns/pf, the increase in L approximately, m/s for all types of tissue. Compared to the conduction velocity in tissue without f, this increase is about for remodeled tissue ( L T, and L T G [ns/pf] f Fig. 4. Longitudinal conduction velocity L for normal, remodeled and fibrotic atrial tissue. f is uniform and the expression of f is indicated by maximum conductance f. The standard deviation of four independent measurements of L along the same fiber is indicated by the error bars., see Fig. 4). Note that although for f above, ;= ns/pf spontaneous diastolic action potentials can occur, the increase in conduction velocity for f below, ns/pf is caused by a higher (more depolarized) diastolic brane potential. For values of f above, ns/pf ectopic pacemaking with BCL less than ms occurred and the pacemaker at the stimulation site was captured (results not shown). D. Regional expression of f Average BCLs for various L T and size of region with expression of f are presented in Table II. Region size is indicated as a percentage of the total tissue. A dash means no ectopic activity occurred. In these simulations, the current generated by the f channels was spread over the entire tissue such that the threshold was not reached in any of the segments. For comparison, f, ns/pf in single cell simulations leads to an average BCL of, * ms. From Table II it can be observed that when f is present in of the tissue, ectopic activity occurs in normal atrial tissue. In tissue with reduced cellular coupling, smaller regions with expression of f lead to ectopic activity. Furthermore, the BCL decreases when cellular coupling decreases. E. Non-uniform expression of f The tissue properties, average BCLs and standard deviation of BCLs for the gradual increase of f (series A) and random distribution of f (series B) are presented in Table III. In both series, the average BCL decreases when coupling is reduced. In case coupling is non-uniformly reduced, the BCLs are even more decreased. The standard deviation of the BCLs is smaller for coupled tissue compared to totally uncoupled tissue ( L T, ). Due to coupling, the segments fire with similar frequencies (frequency entrainment [52]). By inspecting the times of activation we found that when coupling is increased,

7 L L L T 7 TABLE II BASIC CYCLE LENGTH IN MS FOR REGIONAL EXPRESSION OF f T TABLE III BASIC CYCLE LENGTH IN MS RELATED TO CELLULAR COUPLING L T series A series A series B seriesb mean std mean std the segments not only fire with similar frequencies, but also at similar times (waveform entrainment [52]). The larger standard deviation of BCL in less coupled tissue is mainly caused by a few non-synchronized segments that fire at their own basic cycle length. F. Large-scale simulations The brane potentials after the second stimulation are presented for normal atrial tissue in the left column of Fig. 5. Note that although the cardiac tissue is modeled as an irregular brickwall, the curvature of the wavefront is smooth and ellipsoidal, confirming that the segments are small enough to represent normal atrial tissue. The conduction velocities, m/s and T, m/s as well as the anisotropy ratio L5 T, are similar to the results reported by Spach and Boineau ( L, m/s, T, m/s and L5 T, [47]). The brane potentials after the second stimulation for pathological tissue are presented in the center and right column of Fig. 5. Ectopic activity occurs on the right side of the tissue approximately ms after the start of the simulation. This ectopic activity is related to an increased expression of f in that area. It can be observed that at ms more ectopic activity is present in the right column compared to the center column. Since cellular coupling is the only difference between the two simulations, and gap junction remodeling was applied just before the second stimulation at ms simulation time, the state of the two stimulations was exactly equal till ms simulation time. The increase in ectopic activity between these simulations can thus only be explained by the decrease in cellular coupling. In Fig. 6 the brane potentials are shown at several simulation times for the second simulation of pathological tissue. Each of the figures represents a moment of collision of the depolarization wavefront generated by the sinus node on the left hand side and the wavefront generated by ectopic activity on the right hand side. Note that the collision front moves to the right during the course of the simulation and eventually reaches the ectopic area before it can depolarize. After approximately seconds of simulation time, the ectopic pacemaker is captured by the normal pacemaker. The yellow and green spots indicate spontaneous action potentials from uncoupled cells that are not activated by the depolarization wave. These cells are not capable of initiating new depolarization waves, and thus no reentry will occur. IV. DISCUSSION A. Mechanisms leading to atrial arrhythmia Many hypotheses have been proposed that may explain AF initiation and perpetuation, such as structural remodeling [6], [7], [8], [9], [53], electrophysiological remodeling [1], [54], stretch [55], atrial dilation [56], and atrial fibrillation itself ( AF begets AF ) [57]. Recurrence of AF by late phase 3 early afterdepolarizations (EADs) shortly after termination of AF has been reported by Burashnikov and Antzelevitch [58]. Distinct electrophysiological properties between myocytes from the left atrial free wall and the pulmonary veins may contribute to the role of the pulmonary veins in atrial fibrillation [59], [6]. In the present study we investigate the influence of f on the conduction velocity and ectopic activity triggered by diastolic action potentials. B. Hyperpolarization-activated inward currents f is not the only identified hyperpolarization-activated inward current. Ehrlich et al. [61] identified a hyperpolarizationactivated time-dependent K current KH in canine myocytes from the pulmonary veins. Chen et al. [62] observed pacemaker currents in canine PV myocytes which they believed to be f. It is suggested by Ehrlich et al. [61] that these currents may correspond to KH. Hoppe and Beuckelmann found a considerable variability of f expression from cell to cell [22]. Based on a study by Wu et al. [63] identifying f and K1 in atrial cells from the cat, Hoppe and Beuckelmann suggest that diastolic depolarization may not only depend on upregulation of f, but also on downregulation of K1 [22]. We observed that for f smaller than, ;= ns/pf, a steady-state brane potential is reached at which the depolarizing f current is compensated by the rectifying K1 current. Our findings support the suggestion by Hoppe and Beuckelmann that it is the combination of upregulation of f and downregulation of K1 that may lead to diastolic depolarizations [22]. C. The role of f in structurally remodeled tissue Based on our simulation results, we conclude that ectopic activity in regions with an increased expression of f is

8 L T 8 time L T L T, L T, (ms) L T L T L T FF FF, FF,, m/s L, 9* m/s L, m/s, m/s T, m/s T, m/s Fig. 5. Simulation of normal (left column) and pathological (center and right column) atrial tissue with non-uniform expression of f. The brane potential is shown for different simulation times after the second stimulation. Just excited tissue is colored red, depolarized tissue is yellow or green, and repolarized tissue is blue. enhanced by a reduced cellular coupling. Similar observations are reported by Wilders et al. [64] in a study on focal activity related to anisotropy. Cai et al. [52], Winslow et al. [65] and Cloherty et al. [66] observed in simulation studies that little cellular coupling is required for frequency entrainment of sinoatrial node cells. Frequency entrainment was also observed in our simulations of regional and distributed focal activity. However, we also observed that non-uniform uncoupling due to fibrosis and gap junction remodeling leads to lower basic cycle lengths of ectopic activity. We conclude that the combination of a distributed expression of f and uncoupling due to fibrosis and gap junction remodeling might lead to ectopic activity, contributing to arrhythmogenesis in diseases characterized by f overexpression such as heart failure, hypertrophy, and atrial fibrillation [18], [2], [21], [22]. From our simulation results we also conclude that an increased expression of f leads to an increased conduction velocity. Due to the presence of f, the diastolic brane potential rises, such that less current load is needed to reach the threshold. Especially in cases where few cells need to load a larger number of cells (current-to-load mismatch) this mechanism may prevent drastic slowing or even a total block of action potential propagation. We observed that the increase in conduction velocity is similar in normal, remodeled and fibrotic tissue. Compared to the conduction velocity in tissue without f we observed an increase of up to in remodeled and fibrotic tissue. Based on these findings, we propose that an increased expression of f in early stages of cellular uncoupling due to aging or pathology may be a mechanism to prevent conduction slowing and block. Michels et al. [18] hypothesize a therapeutic role for f blockers to modify pathological automaticity. Based on our simulation results, we conclude that blocking of f under pathological conditions might lead to slowing or block of action potential propagation.

9 9 time (ms) 112, FF, L T L T, FF, time (ms) L T L T Fig. 6. Capturing ectopic rhythm. An 8 cm 3 cm sheet of atrial tissue is represented by an irregular brickwall structure. The brane potential is shown for different simulation times. The two black lines indicate how the collision front moves to the right. After approximately 15 seconds of simulation, the wavefront generated by ectopic activity vanishes. Just excited tissue is colored red, depolarized tissue is yellow or green, and repolarized tissue is blue. D. Model validity and limitations In the present study, structural remodeling and active brane behavior ended with a hyperpolarization-activated inward current are brought together o one model. The aim of the model is to study action potential propagation and ectopic activity in normal and in structurally remodeled tissue. Although similar results would have been obtained by a monodomain formulation of the model, we think that, in order to model reduced racellular coupling caused by pathology, the erstitial space should explicitly be modeled as a possible conduction pathway. The validity of the model with respect to the cardiac tissue is discussed in Methods (see also Ref. [4]). Below, we discuss the validity of our model of f, the modified Na kinetics, the ionic concentrations, and the distribution of f expression. 1) f kinetics: We use large, non-physiological, values for the maximum brane conductance f to obtain significant diastolic depolarization and diastolic action potentials. f conductances of this size have not been observed experimentally [2], [21], [22], [23]. However, Michels et al. [18] recently observed that half-maximum activation of single-channel f is in fact within diastolic range of human atrial myocardium. These recent observations are most likely because cell dialysis was omitted in the new study [18]. The f channel availability in human atrial myocytes ( o in our model) reported by Michels et al. [18] in diastolic range is much larger than the measurements reported by DiFrancesco [43] on which o in our model is based. For example, f ns/pf in our model leads to a steady-state brane potential mv, a conductance f f o ns/pf and o (see Fig. 2). However, the channel availability reported by Michels et al. [18] would lead to for, giving a maximum conductance o ns/pf. Porciatti et al. [19] f o f reported a current density of pa/pf at mv for human atrial myocytes. For reversal potential f mv [19], the conductance f f ns/pf. f for mv [19], the maximum Since o conductance f equals f ns/pf in that case. Although the values we have used for f are somewhat larger than the experimentally obtained values, the f current size in diastolic range is similar to the recent findings by Michels et al. [18]. The major findings of our study, namely the facilitation of action potential propagation and possible ectopic activity in =,, =, =, 5, 5,, = =, 5,,,,

10 1 uncoupled tissue, also hold for physiological f when the f kinetics are based on these recent findings. 2) Na kinetics: To obtain diastolic action potentials for large values of f, we adapted the kinetics of Na such that the Na channels open faster when the brane depolarizes. In single cell simulations with the Courtemanche model, action potentials are generated by applying a stimulation current of 2 pa/pf for 2 ms [42]. When slowly depolarizing the brane by applying a small stimulation current for a longer period of time, a minimum current of.5 pa/pf is required to generate an action potential. When smaller stimulation currents are applied, the threshold for the opening of the Na channels is never reached due to the increasing voltage dependent K1 current. K1 reaches a maximum of approximately.5 pa/pf for brane potential ;= mv [42]. By adapting the Na kinetics as described in Methods, action potentials can be generated when applying a stimulation current above.4 pa/pf. The changes of the Na kinetics have no effect on the action potential morphology and duration when the usual stimulation current is applied. 3) Ionic concentrations: The ionic currents of the Courtemanche model eract with racellular Na, K and Ca concentrations [42]. We adapted the model to take the influence of f on Na i and K i o account as described by DiFrancesco and Noble [45]. However, the effect on ionic concentrations of the stimulus current and the racellular currents responsible for loading of the cells are not taken o account. This may lead to a drift in ionic balance when simulating a longer period of time [42], [67]. Indeed, we observed some deviation in the ionic balance after several seconds of simulation time (not shown). However, the drift was marginal and since the longest simulation run lasted only 15 seconds, we do not expect any noticeable influence on our simulation results. 4) Distribution of f expression: An important aspect of our model is the non-uniform expression of f. Porciatti et of human atrial cells. al. [19] observed expression of f in Hoppe and Beuckelmann found f in * to of human atrial myocytes [22]. They report a considerable variability of f size from cell to cell [22]. Sartiani et al. [28] detected f in about of cultured HL-1 cells (adult mouse atrial myocytes). Since little quantitative information is available on the distribution of f expression, we decided to model variability in the expression of f using a normal distribution of f. V. CONCLUSION In our model, an increased expression of f leads to a larger conduction velocity in normal, remodeled and fibrotic tissue. The nominal increase in conduction velocity is similar for all types of tissue. Compared to tissue without f the relative increase in conduction velocity is up to in remodeled and fibrotic tissue. Based on these results, we propose that an increased expression of f in early stages of cellular uncoupling, due to aging or pathology, may be a mechanism to facilitate action potential propagation. Hence, f may prevent conduction slowing and block. We also found that in tissue with a non-uniform expression of f, the basic cycle length of an ectopic pacemaker decreases due to cellular uncoupling. Thus, overexpression of f may lead to ectopic activity, especially in regions with reduced cellular coupling. APPENDIX NUMERICAL INTEGRATION SCHEME A numerical egration scheme to solve the equations (1) through (6) is obtained by roducing the connectivity matrices and and rewriting the equations as a combination of a system of differential equations and a system of linear equations. For a tissue represented by nodes, the matrices and are defined by &! &!!#" if # 5 " 8(: 5 " if otherwise!#" if # 5 " 8(: 5 " if otherwise (19) (2) Observe that the number of non-zero elements on each row of these matrices equals the number of adjacent nodes for the corresponding node plus one. Since this number is in general small compared to the total number of nodes,, and are sparse matrices. Also note that and are constructed such that the sum of all elements in each row equals zero, which makes them singular matrices. The matrix is defined such that for node it holds 4! 8!!, (21) Using (21) to rewrite equations (1) through (6) of the Cellular Bidomain Model we obtain the system of equations ion (22) (23) (24) where, and are vectors of length representing the ernal, ernal and brane potential of all nodes, and vector ion of length represents the ionic brane current of all nodes. Furthermore, and are diagonal matrices with elements and, respectively, on the diagonal. A numerical egration scheme for the equations (22), (23) and (24) is obtained as follows. Let vectors, and denote the brane, ernal and ernal potential of all nodes at time, where represents the simulation time step size. Vector denotes the ionic brane current ion of all nodes at time. ion is obtained from and ion using a modified Euler method as described in Ref. [42]. The values of the potentials on time can be computed by the following explicit numerical scheme: ion (25) (26)

11 4 Hence, first the brane potential on time, (27) is computed using a forward Euler step. N, the ernal potential on time is found by solving the system of linear equations (26). Finally, the new ernal potential is obtained by adding the new brane and ernal potentials. Since is a diagonal matrix, the computation of from Equation (25) requires only one matrix-vector multiplication with sparse matrix. Solving from Equation (26) is, however, more complicated. Due to the singularity of the matrix, no unique solution of this system of equations exists. However, it can be shown that if the tissue is connected, the various solutions of (26) differ only by a constant shift of all ernal potentials. To obtain a unique solution for Equation (26) we roduce one ra equation stating that on each time the sum of ernal potentials equals zero, i.e., where 8 (28) denotes the ernal potential of node at time. Initially, this ra requirement can be satisfied by choosing, for all nodes. The system of differential equations is solved using a forward Euler scheme with time step, ms. The system of linear equations is then solved with an iterative method. Let, and let 3 be the diagonal part of, with reversed sign. Hence, the elements of 3 are given by 3&. & 5 " & 5 ". Since the tissue is connected, each & 5 " 8(: node is connected to at least one other node, which implies that all elements 3) are positive. Hence, the matrix 3 is not singular. Equation (26) can now be reformulated as 3 3, (29) The solution of (29) can be approximated by iterating (3) where denotes the approximation of after iterations. This is in fact Jacobi s method for solving systems of linear equations. As a first approximation of at time, the value of at time is chosen, i.e.,, (31) The stop criterion is based on Equation (5), and defined by (32) for all nodes. The system of linear equations (26) is often solved by the Conjugate Gradient Method (CGM). Although CGM converges faster than Jacobi s method, one iteration of CGM requires more computational effort than one iteration of Jacobi s method. We now compare the computation time spent by our method to the computation time spent by the Conjugate Gradient Method. For both methods we used the solution of the former time step as a first approximation and Equation (32) as stop criterion. We measured the computation time spent by both methods for on ranging from 6 through a sample tissue of cm, cm with varying segment sizes and varying ernal conductances ( L T, and L T, ). As expected, the number of iterations per time step increases faster for Jacobi s method compared to CGM for decreasing. However, for $, the number of iterations used by Jacobi s method is smaller than the number of iterations used by CGM. Jacobi s method uses less computation time per time step for $ ". 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His main area of research is large-scale computer simulations with biomedical applications. His current research topic is modeling and simulation of cardiac electrophysiology and arrhythmia. In his former job at TNO Physics and Electronics Laboratory (TNO-FEL) he was the project leader for several collaborative projects in the areas of Collaborative Virtual Environments and Distributed Simulation, both nationally and ernationally with partners in the industry and academic world. He holds a M.Sc. in Computer Science and a Master of Technological Design in Software Technology, both from the Technische Universiteit Eindhoven. Rikkert Keldermann is a Ph.D. student in the Theoretical Biology group at Utrecht University. He received his B.Sc. and M.Sc. degrees in Biomedical Engineering from the Technische Universiteit Eindhoven in 21 and 24, respectively. His current research project involves modeling and organization of 3D ventricle fibrillation in the human heart. Huub ten Eikelder studied applied Mathematics at Delft University of Technology and received his M.Sc. in February From 1979 until 1983 he worked at the Theoretical Physics group of the Technische Universiteit Eindhoven, where he received his Ph.D. in From 1984 until 21 he worked at the Department of Mathematics and Computer Science of the Technische Universiteit Eindhoven. First he worked in the field of logic and foundations of programming languages. Later he moved to the field of optimization and neural networks. From September 21 he works as assistant professor in the Biomodeling and Bioinformatics group of the Department of Biomedical Engineering of the Technische Universiteit Eindhoven. His current research erest are mathematical modeling of biological systems and computational neuroscience. Theo Arts is part-time professor in modeling of biological structures at the Technische Universiteit Eindhoven and associate professor at the Department of Biophysics, Universiteit Maastricht. In 1978 he received his Ph.D. at Universiteit Maastricht. From he was research scientist at Cedars Sinai Medical Center, Los Angeles, USA. From he was part-time professor at Twente University. From he was the Maastricht coordinator in setting up the Biomedical Engineering School. His present research erests are cardiac mechanics, modeling of the self adapting heart and circulation, and clinical applicable techniques to measure deformation of the heart. Peter Hilbers received the B.Sc. and M.Sc. degrees in mathematics, and the Ph.D. degree in computer science from Groningen University, Groningen, the Netherlands in 1979, 1983, and 1989, respectively. From 1988 to 1989 he was an assistant professor with the Department of Computing Science at Groningen University. From 1989 till 1996 he was a research engineer in the the Department of Mathematics and Systems Engineering at the Koninklijke/Shell-Laboratorium, Amsterdam (Shell Research B.V.). In 1993 he became parttime professor at the Computer Science Department of the Technische Universiteit Eindhoven, where he also became a fulltime professor in In 21 he started a new group on Biomodeling and Bioinformatics in the Biomedical Engineering Department of the Technische Universiteit Eindhoven. His research erests include molecular modeling, parallel and distributed computing, bioinformatics, biomodeling and computer simulations of biomedical processes.