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Am J Physiol Heart Circ Physiol 292: H2832-H2853, 2007. First published February 2, 2007; doi:10.1152/ajpheart.00923.2006
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Mechanoelectric feedback leads to conduction slowing and block in acutely dilated atria: a modeling study of cardiac electromechanics

Nico H. L. Kuijpers,1 Huub M. M. ten Eikelder,1 Peter H. M. Bovendeerd,1 Sander Verheule,2 Theo Arts,3 and Peter A. J. Hilbers1,4

Departments of 1Biomedical Engineering and 2Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven; and Departments of 3Physiology and 4Biophysics, Maastricht University, Maastricht, The Netherlands

Submitted 25 August 2006 ; accepted in final form 25 January 2007


    ABSTRACT
 TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
Atrial fibrillation, a common cardiac arrhythmia, is promoted by atrial dilatation. Acute atrial dilatation may play a role in atrial arrhythmogenesis through mechanoelectric feedback. In experimental studies, conduction slowing and block have been observed in acutely dilated atria. In the present study, the influence of the stretch-activated current (Isac) on impulse propagation is investigated by means of computer simulations. Homogeneous and inhomogeneous atrial tissues are modeled by cardiac fibers composed of segments that are electrically and mechanically coupled. Active force is related to free Ca2+ concentration and sarcomere length. Simulations of homogeneous and inhomogeneous cardiac fibers have been performed to quantify the relation between conduction velocity and Isac under stretch. In our model, conduction slowing and block are related to the amount of stretch and are enhanced by contraction of early-activated segments. Conduction block can be unidirectional in an inhomogeneous fiber and is promoted by a shorter stimulation interval. Slowing of conduction is explained by inactivation of Na+ channels and a lower maximum upstroke velocity due to a depolarized resting membrane potential. Conduction block at shorter stimulation intervals is explained by a longer effective refractory period under stretch. Our observations are in agreement with experimental results and explain the large differences in intra-atrial conduction, as well as the increased inducibility of atrial fibrillation in acutely dilated atria.

atrial fibrillation; excitation-contraction coupling; impulse propagation; stretch-activated current


ATRIAL FIBRILLATION (AF) is a common cardiac arrhythmia (33). An important risk factor for AF is chronic atrial dilatation (38, 56), whereas experimental studies indicate a role of acute atrial dilatation in the initiation of atrial arrhythmia (2, 34, 39, 48, 50). Conduction slowing and shortening of the refractory period in acutely dilated atria have been reported (6, 16, 40). Eijsbouts et al. (10, 11) found, in addition to conduction slowing, an increased occurrence of intra-atrial block. Hu and Sachs (15) and Kohl and Sachs (28) hypothesize that stretch-induced changes in electrophysiological behavior can be explained by stretch-activated channels (SACs). In the present simulation study, we investigate this hypothesis for atrial impulse propagation.

Several models have been proposed to describe SACs based on experimental observations (12, 45, 52, 58, 59). Similar models have been applied in large-scale computer simulations to investigate the effect of stretch on defibrillation (53) and the termination of ventricular tachycardia by means of precordial thump (31). Since cardiomechanics are not considered in these studies, the stretch-activated current (Isac) is not influenced by contraction. Models of the ventricles in which contraction is triggered by electrical activation describe stimulation from the Purkinje system (24, 55) and epicardial stimulation (25, 54). In these studies, mechanical deformation is triggered by electrical activation. However, mechanoelectric feedback, i.e., the effect of mechanical deformation on the electrophysiology, is not considered. To investigate the influence of mechanical deformation on impulse propagation, a strong coupling between cardiomechanics and electrophysiology is required, as proposed elsewhere (32, 35, 36). In these studies, tissue conductivity is directly affected by mechanical deformation, and the amount of Isac is related to local deformation of the cardiac tissue. Physiological details, such as ionic membrane currents, intracellular Ca2+ handling, and cross-bridge formation, are not considered in these models.

In the present study, we investigate the role of Isac in conduction slowing and block as observed in acutely dilated atria. We apply a discrete bidomain model with strong coupling between cardiomechanics and cardiac electrophysiology. Our model describes ionic membrane currents, Ca2+ storage and release from the sarcoplasmic reticulum (SR), and cross-bridge formation. In contrast to all other multicellular models, contractile forces are directly coupled to free Ca2+ concentration, as well as sarcomere length. In our model, the amount of Isac is related to local stretch and may change during contraction. We performed simulations of homogeneous and inhomogeneous cardiac fibers under stretch to quantify the conduction velocity in the presence of Isac. We observed conduction slowing, a longer effective refractory period (ERP), and (unidirectional) conduction block with increasing stretch. Furthermore, we found that contraction of early-activated fiber segments can lead to conduction block in later-activated segments. The observed phenomena are in agreement with experimental observations and provide an explanation for the increased inducibility of AF in acutely dilated atria.


    METHODS
 TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
In the present study, we apply our discrete bidomain model, the cellular bidomain model (29, 30), which describes active membrane behavior, as well as intercellular coupling and interstitial currents, and has been extended to model cardiac tissue mechanics and Isac. We describe the extensions to our model of cardiac electrophysiology, in particular the influence of stretch on fiber conductivity, our model of the Isac, the Ca2+-force relation, the mechanical behavior of a single segment, and the mechanical behavior of a cardiac fiber. Furthermore, the numerical integration scheme is described, and an overview of the simulations is given.

Modeling Cardiac Electrophysiology

In the cellular bidomain model, the cardiac tissue is subdivided into segments, each with its own membrane model describing the ionic membrane currents (29, 30). The state of each segment is defined by the intracellular potential (Vint), the extracellular potential (Vext), and the state of the cell membrane, which is expressed in gating variables and ion concentrations. The membrane potential (Vmem) is defined by

Formula 1(1)
Intracellular and extracellular currents between adjacent segments are related to intracellular and extracellular conductivities and satisfy Ohm's law. Exchange of current between the intracellular and extracellular space occurs as transmembrane current (Itrans)

Formula 2(2)
where {chi} is the surface-to-volume ratio needed to convert Itrans per unit membrane surface to Itrans per unit tissue volume, Cmem is the membrane capacitance, which is typically 1 µF/cm2 for biological membranes (1), and Iion is ionic current (expressed in µA/cm2 membrane surface or pA/pF when Cmem = 1 µF/cm2). Iion depends on Vmem, gating variables, and ion concentrations (see below). Impulse propagation is related to the longitudinal conductivity parameters gint and gext. The bidomain parameters used for the present study are from Henriquez (13) and are based on measurements by Clerc (7) (Table 1).


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Table 1. Model parameters

 
To incorporate Isac, we modified the model of the human atrial action potential (AP) of Courtemanche et al. (8). The total ionic current is given by

Formula 3(3)
where INa is fast inward Na+ current, IK1 is inward rectifier K+ current, Ito is transient outward K+ current, IKur is ultrarapid delayed rectifier K+ current, IKr is rapid delayed rectifier K+ current, IKs is slow delayed rectifier K+ current, ICa,L is L-type Ca2+ current, Ip,Ca is Ca2+ pump current, INaK is Na+-K+ pump current, INaCa is Na+/Ca2+ exchanger current, and Ib,Na and Ib,Ca are background Na+ and Ca2+ currents (8). The model for the ionic and pump currents, including handling of intracellular Ca2+ concentration ([Ca2+]i) by the SR, is adopted from the model of Courtemanche et al. (8).

Influence of Stretch on Fiber Conductivity

The intracellular and extracellular conductivities (gint and gext) may change during stretch or contraction of the fiber. Under stretch, the length of the cells increases and the cross-sectional area decreases, leading to a reduced fiber conductivity. To quantify the changes in gint and gext (mS/cm), we assume that the resistivity of the intracellular space (Rint = 1/gint, {Omega}·cm) is determined partly by the myoplasmic resistivity (Rmyo) and partly by the gap-junctional resistivity (Rjunc)

Formula 4(4)
For the nonstretched fiber, we define gint = gint 0 = 1/Rint 0, Rmyo = Rmyo 0, and Rjunc = Rjunc 0. When the fiber is stretched with stretch ratio {lambda}, cell length increases and cross-sectional area decreases (assuming that cell volume is conserved). Since Rmyo is proportional to the length and inversely proportional to the cross-sectional area of the cell, we obtain

Formula 5(5)
On the basis of the assumption that the total number of gap junctions in the fiber does not change under stretch, the number of gap junctions per length unit decreases proportionally with {lambda}, which leads to

Formula 6(6)
If Eqs. 5 and 6 are combined, gint is related to {lambda} by

Formula 7(7)
For the extracellular domain, we assume that gext is related to gext 0 and {lambda} by

Formula 8(8)
Chapman and Fry (5) determined that 52% of the total resistivity was attributed to gap-junctional resistance in frog myocardial cells (Rjunc 0/Rint 0 = 0.52). Since these cells are longer (131 µm) (5) than human atrial cells (94 µm) (37), we estimate that Rjunc 0/Rint 0 = 0.6 for human atrial myocardium (Table 1).

Isac

On the basis of experimental observations, we assume that Isac in atrial myocytes is a nonselective cation current with a near-linear current-voltage relation (26). The reversal potential is –3.2 mV for rat atrial myocytes (26). In our model, Isac is permeable to Na+, K+, and Ca2+ and is defined by

Formula 9(9)
where Isac,Na, Isac,K, and Isac,Ca represent the Na+, K+, and Ca2+ contributions, respectively, to Isac. These currents are defined by the constant-field Goldman-Hodgkin-Katz current equation (22)

Formula 10(10)

Formula 11(11)

Formula 12(12)
where PNa, PK, and PCa denote the relative permeabilities to Na+, K+, and Ca2+, zNa, zK, and zCa represent the ion valences, and F is Faraday's constant, R is the universal gas constant, and T is temperature (310 K) (8).

The conductance (gsac) depends on {lambda} as follows

Formula 13(13)
where Gsac is the maximum membrane conductance, Ksac is a parameter to define the amount of current when the cell is not stretched [{lambda} = 1, sarcomere length (ls) = 1.78 µm], and {alpha}sac is a parameter to describe the sensitivity to stretch. Ksac and {alpha}sac are from Zabel et al. (58) (Table 1).

The reversal potential (Esac) can be obtained by solving the following equation for Vmem: Isac,Na + Isac,K + Isac,Ca = 0. In the present study, we consider two cases: PNa:PK:PCa = 1:1:1, with Esac = –0.2 mV, and PNa:PK:PCa = 1:1:0, with Esac = –0.9 mV. In both cases, Isac has a near-linear current-voltage relation (Fig. 1).


Figure 1
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Fig. 1. Current-voltage relation for stretch-activated current (Isac) and its Na+, K+, and Ca2+ components (Isac,Na, Isac,K, and Isac,Ca); stretch ratio ({lambda}) = 1.2. Top: Isac permeable to Ca2+. Bottom: Isac not permeable to Ca2+. Gsac, maximum membrane conductance; PNa, PK, and PCa, Na+, K+, and Ca2+ permeability; Vmem, membrane potential.

 
To describe the influence of Isac,Na, Isac,K, and Isac,Ca on intracellular Na+, K+, and Ca2+ concentrations ([Na+]i, [K+]i and [Ca2+]i), respectively, we replace Eqs. 2125 of the model of Courtemanche et al. (8) by

Formula 14(14)

Formula 15(15)

Formula 16(16)

Formula 17(17)

Formula 18(18)
where Cm is the membrane capacitance of a single atrial myocyte (100 pF) (8), F is Faraday's constant, Vi is the intracellular volume (13,668 µm3) (8), Vup and Vrel are the volumes of the SR uptake and release compartments, respectively, Iup,leak, Iup, and Irel represent the SR currents, [Trpn] is troponin concentration, [Cmdn] is calmodulin concentration, and Km is the half-saturation constant. Equation 18 is Eq. 25 in the model of Courtemanche et al. (8) and represents the influence of Ca2+ buffering in the cytoplasm mediated by troponin ([Ca2+]Trpn) and calmodulin ([Ca2+]Cmdn) on [Ca2+]i.

Modeling the Ca2+-Force Relation

Rice et al. (4244) proposed five models of isometric force generation in cardiac myofilaments. To model the Ca2+-force relation in the present study, we apply their model 4, which is based on a functional unit of troponin, tropomyosin, and actin. The binding of Ca2+ to troponin is described by two states: unbound troponin and Ca2+ bound to troponin. Tropomyosin can be in one of six states: nonpermissive with 0 and 1 cross bridges (N0 and N1) and permissive with 0, 1, 2, and 3 cross bridges (P0, P1, P2, and P3). The permissive states refer to tropomyosin for which the accompanying actin binding sites are available for cross bridges to bind and generate force. Transitions between the states are governed by rate functions that depend on [Ca2+]i and ls.

In the model of Courtemanche et al. (8), Ca2+ buffering by troponin is modeled by

Formula 19(19)
where [Ca2+]Trpn is Ca2+-bound troponin concentration, [Trpn]max is total troponin concentration (70 µM) (8), and Km,Trpn is half-saturation constant for troponin (0.5 µM) (8). In the model of Rice et al. (43, 44), the concentration of Ca2+ bound to high-affinity troponin sites is [HTRPNCa] and the dynamics are governed by

Formula 20(20)
where [HTRPN]tot represents the total troponin high-affinity site concentration and khtrpn+ and khtrpn are the Ca2+ on- and off-rates for troponin high-affinity sites (Table 1). The concentration of Ca2+ bound to low-affinity troponin sites is [LTRPNCa], and the dynamics are governed by

Formula 21(21)
where [LTRPN]tot represents the total troponin low-affinity site concentration and kltrpn+ and kltrpn are the Ca2+ on- and off-rates for troponin low-affinity sites (Table 1).

In our model, the Ca2+ transient is computed by the model of Courtemanche et al. (8) using an immediate formulation of Ca2+ binding by troponin (Eq. 19). The resulting Ca2+ transient is used to compute Ca2+ binding to troponin by Eqs. 20 and 21, and [LTRPNCa] is used to compute the tropomyosin rate from nonpermissive to permissive, as in the model of Rice et al. (43, 44). In the present study, we do not consider a feedback mechanism that influences the Ca2+ transient through a change in the affinity of troponin for Ca2+ binding as in model 5 (43, 44). The choice between model 4 and model 5 is motivated in the DISCUSSION.

In model 4, the force generated by the sarcomeres depends on the fraction of tropomyosin in the force-generating states N1, P1, P2, and P3. We use the normalized force (Fnorm), which is defined by

Formula 22(22)
where P1max, P2max, and P3max are defined as in Ref. 44 and {phi}(ls) describes the physical overlap structure of thick and thin filaments within a sarcomere (44). When {phi}(ls) = 1, all myosin heads are able to interact with actin in the single overlap zone; when {phi}(ls) < 1, some of the filaments are in the double or nonoverlap zones. {phi}(ls) is defined by

Formula 23(23)

In Fig. 2, the steady-state Ca2+-force relation is presented for model 4 (44). Fnorm increases with increasing [Ca2+]i and with increasing ls, with a maximum at ls = 2.3 µm. To emphasize the dependence on [Ca2+]i and ls, we will denote Fnorm as a function: Fnorm([Ca2+]i,ls).


Figure 2
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Fig. 2. Steady-state Ca2+-force relation for model 4 from Rice et al. (44). Top: normalized force (Fnorm) vs. intracellular Ca2+ concentration ([Ca2+]i) for sarcomere length (ls) = 1.7–2.3 µm. Bottom: Fnorm vs. ls for [Ca2+]i = 0.3, 0.6, 0.9, and 1.2 µM.

 
Mechanical Behavior of a Single Segment

The mechanical behavior of a single segment in our model is modeled as described by Solovyova et al. (49) by the classical three-element rheological scheme introduced by Hill in 1938 (14). Active force is generated by the contractile element (CE), and passive forces are generated in a series elastic element (SE) and a parallel elastic element (PE; Fig. 3). PE describes the force-length relation when the segment is not stimulated. CE and SE together describe the additional force generated on stimulation of the segment. The element lengths are lCE, lSE, and lPE. The reference lengths, i.e., the lengths at which the segment is at rest and no force is applied, are lCE 0, lSE 0, and lPE 0.


Figure 3
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Fig. 3. Three-element scheme to model mechanical behavior of a single cell. Active force (FCE) is generated by the contractile element (CE), and passive forces (FSE and FPE) are generated in the series elastic element (SE) and the parallel elastic element (PE). ls = lCE denotes sarcomere length, and lSE and lPE are SE and PE lengths. During mechanical equilibrium, FCE = FSE, Fsegment = FSE + FPE, and lPE = lCE + lSE.

 
The force generated by the contractile element (FCE) is defined by

Formula 24(24)
where fCE is a scaling factor, {nu} = –(dls/dt) represents the sarcomere shortening velocity, and Fnorm([Ca2+]i,ls) is Fnorm generated by the sarcomeres. The relation between the generated force and v is Hill's force-velocity relation (14, 17) and appears to be hyperbolic for skeletal and cardiac muscle (3, 9). We model the Hill relation by a function f{nu}({nu}) as proposed by Hunter et al. (17)

Formula 25(25)
where {nu}max is the maximum sarcomere shortening velocity and c{nu} is a constant describing the shape of the hyperbole.

The forces generated in the SE and PE are nonlinearly dependent on their respective lengths lSE and lPE (49) and are defined by

Formula 26(26)
and

Formula 27(27)
where lSE 0 and lPE 0 denote the reference element lengths and fSE, kSE, fPE, and kPE are material constants describing the elasticity of the elements.

From mechanical equilibrium, it follows that FCE must be equal to the force generated in the SE (FSE). The total force generated by the segment (Fsegment) is defined as FSE + FPE. Furthermore, lPE must be equal to the lCE + lSE (Fig. 3). Therefore, during mechanical equilibrium

Formula 28(28)

Formula 29(29)

Formula 30(30)
lCE, lSE, and lPE are related to physiological sarcomere length (ls) and reference sarcomere length (ls 0) by lCE = ls and lCE 0 = ls 0 (49). The reference length of a segment is 0.01 cm and is related to lPE,0 by a scaling factor {xi}. For segment n, we define the reference length ln 0 by

Formula 31(31)
and the actual length ln by

Formula 32(32)
where lPE 0n and lPEn represent the reference length and the actual length of the PE of segment n. The {lambda} for segment n ({lambda}n) is then defined by

Formula 33(33)

The parameters for the three-element mechanical model are obtained from Solovyova et al. (49) (Table 1). In Fig. 4, active force (FSE), passive force (FPE), and total force (Fsegment) are presented for ls = 1.7–2.5 µm and [Ca2+]i = 1.2, 0.9, 0.6, and 0.3 µM. When the sarcomeres generate force, i.e., lSE > 0, lPE = lCE + lSE is larger than ls = lCE. This results in a steeper increase of FPE for increasing ls and is in agreement with the passive force-length relation for intact cardiac muscle measured by Kentish et al. (23).


Figure 4
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Fig. 4. Active force (FSE), passive force (FPE), and total force (Fsegment = FSE + FPE) vs. ls for [Ca2+]i = 1.2, 0.9, 0.6, and 0.3 µM.

 
Mechanical Behavior of a Cardiac Fiber

A cardiac fiber is modeled as a string of segments that are coupled in series. From mechanical equilibrium, it follows that the force Fsegmentn generated by a single segment n, n isin N, is equal to the force generated by the fiber (Ffiber)

Formula 34(34)
If we take into account that ln 0 may be different for each segment n, n isin N, the stretch ratio of the fiber ({lambda}fiber) is defined by

Formula 35(35)
where L denotes the actual fiber length and L0 is the reference length.

In the present study, inhomogeneous cardiac tissue is represented by a 5-cm-long fiber with varying thickness and stiffness. The fiber is composed of 0.01-cm-long segments with 0.01- to 0.1-mm2 cross-sectional area. Tissue conductivity is related to stretch and may vary during the simulation. To enforce nonuniform stretch during the simulations, the diameter and stiffness of the left half of the fiber are varied, while the diameter (0.01 mm2) and stiffness of the right half are normal. Linear interpolation is applied in a 0.5-cm transitional zone in the center of the fiber. Thick tissue is modeled by increasing the diameter of the segments, which affects the electrophysiological and mechanical properties of the tissue. Conductances, membrane surface, and the mechanical parameters fCE, fSE, and fPE are scaled with the increase of the cross-sectional area. To simulate stiff tissue, the mechanical parameter fPE is scaled. Scaling factors for maximum and minimum thickness are denoted by tmax and tmin, respectively, and scaling factors for maximum and minimum stiffness by smax and smin, respectively.

Numerical Integration Scheme

To obtain criteria for the size of individual segments, we apply cable theory and consider subthreshold behavior along a fiber as previously described (30). For the bidomain parameters in Table 1, we obtain a length constant between 0.12 and 0.16 cm for {lambda} = 1.0. When {lambda} is increased to 1.4, the length constant decreases ~15% for Rjunc 0/Rint 0 = 0.6. To obtain accurate simulation results, the fiber is modeled with segments that are 0.01 cm long, which is less than one-tenth of the length constant for {lambda} ≤ 1.4. To solve the equations of the cellular bidomain model, we use a forward Euler scheme with a 0.01-ms time step to compute Vmem and an iterative method to solve the system of linear equations as described in Kuijpers et al. (30). Our method does not require matrix inversions and, therefore, is well suited to solve the system of equations when the conductivities change during the simulations as a result of stretch or contraction.

The ionic membrane currents are computed using a modified Euler method as described by Courtemanche et al. (8). To reduce computation time, the time step changes during the simulation as follows: a 0.01-ms time step is used shortly before and during the upstroke of the AP, and a 0.1-ms time step is used during repolarization and rest. The Ca2+-force relation is computed using a forward Euler method with a fixed 0.1-ms time step, which is also the time step used to compute the cardiac mechanics (see APPENDIX). Local conductivities are adjusted to the local {lambda} whenever the mechanical state is updated.

To compare 0.1-ms (see above) with 0.01-ms time steps, we performed two simulations with the same parameter settings, but with different time steps. The differences in conduction velocity ({theta}), membrane currents, ionic concentrations, and mechanical forces were negligible, but computation time was reduced by 75%.

Simulation Protocol

To illustrate the excitation-contraction coupling in our model, we performed single-cell simulations with constant ls (isosarcometric contraction) and single-cell simulations with constant applied force (isotonic contraction). The influence of Isac on the AP was investigated by application of a constant stretch to a single cell (isometric simulation). The cell was electrically stimulated with a frequency of 1 Hz. For investigation of spontaneous activity under stretch, simulations were performed with increasing stretch, but without electrical stimulation.

The influence of stretch on {theta} was investigated by stimulating the first segment of a 1-cm fiber. The fiber was short, such that contraction of early-activated segments did not affect impulse propagation in later-activated areas. The {lambda}fiber was kept constant during the simulation (isometric fiber contraction). We used longer (5 cm) fibers to investigate the influence of contraction on impulse propagation. Simulations were performed with contraction enabled and with contraction disabled. Disabled contraction was implemented by assuming that [Ca2+]i was equal to its resting value of 0.102 µM (8) when Fnorm([Ca2+]i,ls) was computed. Thickness and stiffness were varied to simulate inhomogeneous cardiac tissue.

All simulations were performed over a 12-s period. Electrical stimulation was performed each 1 s (1 Hz) or each 0.5 s (2 Hz) by application of a stimulus current. In the case of single-cell simulations, a stimulus current of 20 pA/pF was applied for 2 ms as described by Courtemanche et al. (8). In the case of fiber simulations, the leftmost or the rightmost segment was stimulated by application of a stimulus current of 100 pA/pF until the membrane was depolarized. For the 1-cm fiber, the overall {theta} was measured by determination of the moment of excitation of two segments located 1 mm from each of the fiber ends. For the 5-cm (inhomogeneous) fiber, local {theta} was computed for each segment using the excitation time between two segments located 0.5 mm to the left and to the right in the nonstretched fiber.


    RESULTS
 TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
Isosarcometric Contraction

Figure 5 illustrates the relation between the electrophysiology described by the model of Courtemanche et al. (8) and the force-producing states described by the model of Rice et al. (4244). For the Vmem trace in Fig. 5, AP duration (APD) at 50% repolarization (APD50) and APD at 90% repolarization (APD90) are 184 and 304 ms, respectively. The AP amplitude and AP overshoot are 107 and 28 mV, respectively, and the maximum upstroke velocity [(dVmem/dt)max] is 187 V/s. For the Ca2+ transient, resting [Ca2+]i is 0.11 µM, peak [Ca2+]i is 0.87 µM, and time required to return [Ca2+]i to one-half of maximum [Ca2+]i is 178 ms. Since the dynamics of the concentration of Ca2+ bound to low-affinity troponin sites ([LTRPNCa]) are governed by a differential equation (21), the trace of [LTRPNCa] is smooth compared with that of [Ca2+]i.


Figure 5
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Fig. 5. Vmem, [Ca2+]i, concentration of Ca2+ bound to low-affinity troponin sites [LTRPNCa], and fraction of functional units in force-producing state P1, N1, P2, or P3 for ls = 2.3 µm. A stimulus current was applied at 100 ms. Traces from 12th action potential (AP) are shown for stimulation at 1 Hz. Isac was disabled.

 
From the traces of Fnorm and individually normalized traces of Fnorm for ls = 1.7–2.3 µm in Fig. 6, it can be observed that peak force, time to peak force, and relaxation time increase with increasing ls, which is consistent with the experimental data measured by Janssen and Hunter (18).


Figure 6
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Fig. 6. Fnorm for ls = 1.7–2.3 µm (isosarcometric contraction). Top: Fnorm. Bottom: Fnorm individually normalized to maximum Fnorm. A stimulus current was applied at 100 ms. Traces from 12th contraction are shown for stimulation at 1 Hz. Isac was disabled.

 
Isotonic Contraction

In Fig. 7, traces of Fnorm, FCE, ls, and lPE are presented for simulations of isotonic contraction with applied force (Fsegment) of 5–250 mN/mm2. The AP and Ca2+ transient are the same as in Fig. 5. Less time is required to return Fnorm to its resting value than in the case of isosarcometric contraction (Fig. 6, top). This is explained by shortening of the sarcomeres during contraction: a shorter sarcomere yields a lower contractile force (Fig. 2, bottom). The FCE traces exhibit a plateau phase for Fsegment ≤ 25 mN/mm2. For Fsegment = 250 mN/mm2, lPE remains constant, indicating no shortening.


Figure 7
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Fig. 7. Fnorm, FCE, ls and segment length (lPE) for isotonic contractions with applied force (Fsegment) = 5–250 mN/mm2. A stimulus current was applied at 100 ms. Traces from 12th contraction are shown for stimulation at 1 Hz. Isac was disabled.

 
Effect of Isac on AP

Figure 8 illustrates the effect of Isac on the AP. Vmem, Isac, ICa,L, and [Ca2+]i are presented for Isac permeable to Ca2+ and Isac not permeable to Ca2+ for {lambda} = 1.00, 1.10, and 1.20. The cell was stimulated with a frequency of 1 Hz. With increasing {lambda}, repolarization is prolonged and the resting Vmem is depolarized. Isac is small during the plateau phase and larger during repolarization and rest, which is consistent with a reversal potential between 0 and –1 mV. ICa,L is somewhat lowered under stretch, and the Ca2+ transient is increased. The lowered ICa,L is explained by the Ca2+-dependent inactivation of ICa,L (8). Interestingly, whether Isac is permeable or not permeable to Ca2+, the Ca2+ transient increases with increasing stretch. The characteristics for the AP and the Ca2+ transient are presented in Table 2 for Isac permeable to Ca2+ and in Table 3 for Isac not permeable to Ca2+. For Isac permeable to Ca2+, peak [Ca2+]i and the time required for return of [Ca2+]i to one-half of maximum [Ca2+]i is increased ~3%.


Figure 8
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Fig. 8. Vmem, Isac, L-type inward Ca2+ current (ICa,L), and [Ca2+]i for stretch applied to single cells at stretch ratio ({lambda}) = 1.00, 1.10, and 1.20. Left: Isac permeable to Ca2+ (Gsac = 0.015 µm/s). Right: Isac not permeable to Ca2+ (Gsac = 0.010 µm/s). A stimulus current was applied at 100 ms.

 

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Table 2. AP characteristics when Isac is permeable to Ca2+ (Gsac = 0.015 µm/s)

 

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Table 3. AP characteristics when Isac is not permeable to Ca2+ (Gsac = 0.010 µm/s)

 
Stretch-Induced APs

Figure 9 illustrates the effect of increasing {lambda} in the presence of Isac. Vmem, Isac, ICa,L, and [Ca2+]i are presented for {lambda} linearly increasing from 1.00 at 0-ms simulation to 1.25, 1.35, and 1.45 at 200-ms simulation; {lambda} is constant after 200 ms. In both cases, stretch-induced APs are elicited for {lambda} = 1.35 and 1.45. The APs for {lambda} = 1.35 have a low upstroke steepness and are mainly driven by ICa,L. ICa,L increases faster for {lambda} = 1.35 when Isac is permeable to Ca2+, which explains why Vmem reaches its maximum 50 ms earlier than when Isac is not permeable to Ca2+. For Isac permeable to Ca2+ and for Isac not permeable to Ca2+, the sarcoplasmic Ca2+ flux signal for the Ca2+ release current (Irel) is too small to trigger Ca2+ release from the SR. This explains why no Ca2+ transients are observed for {lambda} = 1.35.


Figure 9
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Fig. 9. Vmem, Isac, ICa,L, and [Ca2+]i for stretch applied to single cells. {lambda} was increased from 1.00 at 0-ms stimulation to 1.25, 1.35, and 1.45 at 200 ms; {lambda} was constant after 200 ms. Left: Isac permeable to Ca2+ (Gsac = 0.015 µm/s). Right: Isac not permeable to Ca2+ (Gsac = 0.010 µm/s). No stimulus current was applied.

 
Effect of Rjunc 0/Rint 0 on {theta}

To investigate the influence of Rjunc 0/Rint 0 on {theta}, we simulated impulse propagation along a 1-cm fiber for various Rjunc 0/Rint 0 and {lambda}fiber. Isac was disabled in these simulations (Gsac = 0.0 µm/s). In Fig. 10, the overall {theta} is presented for Rjunc 0/Rint 0 = 0.0–1.0, and {lambda}fiber = 1.0–1.4. For Rjunc 0/Rint 0 = 1.0, {theta} is little affected by increasing {lambda}fiber; for lower values of Rjunc 0/Rint 0, {theta} decreases with increasing {lambda}fiber. The decrease in {theta} is almost linear for Rjunc 0/Rint 0 = 0.4–0.8.


Figure 10
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Fig. 10. Overall conduction velocity ({theta}) for impulse propagation along a 1-cm fiber for relative gap-junctional resistivity (Rjunc 0/Rint 0) = 0.0–1.0. Impulse propagation was initiated by application of a stimulus current to the 1st segment. Isac was disabled.

 
Isometric Fiber Contraction

To investigate the influence of Isac on impulse propagation, we simulated a series of isometric contractions in a 1-cm-long fiber. Gsac and {lambda}fiber were varied (Gsac = 0.0–0.020 µm/s, {lambda}fiber = 1.0–1.4). The leftmost segment was electrically stimulated with a frequency of 1 Hz. In Fig. 11, {theta} and (dVmem/dt)max are presented for various Gsac and {lambda}fiber. The decrease of {theta} with increasing Gsac is accompanied by a decrease of (dVmem/dt)max. When Isac was permeable to Ca2+, block of impulse propagation occurred for {lambda}fiber ≥ 1.35 for Gsac = 0.010 µm/s, {lambda}fiber ≥ 1.25 for Gsac = 0.015 µm/s, and {lambda}fiber ≥ 1.15 for Gsac = 0.020 µm/s. When Isac was not permeable to Ca2+, block of impulse propagation occurred for {lambda}fiber ≥ 1.25 for Gsac = 0.010 µm/s, {lambda}fiber ≥ 1.10 for Gsac = 0.015 µm/s, and {lambda}fiber ≥ 1.05 for Gsac = 0.020 µm/s.


Figure 11
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Fig. 11. {theta} and maximum upstroke velocity [(dVmem/dt)max] for impulse propagation along a 1-cm fiber (Rjunc 0/Rint 0 = 0.6). Gsac and {lambda}fiber were varied. Left: Isac permeable to Ca2+. Right: Isac not permeable to Ca2+. Impulse propagation was initiated by application of a stimulus current to the 1st segment.

 
In Fig. 12, the APs and traces of Isac, ICa,L, and [Ca2+]i are presented for the center segment ({lambda}fiber = 1.00, 1.10, and 1.20). Traces of Isac,Na, Isac,K, and Isac,Ca are shown in Fig. 13. As expected, Isac increases with increasing {lambda} during repolarization and rest. As in the single-cell simulations (Fig. 8), ICa,L decreases and [Ca2+]i increases with increasing {lambda}. Since depolarization is mainly driven by INa, we further examine INa to investigate the cause of the decrease in {theta} and (dVmem/dt)max. The ionic current INa is defined by

Formula 36(36)
where GNa is the maximum INa conductance, ENa is the equilibrium potential for Na+, m is the fast activation variable, and h and j are the fast and slow inactivation variables (8). In Fig. 14, traces of m, h, j, and INa are presented for the center segment ({lambda}fiber = 1.00, 1.10, and 1.20). As {lambda}fiber increases, h and j are lower during rest and explain the lower INa during depolarization; i.e., the membrane is less excitable. Except for Ito and IKur, all ionic currents were similar during the upstroke and shortly after the upstroke (not shown). Ito and IKur were smaller and caused the less prominent notch of the AP (Fig. 12).


Figure 12
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Fig. 12. Vmem, Isac, ICa,L, and [Ca2+]i for the center segment of a 1-cm fiber ({lambda}fiber = 1.00, 1.10, and 1.20). Left: Isac permeable to Ca2+ (Gsac = 0.015 µm/s). Right: Isac not permeable to Ca2+ (Gsac = 0.010 µm/s). A stimulus current was applied to the 1st segment at 100 ms.

 

Figure 13
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Fig. 13. Isac,Na, Isac,K, and Isac,Ca for the center segment of a 1-cm fiber ({lambda}fiber = 1.00, 1.10, and 1.20). Left: Isac permeable to Ca2+ (Gsac = 0.015 µm/s). Right: Isac not permeable to Ca2+ (Gsac = 0.010 µm/s). A stimulus current was applied to the 1st segment at 100 ms.

 

Figure 14
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Fig. 14. Activation gating variable (m), fast inactivation gating variable (h), and slow inactivation gating variable (j) for the fast inward Na+ current (INa) of the center segment of a 1-cm fiber ({lambda}fiber = 1.00, 1.10, and 1.20). Left: Isac permeable to Ca2+ (Gsac = 0.015 µm/s). Right: Isac not permeable to Ca2+ (Gsac = 0.010 µm/s). A stimulus current was applied to the 1st segment at 100 ms. Note different time scale for INa.

 
Impulse Propagation Along a Homogeneous Fiber

To investigate the effect of contraction of early-activated areas on {theta} in later-activated areas, we simulated impulse propagation along a 5-cm-long fiber with {lambda}fiber = 1.00, 1.05, 1.10, 1.15, and 1.20 (Gsac = 0.015 µm/s, PNa:PK:PCa = 1:1:1). All simulations were performed with contraction enabled as well as with contraction disabled. Impulse propagation was initiated by application of a stimulus current to the leftmost segment.

In Fig. 15, traces of Vmem, Isac, [Ca2+]i, and {lambda} are presented for segments located 1.0, 2.5, and 4.0 cm from the stimulation site ({lambda}fiber = 1.15). With contraction enabled, the early-activated segments start contracting, so that {lambda} increases for the later-activated segments, which results in an increased Isac and a depolarized resting Vmem. In Fig. 16, {theta} and (dVmem/dt)max are presented with contraction disabled (tmax = 1, smax = 1) and contraction enabled (tmax = 1, smax = 1).


Figure 15
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Fig. 15. Vmem, Isac, [Ca2+]i, and {lambda} for 1.0-, 2.5-, and 4.0-cm segments of a homogeneous fiber with contraction enabled (left) and contraction disabled (right). Fiber length was 5 cm, Isac was permeable to Ca2+ (Gsac = 0.015 µm/s), and {lambda}fiber = 1.15. Impulse propagation was initiated by application of a stimulus current to the leftmost segment at 0 cm.

 

Figure 16
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Fig. 16. Left stimulation: {theta} and (dVmem/dt)max for impulse propagation along a homogeneous fiber and various inhomogeneous fibers with contraction disabled (top) and enabled (bottom). Isac was permeable to Ca2+ (Gsac = 0.015 µm/s). Fiber length = 5 cm, {lambda}fiber = 1.15. Impulse propagation was initiated by application of a stimulus current to the leftmost segment. tmax and smax, scaling factors for thickness and stiffness, respectively.

 
Impulse Propagation Along an Inhomogeneous Fiber

To investigate the effect of inhomogeneity in tissue properties on {theta}, we simulated impulse propagation along an inhomogeneous 5-cm-long fiber with {lambda}fiber = 1.00, 1.05, 1.10, 1.15, and 1.20 (Gsac = 0.015 µm/s, PNa:PK:PCa = 1:1:1). For the left half of the fiber, tmax = 1–10 and smax = 1–10. For the right half of the fiber, tmin = 1 and smin = 1. Linear interpolation was applied in the central 0.5 cm of the fiber. As described above, all simulations were performed with contraction enabled as well as with contraction disabled. Impulse propagation was initiated by application of a stimulus current to the leftmost or the rightmost segment.

In Fig. 16, {theta} and (dVmem/dt)max are presented for various simulations after stimulation of the leftmost segment ({lambda}fiber = 1.15) with contraction disabled and with contraction enabled. In thick and/or stiff tissue (left half of the fiber), {theta} was larger; in the remaining, more stretched, parts, {theta} was smaller. In areas where the depolarization wave travels from thick tissue to thin tissue (tmax ≥ 5), was increased, which is explained by the smaller amount of charge required by the downstream segments to reach the excitation threshold. Decrease of (dVmem/dt)max and block of impulse propagation occurred in the inhomogeneous fibers when contraction was enabled.

In Fig. 17, {theta} and (dVmem/dt)max are presented for various simulations after stimulation of the rightmost segment ({lambda}fiber = 1.15) with contraction disabled and with contraction enabled. In thick and/or stiff tissue (left half of the fiber), {theta} was larger; in the remaining, more stretched, parts, {theta} was smaller. In areas where the depolarization wave travels from thin tissue to thick tissue (tmax ≥ 5), {theta} was decreased, which is explained by the larger amount of charge required by the downstream segments to reach the excitation threshold. Conduction block was not observed after stimulation from the right. Thus, for {lambda}fiber = 1.15, conduction block was unidirectional when contraction was enabled.


Figure 17
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Fig. 17. Right stimulation: {theta} and (dVmem/dt)max for impulse propagation along a homogeneous fiber and various inhomogeneous fibers with contraction disabled (top) and enabled (bottom). Isac was permeable to Ca2+ (Gsac = 0.015 µm/s). Fiber length = 5 cm, {lambda}fiber = 1.15. Impulse propagation was initiated by application of a stimulus current to the rightmost segment.

 
In Fig. 18, traces of Vmem, Isac, [Ca2+]i, and {lambda} are presented for three segments of an inhomogeneous fiber ({lambda}fiber = 1.15, tmax = 10, smax = 1) with contraction enabled and with contraction disabled. As shown in Fig. 15, the early-activated segments start contracting, causing {lambda} to increase for the later-activated segments, which leads to an increased resting Vmem. The AP of the segment at 4.0 cm had a low upstroke steepness and, similar to the AP for {lambda} = 1.35 in Fig. 9, the Ca2+ transient was absent, such that no contraction occurred. From the rapid decrease in (dVmem/dt)max at ~4.0 cm (Fig. 16, bottom), it can be concluded that this type of AP cannot generate enough current to propagate.


Figure 18
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Fig. 18. Vmem, Isac, [Ca2+]i, and {lambda} for 1.0-, 2.5-, and 4.0-cm segments of an inhomogeneous fiber (tmax = 10, smax = 1) with contraction enabled (left) and disabled (right). Fiber length was 5 cm, Isac was permeable to Ca2+ (Gsac = 0.015 µm/s), and {lambda}fiber = 1.15. Impulse propagation was initiated by application of a stimulus current to the leftmost segment at 0 cm.

 
Short Stimulation Intervals and Unidirectional Block

To investigate the effect of a shorter stimulation interval on impulse propagation, we stimulated the 5-cm fibers with an interval of 500 ms (2 Hz). In Table 4, {theta} for left stimulation at 1 Hz and at 2 Hz are presented for the left half (1.0 -> 2.5 cm) and for the right half (2.5 -> 4.0 cm) of the fiber. Since stimulation at 2 Hz can lead to alternating impulse propagation and conduction block, we distinguish between even (each 1 s) and odd (each 0.5 s) stimulation. The same data are presented in Table 5 for right stimulation.


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Table 4. Conduction velocity for left stimulation

 

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Table 5. Conduction velocity for right stimulation

 
From Tables 4 and 5, it can be concluded that stimulation at 2 Hz in general leads to slower conduction and conduction block at lower {lambda}. This is explained by a longer ERP under stretch (Fig. 14). Figure 19 illustrates the subtle transition from conduction block in the leftmost 0.5 cm every other stimulation to normal impulse propagation every stimulation ({lambda}fiber = 1.05, tmax = 1, smax = 1). In this case, the ERP decreased after each stimulation, such that after stimulation at 2,100 ms, the AP could propagate. The decrease in ERP is visible in Fig. 19 as the increasing INa inactivation gating variables h and j at the moment of stimulation (segment at 0.1 cm). After 2,100 ms, the cells in the fiber are stimulated at a higher frequency, which leads to a shorter APD and a more decreased ERP. Thus, impulse propagation at a higher frequency becomes a stable situation.


Figure 19
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Fig. 19. Vmem, Isac, h, and j for 0.1-, 0.5-, and 1.0-cm segments of a homogeneous fiber stimulated with an interval of 500 ms. Fiber length was 5 cm (tmax = 1, smax = 1), Isac was permeable to Ca2+ (Gsac = 0.015 µm/s), and {lambda}fiber = 1.05. Impulse propagation was initiated by application of a stimulus current to the leftmost segment at 0 cm.

 
When an inhomogeneous fiber is stimulated from the right, conduction block may occur at lower {lambda}. This can be explained by prolongation of the repolarization phase of the AP. Figure 20 illustrates this situation for {lambda}fiber = 1.10, tmax = 10, and smax = 1. The extended repolarization phase of the segment at 4.0 cm (which is close to the stimulation site) is caused by contraction of the later-activated segments in the left half of the fiber.


Figure 20
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Fig. 20. Vmem, Isac, [Ca2+]i, and {lambda} for 1.0-, 2.5-, and 4.0-cm segments of an inhomogeneous fiber stimulated with an interval of 500 ms. Fiber length was 5 cm (tmax = 10, smax = 1), Isac was permeable to Ca2+ (Gsac = 0.015 µm/s), and {lambda}fiber = 1.10. Impulse propagation was initiated by application of a stimulus current to the rightmost segment at 5.0 cm.

 

    DISCUSSION
 TOP
 ABSTRACT
 METHODS
 RESULTS
 DISCUSSION
 APPENDIX
 REFERENCES
 
In our model, contraction of the cardiac fiber is triggered by the Ca2+ transient, which occurs after depolarization of the membrane. By modeling an Isac, contraction of early-activated parts of the fiber leads to stretch in the later-activated parts and influences impulse propagation, APD, and ERP. For increasing levels of applied stretch, we observed conduction block, which can be unidirectional in an inhomogeneous fiber.

Conduction Slowing and ERP

Our model provides two mechanisms to explain conduction slowing as observed in acutely dilated atria (6, 10, 11, 16, 40): 1) the decrease in tissue conductivity due to stretch and 2) a decreased membrane excitability caused by the Isac (Fig. 11). In an experimental study, Eijsbouts et al. (11) reported a decreased {theta} and local conduction block when the right atrium of a rabbit was acutely dilated. They increased atrial pressure from 2 to 9 and 14 cmH2O and measured {lambda} as well as {theta}. With increasing pressure, {theta} first inc