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Mechanoelectric feedback leads to conduction slowing and block in acutely dilated atria: a modeling study of cardiac electromechanics

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 (I_sac) 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 I_sac 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

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 (I_sac) 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 I_sac 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 I_sac 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 I_sac 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 I_sac. 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 I_sac. We describe the extensions to our model of cardiac electrophysiology, in particular the influence of stretch on fiber conductivity, our model of the I_sac, 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 (V_int), the extracellular potential (V_ext), and the state of the cell membrane, which is expressed in gating variables and ion concentrations. The membrane potential (V_mem) is defined by

(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 (I_trans)

(2)

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

(3)

where I_Na is fast inward Na⁺ current, I_K1 is inward rectifier K⁺ current, I_to is transient outward K⁺ current, I_Kur is ultrarapid delayed rectifier K⁺ current, I_Kr is rapid delayed rectifier K⁺ current, I_Ks is slow delayed rectifier K⁺ current, I_Ca,L is L-type Ca²⁺ current, I_p,Ca is Ca²⁺ pump current, I_NaK is Na⁺-K⁺ pump current, I_NaCa is Na⁺/Ca²⁺ exchanger current, and I_b,Na and I_b,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 (g_int and g_ext) 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 g_int and g_ext (mS/cm), we assume that the resistivity of the intracellular space (R_int = 1/g_int, ·cm) is determined partly by the myoplasmic resistivity (R_myo) and partly by the gap-junctional resistivity (R_junc)

(4)

For the nonstretched fiber, we define g_int = g_{int 0} = 1/R_{int 0}, R_myo = R_{myo 0}, and R_junc = R_{junc 0}. When the fiber is stretched with stretch ratio , cell length increases and cross-sectional area decreases (assuming that cell volume is conserved). Since R_myo is proportional to the length and inversely proportional to the cross-sectional area of the cell, we obtain

(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 , which leads to

(6)

If Eqs. 5 and 6 are combined, g_int is related to by

(7)

For the extracellular domain, we assume that g_ext is related to g_{ext 0} and by

(8)

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

I_sac

On the basis of experimental observations, we assume that I_sac 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, I_sac is permeable to Na⁺, K⁺, and Ca²⁺ and is defined by

(9)

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

(10)

(11)

(12)

where P_Na, P_K, and P_Ca denote the relative permeabilities to Na⁺, K⁺, and Ca²⁺, z_Na, z_K, and z_Ca 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 (g_sac) depends on as follows

(13)

where G_sac is the maximum membrane conductance, K_sac is a parameter to define the amount of current when the cell is not stretched [ = 1, sarcomere length (l_s) = 1.78 µm], and _sac is a parameter to describe the sensitivity to stretch. K_sac and _sac are from Zabel et al. (58) (Table 1).

The reversal potential (E_sac) can be obtained by solving the following equation for V_mem: I_sac,Na + I_sac,K + I_sac,Ca = 0. In the present study, we consider two cases: P_Na:P_K:P_Ca = 1:1:1, with E_sac = –0.2 mV, and P_Na:P_K:P_Ca = 1:1:0, with E_sac = –0.9 mV. In both cases, I_sac has a near-linear current-voltage relation (Fig. 1).

View larger version (8K): [in this window] [in a new window]   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 () = 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.

(14)

(15)

(16)

(17)

(18)

where C_m is the membrane capacitance of a single atrial myocyte (100 pF) (8), F is Faraday's constant, V_i is the intracellular volume (13,668 µm³) (8), V_up and V_rel are the volumes of the SR uptake and release compartments, respectively, I_up,leak, I_up, and I_rel represent the SR currents, [Trpn] is troponin concentration, [Cmdn] is calmodulin concentration, and K_m 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. (42–44) 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 l_s.

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

(19)

where [Ca²⁺]_Trpn is Ca²⁺-bound troponin concentration, [Trpn]_max is total troponin concentration (70 µM) (8), and K_m,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

(20)

where [HTRPN]_tot represents the total troponin high-affinity site concentration and k_htrpn⁺ and k_htrpn^– 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

(21)

where [LTRPN]_tot represents the total troponin low-affinity site concentration and k_ltrpn⁺ and k_ltrpn^– 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 (F_norm), which is defined by

(22)

where P1_max, P2_max, and P3_max are defined as in Ref. 44 and (l_s) describes the physical overlap structure of thick and thin filaments within a sarcomere (44). When (l_s) = 1, all myosin heads are able to interact with actin in the single overlap zone; when (l_s) < 1, some of the filaments are in the double or nonoverlap zones. (l_s) is defined by

(23)

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

View larger version (11K): [in this window] [in a new window]   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.

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 l_CE, l_SE, and l_PE. The reference lengths, i.e., the lengths at which the segment is at rest and no force is applied, are l_{CE 0}, l_{SE 0}, and l_{PE 0}.

View larger version (6K): [in this window] [in a new window]   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.

(24)

where f_CE is a scaling factor, = –(dl_s/dt) represents the sarcomere shortening velocity, and F_norm([Ca²⁺]_i,l_s) is F_norm 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() as proposed by Hunter et al. (17)

(25)

where _max is the maximum sarcomere shortening velocity and c is a constant describing the shape of the hyperbole.

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

(26)

and

(27)

where l_{SE 0} and l_{PE 0} denote the reference element lengths and f_SE, k_SE, f_PE, and k_PE are material constants describing the elasticity of the elements.

From mechanical equilibrium, it follows that F_CE must be equal to the force generated in the SE (F_SE). The total force generated by the segment (F_segment) is defined as F_SE + F_PE. Furthermore, l_PE must be equal to the l_CE + l_SE (Fig. 3). Therefore, during mechanical equilibrium

(28)

(29)

(30)

l_CE, l_SE, and l_PE are related to physiological sarcomere length (l_s) and reference sarcomere length (l_{s 0}) by l_CE = l_s and l_{CE 0} = l_{s 0} (49). The reference length of a segment is 0.01 cm and is related to l_PE,0 by a scaling factor . For segment n, we define the reference length l_{n 0} by

(31)

and the actual length l_n by

(32)

where l_{PE 0}ⁿ and l_PEⁿ represent the reference length and the actual length of the PE of segment n. The for segment n (_n) is then defined by

(33)

The parameters for the three-element mechanical model are obtained from Solovyova et al. (49) (Table 1). In Fig. 4, active force (F_SE), passive force (F_PE), and total force (F_segment) are presented for l_s = 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., l_SE > 0, l_PE = l_CE + l_SE is larger than l_s = l_CE. This results in a steeper increase of F_PE for increasing l_s and is in agreement with the passive force-length relation for intact cardiac muscle measured by Kentish et al. (23).

View larger version (13K): [in this window] [in a new window]   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.

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

(34)

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

(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 f_CE, f_SE, and f_PE are scaled with the increase of the cross-sectional area. To simulate stiff tissue, the mechanical parameter f_PE is scaled. Scaling factors for maximum and minimum thickness are denoted by t_max and t_min, respectively, and scaling factors for maximum and minimum stiffness by s_max and s_min, 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 = 1.0. When is increased to 1.4, the length constant decreases 15% for R_{junc 0}/R_{int 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 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 V_mem 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 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 (), 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 l_s (isosarcometric contraction) and single-cell simulations with constant applied force (isotonic contraction). The influence of I_sac 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 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 _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 F_norm([Ca²⁺]_i,l_s) 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 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 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. (42–44). For the V_mem 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 [(dV_mem/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.

View larger version (14K): [in this window] [in a new window]   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.

View larger version (12K): [in this window] [in a new window]   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.

In Fig. 7, traces of F_norm, F_CE, l_s, and l_PE are presented for simulations of isotonic contraction with applied force (F_segment) of 5–250 mN/mm². The AP and Ca²⁺ transient are the same as in Fig. 5. Less time is required to return F_norm 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 F_CE traces exhibit a plateau phase for F_segment 25 mN/mm². For F_segment = 250 mN/mm², l_PE remains constant, indicating no shortening.

View larger version (14K): [in this window] [in a new window]   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.

Figure 8 illustrates the effect of I_sac on the AP. V_mem, I_sac, I_Ca,L, and [Ca²⁺]_i are presented for I_sac permeable to Ca²⁺ and I_sac not permeable to Ca²⁺ for = 1.00, 1.10, and 1.20. The cell was stimulated with a frequency of 1 Hz. With increasing , repolarization is prolonged and the resting V_mem is depolarized. I_sac is small during the plateau phase and larger during repolarization and rest, which is consistent with a reversal potential between 0 and –1 mV. I_Ca,L is somewhat lowered under stretch, and the Ca²⁺ transient is increased. The lowered I_Ca,L is explained by the Ca²⁺-dependent inactivation of I_Ca,L (8). Interestingly, whether I_sac 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 I_sac permeable to Ca²⁺ and in Table 3 for I_sac not permeable to Ca²⁺. For I_sac 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%.

View larger version (16K): [in this window] [in a new window]   Fig. 8. Vmem, Isac, L-type inward Ca2+ current (ICa,L), and [Ca2+]i for stretch applied to single cells at stretch ratio () = 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.

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

View larger version (16K): [in this window] [in a new window]   Fig. 9. Vmem, Isac, ICa,L, and [Ca2+]i for stretch applied to single cells. was increased from 1.00 at 0-ms stimulation to 1.25, 1.35, and 1.45 at 200 ms; 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 R_{junc 0}/R_{int 0} on

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

View larger version (11K): [in this window] [in a new window]   Fig. 10. Overall conduction velocity () 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.

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

View larger version (15K): [in this window] [in a new window]   Fig. 11. and maximum upstroke velocity [(dVmem/dt)max] for impulse propagation along a 1-cm fiber (Rjunc 0/Rint 0 = 0.6). Gsac and 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.

(36)

where G_Na is the maximum I_Na conductance, E_Na 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 I_Na are presented for the center segment (_fiber = 1.00, 1.10, and 1.20). As _fiber increases, h and j are lower during rest and explain the lower I_Na during depolarization; i.e., the membrane is less excitable. Except for I_to and I_Kur, all ionic currents were similar during the upstroke and shortly after the upstroke (not shown). I_to and I_Kur were smaller and caused the less prominent notch of the AP (Fig. 12).

View larger version (16K): [in this window] [in a new window]   Fig. 12. Vmem, Isac, ICa,L, and [Ca2+]i for the center segment of a 1-cm fiber (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.

View larger version (15K): [in this window] [in a new window]   Fig. 13. Isac,Na, Isac,K, and Isac,Ca for the center segment of a 1-cm fiber (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.

View larger version (16K): [in this window] [in a new window]   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 (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.

To investigate the effect of contraction of early-activated areas on in later-activated areas, we simulated impulse propagation along a 5-cm-long fiber with _fiber = 1.00, 1.05, 1.10, 1.15, and 1.20 (G_sac = 0.015 µm/s, P_Na:P_K:P_Ca = 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 V_mem, I_sac, [Ca²⁺]_i, and are presented for segments located 1.0, 2.5, and 4.0 cm from the stimulation site (_fiber = 1.15). With contraction enabled, the early-activated segments start contracting, so that increases for the later-activated segments, which results in an increased I_sac and a depolarized resting V_mem. In Fig. 16, and (dV_mem/dt)_max are presented with contraction disabled (t_max = 1, s_max = 1) and contraction enabled (t_max = 1, s_max = 1).

View larger version (16K): [in this window] [in a new window]   Fig. 15. Vmem, Isac, [Ca2+]i, and 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 fiber = 1.15. Impulse propagation was initiated by application of a stimulus current to the leftmost segment at 0 cm.

View larger version (15K): [in this window] [in a new window]   Fig. 16. Left stimulation: 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, 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.

To investigate the effect of inhomogeneity in tissue properties on , we simulated impulse propagation along an inhomogeneous 5-cm-long fiber with _fiber = 1.00, 1.05, 1.10, 1.15, and 1.20 (G_sac = 0.015 µm/s, P_Na:P_K:P_Ca = 1:1:1). For the left half of the fiber, t_max = 1–10 and s_max = 1–10. For the right half of the fiber, t_min = 1 and s_min = 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, and (dV_mem/dt)_max are presented for various simulations after stimulation of the leftmost segment (_fiber = 1.15) with contraction disabled and with contraction enabled. In thick and/or stiff tissue (left half of the fiber), was larger; in the remaining, more stretched, parts, was smaller. In areas where the depolarization wave travels from thick tissue to thin tissue (t_max 5), was increased, which is explained by the smaller amount of charge required by the downstream segments to reach the excitation threshold. Decrease of (dV_mem/dt)_max and block of impulse propagation occurred in the inhomogeneous fibers when contraction was enabled.

In Fig. 17, and (dV_mem/dt)_max are presented for various simulations after stimulation of the rightmost segment (_fiber = 1.15) with contraction disabled and with contraction enabled. In thick and/or stiff tissue (left half of the fiber), was larger; in the remaining, more stretched, parts, was smaller. In areas where the depolarization wave travels from thin tissue to thick tissue (t_max 5), 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 _fiber = 1.15, conduction block was unidirectional when contraction was enabled.

View larger version (15K): [in this window] [in a new window]   Fig. 17. Right stimulation: 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, fiber = 1.15. Impulse propagation was initiated by application of a stimulus current to the rightmost segment.

View larger version (17K): [in this window] [in a new window]   Fig. 18. Vmem, Isac, [Ca2+]i, and 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 fiber = 1.15. Impulse propagation was initiated by application of a stimulus current to the leftmost segment at 0 cm.

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, 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.

View larger version (21K): [in this window] [in a new window]   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 fiber = 1.05. Impulse propagation was initiated by application of a stimulus current to the leftmost segment at 0 cm.

View larger version (21K): [in this window] [in a new window]   Fig. 20. Vmem, Isac, [Ca2+]i, and 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 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

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 I_sac (Fig. 11). In an experimental study, Eijsbouts et al. (11) reported a decreased 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 as well as . With increasing pressure, first increases and then decreases for normal stimulation (240-ms interval). For fast stimulation (125-ms interval), decreases nonlinearly with increasing pressure (11). In our model, decreases linearly with increasing when no I_sac is present (Fig. 10). It is therefore likely that the nonlinear decrease in observed by Eijsbouts et al. is explained by a reduced excitability of the membrane, rather than a reduced tissue conductivity. Eijsbouts et al. (10, 11) also observed an increase in conduction block when the atrium was stimulated at a higher frequency, which is consistent with our findings (Tables 4 and 5).

Similar to our results, Shaw and Rudy (47) observed slowing of impulse propagation related to a reduced membrane excitability in a simulation study of impulse propagation in ischemic cardiac tissue. The extracellular K⁺ concentration ([K⁺]_o) was increased, which leads to a depolarized V_mem and a reduced (dV_mem/dt)_max. Their simulation results (47) correspond to experimental results (19). Kléber and Rudy (27) explain the decreased in these experiments by a significant Na⁺ channel inactivation. The result is a depressed membrane excitability [reduced (dV_mem/dt)_max], reduced , and, eventually, conduction block (27). Thus their explanation for conduction slowing and block in ischemic tissue is similar to our explanation for conduction slowing and block under stretch.

In an experimental study, Sung et al. (51) observed a decrease in and an increase in APD when end-diastolic pressure was increased in the left ventricle of isolated rabbit hearts. Interestingly, the SAC blocker streptomycin had little effect on and APD (51). Satoh and Zipes (46) measured differences in ERP in the thin atrial free wall and the crista terminalis. Under stretch, the ERP of the thin atrial free wall was increased more than that in the thicker crista terminalis. Satoh and Zipes explain this difference by assuming that the thin free wall is more stretched than the thicker parts. Huang et al. (16) observed slow conduction related to a shorter stimulation interval in dilated atria, but they did not measure a significant change in atrial ERP after dilatation. Conduction slowing in our model at a 500-ms stimulation interval is attributed to a longer ERP under stretch. From these observations, we conclude that experimentally observed changes in conduction and atrial ERP can be explained by I_sac.

Clinical Relevance

AF is associated with hemodynamic or cardiomechanical disorders such as hypertension, mitral valve disease, and cardiac failure (21). Ravelli et al. (41) found that atrial stretch caused by contraction of the ventricles influences atrial flutter cycle length in humans. Experimentally, it has been observed that acute atrial dilatation facilitates the induction and maintenance of AF in rabbit atria (2, 40) and in canine atria (16, 46). Bode et al. (2) report that the SAC blocker Gd³⁺ reduces the stretch-induced vulnerability to AF, confirming that I_sac plays a significant role in the vulnerability to AF in acutely dilated atria.

In the present study, we observed conduction slowing, an increased ERP, and (unidirectional) conduction block with increasing stretch. These phenomena are attributed to I_sac and can lead to alternating impulse propagation and contractions at a stimulation frequency of 2 Hz. Conduction slowing (16), unidirectional block (27), and dispersion in atrial ERP (46) are related to the inducibility of AF. In the present study, these effects begin at = 1.15 and 1.05 for stimulation at 1 and 2 Hz, respectively. Bode et al. (2) gradually increased intra-atrial pressure in rabbit hearts up to 30 cmH₂O. They could not induce AF in the undilated atrium at 0 cmH₂O, but they observed a 50% probability of AF induction at 8.8 cmH₂O (baseline) and at 19.0 cmH₂O (after Gd³⁺), which increased to 100% (baseline) and 90% (after Gd³⁺) when pressure was further increased (2). However, stretch was not measured in that study. Eijsbouts et al. (11) measured = 1.16 ± 0.14 at 9 cmH₂O, confirming our model predictions that the vulnerability to AF is substantially increased when is 1.15.

Model Validity and Limitations

To our best knowledge, our model is the first to integrate cardiac electrophysiology and cardiomechanics with physiological details such as ionic membrane currents, intracellular Ca²⁺ handling, and cross-bridge formation. In our model, changes in impulse propagation under stretch are related to I_sac and to a reduced conductivity. We do not consider other mechanisms that could influence impulse propagation, such as stretch-related function of other membrane channels, autonomic reflexes, and metabolic changes.

The validity of our model largely depends on the validity of the underlying models and parameters. Validity and limitations of the models for the ionic membrane currents, cross-bridge formation, and cardiomechanics are extensively discussed elsewhere (8, 44, 49, 58). Here, we discuss the validity and limitations of the integrated model with respect to the Ca²⁺-force relation, excitation-contraction coupling, stretch and fiber conductivity, I_sac, intracellular ion concentrations, Ca²⁺-troponin binding, and the Ca²⁺ transient.

Ca²⁺-force relation. Rice et al. (44) proposed five models of isometric force generation in cardiac myofilaments. These are constructed assuming different subsets of three putative cooperative mechanisms (44). Model 4 assumes that the binding of a cross bridge increases the rate of formation of neighboring cross bridges and that multiple cross bridges can maintain activation of the thin filament in the absence of Ca²⁺. The model also simulates end-to-end interactions between adjacent troponin and tropomyosin. The hypothesis that cross-bridge binding increases the affinity of troponin for Ca²⁺ is assumed by model 5, but not by model 4.

To choose between model 4 and model 5 for the present study, we studied the Ca²⁺-force relation (see Fig. 2 for model 4) and the isosarcometric twitches (see Fig. 6 for model 4). We found better agreement between the Ca²⁺-force relation obtained by model 4 and the experimental results of Kentish et al. (23), in particular for >1.9-µm-long sarcomeres. When we compared the isosarcometric twitches, we found that, for model 5, the peak force was lower and the latency to peak force was increased for longer sarcomeres. Compared with the experimental data measured by Janssen and Hunter (18), the latency to peak force increased too greatly with sarcomere length. Our findings confirm the finding by Rice et al. (44) that the hypothesis that cross-bridge binding increases the affinity of troponin for Ca²⁺ is not crucial to reproduce the experimental results. Since the twitches obtained by model 4 better resemble the experimental results from Janssen and Hunter, we have chosen model 4 to describe the Ca²⁺-force relation.

Excitation-contraction coupling. To compute F_norm during contraction, Rice et al. (44) used a Ca²⁺ transient with a peak [Ca²⁺]_i of 0.97 µM and 130 ms to return [Ca²⁺]_i to half of maximum [Ca²⁺]_i. Our peak forces are smaller and the traces are less prolonged than those of Rice et al. (44). This is explained by the lower peak value of [Ca²⁺]_i obtained from the model of Courtemanche et al. (8). However, the main characteristics, i.e., increasing peak force, increasing time to peak force, and increasing relaxation time with increasing l_s, are observed in our model and are in agreement with experimental measurements (18). These characteristics are important with respect to the Frank-Starling mechanism, which states that when the amount of blood flowing into the heart increases, the wall becomes more stretched and the cardiac muscle contracts with increased force. Furthermore, isosarcometric twitch duration (Fig. 6, top) is longer than isotonic twitch duration (Fig. 7, top). This is explained by shortening of the sarcomeres during isotonic contraction and is in agreement with experimental results (17).

Stretch and fiber conductivity. In our model, g_int and g_ext are determined by on the basis of the assumption that 60% of R_int is attributed to R_junc. The assumption that R_int is determined by R_myo only is equivalent to R_{junc 0}/R_{int 0} = 0.0 and would lead to a faster decrease in for increasing (Fig. 10). Since intercellular coupling in cardiac tissue is through the gap junctions and since the number of gap junctions does not change when the tissue is acutely stretched, we believe that neglecting the distinction between R_myo and R_junc leads to overestimation of the effect of on conductivity.

I_sac. We model I_sac as a nonselective cation current with E_sac = 0 to –1 mV. Our model of I_sac has a near-linear current-voltage relation, which can be approximated by

(37)

with g_sac = 0.0027 nS/pF for = 1.0, g_sac = 0.0049 nS/pF for = 1.2, and g_sac = 0.0088 nS/pF for = 1.4. Wagner et al. (57) used an externally applied current with g_sac = 0.0083 nS/pF and E_sac –10 mV to elicit an AP in atrial cells from the rat. In our model, stretch-induced APs are elicited for = 1.35 (Fig. 8), which corresponds to g_sac = 0.0076 nS/pF and is in a similar range.

Intracellular ion concentrations. The ionic currents of the model of Courtemanche et al. (8) interact with [Na⁺]_i, [K⁺]_i, and [Ca²⁺]_i (8). To model the influence of I_sac on the intracellular ion concentrations, we assume that SACs are permeable to Na⁺, K⁺, and Ca²⁺. Whether SACs in human atrial cells are permeable to Ca²⁺ is still a matter of debate. To investigate the effect of permeability of SACs for Ca²⁺, we performed simulations in which I_sac was permeable to Ca²⁺ and not permeable to Ca²⁺. Our results show an increase in [Ca²⁺]_i under stretch, whether I_sac is permeable to Ca²⁺ or not permeable to Ca²⁺ (Fig. 8). However, when I_sac is permeable to Ca²⁺, [Ca²⁺]_i is increased 3% compared with I_sac not permeable to Ca²⁺ (Tables 2 and 3). Our findings are in agreement with experimental observations that [Ca²⁺]_i increases in response to stretch (4, 52). Kamkin et al. (20) observed that the behavior of SACs in isolated human atrial cells did not change when a Ca²⁺-free external solution was used, suggesting that the current through the SACs was preferentially carried by Na⁺, rather than by Ca²⁺ (57). In Fig. 13 (left), the current density of I_sac,Ca is only 5% of the current density of I_sac,Na, which explains the finding by Kamkin et al.

Ca²⁺-troponin binding. Binding of Ca²⁺ to troponin is modeled in two different ways: in the model of Courtemanche et al. (8), an immediate formulation (Eq. 19) to describe Ca²⁺ binding to troponin is used; in the model of Rice et al. (44), binding of Ca²⁺ to troponin is modeled by Eqs. 20 and 21. The resulting [LTRPNCa] is then used to compute the force generated by the sarcomeres. Although having two different ways to model binding of Ca²⁺ to troponin is not elegant, one cannot simply replace the immediate formulation proposed by Courtemanche et al. with the formulation proposed by Rice et al. This would be a major modification of the model of Courtemanche et al. requiring adaptation of various parameters related to the Ca²⁺ fluxes and ionic currents, which is beyond the scope of the present study.

Ca²⁺ transient. As shown in Figs. 9 and 18 (left), although there is an AP, the Ca²⁺ transient is absent. The APs for which no Ca²⁺ transient occurred are characterized by a low steepness of the AP upstroke [low (dV_mem/dt)_max] and were mainly driven by I_Ca,L. The absence of the Ca²⁺ transient is explained by the absence of Ca²⁺ release from the junctional SR, which may be caused by the lower (dV_mem/dt)_max. In our model, the effect of the intercellular currents on the ion concentrations is not taken into account. In case the AP is mainly driven by I_Ca,L, [Ca²⁺]_i in the downstream cell may slowly rise because of the Ca²⁺ in the current flow between the cells. This increase in [Ca²⁺]_i may then trigger buffered Ca²⁺ release. One may expect that a similar effect can be obtained when the SACs are permeable to Ca²⁺. Although AP rises faster in the presence of I_sac,Ca ( = 1.35 in Fig. 9), a Ca²⁺ transient was not observed. In simulations where Ca²⁺ transients did occur, the effect of I_sac,Ca was limited to an 3% increase in [Ca²⁺]_i (Tables 2 and 3). On the basis of these observations, we conclude that the permeability of I_sac for Ca²⁺ contributes little to the changes in electrophysiological behavior under stretch.

In conclusion, 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. Our observations are in agreement with experimental results and provide an explanation for the increased inducibility of AF observed in acutely dilated atria.

	APPENDIX

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX REFERENCES

 
Solving the three-element model. The numerical scheme to solve the equations for the three-element mechanical model is based on the scheme described by Solovyova et al. (49). The forces and lengths of CE, SE, and PE are computed by introducing l₁ = l_CE – l_{CE 0} and l₂ = l_PE – l_{PE 0}. Furthermore, it is assumed that l_{PE 0} = l_{CE 0}, from which it follows that l_{SE 0} = l_{PE 0} – l_{CE 0} = 0 µm. Equations 26 and 27 can now be rewritten as

(38)

and

(39)

The mechanical state of each segment is now defined by l₁, l₂, dl₁/dt, dl₂/dt, and F_norm([Ca²⁺]_i,l_s). Each simulation time step F_norm is computed using [Ca²⁺]_i obtained from the model of Courtemanche et al. (8) and l_s of the former time step. Next, l₁ and l₂ are updated using a forward Euler step and dl₁/dt and dl₂/dt of the former time step. New values of dl₁/dt and dl₂/dt are then computed using Eqs. 24 and 25

(40)

from which sarcomere shortening velocity () can be obtained by

(41)

F_CE can be obtained from Eqs. 38 and 28 by

(42)

Since l_s = l_CE and l₁ = l_CE – l_{CE 0}, we obtain for

(43)

from which dl₁/dt follows immediately.

For isometric single-segment simulations, dl₂/dt = 0 and the generated force can be directly computed from Eqs. 38 and 39. For isotonic simulations, dl₂/dt can be obtained from dF_segment/dt using F_segment = F_SE + F_PE and Eqs. 38 and 39

(44)

and, by taking the derivative

(45)

from which dl₂/dt can be computed by

(46)

Note that during isotonic contraction dF_segment/dt = 0.

Computing cardiac fiber mechanics. To obtain a solution for a multiple-segment simulation, we define and by

(47)

and

(48)

Equation 46 can be formulated for each segment n by

(49)

where _n and _n denote and for segment n. Using l₂ⁿ = l_PEⁿ – l_{PE 0}ⁿ and l_n = _nl_PEⁿ (Eq. 32), we obtain for fiber length L

(50)

Using F_segmentⁿ = F_fiber for each segment n N (Eq. 34) and introducing a and b, we obtain

(51)

where a and b are defined by

(52)

and

(53)

Finally, using the definition of _fiber = L/L₀ (Eq. 35) and Eq. 51, we obtain

(54)

from which follows

(55)

For isotonic simulations, F_fiber is constant and dF_fiber/dt = 0. Using F_segmentⁿ = F_fiber (Eq. 34), dl₂ⁿ/dt can be obtained from Eq. 46 for each segment n N. For isometric simulations, _fiber is constant and d_fiber/dt = 0. In the isometric case, F_fiber and its derivative dF_fiber/dt can be obtained from Eq. 55.

In summary, the mechanical state of segment n is described by l₁ⁿ, l₂ⁿ, dl₁ⁿ/dt, dl₂ⁿ/dt, and F_normⁿ. F_normⁿ is dependent on [Ca²⁺]_i obtained from the model of Courtemanche et al. (8) and the sarcomere length l_s = l_{s 0} + l₁ⁿ. For each time step, l₁ⁿ and l₂ⁿ are computed using a forward Euler step, and dl₁ⁿ/dt and dl₂ⁿ/dt, respectively. However, initial values for l₁ⁿ and l₂ⁿ remain to be defined. Initially, it is assumed that the electrophysiological state of all segments is resting; i.e., V_mem is –81 mV and [Ca²⁺]_i is 0.102 µM (8). For such low [Ca²⁺]_i, F_norm is small and we assume F_normⁿ = 0 for each segment n. Thus, F_CEⁿ = 0, and since F_SEⁿ = F_CEⁿ, the force generated by the segment must come from the PE, i.e., F_PEⁿ = F_segmentⁿ. Since F_SEⁿ = F_CEⁿ = 0, l₁ⁿ = l₂ⁿ (Eq. 38). For homogeneous tissue, the material properties k_PE and f_PE are equal for all segments. For isotonic simulations, l₂ⁿ can be obtained from Eq. 39 by

(56)

For isometric simulations, l₂ⁿ can be directly obtained from the initial stretch ratio _fiber by

(57)

For inhomogeneous tissue, it is assumed that, initially, _fiber = 1 and F_fiber = 0. In that case, l₁ⁿ = l₂ⁿ = 0 for all segments n. During the simulation, stretch (isometric simulation) or force (isotonic simulation) is slowly increased until the desired values are reached. This process typically requires 200 ms of simulation. Finally, it is assumed that, initially

(58)

for homogeneous and inhomogeneous tissue.

	FOOTNOTES

 

Address for reprint requests and other correspondence: N. H. L. Kuijpers, Dept. of Biomedical Engineering, Eindhoven Univ. of Technology, PO Box 513, 5600 MB, Eindhoven, The Netherlands (e-mail: N.H.L.Kuijpers{at}tue.nl)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

	REFERENCES

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX REFERENCES

 

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