INTRODUCTION

TOP ABSTRACT INTRODUCTION MODEL DESCRIPTION RESULTS DISCUSSION APPENDIX REFERENCES

ATRIAL ARRHYTHMIAS such as atrial fibrillation (AF) represent common clinical problems that remain difficult to treat. Experimental dog models have been used widely to study atrial arrhythmia mechanisms in vivo (26). In vivo studies have provided useful insights into the role of atrial tachycardia-induced remodeling (36), electrical heterogeneity (10), and pathological alterations (3, 4) in promoting the occurrence of atrial reentrant arrhythmias. More recently, patch clamp studies in isolated canine atrial myocytes have clarified the properties of a variety of ionic currents in dog atrium (38-40) and suggested ionic mechanisms for regional variations in action potential (AP) properties (11) and tachycardia-induced AP remodeling (38). It is of great interest and potential importance to link these two levels of observation to determine the ionic mechanisms that control the occurrence of arrhythmias in vivo.

Pharmacological tools can be used to evaluate the roles of individual ionic currents (38-40). However, the lack of specificity of pharmacological probes is an important limitation. In addition, changes in one ionic current alter the voltage-time trajectory of the AP and therefore secondarily affect other currents. Therefore, the changes in the AP caused by even a specific ion current inhibitor cannot be interpreted as reflecting the role of only that current. One approach to obtaining a more precise appreciation of the ionic determinants of the AP is to create a mathematical model incorporating realistic biophysical simulations of the ionic processes believed to be involved. The model can be interrogated to assess its agreement with physiological behavior in a variety of conditions to determine its applicability, and the role of individual components can be tested by modifying them and evaluating the resulting impact on the AP. The resulting simulations can then be compared to directly measured changes under experimental conditions.

The first mathematical model of the AP was developed by Hodgkin and Huxley (16) to simulate the electrical behavior of the squid giant axon. Since then, mathematical models of APs based on formulations of ionic currents, pumps, and exchangers have provided insights into properties of rabbit atrial (21) and sinoatrial node (8), guinea pig ventricular (22), bullfrog atrial (29), Purkinje fiber (24), and canine ventricular (37) APs. More recently, models of the human atrial AP have been published (6, 27). These models can account for a wide range of important behaviors and have been used to analyze the effects on human atrial electrophysiology of tachycardia-induced remodeling and selective K⁺ channel blockade (7).

There is no published mathematical model of the canine atrial AP. Such a model would be valuable to interpret better the extensive information available about atrial arrhythmias in vivo in the dog and to consolidate the increasing body of knowledge regarding canine atrial ionic mechanisms. The objectives of the present study were 1) to develop a mathematical model of the canine atrial AP using information obtained from the direct measurement of ionic currents in canine atrial myocytes, and 2) to evaluate the agreement between model APs and APs from different experimental paradigms including varying activation rate, discrepant locations of recording in the atrium, and electrical remodeling caused by long-term atrial tachycardia.

R	Gas constant
T	Temperature
F	Faraday constant
Cm	Membrane capacitance
AP	Action potential
Vmax	Maximal AP upstroke velocity
APA	AP amplitude
APO	AP overshoot
APD	AP duration
APD90	APD to 90% repolarization
APD95	APD to 95% repolarization
AF	Atrial fibrillation
SR	Sarcoplasmic reticulum
JSR	Junctional SR, SR release compartment
NSR	Network SR, SR uptake compartment
Vcell	Cell volume
Vi	Intracellular volume
Vup	NSR volume
Vrel	JSR volume
[X]o	Extracellular concentration of ion X
[X]i	Intracellular concentration of ion X
Cmdn	Calmodulin, sarcoplasmic Ca2+ buffer
Trpn	Troponin, sarcoplasmic Ca2+ buffer
Csqn	Calsequestrin, JSR Ca2+ buffer
[Ca2+]rel	Ca2+ concentration in JSR
[Ca2+]up	Ca2+ concentration in NSR
[Ca2+]Cmdn	Ca2+-bound calmodulin concentration
[Ca2+]Trpn	Ca2+-bound troponin concentration
[Ca2+]Csqn	Ca2+-bound calsequestrin concentration
EX	Equilibrium potential for ion X
Iion	Total sarcolemmal ionic current
Istim	Stimulus current
x	Forward rate constant for gating variable x
x	Reverse rate constant for gating variable x
x	Time constant for gating variable x
x	Steady-state relation for gating variable x
INa	Fast inward Na+ current
gNa	Maximal INa conductance
m	INa activation variable
h	INa fast inactivation variable
j	INa slow inactivation variable
IK1	Inward rectifier K+ current
gK1	Maximal IK1 conductance
Ito	Transient outward K+ current
gto	Maximal Ito conductance
oa	Ito activation variable
oi	Ito inactivation variable
IKur,d	Ultrarapid delayed rectifier K+ current
gKur,d	Maximal IKur,d conductance
ua	IKur,d activation variable
ui	IKur,d inactivation variable
IKr	Rapid delayed rectifier K+ current
gKr	Maximal IKr conductance
xr	IKr activation variable
IKs	Slow delayed rectifier K+ current
gKs	Maximal IKs conductance
xs	IKs activation variable
ICa	Inward Ca2+ current
gCa	Maximal ICa conductance
d	ICa activation variable
f	ICa voltage-dependent inactivation variable
fCa	ICa Ca2+-dependent inactivation variable
INaCa	Na+/Ca2+ exchanger current
INaCa(max)	INaCa scaling factor
Ip,Ca	Sarcoplasmic Ca2+ pump current
Ip,Ca(max)	Maximal Ip,Ca
INaK	Na+-K+ pump current
fNaK	Voltage-dependence parameter for INaK
	[Na+]o-dependence parameter for INaK
Km,Na(i)	[Na+]i half-saturation constant for INaK
Km,K(o)	[K+]o half-saturation constant for INaK
ICl,Ca	Ca2+-activated Cl current
qCa	Ca2+ flux-dependent activation gating variable for ICl,Ca
Km,Na	[Na+]o saturation constant for INaCa
Km,Ca	[Ca2+]o saturation constant for INaCa
ksat	Saturation constant for INaCa
Ib,Na	Background Na+ current
gb,Na	Maximal Ib,Na conductance
Ib,Ca	Background Ca2+ current
gb,Ca	Maximal Ib,Ca conductance
Irel	Ca2+ release current from the JSR
krel	Maximal Ca2+ release rate for Irel
u	Activation gating variable for Irel
v	Ca2+ flux-dependent inactivation gating variable for Irel
w	Voltage-dependent inactivation gating variable for Irel
Fn	Sarcoplasmic Ca2+ flux signal for Irel
Iup	Ca2+ uptake current into the NSR
Iup(max)	Maximal Ca2+ uptake rate for Iup
[Ca2+]up(max)	Maximal Ca2+ concentration in NSR
Itr	Ca2+ transfer current from NSR to JSR
tr	Ca2+ transfer time constant
Iup,leak	Ca2+ leak current from the NSR
[Cmdn]max	Total calmodulin concentration in myoplasm
[Trpn]max	Total troponin concentration in myoplasm
[Csqn]max	Total calsequestrin concentration in JSR
z	Valence

	MODEL DESCRIPTION

TOP ABSTRACT INTRODUCTION MODEL DESCRIPTION RESULTS DISCUSSION APPENDIX REFERENCES

The cell membrane was modeled as a capacitor connected in parallel with variable resistances (ion channels) and batteries (driving forces) following the Hodgkin-Huxley formalism for an excitable membrane (16). In this formalism, each ionic current is proportional to the product of the driving force and the appropriate membrane conductance, which is in turn governed by gating functions with activation and inactivation variables. The activation and inactivation variables are represented as probabilities of ion channel opening, varying between 0 and 1, and are generally voltage and time dependent. The rate of change in the transmembrane potential (V) is given by

(1)

where I_ion and I_stim are the total transmembrane ionic current and stimulus current, respectively, and C_m is the total membrane capacitance. The total transmembrane ionic current is given by

(2)

where I_Na is the fast Na⁺ current, I_K1 is the inward rectifier K⁺ current, I_to is the transient outward K⁺ current, I_Kur,d is the dog ultrarapid delayed rectifier K⁺ current, I_Kr and I_Ks are the rapid and slow delayed rectifier K⁺ current components, respectively, I_Ca is the L-type Ca²⁺ current, I_Cl,Ca is the Ca²⁺-activated Cl current, I_NaCa is the Na⁺/Ca²⁺ exchanger current, I_NaK is the Na⁺-K⁺ pump current, and I_b,Na and I_b,Ca are the background Na⁺ and Ca²⁺ currents, respectively.

The model constantly monitors intracellular concentrations of Na⁺, K⁺, Ca²⁺, and Cl and maintains constant extracellular concentrations. Figure 1 is a schematic representation of the ionic components of the model, which include membrane currents, pumps, and exchangers. The intracellular space includes network (NSR) and junctional compartments of the sarcoplasmic reticulum (JSR) that play a role in the intracellular handling of Ca²⁺ (uptake from and release to the myoplasm, respectively). Unless otherwise noted, physical units are as follows: time (t) is in milliseconds, V is in millivolts, C_m is in picofarads, current densities are in picoamperes per picofarad, conductances (g) are in nanosiemens per picofarad, and concentrations are in millimoles per liter.

View larger version (21K): [in this window] [in a new window]   Fig. 1.   Schematic representation of the canine atrial model myocyte. Intracellular compartments (outlined in dashed lines) indicate the intracellular pools of ion species. The ion concentration in each pool is affected by ionic currents, pumps, and exchangers. Rectangular boxes indicate sarcolemmal ion currents; crossed circles indicate pumps and exchangers. The sarcoplasmic reticulum is divided into two compartments: the Ca2+-release compartment, or junctional sarcoplasmic reticulum (JSR), and the Ca2+-uptake compartment, or network sarcoplasmic reticulum (NSR). See Glossary for definitions.

The computer software encoding the model was written in C with the use of double-precision arithmetic. All simulations were performed on a Dell OptiPlex GX1 personal computer with an Intel Pentium II processor at 450 Hz using a RedHat Linux operating system. Time derivatives were integrated with a modified Euler method employing fixed time steps of 0.005 ms. This method ensured that the largest change in transmembrane potential over a single time iteration did not exceed 0.25 mV (33). To ensure convergence of solutions using this approach, simulations were also obtained with time steps of 0.001 ms (reduced by 80%). The differences between the APs obtained with time steps of 0.001 and 0.005 ms varied between 0.0031 ± 0.016 mV (mean ± SD) and did not exceed 0.23 mV at any point during the AP. The full system of equations is given in the APPENDIX, and model constants are given in Table 1.

Membrane Currents

Fast Na+ current. The model implemented I_Na using the modification of the Ebihara-Johnson model (9) proposed by Luo and Rudy (22) and applied previously in our human atrial AP model (6). Model I_Na is given by

(3)

where g_Na is the maximal Na⁺ conductance, m is the activation variable, and h and j are the inactivation variables.

Transient outward K+ current. We formulated I_to on the basis of results obtained from our laboratory in isolated canine atrial myocytes (38, 39). Figure 2A shows steady-state values for I_to inactivation and activation in the model. Experimental values from Yue et al. (39) were fit to the functions given in the APPENDIX by using a nonlinear least-squares algorithm (23). Figure 2B shows corresponding model and experimental time constants. The activation time constants were bell shaped, with a maximum of 5.9 ms at 27 mV. Experimental values for onset and removal of inactivation are represented by open and filled symbols, respectively. The current is given by

(4)

where g_to is the maximal I_to conductance, o_a is the activation variable, and o_i is the inactivation variable. Figure 2C shows the simulated response of I_to to a voltage-clamp pulse protocol (inset). The g_to was adjusted to obtain current amplitudes that agreed with experimental observations. The peak current-voltage (I-V) relationship (Fig. 2D) obtained by the model shows excellent agreement with the experimental values reported by Yue et al. (39).

View larger version (23K): [in this window] [in a new window]   Fig. 2.   Transient outward K+ current (Ito). A: activation (dashed line) and inactivation (solid line) steady-state values and corresponding experimental data (symbols). B: time constant values for activation (dashed line) and inactivation (solid line) and corresponding experimental data (symbols). , Removal of inactivation determined experimentally. C: Ito response to voltage steps in the model (inset, pulse protocol). D: peak current-voltage (I-V) relation for model (line) and in experiments (symbols).

Dog ultrarapid delayed rectifier K+ current. The current is given by

(5)

(6)

where g_Kur,d is the voltage-dependent maximal conductance, and u_a and u_i are the activation and inactivation variables, respectively. Figure 3, A and B, shows steady-state values and time constants for activation and inactivation of I_Kur,d. The activation time constants shown in Fig. 3B were fit to values reported by Yue et al. (40). Inactivation of I_Kur,d is very slow. In the absence of canine-specific measurements for I_Kur,d inactivation at 37°C, we used the inactivation steady-state values and time constants for human atrial myocytes reported by our laboratory (34) and used previously in our human AP model (6). Figure 3C shows I_Kur,d elicited by a simulated voltage-clamp pulse protocol. The peak I-V relationship in Fig. 3D shows good agreement with data from Yue et al. (40).

View larger version (22K): [in this window] [in a new window]   Fig. 3.   Ultrarapid delayed rectifier K+ current (IKur,d). A: activation (dashed line) and inactivation (solid line) steady-state values and corresponding experimental data (symbols). B: time constant values for activation (dashed line) and inactivation (solid line). Triangles represent experimental values for activation (at and above 10 mV) and deactivation (below 10 mV). C: IKur,d response to voltage steps in the model (inset, pulse protocol). D: peak I-V relation for model (line) and in experiments (symbols).

Rapid and slow K+ delayed rectifier current components. The currents are given by

(7)

(8)

where g_Kr and g_Ks are the maximal current densities and x_r and x_s are the activation variables for the respective current components. Voltage-dependent activation steady states are shown in Fig. 4A; experimental values are from Li et al. (19). Figure 4B shows the activation time constants along with experimental data. Figure 4, C and D, shows model currents elicited by the pulse protocol (inset). Figure 4, E and F, shows the peak I-V relationships provided by the model and obtained from the experimental data of Li et al. (19).

View larger version (24K): [in this window] [in a new window]   Fig. 4.   Rapid (IKr) and slow components (IKs) of the delayed rectifier K+ current. A: steady-state activation for IKr (solid line) and IKs (dashed line). Corresponding experimental values are shown with symbols. B: time constants for IKr (solid line) and IKs (dashed line). Experimental data (symbols) show time constants for activation and deactivation at more positive and negative potentials, respectively. Open symbols represent data obtained during voltage steps; filled symbols represent data from tail currents. C and D: responses to voltage steps in the model (inset, pulse protocol). E and F: peak I-V relations for model (lines) and in experiments (symbols).

Inward rectifier K+ current. The model follows the formulation of I_K1 in our human AP model (6). Maximal conductance (g_K1) was adjusted to obtain a peak I-V relationship corresponding to experimental results (38). The voltage-dependent, time-independent current is given by

(9)

Ca²+ current. We based our formulation of I_Ca on experimental data (38) and previous AP models (6, 22). The current is given by

(10)

where g_Ca is the maximal conductance. The formulation includes voltage-dependent activation (d) and voltage- and Ca²⁺-dependent inactivation (f and f_Ca). The computed equilibrium potential for Ca²⁺ (Eq. A10) is between 100 and 130 mV, which is inconsistent with experimental data for I_Ca reversal (11, 19, 38), in accordance with previous observations. The reversal potential for I_Ca was therefore fixed at 65 mV as in previous models (6, 21). The steady-state activation and inactivation variables (Fig. 5A) and the time constants for voltage-dependent inactivation (_f) (Fig. 5B) were fit to experimental values (38). In the absence of reported activation time constants specific to canine atria, we adopted the formulation of Luo and Rudy (22). Figure 5C shows steady-state Ca²⁺-dependent I_Ca inactivation. Figure 5E shows I_Ca elicited by a simulated pulse protocol. The peak I-V relationship obtained from the model (Fig. 5F) is U-shaped, reversing at 65 mV, and agrees with experimental data (38). Ca²⁺ transients elicited at different rates are shown in Fig. 5D.

View larger version (32K): [in this window] [in a new window]   Fig. 5.   Sarcolemmal Ca2+ current (ICa). A: voltage-dependent activation (dashed line) and inactivation (solid line) steady-state values and corresponding experimental data (symbols). B: time constant values for activation (dashed line) and inactivation (solid line) and corresponding experimental data (symbols). C: intracellular Ca2+ concentration ([Ca2+]i)-dependent inactivation at steady state. D: Ca2+ transients generated during activity at 0.5, 1, 2, and 4 Hz. E: ICa response to voltage steps (inset, pulse protocol). F: peak I-V relation for model (line) and experiment (symbols).

Ca²+-activated Cl current. The detailed mechanism of I_Cl,Ca activation remains poorly understood, although direct activation by free cytoplasmic Ca²⁺ appears central. Factors influencing I_Cl,Ca include channel density, colocalization with sarcolemmal Ca²⁺ channels, Ca²⁺ load in the sarcoplasmic reticulum (SR), and Ca²⁺-induced Ca²⁺ release (5). In the model, activation of I_Cl,Ca was initiated by Ca²⁺ flux into the cell, determining the increase in intracellular Ca²⁺ concentration ([Ca²⁺]_i) in a localized region between the closely coupled sarcolemmal I_Ca and the Ca²⁺ release channels of the SR (18, 41). The current is given by

(11)

where g_Cl,Ca is the maximal current conductance and q_Ca represents the Ca²⁺ flux-dependent activation variable as defined in Fig. 6A. In response to depolarizing pulses, model-derived I_Cl,Ca (Fig. 6B) decayed fully within 20 ms of the onset of depolarization. The peak I-V relationship (Fig. 6C) resembles experimental values reported by Yue et al. (38).

View larger version (10K): [in this window] [in a new window]   Fig. 6.   Ca2+-activated Cl current (ICl,Ca). A: ICl,Ca activation at steady state (qCa) depends on a sarcolemmal Ca2+ flux signal (Fn). B: response to voltage steps (inset, pulse protocol). C: peak I-V relation for model (line) and in experiments (symbols).

Na+-K⁺ pump current and Na⁺/Ca²⁺ exchanger current. We implemented the Na⁺-K⁺ pump (Eqs. A57-A59) by following previous models (6, 22). The parameter for maximal current [I_NaK(max)] was adjusted to maintain stable intracellular ion concentrations at rest. The exchanger current formulation (Eq. A60) similarly followed previously published formulations (6, 22).

Background Na+ and Ca²⁺ currents. The amplitudes of the background currents were adjusted to maintain stable resting intracellular ion concentrations (Eqs. A61 and A62).

Ca²+ pump current. We included a sarcolemmal Ca²⁺ pump current (I_p,Ca) formulation (Eq. A63) by following previous models (6, 22).

SR Ca²+ handling. We implemented SR Ca²⁺ handling (Eqs. A64-A73) using the two-compartment model (NSR and JSR) of Luo and Rudy (22) with a modification proposed by Friedman (12). The NSR subserves Ca²⁺ uptake, whereas the JSR governs release. As depicted in Fig. 1, the Ca²⁺ uptake current (I_up) moves Ca²⁺ from the intracellular space to the network compartment, whereas the Ca²⁺ release current (I_rel) releases Ca²⁺ from the junctional compartment to the myoplasm. A transfer current (I_tr) moves Ca²⁺ from the NSR to the JSR. Ca²⁺ release from the junctional compartment is induced by Ca²⁺ flux into the myoplasm, with close coupling between sarcolemmal I_Ca channels and I_rel channels of the SR. The release current is thus a transient Ca²⁺ flux with a negative feedback that contributes to its rapid inactivation following a brief opening. Activation and inactivation gating for I_rel were implemented as previously described (6).

Myoplasmic and SR Ca²+ buffers. Ca²⁺ buffering is mediated by calmodulin and troponin in the myoplasm and by calsequestrin in the SR release compartment. Ca²⁺ binding by intracellular buffers was modeled as a dynamic process, as described in the Lindblad et al. (21) rabbit atrial myocyte model. Buffer concentrations and binding constants (see Table 1) were obtained from Luo and Rudy (22) and Rasmusson et al. (29).

	RESULTS

TOP ABSTRACT INTRODUCTION MODEL DESCRIPTION RESULTS DISCUSSION APPENDIX REFERENCES

Canine AP Model

View larger version (25K): [in this window] [in a new window]   Fig. 7.   Model action potential (AP) (top left), underlying membrane currents (INa, ICa, Ito, IKr, INaCa, IKur,d, IKs, INaK, ICl,Ca, and IK1), and [Ca2+]i transient during AP (bottom right). For reference, dashed lines show the time course of the AP to compare with corresponding currents. Solid arrows on the y-axes indicate the zero reference (0 mV for the AP, 0 pA/pF for currents, 0 µM for [Ca2+]i). The dashed arrow on the y-axis for INaK indicates the baseline value of 0.13 pA/pF. Recordings were taken during the 10th pulse from rest during activity at 1 Hz.

Variability of AP Morphology

View larger version (29K): [in this window] [in a new window]   Fig. 8.   Effect of varying Ito (A), IKur,d (B), IKr (C), IK1 (D), ICl,Ca (E), and ICa conductance (F) on action potential morphology. Lines correspond to conductances (g; solid lines), conductance adjusted to 10% (dotted line), conductance at 25, 50, 75, 90, and 100% of control (heavy line), and conductance at 110, 125, 150, 200, 250, and 300% of control (dashed line). Each AP waveform represents the 10th AP waveform after pacing from rest at 1 Hz.

In general, AP duration (APD) increases when the conductance of a repolarizing current decreases (Fig. 8, A-E); however, the nature and degree of prolongation are different for each ionic current. For I_to, the degree of APD prolongation saturates with decreasing conductance (Fig. 8A), reaching a 17% prolongation of APD₉₅ at a 25% conductance reduction. For I_Kur,d, the plateau of the AP is raised as conductance decreases, accentuating spike and dome morphology and prolonging APD (Fig. 8B). For both I_to and I_Kur,d, an increase in conductance progressively decreases APD and produces triangular APs. I_Kr shows strong inward rectification positive to 0 mV, limiting its role in the early phases of the AP (Fig. 8C); however, changes in I_Kr produce important alterations in repolarization during late phase 2 and phase 3. Because of its slow activation and relatively small density, changes in I_Ks do not alter APD₉₅ by more than 5 ms in the model (data not shown). For I_K1, small decreases in conductance (<50%) increase APD, with little change in the resting potential (Fig. 8D). Larger decreases produce substantial depolarization, whereas increases in conductance reduce APD and cause very slight hyperpolarization. The contribution of I_Cl,Ca to outward current during phase 1 is ~5% of the combined densities of I_to and I_Kur,d; thus a 90% reduction in I_Cl,Ca conductance causes only a 5-ms increase in APD₉₅, whereas a 300% increase in conductance decreases APD₉₅ by ~10 ms (Fig. 8E). A reduction in I_Ca progressively lowers the plateau and shortens APD, whereas an increase in conductance raises the plateau, accentuates spike and dome morphology, and prolongs APD (Fig. 8F).

APD Adaptation to Rate

View larger version (8K): [in this window] [in a new window]   Fig. 9.   Rate adaptation. A: model APs at frequencies of 1 [longest action potential duration (APD)], 2, and 3 Hz. B: representative APs obtained from an isolated canine atrial myocyte at pacing frequencies of 1 (longest APD), 2, and 3 Hz.

To identify the mechanism of rate adaptation in the model, we examined the effects of rate-dependent changes of individual selected currents on AP morphology. The variables characterizing a specific current at 4 Hz were substituted for those at 1 Hz, and a model AP was generated with the conditions for all other currents set as they were at 1 Hz. The resulting AP reflects the change that can be attributed to the rate-dependent response of the current under evaluation when the frequency is increased from 1 to 4 Hz, as illustrated in Fig. 10. When the activation variables for I_Kr and I_Ks at 4 Hz are substituted into the model at 1 Hz, there is no significant AP change. Substitution of the voltage-dependent inactivation variable for I_Ca (Fig. 10; denoted as f in Eq. 10) accounts for over 50% of the observed rate-dependent AP abbreviation. The addition of [Ca²⁺]_i-dependent I_Ca inactivation (by substitution of the [Ca²⁺]_i values at 4 Hz) to voltage-dependent I_Ca inactivation accounts virtually completely for AP rate adaptation.

View larger version (17K): [in this window] [in a new window]   Fig. 10.   Role of IK, ICa, and [Ca2+] in adaptation of AP to changes in rate. Model APs at 1 Hz (dashed line) and 4 Hz (dotted line) are shown. Traces with symbols indicate the contribution of specified variables to rate adaptation (see text for details). , Activation of IKr and IKs combined; , voltage-dependent inactivation of ICa alone; , voltage-dependent ICa inactivation and [Ca2+].

Regional Heterogeneity

View larger version (9K): [in this window] [in a new window]   Fig. 11.   Relationship between experimentally observed regional AP heterogeneity and model predictions based on measured variations in ionic currents. Model APs at 1 Hz (A) are compared with digitally averaged APs at 1 Hz from Feng et al. (11) (B) in four regions of the canine right atria. PM, pectinate muscle; CT, crista terminalis; APG, appendage; AVR, atrioventricular ring region.

Tachycardia-Induced Atrial Ionic Remodeling

Figure 12 shows the resulting model APs at 1 and 2 Hz (Fig. 12A) and corresponding experimental recordings (right). The model shows APD abbreviation and loss of rate adaptation that is qualitatively similar to experimental observations. Quantitatively, rate-dependent APD adaptation was essentially abolished in myocytes of dogs subjected to rapid atrial pacing for 42 days, but still occurred in the model. The implementation of an additional reduction (to 10% of control values) of I_Ca conductance (Fig. 12, inset) reproduced the experimentally observed loss of rate adaptation.

View larger version (13K): [in this window] [in a new window]   Fig. 12.   Model APs (A) and recordings from a representative canine atrial myocyte from Yue et al. (40) (B) at frequencies of 1 and 2 Hz. Myocytes are from dogs subjected to 0, 1, 7, and 42 days of rapid atrial pacing (P0, P1, P7, and P42, respectively). Adaptation to rate is abolished with reduced ICa. Inset: model P42 simulations paced at 1 and 2 Hz with an additional 20% reduction in ICa.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MODEL DESCRIPTION RESULTS DISCUSSION APPENDIX REFERENCES

We have developed a mathematical model of the canine atrial AP. The model produces APs that are consistent with experimental observations under a variety of conditions and permits critical analysis of the mechanisms and interactions of ionic currents that underlie AP properties in different situations.

Behavior of the Model AP

Variability of AP morphology. APs in isolated myocytes are known to exhibit a wide range of morphologies, even among cells isolated from small myocardial specimens. Wang et al. (35) identified three principal AP morphologies in atrial myocytes isolated from small human right atrial samples and attributed morphological variability to differences in the relative densities of repolarizing currents. Type 1 APs had a spike-and-dome waveform; type 2, an elevated, level plateau phase; and type 3, a triangular AP with little to no plateau region. Our sensitivity analysis (Fig. 8) demonstrates the effects of varying current densities on AP morphology and is consistent with the notion that the relative densities of currents may be responsible for differences in AP morphology.

Rate adaptation of the model AP. Loss of atrial AP rate adaptation has been linked to electrical remodeling due to AF (13, 36) and vulnerability to AF induction in humans (1). Consistent with experimental observations (38), changes in I_Ca were central to rate-dependent AP adaptation in the model (Fig. 10). Unlike our human atrial AP model (6), I_K does not play a direct role in rate adaptation over 1-4 Hz in the present model. This discrepancy is likely due to the shorter and less positive plateau in the present model, because the amplitude of I_K (particularly I_Kr) is very sensitive to changes in the time the cell remains at voltages positive to the activation threshold (approximately 30 mV).

AP Model Applications

Regional heterogeneity. Heterogeneity of atrial refractoriness appears to play an important role in AF maintenance (10). Simulations of experimentally recorded regional differences in ionic conductances (11) produced APs with durations and morphologies closely resembling experimental observations (Fig. 11). Our model thus supports an ionic substrate as the basis for regional AP heterogeneity and constitutes the first quantitative effort of which we are aware to determine whether experimentally observed regional variations in ionic currents can account for regional AP differences.

AF-induced electrical remodeling. The discovery of AF-induced remodeling (36) was an important advance in our understanding of AF mechanisms. Atrial tachycardia-induced ionic changes have been observed and suggested to account for the refractoriness alterations that are the hallmark of remodeling (38). Incorporating the ionic changes reported for remodeling in the model qualitatively reproduced experimental AP alterations (Fig. 12), indicating that the ionic alterations are likely an important contributor to AP remodeling. However, neither the overall AP abbreviation nor the reduction in AP rate adaptation was as striking in the model as those described experimentally. These findings suggest that the reported changes in I_Ca and I_to may be insufficient to account for tachycardia-induced changes in AP morphology or may be underestimated. Studies in dogs (14) and goats (31, 32) have suggested that electrical remodeling may be mediated by intracellular Ca²⁺ overload. Thus other ion transport processes, including the Na⁺/Ca²⁺ exchanger (28), the Na⁺-K⁺ pump, or the proton pump, which have not been studied experimentally in tachycardia-induced remodeling, may be involved. Inefficient coupling between sarcolemmal I_Ca and SR Ca²⁺ release is associated with hypertrophy in rat ventricular myocytes (15, 30) and could be involved. Alternatively, the lack of full agreement with experimental findings may be due to inadequate expression by the model of changes in ionic processes at rapid rates.

Comparison With Other Atrial AP Models

Lindblad et al. (21) described a mathematical model of the rabbit atrial cell that incorporates intracellular Ca²⁺ handling by the SR and provides a more complete formulation of repolarizing K⁺ currents, including I_to, I_Kr, and I_Ks. Unlike the present model, I_Ca inactivation is unaffected by intracellular Ca²⁺. The AP is very sensitive to rate changes over the range between 0.2 and 3 Hz because of the rate sensitivity of I_to and its critical role in repolarization. Although that model faithfully reproduces properties of the rabbit atrial AP, species differences (particularly in the relative roles of I_to and I_K) limit its applicability to the dog.

Models of the human atrial AP have been published by Nygren et al. (27) and our group (6). Both models include similar ionic currents. Human atrial myocytes display a longer APD (by ~60%) than in the dog. I_Ca activates and inactivates faster in the dog model, reducing plateau voltage and duration, and consequently APD, in agreement with experimental AP recordings. The reduced plateau potential in the dog model also results in less contribution to repolarization from I_K.

Potential Limitations

The mechanism of activation of I_Cl,Ca is not yet fully understood. In the model, we formulated I_Cl,Ca with a Ca²⁺-dependent activation scheme that closely reproduced observed peak current-voltage relationships (Fig. 6). There is evidence that, in some experimental paradigms, reductions in I_Ca are not accompanied by altered I_Cl,Ca (38), suggesting that I_Cl,Ca is regulated by additional factors not considered in the present model. Furthermore, the mechanisms governing intracellular Cl homeostasis are unclear. In the present model, we did not include a regulatory mechanism to counterbalance the outward movement of Cl via I_Cl,Ca. Within the framework of the AP, however, I_Cl,Ca is of limited importance (Figs. 7 and 8E), and the accumulation of intracellular Cl occurs very slowly at 1 Hz (~1.6 mM/h or 6.43 mM in 4 h).

Alterations in I_Ks produced little change in AP morphology or duration. Delayed rectifier currents are particularly sensitive to damage during cell isolation (39), so the limited role of I_Ks in the model may be due to an underestimation of the current amplitude. Therefore, the finding that I_Kr and I_Ks play little role in AP rate adaptation should be interpreted with caution. Further experimental and theoretical work are needed to clarify better the properties and roles of I_Kr and I_Ks under physiological conditions.

Potential Significance

AF is the most frequently encountered sustained arrhythmia in clinical practice. Key features associated with AF include loss of rate adaptation (1, 2) and increased heterogeneity of refractory periods (10, 26). Our model reproduces observed AP rate adaptation (Fig. 9) and thereby gives potential insights into underlying mechanisms. These investigations point to Ca²⁺ release from the SR, changes in intracellular [Ca²⁺], and I_Ca inactivation as important mediators of rate adaptation (Fig. 10) and support the role of an ionic substrate for regional AP heterogeneity as suggested by Feng et al. (11).