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Am J Physiol Heart Circ Physiol 282: H516-H530, 2002. First published October 4, 2001; doi:10.1152/ajpheart.00612.2001
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Vol. 282, Issue 2, H516-H530, February 2002

Ionic mechanism of electrical alternans

Jeffrey J. Fox1,2, Jennifer L. McHarg1, and Robert F. Gilmour Jr1

1 Department of Biomedical Sciences and 2 Department of Physics, Cornell University, Ithaca, New York 14853-6401


    ABSTRACT
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES

Although alternans of action potential duration (APD) is a robust feature of the rapidly paced canine ventricle, currently available ionic models of cardiac myocytes do not recreate this phenomenon. To address this problem, we developed a new ionic model using formulations of currents based on previous models and recent experimental data. Compared with existing models, the inward rectifier K+ current (IK1) was decreased at depolarized potentials, the maximum conductance and rectification of the rapid component of the delayed rectifier K+ current (IKr) were increased, and IKr activation kinetics were slowed. The slow component of the delayed rectifier K+ current (IKs) was increased in magnitude and activation shifted to less positive voltages, and the L-type Ca2+ current (ICa) was modified to produce a smaller, more rapidly inactivating current. Finally, a simplified form of intracellular calcium dynamics was adopted. In this model, APD alternans occurred at cycle lengths = 150-210 ms, with a maximum alternans amplitude of 39 ms. APD alternans was suppressed by decreasing ICa magnitude or calcium-induced inactivation and by increasing the magnitude of IK1, IKr, or IKs. These results establish an ionic basis for APD alternans, which should facilitate the development of pharmacological approaches to eliminating alternans.

action potential duration restitution; calcium current; potassium currents


    INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES

THE DURATION of the cardiac action potential is determined in large part by the preceding diastolic interval. This relationship between action potential duration and diastolic interval, known as the action potential duration restitution relation, is an important determinant of cardiac dynamics (17). In particular, if the slope of the restitution relation is >= 1, an alternation of action potential duration, or electrical alternans, commonly develops during high-frequency pacing (2, 8).

It has been suggested that rate-dependent electrical alternans may be a precursor to the development of ventricular arrhythmias, particularly ventricular fibrillation (VF) (6, 10, 19, 22). In support of this idea, several recent experiments (5, 11, 23) have shown that when the slope of the restitution relation is >= 1, rapid pacing induces both alternans and fibrillation in isolated ventricles. If the slope of the restitution relation is reduced to <1, neither electrical alternans nor fibrillation occurs (5, 11, 12, 23). Unfortunately, the interventions used to date to suppress alternans and fibrillation [high-dose calcium channel blockers (23), hyperkalemia (12), and bretylium (5)] have limited clinical utility. More effective means of suppressing alternans need to be identified, a process that would be facilitated by a more complete understanding of the ionic basis for alternans.

One approach to determining the ionic basis for alternans is to use a computer model, several of which have been developed. For example, Luo and Rudy (15, 16), using data obtained primarily from guinea pig myocytes, developed a comprehensive ionic model (LR1) that subsequently was updated (LRd) to include formulations for the rapid and slow components of the delayed rectifier K+ current (IKr and IKs, respectively). Recently, Winslow et al. (26) modified the LRd model using data for ionic currents obtained from canine ventricular myocytes (CVM) and a formulation for calcium dynamics developed originally in guinea pig myocardial cells (9). An alternative formulation for calcium dynamics has been proposed by Chudin et al. (1) in their modification of the LR1 model.

Each of the models described above has limitations with respect to the study of the ionic basis for electrical alternans. The Winslow and LRd models do not produce sustained alternans at rapid pacing rates, whereas the Chudin model, which does generate electrical alternans, lacks formulations for repolarizing K+ currents likely to contribute importantly to alternans [IKr, IKs, and the transient outward K+ current (Ito)].

Given that a complete ionic model that generates electrical alternans is not currently available, we set out to develop such a model, guided by the results obtained from our experimental studies in the canine ventricle (11, 23). Our initial objectives were to develop an ionic model of the CVM that exhibits stable electrical alternans and to use the model to identify the ionic currents responsible for alternans. Once the relevant ionic currents were identified, we then manipulated these currents to eliminate alternans. Our expectation is that the same ionic manipulations that suppress alternans in the ionic model will suppress fibrillation in vivo, in which case the results of the present study may suggest novel approaches to the prevention of VF.

Glossary


 alpha h   Voltage-dependent h gate parameter
 alpha j   Voltage-dependent j gate parameter
 alpha m   Voltage-dependent m gate parameter
 alpha Xto   Voltage-dependent Xto gate parameter
 beta h   Voltage-dependent h gate parameter
 beta i   Myoplasmic buffering factor
 beta j   Voltage-dependent j gate parameter
 beta m   Voltage-dependent m gate parameter
 beta SR   Sarcoplasmic reticulum buffering factor
 beta Xto   Voltage-dependent Xto gate parameter
 gamma    Sarcoplasmic reticulum Ca2+-dependent Jrel factor
 eta    Controls voltage dependence of INaCa
 sigma    Extracellular Na+ INaK factor
 tau d   ICa activation time constant
 tau f   ICa inactivation time constant
 tau fCa   Ca2+-dependent ICa inactivation time constant
 tau Kr   IKr activation time constant
 tau Ks   IKs activation time constant
Acap   Capacitive membrane area
APD   Action potential duration
BCL   Basic cycle length
Csc   Specific membrance capacity
 Delta Camax   Maximum change in Ca2+
 Delta Camin   Minimum change in Ca2+
[Ca2+]i   Intracellular Ca2+ concentration
[Ca2+]o   Extracellular Ca2+ concentration
[Ca2+]SR   Sarcoplasmic reticulum Ca2+ concentration
[CMDN]tot   Total calmodulin concentration
[CSQN]tot   Total calsequestrin concentration
CVM   Canine ventricular myocyte
d   ICa activation gate
dinfinity    Steady-state ICa activation
DI   Diastolic interval
ECa   Ca2+ equilibrium potential
EK   K+ equilibrium potential
EKs   IKs equilibrium potential
ENa   Na+ equilibrium potential
f   ICa inactivation gate
finfinity    Steady-state ICa inactivation
f<UP><SUB>Ca</SUB><SUP>∞</SUP></UP>   Steady-state Ca2+-dependent ICa inactivation
fCa   Ca2+-dependent ICa inactivation gate
fNaK   Voltage-dependent INaK factor
F   Faraday constant
<A><AC>G</AC><AC>&cjs1171;</AC></A>Cab   Peak ICab conductance
<A><AC>G</AC><AC>&cjs1171;</AC></A>K1   Peak IK1 conductance
<A><AC>G</AC><AC>&cjs1171;</AC></A>Kp   Peak IKp conductance
<A><AC>G</AC><AC>&cjs1171;</AC></A>Kr   Peak IKr conductance
<A><AC>G</AC><AC>&cjs1171;</AC></A>Ks   Peak IKs conductance
<A><AC>G</AC><AC>&cjs1171;</AC></A>Na   Peak INa conductance
<A><AC>G</AC><AC>&cjs1171;</AC></A>to   Peak Ito conductance
h   Fast INa inactivation gate
ICa   L-type Ca2+ channel current
<A><AC>I</AC><AC>&cjs1171;</AC></A>Ca   Maximal ICa
ICab   Ca2+ background current
ICahalf   <A><AC>I</AC><AC>&cjs1171;</AC></A>Ca level that reduces PCaK by one-half
ICaK   K+ current through the L-type Ca2+ channel
IK1   Inward rectifier K+ current
IKp   Plateau K+ current
IKr   Rapid component of the delayed rectifier K+ current
IKs   Slow component of the delayed rectifier K+ current
INa   Na+ current
INab   Na+ background current
INaCa   Na+/Ca2+ exchange current
INaK   Na+-K+ pump current
<A><AC>I</AC><AC>&cjs1171;</AC></A>NaK   Maximal INaK
IpCa   Sarcolemmal Ca2+ pump current
<A><AC>I</AC><AC>&cjs1171;</AC></A>pCa   Maximal IpCa
Istim   Stimulus current
Ito   Transient outward K+ current
j   Slow INa inactivation gate
Jleak   Leakage Ca2+ flux from the sarcoplasmic reticulum
Jrel   Release Ca2+ flux from the sarcoplasmic reticulum
Jup   Uptake Ca2+ flux to the sarcoplasmic reticulum
JSR   Junctional sarcoplasmic reticulum
kNaCa   Scaling factor for INaCa
ksat   INaCa saturation factor for INaCa
K<UP><SUB>1</SUB><SUP>∞</SUP></UP>   Steady-state IK1 activation
KKp   IKp activation
K<UP><SUB><IT>m</IT></SUB><SUP>CMDN</SUP></UP>   Ca2+ half-saturation constant for calmodulin
K<UP><SUB><IT>m</IT></SUB><SUP>CSQN</SUP></UP>   Ca2+ half-saturation constant for calsequestrin
KmCa   Ca2+ half-saturation constant for INaCa
KmfCa   Ca2+ half-saturation constant for fCa
KmK1   K+ half-saturation constant for IK1
KmKo   K+ half-saturation constant for INaK
KmNa   Na+ half-saturation constant for INaCa
KmNai   Na+ half-saturation constant for INaK
KmpCa   Half-saturation constant for IpCa
Kmup   Ca2+ half-saturation constant for Jup
[K+]i   Intracellular K+ concentration
[K+]o   Extracellular K+ concentration
LR1   Luo and Rudy model
LRd   Updated Luo and Rudy model
m   INa activation gate
[Na+]i   Intracellular Na+ concentration
[Na+]o   Extracellular Na+ concentration
NSR   Nonjunctional sarcoplasmic reticulum
PCa   L-type Ca2+ channel permeability to Ca2+
PCaK   L-type Ca2+ channel permeability to K+
Pleak   Ca2+ leakage permability between the sarcoplasmic reticulum and the myoplasm
Prel   Ca2+ maximal release permeability from the sarcoplasmic reticulum
R   Ideal gas constant
SR   Sarcoplasmic reticulum
t   Time
T   Temperature
V   Voltage
 Delta Vmax   Maximum change in voltage
 Delta Vmin   Minimum change in voltage
Vmyo   Myoplasmic volume
VSR   Sarcoplasmic reticulum volume
Vup   Maximal Ca2+ uptake to the sarcoplasmic reticulum
VF   Ventricular fibrillation
XKr   IKr activation gate
X<UP><SUB>Kr</SUB><SUP>∞</SUP></UP>   Steady-state IKr activation
XKs   IKs activation gate
X<UP><SUB>Ks</SUB><SUP>∞</SUP></UP>   Steady-state IKs activation
Xto   Ito activation gate
Yto   Ito inactivation gate


    MATERIALS AND METHODS
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES

To study the ionic mechanism of electrical alternans in canine myocytes, we constructed a CVM model using appropriate formulations of ionic currents from the LRd, Winslow, and Chudin models, altered as necessary to fit experimental voltage-clamp data from CVM. It has been well established that cellular electrical properties in the canine ventricle vary, both between right and left ventricles and within a given ventricle, according to whether a cell resides in the epicardium, endocardium, or midmyocardium (13, 14). Because the Winslow model is the only existing ionic model based on the electrical properties of the canine ventricle, we elected to use that model as the basis for the CVM model. Consequently, the CVM model, like its predecessor, recreates the midmyocardial or M cell action potential. Further alterations of various currents, including IKs, Ito and INaCa, would be required to model the electrical activity of canine endocardial and epicardial myocytes (13, 29).

The CVM model contains the following ionic current formulations
<FR><NU>d<IT>V</IT></NU><DE>d<IT>t</IT></DE></FR><IT>=</IT>−(<IT>I</IT><SUB>stim</SUB><IT>+I</IT><SUB>Na</SUB><IT>+I</IT><SUB>Kl</SUB><IT>+I</IT><SUB>Kr</SUB><IT>+I</IT><SUB>Ks</SUB><IT>+I</IT><SUB>to</SUB><IT>+I</IT><SUB>Kp</SUB><IT>+I</IT><SUB>NaK</SUB>

<IT>+I</IT><SUB>NaCa</SUB><IT>+I</IT><SUB>Nab</SUB><IT>+I</IT><SUB>Cab</SUB><IT>+I</IT><SUB>pCa</SUB><IT>+I</IT><SUB>Ca</SUB><IT>+I</IT><SUB>CaK</SUB>)

Stimulus current. Istim used to drive the model was a square wave pulse consisting of -80 µA/µF of current for 1 ms.

Sodium current. INa was the same as that used in the Winslow model (26) except that the discontinuities in the h and j gate formulations were removed.
<AR><R><C>I<SUB>Na</SUB><IT>=<A><AC>G</AC><AC>&cjs1171;</AC></A></IT><SUB>Na</SUB><IT>m</IT><SUP>3</SUP><IT>hj</IT>(<IT>V−E</IT><SUB>Na</SUB>)</C><C>  &agr;<SUB>m</SUB>=0.32 <FR><NU><IT>V+</IT>47.13</NU><DE>1<IT>−e</IT><SUP><IT>−</IT>0.1(<IT>V+</IT>47.13)</SUP></DE></FR></C></R><R><C><FR><NU>d<IT>m</IT></NU><DE>d<IT>t</IT></DE></FR><IT>=&agr;<SUB>m</SUB></IT>(1<IT>−m</IT>)<IT>−&bgr;<SUB>m</SUB>m</IT></C><C>  &bgr;<SUB>m</SUB>=0.08<IT>e</IT><SUP>−<IT>V/</IT>11</SUP></C></R><R><C><FR><NU>d<IT>h</IT></NU><DE>d<IT>t</IT></DE></FR><IT>=&agr;<SUB>h</SUB></IT>(1<IT>−h</IT>)<IT>−&bgr;<SUB>h</SUB>h</IT></C><C>  &agr;<SUB>h</SUB>=0.135<IT>e</IT><SUP>(<IT>V</IT>+80)<IT>/</IT>−6.8</SUP></C></R><R><C><FR><NU>d<IT>j</IT></NU><DE>d<IT>t</IT></DE></FR><IT>=&agr;<SUB>j</SUB></IT>(1<IT>−j</IT>)<IT>−&bgr;<SUB>j</SUB>j</IT></C><C>  &bgr;<SUB>h</SUB>=<FR><NU>7.5</NU><DE>1<IT>+e</IT><SUP><IT>−</IT>0.1(<IT>V+</IT>11)</SUP></DE></FR></C></R><R><C>E<SUB>Na</SUB><IT>=</IT><FR><NU><IT>R</IT>T</NU><DE><IT>F</IT></DE></FR> ln<FENCE><FR><NU>[Na<SUP><IT>+</IT></SUP>]<SUB>o</SUB></NU><DE>[Na<SUP><IT>+</IT></SUP>]<SUB>i</SUB></DE></FR></FENCE></C><C>  &agr;<SUB>j</SUB>=<FR><NU>0.175<IT>e</IT><SUP>(<IT>V</IT>+100)/−23</SUP></NU><DE>1<IT>+e</IT><SUP>0.15(<IT>V+</IT>79)</SUP></DE></FR></C></R><R><C></C><C>  &bgr;<SUB>j</SUB>=<FR><NU>0.3</NU><DE>1<IT>+e</IT><SUP><IT>−</IT>0.1(<IT>V+</IT>32)</SUP></DE></FR></C></R></AR>

Inward rectifier K+ current. IK1 was formulated to agree with data from Freeman et al. (4). These data indicate a smaller outward current at depolarized potentials than is seen in the Winslow model
I<SUB>K1</SUB><IT>=<A><AC>G</AC><AC>&cjs1171;</AC></A></IT><SUB>K1</SUB><IT>K</IT><SUP><IT>∞</IT></SUP><SUB>1</SUB> <FR><NU>[K<SUP><IT>+</IT></SUP>]<SUB>o</SUB></NU><DE>[K<SUP><IT>+</IT></SUP>]<SUB>o</SUB><IT>+K</IT><SUB><IT>m</IT>K1</SUB></DE></FR> (<IT>V−E</IT><SUB>K</SUB>)

K<SUP>∞</SUP><SUB>1</SUB>=<FR><NU>1</NU><DE>2+e<SUP>1.62F/(RT) (<IT>V−E</IT><SUB>K</SUB>)</SUP></DE></FR>

E<SUB>K</SUB><IT>=</IT><FR><NU><IT>R</IT>T</NU><DE><IT>F</IT></DE></FR>  ln <FENCE><FR><NU>[K<SUP><IT>+</IT></SUP>]<SUB>o</SUB></NU><DE>[K<SUP><IT>+</IT></SUP>]<SUB>i</SUB></DE></FR></FENCE>

Rapid component of the delayed rectifier K+ current. IKr was fit to the data from Gintant (7). In particular, we reproduced the voltage-clamp experiment used to generate Fig. 2 in his paper. The Winslow formulation of the current was altered to increase rectification, slow kinetics at depolarized potentials, and increase maximum conductance
I<SUB>Kr</SUB><IT>=<A><AC>G</AC><AC>&cjs1171;</AC></A></IT><SUB>Kr</SUB><IT>R</IT>(<IT>V</IT>)<IT>X</IT><SUB>Kr</SUB><RAD><RCD> <FR><NU>[K<SUP><IT>+</IT></SUP>]<SUB>o</SUB></NU><DE>4</DE></FR></RCD></RAD> (<IT>V−E</IT><SUB>K</SUB>)

<FR><NU>d<IT>X</IT><SUB>Kr</SUB></NU><DE>d<IT>t</IT></DE></FR><IT>=</IT><FR><NU><IT>X</IT><SUP><IT>∞</IT></SUP><SUB>Kr</SUB><IT>−X</IT><SUB>Kr</SUB></NU><DE><IT>&tgr;</IT><SUB>Kr</SUB></DE></FR>

R(V)=<FR><NU>1</NU><DE>1+2.5e<SUP>0.1(V+28)</SUP></DE></FR>

&tgr;<SUB>Kr</SUB><IT>=</IT>43<IT>+</IT><FR><NU>1</NU><DE><IT>e</IT><SUP><IT>−</IT>5.495<IT>+</IT>0.1691<IT>V</IT></SUP><IT>+e</IT><SUP><IT>−</IT>7.677<IT>−</IT>0.0128<IT>V</IT></SUP></DE></FR>

X<SUP><IT>∞</IT></SUP><SUB>Kr</SUB><IT>=</IT><FR><NU>1</NU><DE>1<IT>+e</IT><SUP><IT>−</IT>2.182<IT>−</IT>0.1819<IT>V</IT></SUP></DE></FR>

Slow component of the delayed rectifier K+ current. IKs was fit to data from Varro et al. (25), specifically the results shown in Fig. 2 of their paper. The Winslow model was altered to increase the magnitude of the current and shift activation to less positive voltages
I<SUB>Ks</SUB><IT>=<A><AC>G</AC><AC>&cjs1171;</AC></A></IT><SUB>Ks</SUB><IT>X</IT><SUP>2</SUP><SUB>Ks</SUB>(<IT>V−E</IT><SUB>Ks</SUB>)

E<SUB>Ks</SUB><IT>=</IT><FR><NU><IT>R</IT>T</NU><DE>F</DE></FR>  ln <FENCE><FR><NU>[K<SUP><IT>+</IT></SUP>]<SUB>o</SUB><IT>+</IT>0.01833[Na<SUP><IT>+</IT></SUP>]<SUB>o</SUB></NU><DE>[K<SUP><IT>+</IT></SUP>]<SUB>i</SUB><IT>+</IT>0.01833[Na<SUP><IT>+</IT></SUP>]<SUB>i</SUB></DE></FR></FENCE>

<FR><NU>d<IT>X</IT><SUB>Ks</SUB></NU><DE>d<IT>t</IT></DE></FR><IT>=</IT><FR><NU><IT>X</IT><SUP><IT>∞</IT></SUP><SUB>Ks</SUB><IT>−X</IT><SUB>Ks</SUB></NU><DE><IT>&tgr;</IT><SUB>Ks</SUB></DE></FR>

X<SUP><IT>∞</IT></SUP><SUB>Ks</SUB><IT>=</IT><FR><NU>1</NU><DE>1<IT>+e</IT><SUP>(<IT>V</IT>−16)<IT>/</IT>−13.6</SUP></DE></FR>

&tgr;<SUB>Ks</SUB><IT>=</IT><FR><NU>1</NU><DE><FR><NU>0.0000719(<IT>V−</IT>10)</NU><DE>1<IT>−e</IT><SUP><IT>−</IT>0.148(<IT>V−</IT>10)</SUP></DE></FR><IT>+</IT><FR><NU>0.000131(<IT>V−</IT>10)</NU><DE><IT>e</IT><SUP>0.0687(<IT>V−</IT>10)</SUP><IT>−</IT>1</DE></FR></DE></FR>

Transient outward K+ current. Ito in the model was the same as that in the Winslow model
<AR><R><C>I<SUB>to</SUB><IT>=<A><AC>G</AC><AC>&cjs1171;</AC></A></IT><SUB>to</SUB><IT>X</IT><SUB>to</SUB><IT>Y</IT><SUB>to</SUB>(<IT>V−E</IT><SUB>K</SUB>)</C><C>  &agr;<SUB>X<SUB>to</SUB></SUB><IT>=</IT>0.04516<IT>e</IT><SUP>0.03577<IT>V</IT></SUP></C></R><R><C> </C><C>  &bgr;<SUB>X<SUB>to</SUB></SUB><IT>=</IT>0.0989<IT>e</IT><SUP><IT>−</IT>0.06237<IT>V</IT></SUP></C></R><R><C><FR><NU>d<IT>X</IT><SUB>to</SUB></NU><DE>d<IT>t</IT></DE></FR><IT>=&agr;</IT><SUB><IT>X</IT><SUB>to</SUB></SUB>(1<IT>−X</IT><SUB>to</SUB>)<IT>−&bgr;</IT><SUB><IT>X</IT><SUB>to</SUB></SUB><IT>X</IT><SUB>to</SUB></C><C>  &agr;<SUB>Y<SUB>to</SUB></SUB><IT>=</IT><FR><NU>0.005415<IT>e</IT><SUP>(<IT>V</IT>+33.5)<IT>/</IT>−5</SUP></NU><DE>1<IT>+</IT>0.051335<IT>e</IT><SUP>(<IT>V</IT>+33.5)<IT>/</IT>−5</SUP></DE></FR></C></R><R><C><FR><NU>d<IT>Y</IT><SUB>to</SUB></NU><DE>d<IT>t</IT></DE></FR><IT>=&agr;</IT><SUB><IT>Y</IT><SUB>to</SUB></SUB>(1<IT>−Y</IT><SUB>to</SUB>)<IT>−&bgr;</IT><SUB><IT>Y</IT><SUB>to</SUB></SUB><IT>Y</IT><SUB>to</SUB></C><C>  &bgr;<SUB>Y<SUB>to</SUB></SUB><IT>=</IT><FR><NU>0.005415<IT>e</IT><SUP>(<IT>V</IT>+33.5)<IT>/</IT>5</SUP></NU><DE>1<IT>+</IT>0.051335<IT>e</IT><SUP>(<IT>V</IT>+33.5)<IT>/</IT>5</SUP></DE></FR></C></R></AR>

Plateau K+ current. IKp was the same as that in the Winslow model
I<SUB>Kp</SUB><IT>=<A><AC>G</AC><AC>&cjs1171;</AC></A></IT><SUB>Kp</SUB><IT>K</IT><SUB>Kp</SUB>(<IT>V−E</IT><SUB>K</SUB>)

K<SUB>Kp</SUB><IT>=</IT><FR><NU>1</NU><DE>1<IT>+e</IT><SUP>(7.488−<IT>V</IT>)<IT>/</IT>5.98</SUP></DE></FR>

Na+-K+ pump current. INaK was the same as that in the LRd model
I<SUB>NaK</SUB><IT>=</IT><IT><A><AC>I</AC><AC>&cjs1171;</AC></A></IT><SUB>NaK</SUB><IT>f</IT><SUB>NaK</SUB> <FR><NU>1</NU><DE>1<IT>+</IT><FENCE><FR><NU><IT>K</IT><SUB><IT>m</IT>Nai</SUB></NU><DE>[Na<SUP><IT>+</IT></SUP>]<SUB>i</SUB></DE></FR></FENCE><SUP>1.5</SUP></DE></FR> <FR><NU>[K<SUP><IT>+</IT></SUP>]<SUB>o</SUB></NU><DE>[K<SUP><IT>+</IT></SUP>]<SUB>o</SUB><IT>+K</IT><SUB><IT>m</IT>Ko</SUB></DE></FR>

f<SUB>NaK</SUB><IT>=</IT><FR><NU>1</NU><DE>1<IT>+</IT>0.1245<IT>e</IT><SUP>−0.1<IT>VF/</IT>(<IT>R</IT>T)</SUP><IT>+</IT>0.0365<IT>&sfgr;e</IT><SUP>−<IT>VF/</IT>(<IT>R</IT>T)</SUP></DE></FR>

&sfgr;=<FR><NU>1</NU><DE>7</DE></FR> (e<SUP>[Na<SUP>+</SUP>]<SUB>o</SUB><IT>/</IT>67.3</SUP><IT>−</IT>1)

Na+/Ca2+ exchange current, sarcolemmal pump current, and Ca2+ and Na+ background currents. INaCa, IpCa, ICab, and INab were the same as those in the Winslow model
I<SUB>Nab</SUB><IT>=<A><AC>G</AC><AC>&cjs1171;</AC></A></IT><SUB>Nab</SUB>(<IT>V−E</IT><SUB>Na</SUB>)

I<SUB>NaCa</SUB><IT>=</IT><FR><NU><IT>k</IT><SUB>NaCa</SUB></NU><DE><IT>K</IT><SUP>3</SUP><SUB><IT>m</IT>Na</SUB><IT>+</IT>[Na<SUP><IT>+</IT></SUP>]<SUP>3</SUP><SUB>o</SUB></DE></FR> <FR><NU>1</NU><DE><IT>K</IT><SUB><IT>m</IT>Ca</SUB><IT>+</IT>[Ca<SUP>2<IT>+</IT></SUP>]<SUB>o</SUB></DE></FR> <FR><NU>1</NU><DE>1<IT>+k</IT><SUB>sat</SUB><IT>e</IT><SUP><IT>VF</IT>(<IT>&eegr;</IT>−1)<IT>/</IT>(<IT>R</IT>T)</SUP></DE></FR> 

<IT>×</IT>[<IT>e</IT><SUP><IT>VF&eegr;/</IT>(<IT>R</IT>T)</SUP>[Na<SUP><IT>+</IT></SUP>]<SUP>3</SUP><SUB>i</SUB>[Ca<SUP>2<IT>+</IT></SUP>]<SUB>o</SUB><IT>−e</IT><SUP><IT>VF</IT>(<IT>&eegr;</IT>−1)<IT>/</IT>(<IT>R</IT>T)</SUP> [Na<SUP><IT>+</IT></SUP>]<SUP>3</SUP><SUB>o</SUB> [Ca<SUP>2<IT>+</IT></SUP>]<SUB>i</SUB>]

I<SUB>pCa</SUB><IT>=<A><AC>I</AC><AC>&cjs1171;</AC></A></IT><SUB>pCa</SUB> <FR><NU>[Ca<SUP>2<IT>+</IT></SUP>]<SUB>i</SUB></NU><DE><IT>K</IT><SUB><IT>m</IT>pCa</SUB><IT>+</IT>[Ca<SUP>2<IT>+</IT></SUP>]<SUB>i</SUB></DE></FR>

I<SUB>Cab</SUB><IT>=<A><AC>G</AC><AC>&cjs1171;</AC></A></IT><SUB>Cab</SUB>(<IT>V−E</IT><SUB>Ca</SUB>)

E<SUB>Ca</SUB><IT>=</IT><FR><NU><IT>R</IT>T</NU><DE>2<IT>F</IT></DE></FR>  ln <FENCE><FR><NU>[Ca<SUP>2<IT>+</IT></SUP>]<SUB>o</SUB></NU><DE>[Ca<SUP>2<IT>+</IT></SUP>]<SUB>i</SUB></DE></FR></FENCE>

L-type Ca2+ channel current. ICa in the model was a modified version of that found in the LRd model. A time-dependent, enhanced Ca2+-induced inactivation was used, as well as a decrease in the current magnitude. These changes produced a smaller, more rapidly inactivating Ca2+ current, in agreement with experimental observations by A. C. Zygmunt (personal communication)
I<SUB>Ca</SUB><IT>=<A><AC>I</AC><AC>&cjs1171;</AC></A></IT><SUB>Ca</SUB><IT>fdf</IT><SUB>Ca</SUB><IT>  f<SUP>∞</SUP>=</IT><FR><NU>1</NU><DE>1<IT>+e</IT><SUP>(<IT>V</IT>+12.5)<IT>/</IT>5</SUP></DE></FR>

<A><AC>I</AC><AC>&cjs1171;</AC></A><SUB>Ca</SUB><IT>=</IT><FR><NU><IT><A><AC>P</AC><AC>&cjs1171;</AC></A></IT><SUB>Ca</SUB></NU><DE><IT>C</IT><SUB>sc</SUB></DE></FR> <FR><NU>4<IT>VF</IT><SUP>2</SUP></NU><DE><IT>R</IT>T</DE></FR> <FR><NU><AR><R><C>[Ca<SUP>2<IT>+</IT></SUP>]<SUB>i</SUB><IT>e</IT><SUP>2<IT>VF/</IT>(<IT>R</IT>T)</SUP></C></R><R><C><IT>−</IT>0.341[Ca<SUP>2<IT>+</IT></SUP>]<SUB>o</SUB></C></R></AR></NU><DE><IT>e</IT><SUP>2<IT>VF/</IT>(<IT>R</IT>T)</SUP><IT>−</IT>1</DE></FR><IT> &tgr;<SUB>f</SUB>=</IT>30<IT>+</IT><FR><NU>200</NU><DE>1<IT>+e</IT><SUP>(<IT>V</IT>+20)<IT>/</IT>9.5</SUP></DE></FR>

<FR><NU>d<IT>f</IT></NU><DE>d<IT>t</IT></DE></FR><IT>=</IT><FR><NU><IT>f<SUP>∞</SUP>−f</IT></NU><DE><IT>&tgr;<SUB>f</SUB></IT></DE></FR><IT>  d<SUP>∞</SUP>=</IT><FR><NU>1</NU><DE>1<IT>+e</IT><SUP>(<IT>V</IT>+10)<IT>/</IT>−6.24</SUP></DE></FR>

<FR><NU>d<IT>d</IT></NU><DE>d<IT>t</IT></DE></FR><IT>=</IT><FR><NU><IT>d<SUP>∞</SUP>−d</IT></NU><DE><IT>&tgr;<SUB>d</SUB></IT></DE></FR><IT>  &tgr;<SUB>d</SUB>=</IT><FR><NU>1</NU><DE><AR><R><C><FR><NU>0.25<IT>e</IT><SUP><IT>−</IT>0.01<IT>V</IT></SUP></NU><DE>1<IT>+e</IT><SUP><IT>−</IT>0.07<IT>V</IT></SUP></DE></FR></C></R><R><C><IT> +</IT><FR><NU>0.07<IT>e</IT><SUP><IT>−</IT>0.05(<IT>V+</IT>40)</SUP></NU><DE>1<IT>+e</IT><SUP>0.05(<IT>V+</IT>40)</SUP></DE></FR></C></R></AR></DE></FR>

<FR><NU>d<IT>f</IT><SUB>Ca</SUB></NU><DE>d<IT>t</IT></DE></FR><IT>=</IT><FR><NU><IT>f</IT><SUP><IT>∞</IT></SUP><SUB>Ca</SUB><IT>−f</IT><SUB>Ca</SUB></NU><DE><IT>&tgr;</IT><SUB><IT>f</IT><SUB>Ca</SUB></SUB></DE></FR><IT>  f</IT><SUP><IT>∞</IT></SUP><SUB>Ca</SUB><IT>=</IT><FR><NU>1</NU><DE>1<IT>+</IT><FENCE><FR><NU>[Ca<SUP>2<IT>+</IT></SUP>]<SUB>i</SUB></NU><DE><IT>K</IT><SUB><IT>mf</IT><SUB>Ca</SUB></SUB></DE></FR></FENCE><SUP>3</SUP></DE></FR>

  &tgr;<SUB>f<SUB>Ca</SUB></SUB><IT>=</IT>30

K+ current through the L-type Ca2+ channel. ICaK was also a modified version of the LRd formulation
I<SUB>CaK</SUB><IT>=</IT><FR><NU><IT><A><AC>P</AC><AC>&cjs1171;</AC></A></IT><SUB>CaK</SUB></NU><DE><IT>C</IT><SUB>sc</SUB></DE></FR> <FR><NU><IT>fdf</IT><SUB>Ca</SUB></NU><DE>1<IT>+</IT><FR><NU><IT><A><AC>I</AC><AC>&cjs1171;</AC></A></IT><SUB>Ca</SUB></NU><DE><IT>I</IT><SUB>Cahalf</SUB></DE></FR></DE></FR> <FR><NU>1,000<IT>VF</IT><SUP>2</SUP></NU><DE><IT>R</IT>T</DE></FR> <FR><NU>[K<SUP><IT>+</IT></SUP>]<SUB>i</SUB><IT>e</IT><SUP><IT>VF/</IT>(<IT>R</IT>T)</SUP><IT>−</IT>[K<SUP><IT>+</IT></SUP>]<SUB>o</SUB></NU><DE><IT>e</IT><SUP><IT>VF/</IT>(<IT>R</IT>T)</SUP><IT>−</IT>1</DE></FR>

Calcium handling. A modified form of the intracellular calcium dynamics from Chudin et al. (1) was used. We included buffering from calmodulin in the cytoplasm and calsequestrin in the SR, omitted spontaneous release of calcium from the SR, and combined the concentrations of calcium in the JSR and NSR into a single variable
<FR><NU>d[Ca<SUP>2<IT>+</IT></SUP>]<SUB>i</SUB></NU><DE>d<IT>t</IT></DE></FR><IT>=&bgr;<SUB>i</SUB></IT><FENCE><IT>J</IT><SUB>rel</SUB><IT>+J</IT><SUB>leak</SUB><IT>−J</IT><SUB>up</SUB><IT>−</IT><FR><NU><IT>A</IT><SUB>Cap</SUB><IT>C</IT><SUB>sc</SUB></NU><DE>2<IT>FV</IT><SUB>myo</SUB></DE></FR></FENCE>

<IT>×</IT>(<IT>I</IT><SUB>Ca<SUP>+</SUP></SUB><IT>I</IT><SUB>Cab</SUB><IT>+I</IT><SUB>pCa</SUB><IT>−</IT>2<IT>I</IT><SUB>NaCa</SUB>))

&bgr;<SUB>i</SUB>=<FENCE>1+<FR><NU>[CMDN]<SUB>tot</SUB><IT>K</IT><SUP>CMDN</SUP><SUB><IT>m</IT></SUB></NU><DE>(<IT>K</IT><SUP>CMDN</SUP><SUB><IT>m</IT></SUB><IT>+</IT>[Ca<SUP>2<IT>+</IT></SUP>]<SUB>i</SUB>)<SUP>2</SUP></DE></FR></FENCE><SUP><IT>−</IT>1</SUP>

J<SUB>rel</SUB><IT>=<A><AC>P</AC><AC>&cjs1171;</AC></A></IT><SUB>rel</SUB><IT>fdf</IT><SUB>Ca</SUB> <FR><NU><IT>&ggr;</IT>[Ca<SUP>2<IT>+</IT></SUP>]<SUB>SR</SUB><IT>−</IT>[Ca<SUP>2<IT>+</IT></SUP>]<SUB>i</SUB></NU><DE>1<IT>+</IT>1.65<IT>e</IT><SUP><IT>V&cjs0823;  </IT>20</SUP></DE></FR>

&ggr;=<FR><NU>1</NU><DE>1+<FENCE><FR><NU>2,000</NU><DE>[Ca<SUP>2<IT>+</IT></SUP>]<SUB>SR</SUB></DE></FR></FENCE><SUP>3</SUP></DE></FR>

J<SUB>up</SUB><IT>=</IT><FR><NU><IT>V</IT><SUB>up</SUB></NU><DE>1<IT>+</IT><FENCE><FR><NU><IT>K</IT><SUB><IT>m</IT><SUB>up</SUB></SUB></NU><DE>[Ca<SUP>2<IT>+</IT></SUP>]<SUB>i</SUB></DE></FR></FENCE><SUP>2</SUP></DE></FR>

J<SUB>leak</SUB><IT>=<A><AC>P</AC><AC>&cjs1171;</AC></A></IT><SUB>leak</SUB>([Ca<SUP>2<IT>+</IT></SUP>]<SUB>SR</SUB><IT>−</IT>[Ca<SUP>2<IT>+</IT></SUP>]<SUB>i</SUB>)

<FR><NU>d[Ca<SUP>2<IT>+</IT></SUP>]<SUB>SR</SUB></NU><DE>d<IT>t</IT></DE></FR><IT>=&bgr;</IT><SUB>SR</SUB>(<IT>J</IT><SUB>up</SUB><IT>−J</IT><SUB>leak</SUB><IT>−J</IT><SUB>rel</SUB>) <FR><NU><IT>V</IT><SUB>myo</SUB></NU><DE><IT>V</IT><SUB>SR</SUB></DE></FR>

&bgr;<SUB>SR</SUB><IT>=</IT><FENCE>1<IT>+</IT><FR><NU>[CSQN]<SUB>tot</SUB><IT>K</IT><SUP>CSQN</SUP><SUB><IT>m</IT></SUB></NU><DE>(<IT>K</IT><SUP>CSQN</SUP><SUB><IT>m</IT></SUB><IT>+</IT>[Ca<SUP>2<IT>+</IT></SUP>]<SUB>SR</SUB>)<SUP>2</SUP></DE></FR></FENCE><SUP><IT>−</IT>1</SUP>

Numerical methods. The equations listed above were solved using parameter values and initial conditions found in Table 1. The simulations were run on Macintosh G3 and G4 computers using a program written in C. The numerical integration scheme was similar to that used in Luo and Rudy (15, 16) and in Rush and Larsen (24). Briefly, the time steps of integration were made small enough so that the changes in voltage and in calcium concentrations remained below maximum values, Delta Vmax and Delta Camax. If the changes in voltage and calcium concentration were below a minimum value (Delta Vmin and Delta Camin), the time step was increased. By keeping the changes in voltage small, we could solve the linear gate variable equations exactly during each time step. We used the following values: Delta Vmax = 0.8 mV, Delta Vmin = 0.2 mV, Delta Camax = 1.067 × 10-2 µM, and Delta Camin=2.67 × 10-3 µM. (See Refs. 15, 16, and 24 for more details.) The other time-dependent variables in the model were solved using the adaptive fourth-order Runge-Kutta method (21). The errors were normalized as described in Jafri et al. (9). We used a maximum error of 1 × 10-6, a minimum time step of 0.005 ms, and a maximum time step of 0.5 ms. During the stimulus, the step size was fixed at 0.005 ms.

                              
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Table 1.   Parameters and initial conditions

To further increase computational speed, lookup tables were used to avoid repeatedly calculating exponentials and other computationally expensive functions. The lookup tables were calculated once before each simulation for 15,000 values of voltages ranging from -100 to +100 mV. Values of voltages lying between the indexes of the lookup table were calculated using linear interpolation. To check that these numerical techniques did not affect the accuracy of the simulation, simulations also were run using no lookup tables, with a maximum time step of 0.1 ms. The action potential durations throughout a pacedown from a pacing cycle length of 400 ms to a cycle length of 90 ms differed by <1% between the two simulations.

Restitution relations were generated using the procedure described in Koller et al. (11), where action potential duration was expressed as a function of the preceding diastolic interval. The magnitude of action potential duration alternans was defined as the difference in action potential duration between two consecutive action potentials. Action potential duration was measured to 95% of repolarization.


    RESULTS
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES

Action potential and ionic currents. Figure 1 illustrates the action potentials, ionic currents, and Ca2+ transients generated by the CVM model at a pacing cycle length of 400 ms. The action potential (Fig. 1A) was characterized by the familiar spike-and-dome morphology of canine midmyocardial cells. ICa (Fig. 1B) was of smaller magnitude and inactivated more rapidly than ICa in previous models, in agreement with the recent experimental observations of A. C. Zygmunt (private communication). The time course and magnitude of [Ca2+]i (Fig. 1D) was similar to experimental results reported previously (1, 26), indicating that the simplified calcium handling in the CVM model generated realistic Ca2+ transients.


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Fig. 1.   Action potentials, ionic currents, and Ca2+ transients generated by the CVM model after 50 beats at a cycle length of 400 ms. A: action potentials; B: ICa; C: fCa; D: [Ca2+]i; E: INaCa; F: IKr; G: IKs. See Glossary for abbreviations.

As shown in Fig. 1F, IKr increased significantly toward the end of plateau, in good agreement with the data from Gintant (7). In contrast, IKs was too small to contribute significantly to repolarization at this cycle length (Fig. 1G, note the current scale compared with Fig. 1F), primarily because of its very slow recovery from deactivation (25).

Electrical alternans. The CVM model generated electrical alternans at physiologically relevant pacing cycle lengths. Figure 2 shows the action potential and selected plateau currents at a cycle length of 180 ms, where the CVM model produced stable alternans of large magnitude. Note that ICa, fCa, and the Ca2+ transient were significantly different between the long and short action potentials, whereas peak IKr and peak inward INaCa were not. IKs varied in magnitude between the long and short action potentials, but the peak current magnitude remained small.


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Fig. 2.   Action potentials, ionic currents, and Ca2+ transients generated by the CVM model after 50 beats at a cycle length of 180 ms. A: action potentials; B: ICa; C: fCa; D: [Ca2+]i; E: INaCa; F: IKr; G: IKs.

Figure 3 shows the relationship between action potential duration and the pacing cycle length over the range of cycle lengths that produced electrical alternans (400-90 ms; Fig. 3A) and over a wider range of cycle lengths (8,000-90 ms; Fig. 3C). Action potentials generated at several different pacing cycle lengths are shown in Fig. 3D. The model generated electrical alternans over a wide range of pacing cycle lengths, in association with a region of the restitution relation having a slope equal to 1 (Fig. 3B). At cycle lengths <150 ms, alternans was absent. The initial increase in alternans magnitude as the pacing cycle length was shortened, followed by a subsequent decrease in alternans magnitude with a further shortening of the cycle length, is in good agreement with experimental data (11).


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Fig. 3.   Action potentials generated by the CVM model at pacing cycle lengths of 8,000-90 ms. Two-to-one block occurred at a cycle length of 80 ms. A: action potential duration (APD) plotted as a function of the basic cycle length (BCL) of pacing over a BCL range of 90-400 ms. B: APD restitution, where APD is plotted as a function of the diastolic interval (DI) for DI <210 ms. The solid line has a slope of 1. Note that alternans occurred where the slope of the restitution relation was >= 1. C: APD as a function of BCL over a BCL range of 90-8,000 ms. D: examples of action potentials at BCL = 300, 500, 1,000, 2,000, 4,000, and 8,000 ms. Over this range of BCL, resting membrane potential = -94 mV, action potential amplitude = 139 mV, overshoot = 45 mV, and maximum dV/dt = 278-280 V/s. See Glossary for abbreviations.

Role of plateau Na+ and Ca2+ currents in alternans. The large difference in ICa between the long and short action potentials shown in Fig. 2 suggests that ICa contributes significantly to the development of alternans. Experiments using calcium channel blockers also have indicated that ICa may mediate alternans (23). To simulate the effects of a generic calcium channel blocker in the model, we decreased the magnitude of ICa by 20%. Figure 4 shows the action potential and plateau currents in the decreased ICa model at a pacing cycle length of 180 ms. No alternans of ICa or action potential duration occurred at this or any other pacing cycle length. As expected, the restitution relation lacked a region of slope equal to 1 (Fig. 5A).


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Fig. 4.   Action potentials, ionic currents, and Ca2+ transients generated by the reduced ICa CVM model at a pacing cycle length of 180 ms. A: action potentials; B: ICa; C: fCa; D: [Ca2+]i; E: INaCa; F: IKr; G: IKs.



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Fig. 5.   Relationship between APD and DI in the CVM model after reducing PCa by 20% (A), increasing <A><AC>G</AC><AC>&cjs1171;</AC></A>K1 by 7% (B), increasing <A><AC>G</AC><AC>&cjs1171;</AC></A>Kr by 62% (C), and increasing <A><AC>G</AC><AC>&cjs1171;</AC></A>Ks by 14.3-fold (D). The solid line has a slope of 1. See Glossary for abbreviations.

The elimination of alternans in the reduced ICa model was mediated primarily by alterations of calcium-induced inactivation of ICa and the resultant changes in action potential duration (Fig. 6A). After a long diastolic interval, calcium-induced inactivation recovered to a nearly maximal value, which resulted in a large ICa during the next action potential and a correspondingly long action potential duration. Because of the long action potential duration, the next diastolic interval was shortened. Consequently, the calcium-induced inactivation gate did not recover fully by the time the next stimulus was applied. The subsequent ICa was smaller, causing a shorter action potential duration. A long diastolic interval followed the short action potential duration, and the cycle repeated.


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Fig. 6.   Relationship among the kinetics of the calcium-induced inactivation gate (fCa), APD, DI, and the time course of ICa in the normal CVM model (A) and in the reduced ICa model (B) at a pacing cycle length of 180 ms. See text for discussion and Glossary for abbreviations.

When ICa was diminished, the action potential duration was shortened, resulting in a prolongation of diastolic interval (Fig. 6B). The longer diastolic interval allowed for complete recovery of fCa. Consequently, ICa was constant for each action potential, although reduced in magnitude.

According to the scenario described above, not only should a reduction of ICa decrease alternans magnitude, but an increase in ICa should increase alternans magnitude. To test this hypothesis, the magnitude of ICa was varied, and the resultant magnitude of action potential duration alternans was measured. As shown in Fig. 7, alternans magnitude was proportional to the magnitude of ICa. In addition, alternans magnitude could be altered predictably by varying the time constant for calcium-induced inactivation (tau fCa), where decreasing tau fCa eliminated alternans of ICa and action potential duration, secondary to a reduction in the magnitude of ICa, and increasing tau fCa had the opposite effects.


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Fig. 7.   Dose-response relationships between ionic current magnitude and alternans magnitude in the CVM model. Shown are the maximum magnitudes of APD alternans as a function of a particular model parameter. open circle , Control parameter value. Left (from top to bottom): maximum INa conductance, INaCa, maximum ICa permeability, time constant for fCa, and maximum conductance for Ito. Right (from top to bottom): maximum conductance for IKr, time constant for IKr, maximum conductance for IKs, maximum conductance for IK1, and maximum conductance for IKp. See Glossary for abbreviations.

The magnitude of action potential duration alternans also could be altered by changing the magnitude of INa and INaCa (Fig. 7). As INa was increased (by increasing <A><AC>G</AC><AC>&cjs1171;</AC></A>Na), alternans magnitude decreased. Conversely, alternans magnitude was increased after a reduction of INa. Both increases and decreases of INaCa, secondary to alterations of kNaCa, reduced the magnitude of action potential duration alternans.

Role of repolarizing K+ currents in alternans. The effects of altering Ito, IKp, IK1, IKr, and IKs on alternans also were determined (Fig. 7). The magnitude of each of the currents was increased individually until alternans no longer occurred during pacing at any cycle length. Elimination of alternans occurred after increasing Ito by >= 10%, IK1 by >= 7%, or IKr by >= 62%. A substantially greater increase in the magnitude of IKs or IKp was required to eliminate alternans. Decreasing the magnitude each of the K+ currents increased the magnitude of action potential duration alternans with the exception of Ito, where decreasing the magnitude of the current decreased the alternans magnitude.

Increasing IK1, IKr, or IKs reduced action potential duration from a control value of 220 ms to 211, 211 and 197 ms, respectively, at a pacing cycle length of 1,000 ms. Despite the reduction in action potential duration, the magnitudes of ICa and the Ca2+ transient were minimally affected, both at short pacing cycle lengths (compare Figs. 2 and 8) and at a cycle length of 1,000 ms: peak ICa magnitudes for control and elevated IK1, IKr, and IKs were -1.57, -1.57, -1.57, and -1.58 pA/pF, respectively, and peak [Ca2+]i magnitudes were 2.15, 2.10, 2.12, and 2.04 µM, respectively.


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Fig. 8.   [Ca2+]i (left) and ICa (right) in the CVM model after increasing IK1 (A and B), IKr (C and D), or IKs (E and F). See Glossary for abbreviations.


    DISCUSSION
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
REFERENCES

We developed an ionic model of the canine ventricular muscle cell that generates physiologically realistic action potential duration alternans characterized by a large magnitude and a wide range of pacing cycle lengths over which they appear. Action potential duration alternans was caused primarily by an alternans of ICa, where the latter resulted from the time-dependent behavior of the calcium-induced inactivation gate, fCa. Alternans was suppressed by reducing the magnitude of ICa as well as by increasing the magnitude of selected repolarizing K+ currents. Although the CVM model has some limitations, as discussed below, it is the first ionic model of the CVM that reproduces physiological alternans at rapid pacing rates. As such, it provides a useful simulation tool for studying the complicated interactions of cardiac membrane currents.

Role of ICa in alternans. The development of action potential duration alternans required that 1) the duration of the action potential have a sensitive dependence on ICa and 2) the recovery of ICa have a sensitive dependence on diastolic interval. The first condition applied so long as there was a relative balance of repolarizing K+ current and ICa during the action potential plateau. The second condition was manifest during pacing at short cycle lengths, where partial recovery of ICa after short diastolic intervals resulted in short action potential durations, followed by long diastolic intervals. Nearly complete recovery of ICa after long diastolic intervals produced action potentials with long durations, followed by short diastolic intervals. By this mechanism, a self-perpetuating sequence of long-short action potential durations was established. A similar mechanism likely contributed to action potential duration alternans in previously published ionic models (1, 22), although alternans of ICa was not specifically reported in those studies.

After the magnitude of ICa was reduced by decreasing PCa or increasing calcium-induced inactivation, the balance of ICa and repolarizing K+ currents was shifted in favor of the repolarizing currents, resulting in shorter action potential durations. The resultant longer diastolic intervals allowed for complete recovery of ICa, albeit to a lesser magnitude, during pacing at cycle lengths that induced alternans in the control model. At even shorter pacing cycle lengths, diastolic intervals were too short to allow full recovery of ICa. However, because of the reduced magnitude of ICa and rate-dependent accumulation of incompletely deactivated K+ current, the dependence of action potential duration on ICa was minimized and action potential durations remained consistently short. A similar mechanism accounts for the attenuation of alternans in the control model at very short pacing cycle lengths (Fig. 3).

Role of repolarizing K+ currents in alternans. Beat-to-beat alterations of IK1, IKr, and IKs appeared to play a minor role in mediating alternans. As expected, IK1, which has no time dependence, displayed no beat-to-beat variations in magnitude, whereas the beat-to-beat changes in IKs were too small to affect action potential duration appreciably at short pacing cycle lengths. Total IKr also alternated during alternans; however, peak IKr did not, suggesting that alternation of IKr resulted from alternans of action potential duration rather than vice versa.

Although the beat-to-beat variations of IK1, IKr, and IKs did not contribute appreciably to alternans, increasing any one of these currents sufficiently suppressed alternans. The mechanism for this effect was analogous to that described in Role of ICa in alternans for the suppressant effects of reducing ICa on alternans. With elevation of IK1, IKr, or IKs, the balance of repolarizing K+ currents and ICa during the action potential plateau was skewed, resulting in consistently short action potential durations. Consequently, the pattern of action potential duration and diastolic interval was similar to that shown in Fig. 6B except that ICa not only recovered fully but also achieved a larger magnitude.

Implications. Decreasing the magnitude of ICa, either experimentally (23) or in an ionic model (22), has been shown to eliminate alternans and to convert VF into a periodic rhythm. However, this approach clearly is not useful clinically because decreasing ICa decreases the Ca2+ transient, thereby reducing contractile force. With the use of the CVM model to explore other methods for eliminating alternans, we found that alternans was suppressed by increasing the magnitude of three repolarizing K+ currents: IK1, IKr, and IKs.

Given that increasing IK1, IKr, and IKs decreased action potential duration, we determined whether such shortening truncated ICa, in which case increasing K+ conductance might have the same clinical limitation as decreasing Ca2+ conductance. However, ICa was minimally affected both at short and at long pacing cycle lengths, as was the Ca2+ transient. Consequently, it is possible, at least in the CVM model, to increase K+ conductance to the point of suppressing alternans without reducing contractility.

These simulation results suggest a novel strategy for treating ventricular tachyarrhythmias. Previous attempts at treatment of such arrhythmias with pharmacological agents have been largely unsuccessful. In particular, class III antiarrhythmic drugs, which are designed to block K+ currents, have been shown to be proarrhythmic (20). The CVM simulations suggest that a new class of drugs designed to increase the magnitude of selected outward currents may be useful in preventing alternans and, therefore, in preventing the development of arrhythmias such as VF. It should be emphasized, however, that only those K+ channel agonists that reduce the slope of the action potential duration restitution relation are expected to suppress VF. Drugs such as ATP-sensitive K+ channel current agonists, which markedly increase outward K+ current and shorten action potential duration, increase the slope of the restitution relation and, presumably by that mechanism, facilitate the induction of VF (27).

Limitations. While the CVM model successfully reproduces alternans, it has several limitations. First, the formulation of ICa is based solely on the qualitative characteristics of ICa. To improve the model, ICa should conform to the results of quantitative voltage-clamp experiments, where the latter ideally have been conducted under circumstances that preserve the native behavior of ICa during pacing at short cycle lengths (e.g., no buffering of [Ca2+]i or washout of the intracellular space). Second, although the simplified calcium handling in the model reproduces physiological Ca2+ transients, it ignores several of the details of calcium release from the SR. Further work needs to be done to incorporate detailed calcium handling mechanisms such as those in the Winslow model (26). Third, the model does not include the late Na+ current, which may contribute significantly to plateau duration (28). A formulation of this current that agrees with voltage-clamp experiments also needs to be included to complete the model. Finally, it has been shown that transmural heterogeneity of the heart is caused by differences in Ito, IKs, INaCa, and the late Na+ current in endocardial, midmyocardial, and epicardial canine heart cells (13, 14, 27, 29). We hope in the future to develop specific models for canine endocardium, midmyocardium, and epicardium cells that will take these differences into account.


    ACKNOWLEDGEMENTS

These studies were supported by Integrative Graduate Education and Research Trainership Program in Nonlinear Systems National Science Foundation Grant DGE-9870631, by National Heart, Lung, and Blood Institute Grant HL-84536, and by a grant-in-aid from the American Heart Association, New York State Affiliate.


    FOOTNOTES

Address for reprint requests and other correspondence: R. F. Gilmour, Jr., Dept. of Biomedical Sciences, T7 012C VRT, Cornell Univ., Ithaca, NY 14853-6401 (E-mail: rfg2{at}cornell.edu).

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.

10.1152/ajpheart.00612.2001

Received 12 July 2001; accepted in final form 26 September 2001.


    REFERENCES
TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
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Am J Physiol Heart Circ Physiol 282(2):H516-H530
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