Luo-Rudy model development

Development of the Luo-Rudy (LR) model of the mammalian cardiac ventricular cell.

Helpful comments for the implementor / user.

Last update: November 2001

Contents

    Model development
    Studies using the LRd model
    Developments in project
    Errata
    Clickable flowchart
    References
    References with links to journal websites

Model development:
LR91 (1991), LRd94 (1994), LRd95 (1995), LRd99 (1999), and LRd00 (2000).

LR91:

Luo C.H., Rudy Y. A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction. Circ Res 68:1501-26, 1991

First formulation by Luo and Rudy, inspired by Beeler & Reuter (1977).
The model implements six transmembrane currents and, like the Beeler-Reuter model, takes into account concentration changes of intracellular Ca²⁺ only.

The transmembrane currents are:

I_Na: Na⁺ inward current. Formulation according to Beeler & Reuter, with modifications proposed by Haas et al. (1971) and Ebihara & Johnson (1980), with adjustments.
I_si: Slow (Ca²⁺) inward current. Formulation of Beeler & Reuter.
I_K: Time-dependent K⁺ current (delayed rectifier). Formulation of Beeler & Reuter, with modifications.
I_K1: Time-independent K⁺current. Original formulation.
I_Kp: Plateau K⁺current. Original formulation.
I_b: Background current.Original formulation.

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LRd94: ("d" indicates "dynamic")

Luo C.H., Rudy Y. A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes. Circ Res 74:1071-96, 1994
Luo C.H. Rudy, Y. A dynamic model of the cardiac ventricular action potential. II. Afterdepolarizations, triggered activity, and potentiation. Circ Res 74:1097-113, 1994

Major extension of the LR91 model. Serves as a basis for all subsequent studies. Includes formulation for most of the sarcolemmal currents, pumps and exchangers. Implements cell compartmentalization (myoplasm, junctional and nonjunctional sarcoplasmic reticulum), Ca²⁺ buffers in the myoplasm (troponin, calmodulin) and in the junctional sarcoplasmic reticulum (calsequestrin), and calcium-induced Ca²⁺ release. It takes into account myoplasmic concentration changes of Na⁺ and K⁺ as well as Ca²⁺ concentration changes in all three compartments. Sarcolemmal currents are normalized to cell membrane capacitance and expressed in µA/µF, not in µA/cm² (as in LR91 and Beeler-Reuter models). In the initial work, Ca²⁺ buffering was computed using Steffensen's iterative method. Later, buffering was computed analytically (see LRd95).

Sarcolemmal currents:

Currents from LR91 and specific changes:

I_Na: Reduction of gNa_max from 23 mS/cm² to 16 mS/µF.
I_Ca,L: L-type Ca²⁺ inward current. Replaces I_si (which becomes obsolete) used in LR91. Original new formulation. Note erratum below.
I_K: Square-dependence on activation gate x was incorporated.
I_K1: gK1_max at [K⁺]_o = 5.4 mmol/L was increased from 0.6047 mS/cm² to 0.75 mS/µF.
I_Kp: No changes
I_b: Replaced by I_Na,b and I_Ca,b (see "New currents" below) and therefore becomes obsolete.

New currents:

I_NaCa: Na⁺/Ca²⁺ exchanger current. Formulation according to Di Francesco & Noble (1985), with adjustments.
I_NaK: Na⁺/K⁺ ATPase current. Original formulation, inspired by Di Francesco & Noble (1985) and Rasmusson et al. (1990).
I_pCa: Ca²⁺ pump. Original formulation.
I_Ca,b: Ca²⁺ background current. Together with I_Na,b, replaces I_b from LR91, which becomes obsolete. Original formulation.
I_Na,b: Na⁺ background current. Together with I_Ca,b, replaces I_b from LR91, which becomes obsolete. Original formulation.

Intracellular calcium fluxes:

I_rel,CICR: Ca²⁺-induced Ca²⁺ release (CICR) from the junctional sarcoplasmic reticulum (JSR). Original formulation. Triggered by Ca²⁺ entry during 2 ms starting from the time of occurrence of dV/dt_max. CICR is graded (increases with increasing Ca²⁺ entry) but involves a threshold (no release for small entry of Ca²⁺, below a given threshold).
I_up: Ca²⁺ uptake into the nonjunctional sarcoplasmic reticulum (NSR). Original formulation.
I_leak: Ca²⁺ leakage from the NSR. Original formulation.
I_tr: Translocation of Ca²⁺ from the NSR to the JSR. Original formulation.

Processes specifically used to model pathophysiological conditions (not used in other studies unless explicitely stated):

Used to model cell behavior under Ca²⁺-overload conditions (resting diastolic [Ca²⁺]_{myoplasmic,free}>0.3 µmol/L):

I_ns(Ca): Non specific Ca²⁺-activated sarcolemmal current. Original formulation.
I_rel,spont: Spontaneous Ca²⁺ release from the JSR . Original formulation. Triggered by a level of buffered Ca²⁺ in the JSR exceeding a given threshold.

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LRd95:

Zeng J., Laurita K.R., Rosenbaum D.S., Rudy Y. Two components of the delayed rectifier K+ current in ventricular myocytes of the guinea pig type. Theoretical formulation and their role in repolarization. Circ Res 77:140-52, 1995

Incorporation of the two components (rapid and slow) of the delayed rectifier K⁺ current. Introduction of an analytical method to compute Ca²⁺ buffering (based on solving polynomial equations of 2^nd and 3^rd degrees), replacing Steffensen's iterative method used in LRd94.

Specific changes compared with LRd94:

I_K: I_K from LRd94 is replaced with I_Kr and I_Ks (see "New currents" below) and therefore becomes obsolete.
I_Kp: gKp_max decreased from 0.0183 to 0.00552 mS/µF.
I_Ca,L: Hill coefficient (exponent) in gate f_Ca changed from 2 to 1.

New currents:

I_Kr: Rapid component of the delayed rectifier K⁺ current. Original formulation. Maximal conductance is [K⁺]_o-dependent.
I_Ks: Slow component of the delayed rectifier K⁺ current. Original formulation. Maximal conductance is Ca²⁺-dependent.
I_Ca,T: T-type Ca²⁺ current. Original formulation.

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LRd99:

Viswanathan P.C., Shaw R.M., Rudy Y. Effects of IKr and IKs heterogeneity on action potential duration and its rate dependence: a simulation study. Circulation 99:2466-74, 1999

Refinement of I_Ks, CICR (graded release without threshold) and formulation for three different cell types: epi-, mid- and endocardial. The default model cell (control), used in subsequent studies, is epicardial unless stated otherwise.

Specific changes compared with LRd95 (for the control epicardial cell):

I_Ks: Incorporation of a second x_s gate (x_s2). The first x_s gate (x_s1) is the same as the x_s gate in LRd95. Reformulation of gKs_max and its Ca²⁺-dependence.
I_rel,CICR: Reformulation of G_rel by adding a cubic tail to its initial formulation which involved a threshold (no CICR at all for a small entry of Ca²⁺). With this formulation, CICR always occurs (graded response even for a small entry of Ca²⁺).

Unpublished changes:

I_NaK: Change of INa,K_max from 1.5 to 2 µA/µF and of Hill exponent from 1.5 to2.
I_Na,b: Increase of gNa,b_max from 0.00141 to 0.004 mS/µF.

Specific formulation for mid- and endocardial cells:

I_Ks: In the control epicardial cell, the scaling constant of gKs_max is 0.433. To model mid- and endocardial cells, this constant is changed to 0.125 and 0.289, respectively. This models an I_Ks density ratio of about 23:7:15 (exactly: 0.433:0.125:0.289) in epi-/mid-/endocardial cells.

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LRd00:

Faber G.M., Rudy Y. Action potential and contractility changes in [Na(+)](i) overloaded cardiac myocytes: a simulation study. Biophys J 78:2392-404, 2000

Reformulation of CICR and I_NaCa. Formulation of the Na⁺-activated K⁺ current, used to model cell behavior under Na⁺ overload conditions.

A sample source program code (in C/C++) can be found online.

Specific changes compared with LRd99:

I_NaCa: Reformulation according to Varghese & Sell (1997).
I_NaK: Increase of INa,K_max from 2 to 2.25 µA/µF.
I_up: Iup_max increased from 0.005 to 0.00875 mmol/L/ms.
I_rel,CICR: Original reformulation. Triggered by Ca²⁺ entry starting from the time of occurrence of dV/dt_max. CICR is graded, without threshold.

Processes specifically used to model pathophysiological conditions (not used in other studies unless explicitely stated):

Used to model cell behavior under Na⁺-overload conditions ([Na⁺]_i >10 mmol/L):

I_K(Na): Na⁺-activated K⁺ current. Original formulation.

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Studies using the LRd model

These studies used one of the versions of the LRd model but did not introduce perennial changes. So far, proposed changes to the model were used only in the specific contribution (unless stated otherwise). Many of the changes were used to evaluate processes under pathophysiological conditions.

These studies are represented by gray boxes in the flowchart.

Shaw R.M., Rudy Y. The vulnerable window for unidirectional block in cardiac tissue: characterization and dependence on membrane excitability and intercellular coupling. J Cardiovasc Electrophysiol 6:115-31, 1995

LRd94. Study of vulnerable window for unidirectional block in multicellular fibers of LRd cells. Zeng J., Rudy Y. Early afterdepolarizations in cardiac myocytes: mechanism and rate dependence. Biophys J 68:949-64, 1995 LRd94. Mechanistic study of plateau early afterdepolarizations (EADs) and delayed afterdepolarizations (DADs) induced by simulated pharmacological interventions (cesium, Bay K8644, isoproterenol). In this study, a pericellular cleft space was modeled, with a slow diffusion time constant (1 s) for extracellular ions between the cleft and the extracellular bulk medium. This cleft formulation was reused once in the study of Faber et al., 2000, and is found in the sample source code of the LRd00 model (online). This cleft is not an integral part of the LRd model, but can be added as needed. Shaw R.M., Rudy Y. Electrophysiologic effects of acute myocardial ischemia: a theoretical study of altered cell excitability and action potential duration. Cardiovasc Res 35:256-72, 1997 LRd95. Formulation for mechanisms involved during acute ischemia (hyperkalemia, anoxia, acidosis, I_K,ATP). Shaw R.M., Rudy Y. Electrophysiologic effects of acute myocardial ischemia. A mechanistic investigation of action potential conduction and conduction failure. Circ Res 80:124-38, 1997 LRd95. Study of conduction during ischemia (ischemic cell formulation: see above). Shaw R.M., Rudy Y. Ionic mechanisms of propagation in cardiac tissue. Roles of the sodium and L-type calcium currents during reduced excitability and decreased gap junction coupling. Circ Res 81:727-41, 1997 LRd95. Study of conduction during reduced excitability and gap junctional coupling. Formulation of the safety factor. Clancy C.E., Rudy Y. Linking a genetic defect to its cellular phenotype in a cardiac arrhythmia. Nature 400:566-9, 1999 LRd95. Formulation of a Markovian model of wild-type and mutant INa. The wild-type formulation was reused in the subsequent studies of Clancy et al. Viswanathan, P.C., Rudy Y. Pause induced early afterdepolarizations in the long QT syndrome: a simulation study. Cardiovasc Res 42:530-42, 1999 LRd99. Single-cell study of EADs. Hodgkin-Huxley formulation for the LQT 1, LQT 2 and LQT 3 syndromes. Viswanathan P.C. Rudy Y.Cellular arrhythmogenic effects of congenital and acquired long-QT syndrome in the heterogeneous myocardium. Circulation 101:1192-8, 2000 LRd99. Study of EADs in conducted impulses. Formulation for LQT syndromes. Wang Y., Rudy Y. Action potential propagation in inhomogeneous cardiac tissue: safety factor considerations and ionic mechanism. Am J Physiol 278:H1019-29, 2000 LRd99. Study of conduction in discontinuous tissue containing structural heterogeneities. Extension of safety factor definition. Hund T.J., Otani N.F., Rudy Y. Dynamics of action potential head-tail interaction during reentry in cardiac tissue: ionic mechanisms. Am J Physiol 279:H1869-79, 2000 LRd99. Study of long-term reentry in rings of LRd cells. Hund T.J., Rudy Y. Determinants of excitability in cardiac myocytes: mechanistic investigation of memory effect. Biophys J 79:3095-104, 2000 LRd00. Study of effects of long-term concentration changes in paced LRd cells. Clancy C.E., Rudy Y. Cellular consequences of HERG mutations in the long QT syndrome: precursors to sudden cardiac death. Cardiovasc Res 50:301-13, 2001 LRd99 with Markovian INa. Formulation of a Markovian model of wild-type and mutant IKr. Hund T.J., Kucera J.P., Otani N.F., Rudy Y. Charge conservation and long-term steady-state in the Luo-Rudy dynamic cell model. Biophys J, 81:3324-31, 2001 LRd00. Study of ionic charge conservation and model stability during long periods of pacing. Kucera J.P., Rudy Y. Mechanistic insights into very slow conduction in branching cardiac tissue: a model study. Circ Res 89:799-806, 2001 LRd00. Study of conduction in branching tissue structures. Clancy C. E., Rudy Y. A Na+ channel mutation that causes both Brugada and long QT syndrome phenotypes. Circulation :in revision 2001 LRd99 with Markovian INa Incorporation of the transient outward current (I_to) and study of a Na⁺ channel mutation that cause both LQT and Brugada syndromes. Gima K., Rudy Y. Ionic basis of electrocardiographic waveforms. :in preparation 2001 LRd00 with CICR from LRd99. Incorporation of the transient outward current (I_to) and I_K,ATP (Shaw et al., 1997). Study of the ionic basis of the T and J waves. Simulation of ST segment and T wave morphology during ischemia and in the LQT and Brugada syndromes. Back to top of page

Developments in project

Incorporation of I_to.
Markovian formulation for I_Ca,L, I_Ks and CICR.
Development of a model based on canine ionic currents.

Errata

I_Ca,L (LRd94, Luo and Rudy, 1994):
In Circ Res 74:1071-96, 1994, the equation for the steady-state of activation gate f of I_Ca,L is erroneous. The correct equation is in Circ Res 74:1097-113, 1994.

I_K(ATP) (Shaw and Rudy, 1997):
In Cardiovasc Res 35:256-72, 1997, k_0.5 (k_1/2) in the equation for P_ATP should be 0.250 µmol/L, not 0.250 mmol/L. The correct value is in Circ Res 80:124-38, 1997.

Markovian I_Na (Clancy and Rudy, 1999):
In Nature, 400:566-9, 1999, some equations for the rate constants are erroneous. The entire set of correct equations can be found online (http://www.cwru.edu/med/CBRTC/LRdOnline/markovina.htm).

I_K(Na) (LRd00, Faber and Rudy, 2000):
In Biophys J, 78:2392-404, 2000, the unit for gK(Na)_max should be µS/µF, not µS/cm².

Flowchart

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References

Beeler, G. W., and H. Reuter. 1977. Reconstruction of the action potential of ventricular myocardial fibres. J Physiol. 268:177-210.
Clancy, C. E., and Y. Rudy. 1999. Linking a genetic defect to its cellular phenotype in a cardiac arrhythmia. Nature. 400:566-9.
Clancy, C. E., and Y. Rudy. 2001a. Cellular consequences of HERG mutations in the long QT syndrome: precursors to sudden cardiac death. Cardiovasc Res. 50:301-13.
Clancy, C. E., and Y. Rudy. 2001b. A Na+ channel mutation that causes both Brugada and long QT syndrome phenotypes. Circulation:in revision.
DiFrancesco, D., and D. Noble. 1985. A model of cardiac electrical activity incorporating ionic pumps and concentration changes. Phil Trans R Soc Lond Biol. 307:353-98.
Ebihara, L., and E. A. Johnson. 1980. Fast sodium current in cardiac muscle. A quantitative description. Biophys J. 32:779-90.
Faber, G. M., and Y. Rudy. 2000. Action potential and contractility changes in [Na(+)](i) overloaded cardiac myocytes: a simulation study. Biophys J. 78:2392-404.
Gima, K., and Y. Rudy. 2001. Ionic basis of electrocardiographic waveforms. :in preparation.
Haas, H. G., R. Kern, H. M. Einwachter, and M. Tarr. 1971. Kinetics of Na inactivation in frog atria. Pflügers Arch Eur J Physiol. 323:141-57.
Hund, T. J., J. P. Kucera, N. F. Otani, and Y. Rudy. 2001. Charge conservation and long-term steady-state in the Luo-Rudy dynamic cell model. Biophys J. 81:3324-31.
Hund, T. J., N. F. Otani, and Y. Rudy. 2000. Dynamics of action potential head-tail interaction during reentry in cardiac tissue: ionic mechanisms. Am J Physiol Heart Circ Physiol. 279:H1869-79.
Hund, T. J., and Y. Rudy. 2000. Determinants of excitability in cardiac myocytes: mechanistic investigation of memory effect. Biophys J. 79:3095-104.
Kucera, J. P., and Y. Rudy. 2001. Mechanistic insights into very slow conduction in branching cardiac tissue: a model study. Circ Res. 89:799-806.
Luo, C. H., and Y. Rudy. 1991. A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction. Circ Res. 68:1501-26.
Luo, C. H., and Y. Rudy. 1994a. A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes. Circ Res. 74:1071-96.
Luo, C. H., and Y. Rudy. 1994b. A dynamic model of the cardiac ventricular action potential. II. Afterdepolarizations, triggered activity, and potentiation. Circ Res. 74:1097-113.
Rasmusson, R. L., J. W. Clark, W. R. Giles, K. Robinson, R. B. Clark, E. F. Shibata, and D. L. Campbell. 1990. A mathematical model of electrophysiological activity in a bullfrog atrial cell. Am J Physiol Heart Circ Physiol. 259:H370-89.
Shaw, R. M., and Y. Rudy. 1995. The vulnerable window for unidirectional block in cardiac tissue: characterization and dependence on membrane excitability and intercellular coupling. J Cardiovasc Electrophysiol. 6:115-31.
Shaw, R. M., and Y. Rudy. 1997a. Electrophysiologic effects of acute myocardial ischemia. A mechanistic investigation of action potential conduction and conduction failure. Circ Res. 80:124-38.
Shaw, R. M., and Y. Rudy. 1997b. Electrophysiologic effects of acute myocardial ischemia: a theoretical study of altered cell excitability and action potential duration. Cardiovasc Res. 35:256-72.
Shaw, R. M., and Y. Rudy. 1997c. Ionic mechanisms of propagation in cardiac tissue. Roles of the sodium and L-type calcium currents during reduced excitability and decreased gap junction coupling. Circ Res. 81:727-41.
Varghese, A., and G. R. Sell. 1997. A conservation principle and its effect on the formulation of Na-Ca exchanger current in cardiac cells. J Theor Biol. 189:33-40.
Viswanathan, P. C., and Y. Rudy. 1999. Pause induced early afterdepolarizations in the long QT syndrome: a simulation study. Cardiovasc Res. 42:530-42.
Viswanathan, P. C., and Y. Rudy. 2000. Cellular arrhythmogenic effects of congenital and acquired long-QT syndrome in the heterogeneous myocardium. Circulation. 101:1192-8.
Viswanathan, P. C., R. M. Shaw, and Y. Rudy. 1999. Effects of IKr and IKs heterogeneity on action potential duration and its rate dependence: a simulation study. Circulation. 99:2466-74.
Wang, Y., and Y. Rudy. 2000. Action potential propagation in inhomogeneous cardiac tissue: safety factor considerations and ionic mechanism. Am J Physiol Heart Circ Physiol. 278:H1019-29.
Zeng, J., K. R. Laurita, D. S. Rosenbaum, and Y. Rudy. 1995. Two components of the delayed rectifier K+ current in ventricular myocytes of the guinea pig type. Theoretical formulation and their role in repolarization. Circ Res. 77:140-52.
Zeng, J., and Y. Rudy. 1995. Early afterdepolarizations in cardiac myocytes: mechanism and rate dependence. Biophys J. 68:949-64.

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