Research

HISTORY OF MODEL DEVELOPMENT

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Contents

Model Development
Studies using the LRd model

Errata

Development Flowchart

Model Development:

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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:

INa : Na+ inward current. Formulation according to Beeler & Reuter, with modifications proposed by Haas et al. (1971) and Ebihara & Johnson (1980), with adjustments.

Isi : Slow (Ca2+ ) inward current. Formulation of Beeler & Reuter.

IK :Time-dependent K+ current (delayed rectifier). Formulation of Beeler & Reuter, with modifications.

IK1 :Time-independent K+ current. Original formulation.

IKp : Plateau K+ current. Original formulation.

Ib : 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-inducedCa2+ 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:

INa : Reduction of gNamax from 23 mS/cm2 to 16 mS/µF.

ICa,L : L-type Ca2+ inward current. Replaces I si (which becomes obsolete) used in LR91 . Original new formulation. Note erratum below .

IK : Square-dependence on activation gate x was incorporated.

IK1 : gK1 max at [K+]o = 5.4 mmol/L was increased from 0.6047 mS/cm2 to 0.75 mS/µF.

IKp : No changes

Ib : Replaced by INa,b and ICa,b (see "New currents" below) and therefore becomes obsolete.

New currents:

INaCa : Na+ /Ca2+ exchanger current. Formulation according to Di Francesco & Noble (1985), with adjustments.

INaK : Na+ /K+ ATPase current. Original formulation, inspired by Di Francesco & Noble (1985) and Rasmusson et al. (1990) .

IpCa : Ca2+ pump. Original formulation.

ICa,b : Ca2+ background current. Together with INa,b , replaces Ib from LR91 , which becomes obsolete. Original formulation.

INa,b : Na+ background current. Together with ICa,b , replaces Ib from LR91 , which becomes obsolete. Original formulation.

Intracellular calcium fluxes:

Irel,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/dtmax . CICR is graded (increases with increasing Ca2+ entry) but involves a threshold (no release for small entry of Ca2+ , below a given threshold) .

Iup : Ca2+ uptake into the nonjunctional sarcoplasmic reticulum (NSR). Original formulation.

Ileak : Ca2+ leakage from the NSR. Original formulation.

Itr : 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):

Ins(Ca) : Non specific Ca2+-activated sarcolemmal current. Original formulation.

Irel,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 2nd and 3rd degrees), replacing Steffensen's iterative method used in LRd94 .

Specific changes compared with LRd94:

IK : IK from LRd94 is replaced with IKr and IKs (see "New currents" below) and therefore becomes obsolete.

IKp : gKpmax decreased from 0.0183 to 0.00552 mS/µF.

ICa,L : Hill coefficient (exponent) in gate fCa changed from 2 to 1.

New currents:

IKr : Rapid component of the delayed rectifier K+ current. Original formulation. Maximal conductance is [K+]o -dependent.

IKs : Slow component of the delayed rectifier K+ current. Original formulation. Maximal conductance is Ca2+ -dependent.

ICa,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 IKs , 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):

IKs : Incorporation of a second xs gate ( xs2 ). The first xs gate ( xs1 ) is the same as the xs gate in LRd95 . Reformulation of gKsmax and its Ca2+ -dependence.

Irel,CICR : Reformulation of Grel 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:

INaK : Change of INa,Kmax from 1.5 to 2 µA/µF and of Hill exponent from 1.5 to2.

INa,b : Increase of gNa,bmax from 0.00141 to 0.004 mS/µF.

Specific formulation for mid- and endocardial cells:

IKs : In the control epicardial cell, the scaling constant of gKsmax is 0.433. To model mid- and endocardial cells, this constant is changed to 0.125 and 0.289, respectively. This models an IKs 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 INaCa . 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 :

INaCa : Reformulation according to Varghese & Sell (1997).

INaK : Increase of INa,Kmax from 2 to 2.25 µA/µF.

Iup : Iupmax increased from 0.005 to 0.00875 mmol/L/ms.

Irel,CICR : Original reformulation. Triggered by Ca2+ entry starting from the time of occurrence of dV/dtmax . 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):

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

How to calculate changes in [Ca 2+]i:

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LRd07:

Livshitz L.M., Rudy Y. Regulation of Ca2+ and electrical alternans in cardiac myocytes: role of CaMKII and repolarizing currents. Am J Physiol Heart Circ Physiol 292:H2854-H2866, 2007

Livshitz and Rudy modified LRd, incorporating new findings in the relationship between Ca-transient, ICa(L) and SR Ca loading. This version of LRd shows AP and Ca-transient alternans at very fast pacing.

For more on this paper click here.

Matlab code available: LRd07 (click here) and HRd07 (click here)


Errata

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

IK(ATP) (Shaw and Rudy, 1997):
In Cardiovasc Res 35:256-72, 1997, k0.5 (k1/2 ) in the equation for PATP should be 0.250 µmol/L, not 0.250 mmol/L. The correct value is in Circ Res 80:124-38, 1997.

Markovian INa (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.

(Viswanathan, Shaw and Rudy 1999): GKr and GKs conductances are mismatched in the legend to Fig. 2A. To reproduce figure 2A, GKr should be 0.028 and GKs should be 0.65.

IK(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.

Irel ( LRd00 , Faber and Rudy, 2000):
In Biophys J, 78:2392-404, 2000, the value for t should be set to zero at the time of dICa,total/dtmax, not dV/dtmax.

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Development Flowchart


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