The Force-frequency Relationship: Insights from Mathematical Modeling

Overview

Journal Adv Physiol Educ

Publisher American Physiological Society

Specialties Medical Education
Physiology

Date 2013 Mar 9

PMID 23471245

Citations 8

Authors

Jose L Puglisi

Jorge A Negroni

Ye Chen-Izu

Donald M Bers

Affiliations

Soon will be listed here.

Abstract

The force-frequency relationship has intrigued researchers since its discovery by Bowditch in 1871. Many attempts have been made to construct mathematical descriptions of this phenomenon, beginning with the simple formulation of Koch-Wesser and Blinks in 1963 to the most sophisticated ones of today. This property of cardiac muscle is amplified by β-adrenergic stimulation, and, in a coordinated way, the neurohumoral state alters both frequency (acting on the sinoatrial node) as well as force generation (modifying ventricular myocytes). This synchronized tuning is needed to meet new metabolic demands. Cardiac modelers have already linked mechanical and electrical activity in their formulations and showed how those activities feedback on each other. However, now it is necessary to include neurological control to have a complete description of heart performance, especially when changes in frequency are involved. Study of arrhythmias (or antiarrhythmic drugs) based on mathematical models should incorporate this effect to make useful predictions or point out potential pharmaceutical targets.

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References

Manring A, HOLLANDER P . The interval-strength relationship in mammalian atrium: a calcium exchange model. I. Theory. Biophys J. 1971; 11(6):483-501. PMC: 1484006. DOI: 10.1016/S0006-3495(71)86230-6. View

Kuzumoto M, Takeuchi A, Nakai H, Oka C, Noma A, Matsuoka S . Simulation analysis of intracellular Na+ and Cl- homeostasis during beta 1-adrenergic stimulation of cardiac myocyte. Prog Biophys Mol Biol. 2007; 96(1-3):171-86. DOI: 10.1016/j.pbiomolbio.2007.07.005. View

Luo C, Rudy Y . A dynamic model of the cardiac ventricular action potential. II. Afterdepolarizations, triggered activity, and potentiation. Circ Res. 1994; 74(6):1097-113. DOI: 10.1161/01.res.74.6.1097. View

Negroni J, Lascano E . Simulation of steady state and transient cardiac muscle response experiments with a Huxley-based contraction model. J Mol Cell Cardiol. 2008; 45(2):300-12. DOI: 10.1016/j.yjmcc.2008.04.012. View

Lab M . Contraction-excitation feedback in myocardium. Physiological basis and clinical relevance. Circ Res. 1982; 50(6):757-66. DOI: 10.1161/01.res.50.6.757. View

Takeuchi A, Tatsumi S, Sarai N, Terashima K, Matsuoka S, Noma A . Ionic mechanisms of cardiac cell swelling induced by blocking Na+/K+ pump as revealed by experiments and simulation. J Gen Physiol. 2006; 128(5):495-507. PMC: 2151580. DOI: 10.1085/jgp.200609646. View

Miura T, Miyazaki S, Guth B, Kambayashi M, ROSS Jr J . Influence of the force-frequency relation on left ventricular function during exercise in conscious dogs. Circulation. 1992; 86(2):563-71. DOI: 10.1161/01.cir.86.2.563. View

Rice J, Wang F, Bers D, de Tombe P . Approximate model of cooperative activation and crossbridge cycling in cardiac muscle using ordinary differential equations. Biophys J. 2008; 95(5):2368-90. PMC: 2517033. DOI: 10.1529/biophysj.107.119487. View

Kambayashi M, Miura T, Oh B, Rockman H, Murata K, ROSS Jr J . Enhancement of the force-frequency effect on myocardial contractility by adrenergic stimulation in conscious dogs. Circulation. 1992; 86(2):572-80. DOI: 10.1161/01.cir.86.2.572. View

10.

Bers D . Cardiac excitation-contraction coupling. Nature. 2002; 415(6868):198-205. DOI: 10.1038/415198a. View

11.

Luo C, Rudy Y . A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction. Circ Res. 1991; 68(6):1501-26. DOI: 10.1161/01.res.68.6.1501. View

12.

DeSantiago J, Maier L, Bers D . Frequency-dependent acceleration of relaxation in the heart depends on CaMKII, but not phospholamban. J Mol Cell Cardiol. 2002; 34(8):975-84. DOI: 10.1006/jmcc.2002.2034. View

13.

Rice J, de Tombe P . Approaches to modeling crossbridges and calcium-dependent activation in cardiac muscle. Prog Biophys Mol Biol. 2004; 85(2-3):179-95. DOI: 10.1016/j.pbiomolbio.2004.01.011. View

14.

Puglisi J, Bers D . LabHEART: an interactive computer model of rabbit ventricular myocyte ion channels and Ca transport. Am J Physiol Cell Physiol. 2001; 281(6):C2049-60. DOI: 10.1152/ajpcell.2001.281.6.C2049. View

15.

R Shannon T, Wang F, Puglisi J, Weber C, Bers D . A mathematical treatment of integrated Ca dynamics within the ventricular myocyte. Biophys J. 2004; 87(5):3351-71. PMC: 1304803. DOI: 10.1529/biophysj.104.047449. View

16.

Hajdu S, SZENT-GYORGYI A . Action of DOC and serum on the frog heart. Am J Physiol. 1952; 168(1):159-70. DOI: 10.1152/ajplegacy.1951.168.1.159. View

17.

Hilgemann D, Noble D . Excitation-contraction coupling and extracellular calcium transients in rabbit atrium: reconstruction of basic cellular mechanisms. Proc R Soc Lond B Biol Sci. 1987; 230(1259):163-205. DOI: 10.1098/rspb.1987.0015. View

18.

Razumova M, Bukatina A, Campbell K . Stiffness-distortion sarcomere model for muscle simulation. J Appl Physiol (1985). 1999; 87(5):1861-76. DOI: 10.1152/jappl.1999.87.5.1861. View

19.

ROSS Jr J . Adrenergic regulation of the force-frequency effect. Basic Res Cardiol. 1998; 93 Suppl 1:95-101. DOI: 10.1007/s003950050229. View

20.

Noble D . A modification of the Hodgkin--Huxley equations applicable to Purkinje fibre action and pace-maker potentials. J Physiol. 1962; 160:317-52. PMC: 1359535. DOI: 10.1113/jphysiol.1962.sp006849. View