Heart Vessels (1992) 7:200-205
Heart andVesse] s © Springer-Verlag1992
Cardiac muscle fiber force versus length determined by a cardiac muscle crossbridge model Tad W. Taylor 1'2 Yoichi G o t o 1, and Hiroyuki Suga 3 1National Cardiovascular Center Research Institute, 5-7-1 Fujishirodai, Suita, Osaka, 565 Japan 2On leave from the Yale School of Medicine 367 Cedar St., New Haven, CT 06510 USA 3Department of PhysiologyII, Okayama University Medical School, 2-5-1, Shikatacho, Okayama, 700 Japan
Summary. A mathematical model incorporating Huxley's sliding filament crossbridge muscle model coupled with parallel and series elastic components was simulated to examine force-length relations under different external calcium concentrations. Several researchers have determined experimentally in both papillary muscle preparations and in situ heart experiments that the calcium concentration (or effective concentration from inotropic agents) will affect the strength and convexity of the cardiac muscle fiber force-length relations. Simulations were performed over a several-order-of-magnitude fange of calcium concentrations in isometric contractions and these showed that the force-length curve convexity was changed. Simulation results demonstrated that increasing the stiffness in the model contractile element or series elasticity element did not change the forcelength convexity. Increasing the series elasticity element stiffness did slightly change the shape of the forcelength curve. The model predicts that the curve convexity changes as a result of the calcium-troponin interactions. Key words: Heart - Mathematical model - Crossbridge kinetics - Computer simulation - Calcium level
Introduction The purpose of this study was to apply Huxley's sliding filament crossbridge muscle model to investigate the behavior of cardiac force versus muscle fiber length
Address correspondence to: H. Suga
Received March 13, 1992; revision received June 17, 1992; accepted July 24, 1992.
relations under different calcium concentrations. The shape of the force-versus-length curve is of interest to cardiac physiologists who wish to calculate pressurevolume area diagrams to correlate with the cardiac oxygen consumption [1]. Jewell [2] stated that the shape of the relation between tension and sarcomere length in cardiac papillary muscle suggests lengthdependent activation. Lakatta and Henderson [3] stated that variations in the calcium concentration in the bathing solution could cause changes in forcelength curves and alter the shape of the force-length relations. Ter Keurs et al. [4] examined the effect of different extracellular calcium concentrations on the relations between tension and sarcomere length obtained with contractions performed at constant muscle length and constant sarcomere length, They used rat trabecular muscle under extracellular calcium concentrations varying from 0.25 to 2.5 mM, and found that at calcium levels of 0.5 mM the force-length relation was linear, while increasing the calcium concentration to 2.5 mM caused the force-length relation to become concave downward. Allen and Kentish [5] suggested that the steepness of the cardiac force-length relation arises because the degree of activation of the cardiac myofibrils by calcium increases as the muscle length increases. They cite two processes occurring: (1) the myofibril calcium sensitivity increases with increasing muscle length and (2) the amount of calcium supplied to the myofibrils during systole increases with increasing muscle length. They suggested that the first effect is the strongest. Burkhoff et al. [6] investigated the contractilitydependent curvilinearity of end-systolic pressurevolume relations and determined that changing the contractility changed the slope and the curvilinearity of the end-systolic pressure-volume relation. Previous investigators had used papillary muscle preparations; Burkhoff et al. altered the contractility of a whole ventricle by injecting positive and negative inotropic
T.W. Taylor et al.: Muscle force versus length
201
agents into t h e c o r o n a r y circulation a n d w e r e also a b l e to c h a n g e the p r e s s u r e - v o t u m e r e l a t i o n s f r o m b e i n g c o n c a v e u p w a r d to c o n c a v e d o w n w a r d as the contractile state was i n c r e a s e d .
The CE is represented by Huxley's sliding filament muscle sarcomere model [11]. The interaction of actin and myosin molecules, through the myosin crossbridge, generates muscle fiber tension. Actin and myosin kinetics from Panerai are given by: f A + M~.~A-M g
Materials and methods A Maxwell-type three-component Hill model was used to represent the papillary muscle. The structure shown in Fig. 1 was assumed, in which muscle fiber consists of a contractile element (CE), which is freely extensible at rest; a series elasticity element (SE), in series with the CE; and also a parallel elasticity element (PE) which gives the resting elasticity. Simulations were performed analogously to those of Wong [7-10] using the model illustrated in Fig. 1. Both the SE and PE were characterized by non-linear expo'nential functions, while the CE was represented by Huxley's slidingfilament model [11]. The calcium handling and troponin binding were modeled similarly to Panerai [12] with modifications for the Maxwell-type model implemented; Panerai used a Voigt-type model in his analysis. In the calciumtroponin binding analysis, it was assumed that one level of calcium was present regardless of the length in the development of the force-length relations, and calcium levels were independently varied for a given set of force-length curves. The PE and SE behavior are described by exponential relations: Ppc = Po[exp{Kp( L - Lo)} - 1]
(1)
Pso = Pl[exp(K~L,~) - 1]
(2)
and
where Po, Kp, and Ks are element constants; Pl is the preload, determined initially from Eq. 1 as done by Wong [7-10]; L o is the initial fiber length; L is the total fiber length; and Lse is the SE length as a function of time t.
(3)
where M and A are myosin and actin; A-M represents attached crossbridges; f is the association rate constant, or the probability that a myosin crossbridge will bind to an actin molecule; and g represents the dissociation rate eonstant, or the probability that an attached crossbridge will become detached. Wong [7-10] used the expressions for f(x) and g(x), where x is the distance between the myosin head equilibrium position and the reactive site: X < 0,
0
Heart andVesse] s © Springer-Verlag1992
Cardiac muscle fiber force versus length determined by a cardiac muscle crossbridge model Tad W. Taylor 1'2 Yoichi G o t o 1, and Hiroyuki Suga 3 1National Cardiovascular Center Research Institute, 5-7-1 Fujishirodai, Suita, Osaka, 565 Japan 2On leave from the Yale School of Medicine 367 Cedar St., New Haven, CT 06510 USA 3Department of PhysiologyII, Okayama University Medical School, 2-5-1, Shikatacho, Okayama, 700 Japan
Summary. A mathematical model incorporating Huxley's sliding filament crossbridge muscle model coupled with parallel and series elastic components was simulated to examine force-length relations under different external calcium concentrations. Several researchers have determined experimentally in both papillary muscle preparations and in situ heart experiments that the calcium concentration (or effective concentration from inotropic agents) will affect the strength and convexity of the cardiac muscle fiber force-length relations. Simulations were performed over a several-order-of-magnitude fange of calcium concentrations in isometric contractions and these showed that the force-length curve convexity was changed. Simulation results demonstrated that increasing the stiffness in the model contractile element or series elasticity element did not change the forcelength convexity. Increasing the series elasticity element stiffness did slightly change the shape of the forcelength curve. The model predicts that the curve convexity changes as a result of the calcium-troponin interactions. Key words: Heart - Mathematical model - Crossbridge kinetics - Computer simulation - Calcium level
Introduction The purpose of this study was to apply Huxley's sliding filament crossbridge muscle model to investigate the behavior of cardiac force versus muscle fiber length
Address correspondence to: H. Suga
Received March 13, 1992; revision received June 17, 1992; accepted July 24, 1992.
relations under different calcium concentrations. The shape of the force-versus-length curve is of interest to cardiac physiologists who wish to calculate pressurevolume area diagrams to correlate with the cardiac oxygen consumption [1]. Jewell [2] stated that the shape of the relation between tension and sarcomere length in cardiac papillary muscle suggests lengthdependent activation. Lakatta and Henderson [3] stated that variations in the calcium concentration in the bathing solution could cause changes in forcelength curves and alter the shape of the force-length relations. Ter Keurs et al. [4] examined the effect of different extracellular calcium concentrations on the relations between tension and sarcomere length obtained with contractions performed at constant muscle length and constant sarcomere length, They used rat trabecular muscle under extracellular calcium concentrations varying from 0.25 to 2.5 mM, and found that at calcium levels of 0.5 mM the force-length relation was linear, while increasing the calcium concentration to 2.5 mM caused the force-length relation to become concave downward. Allen and Kentish [5] suggested that the steepness of the cardiac force-length relation arises because the degree of activation of the cardiac myofibrils by calcium increases as the muscle length increases. They cite two processes occurring: (1) the myofibril calcium sensitivity increases with increasing muscle length and (2) the amount of calcium supplied to the myofibrils during systole increases with increasing muscle length. They suggested that the first effect is the strongest. Burkhoff et al. [6] investigated the contractilitydependent curvilinearity of end-systolic pressurevolume relations and determined that changing the contractility changed the slope and the curvilinearity of the end-systolic pressure-volume relation. Previous investigators had used papillary muscle preparations; Burkhoff et al. altered the contractility of a whole ventricle by injecting positive and negative inotropic
T.W. Taylor et al.: Muscle force versus length
201
agents into t h e c o r o n a r y circulation a n d w e r e also a b l e to c h a n g e the p r e s s u r e - v o t u m e r e l a t i o n s f r o m b e i n g c o n c a v e u p w a r d to c o n c a v e d o w n w a r d as the contractile state was i n c r e a s e d .
The CE is represented by Huxley's sliding filament muscle sarcomere model [11]. The interaction of actin and myosin molecules, through the myosin crossbridge, generates muscle fiber tension. Actin and myosin kinetics from Panerai are given by: f A + M~.~A-M g
Materials and methods A Maxwell-type three-component Hill model was used to represent the papillary muscle. The structure shown in Fig. 1 was assumed, in which muscle fiber consists of a contractile element (CE), which is freely extensible at rest; a series elasticity element (SE), in series with the CE; and also a parallel elasticity element (PE) which gives the resting elasticity. Simulations were performed analogously to those of Wong [7-10] using the model illustrated in Fig. 1. Both the SE and PE were characterized by non-linear expo'nential functions, while the CE was represented by Huxley's slidingfilament model [11]. The calcium handling and troponin binding were modeled similarly to Panerai [12] with modifications for the Maxwell-type model implemented; Panerai used a Voigt-type model in his analysis. In the calciumtroponin binding analysis, it was assumed that one level of calcium was present regardless of the length in the development of the force-length relations, and calcium levels were independently varied for a given set of force-length curves. The PE and SE behavior are described by exponential relations: Ppc = Po[exp{Kp( L - Lo)} - 1]
(1)
Pso = Pl[exp(K~L,~) - 1]
(2)
and
where Po, Kp, and Ks are element constants; Pl is the preload, determined initially from Eq. 1 as done by Wong [7-10]; L o is the initial fiber length; L is the total fiber length; and Lse is the SE length as a function of time t.
(3)
where M and A are myosin and actin; A-M represents attached crossbridges; f is the association rate constant, or the probability that a myosin crossbridge will bind to an actin molecule; and g represents the dissociation rate eonstant, or the probability that an attached crossbridge will become detached. Wong [7-10] used the expressions for f(x) and g(x), where x is the distance between the myosin head equilibrium position and the reactive site: X < 0,
0
