Canine Smooth Muscle Contraction Model.

Smooth muscle is found extensively in the human body, including in blood vessels, airways, the gastrointestinal tract, and the urinary bladder. Although the contractile proteins of smooth muscle are very similar to those of striated muscle, smooth muscle's contractile mechanism has not been studied as extensively as those for cardiac and skeletal muscle. Previous studies developed a lumped model of muscle contraction and applied it to cardiac muscle and to skeletal muscle. In this study, this model is used to quantitatively describe the contractile properties of canine smooth muscle, using data from the literature. Results show that a single equation relating muscle force to muscle length and time, and a single set of model parameters, is able to describe smooth muscle's passive and active isometric forces, isometric twitch contractions, isotonic contractions, and an inverse force-velocity relation. The latter arises from the model without assumption of a particular force-velocity curve embodied as a contractile element. This new constitutive relation may be used to describe smooth muscle within larger physiological models, for instance to describe blood vessel constriction or urinary bladder function.

Modeling Mouse Soleus Muscle Contraction.

Models of muscle contraction are typically based on a measured force-velocity relation embodied as Hill's contractile element [1]. Adopting a particular force-velocity relation dictates the muscle's mechanical properties. Dynamic crossbridge based models, such as Huxley's [2], typically focus on ultrastructural mechanics. This study adapts a dynamic lumped model of cardiac muscle contraction [3] for description of mouse soleus skeletal muscle. This compact, dynamic model exhibits the main features of skeletal muscle contraction with few assumptions. The main differences between cardiac and skeletal muscle dynamics are described. This approach gives one equation and set of parameters capable of modeling isometric and isotonic contractions, skeletal muscle's force-length relation, variations in contractility, and the force-velocity relation. This new constitutive equation may be useful for modeling striated muscle as part of larger biomechanical models.

Functional Requirements of a Mathematical Model of Muscle Contraction.

Muscle modeling has a long history. Models typically belong to the class of lumped models built around Hill's contractile element, or to crossbridge models at the ultrastructure level. Lumped models built on the contractile element are not sufficiently dynamic, since the muscle's contractile properties are set a priori by the force-velocity relation. We have shown that the force-velocity relation is not a unique descriptor of muscle's contractile state [1]. Alternatively, description of muscle as a time and length dependent force generator may be used to model muscle dynamics. This paper defines the functional requirements of a mathematical model of muscle contraction and evaluates how closely a lumped model based on ultrastructural dynamics resembles muscle. This single model was found able to describe isometric and isotonic contractions, the force-length relation, the force-velocity relation, and changes in contractile state and contraction rate. This approach may help link macro contractile properties, such as the force-velocity relation, to micro dynamics of crossbridge bonds. It may also be useful as a component of larger physiological models and simulations.

Modeling muscle's nonlinear viscoelastic dynamics.

Muscle's contractile properties are complicated by its viscoelastic properties. Failure of early viscoelastic muscle models led to Hill's force-velocity relation embodied as the contractile element. Adopting a particular force-velocity relation to describe muscle is neither easy, nor unique [1]. Time-varying elastance based models of the left ventricle have been popular since the idea was presented in 1969 [2]. This paper investigates adoption of the time-varying elastance concept to describe the viscoelastic properties of muscle. It will be shown that a time-varying elastance must be extended to a time, length, and velocity dependent elastance. Results show how a generalized force generator description of muscle [3] may be used to realistically model muscle's viscoelasticity.

Cardiac muscle strip model parameters and muscle elastance.

A recent functional model of the left ventricle as a pressure generator that is time and volume dependent was adapted to describe the mechanical aspects of heart muscle contraction. Muscle's complex dynamics develop from a single equation based on the formation and relaxation of crossbridge bonds. Muscle is modeled as a force generator that is time and length dependent. This equation permits the calculation of muscle elastance via Em = ∂fm/∂lm from muscle force and length, both as functions of time. This muscle model is defined independently from load properties, and elastance is dynamic and reflects changing numbers of crossbridge bonds. The model parameters were extracted from measured force and length data from cat papillary muscle experiments in the literature. The purpose of this paper is to present in some detail how to describe a particular muscle strip from measured data. The resulting model is tested under a wide range of mechanical conditions, such as isometric and isotonic contractions for normal and varied inotropic state, and muscle velocity is computed for different loads. Computed results compare favorably with similar measurements from the literature. The resulting lumped muscle model is a compact, yet comprehensive functional description of muscle dynamics.

Left ventricular model parameters and cardiac rate variability.

A recent functional model of the left ventricle characterizes the ventricle's contractile state with parameters, rather than variables. The ventricle is treated as a pressure generator that is time and volume dependent. The heart's complex dynamics develop from a single equation based on the formation and relaxation of crossbridge bonds within underlying heart muscle. This equation permits the calculation of ventricular elastance via E(v) = ∂p(v)/∂V(v). This heart model is defined independently from load properties, and ventricular elastance is dynamic and reflects changing numbers of crossbridge bonds. The model parameters were extracted from measured pressure and volume data from isolated canine hearts. The purpose of this paper is to present in some detail how to describe a particular canine left ventricle from measured data. The model is also extended to include heart rate variability, which arises naturally from the model structure. Computed results compare favorably with measurements both in this study and from the literature.

Functional requirements of a mathematical model of the heart.

Functional descriptions of the heart, especially the left ventricle, are often based on the measured variables pressure and ventricular outflow, embodied as a time-varying elastance. The fundamental difficulty of describing the mechanical properties of the heart with a time-varying elastance function that is set a priori is described. As an alternative, a new functional model of the heart is presented, which characterizes the ventricle's contractile state with parameters, rather than variables. Each chamber is treated as a pressure generator that is time and volume dependent. The heart's complex dynamics develop from a single equation based on the formation and relaxation of crossbridge bonds. This equation permits the calculation of ventricular elastance via E(v) = partial differentialp(v)/ partial differentialV(v). This heart model is defined independently from load properties, and ventricular elastance is dynamic and reflects changing numbers of crossbridge bonds. In this paper, the functionality of this new heart model is presented via computed work loops that demonstrate the Frank-Starling mechanism and the effects of preload, the effects of afterload, inotropic changes, and varied heart rate, as well as the interdependence of these effects. Results suggest the origin of the equivalent of Hill's force-velocity relation in the ventricle.

Defining muscle elastance as a parameter.

Functional descriptions of striated muscle are often based on the measured variables force and initial velocity of shortening, embodied as Hill's contractile element. The fundamental difficulty of describing the mechanical properties of muscle with a force-velocity relation that is set a priori, and the practical problem of the act of measurement changing muscle's force-velocity relation or elastance curve, are described. As an alternative, a new model of muscle contraction is presented, which characterizes muscle's contractile state with parameters, rather than variables. Muscle is treated as a force generator that is time, length, and velocity dependent. Muscle dynamics develop from a single equation based on the formation and relaxation of crossbridge bonds. This analytical function permits the calculation of muscle elastance via E(m)=[abstract: see text]. This new muscle model is defined independently from load properties, and muscle elastance is dynamic and reflects changing numbers of crossbridge bonds. This parameter is more representative of the mechanical properties of muscle than are variables such as muscle force and shortening velocity.

A paradigm for quantifying ventricular contraction.

The left ventricle may be described as a time, volume and flow dependent pressure generator. First, isovolumic pressure is measured at various end-diastolic volumes. Next, pressure is adjusted to account for small changes accompanying ejection, denoted the ejection effect. The resulting analytical function can describe pressure generation and ventricular outflow of the ventricle under a wide range of contractile and vascular conditions. This paradigm is unique in separating isovolumic from ejecting ventricular properties, as well as ventricular from vascular conditions.