Generalized distribution-moment approximation for kinetic theories of muscular contraction

Crossbridge theory, originally developed by A.F. Huxley more than 60 years ago to explain the behaviour of striated muscle, has since evolved to encompass many different muscle types and behaviours. The governing equations are generally linear hyperbolic partial differential equations, or systems thereof, describing the evolution of probability density functions. Importantly, the macroscopic behaviour is often described not in terms of these distributions themselves, but rather in terms of their first few moments. Motivated by this observation, G.I. Zahalak proposed the distribution-moment approximation to describe the evolution of these moments alone. That work assumed a Gaussian underlying distribution, and was observed to provide reasonable approximation of the moments despite the non-Gaussian character of the underlying distribution. Here we propose two variations on the distribution-moment approximation: (i) a generalized N-moment approximation based on the Gram-Charlier A-series representation, and (ii) perhaps the simplest possible approximation based on a uniform distribution. Study of these variations suggests that Zahalak's original contention may be correct: approximations based on higher order moments may not be worth their complexity. However, the simplified variation shows more promise, with similar accuracy in approximating the moments yet reduced complexity in the derivation of the approximation.