From sarcomere to cell: an efficient algorithm for linking mathematical models of muscle contraction

Two classes of mathematical framework have previously been developed to model active tension generation in contracting muscle. Cross-bridge models of muscle are biophysically based but computationally expensive to solve, and thus unsuitable for embedding in spatially distributed continuum representations. Fading memory models are computationally efficient but provide limited biophysical insight. In this study a novel computational method is proposed for coupling these two frameworks such that biophysical events can be determined and computational tractability maintained. Within the cross-bridge model, the functional forms of the distribution of cross-bridges, as a function of strain in each state, are approximated using the distribution moment approach. Using the variables of area, mean and standard deviation of each distribution, analytic expressions are developed to calculate the temporal dynamics of stiffness, tension and energy. A root finding method is employed to adjust the variables such that the temporal dynamics of the cross-bridge model match those of an equivalent fading memory model. The method is demonstrated for sinusoidal perturbations in length at two frequencies, with an approximate 30-fold increase in computational efficiency over a conventional technique for finding a solution to the cross-bridge model.