On the comparison of different muscle model dynamics using variational integrators

Hill’s muscle models are used for a variety of applications in analytic and nu- merical simulation of biomechanical systems consisting of multiple muscles. The rheological method of using lumped elements to understand muscle force pro- duction is easier than Huxley model, with reasonable accuracy. However, it is essential to understand the functional behavior of muscle models formed from different compositions of these elements and the resulting force response. This thesis aims to understand these nuance differences, while solving forward dynamics problems of single muscle systems with variational integrators

The thesis progresses in a manner to first understand the root of muscle function, by introducing its architecture, and graduating towards the active and passive systems of force development. This results in the mathematical models, which best approximate the muscle behavior. Also, a structure preserving time-stepping scheme is derived to solve the discrete Euler-Lagrange equations of motion. Such a variational integrator is consistent in angular momentum and the results have good energy behavior, as demonstrated through an example.

The combination of the forward dynamics problem involving a muscle and using a variational integrator ensues for the combinations in different arrangements of Hill’s elements, viz. the contractile component (CC), the parallel elastic component (PEC) and the serial elastic component (SEC). The models are composed of the CC, with either or both the PEC and the SEC in different arrangements. A comparison can be drawn with the muscle force as an output to the system, to understand the influence of the different elements. Also, the modular algorithm for this problem allows for extension to solve for different initial conditions, like variable activation etc. This method can also be extended to employ damping elements to model the muscle more accurately and capture the physics of force production.

The thesis concludes with a qualitative comparison of the dynamic behavior of the different muscle models, with a vertical muscle-mass system example. It recommends to utilize the CC with at least one of the PEC or the SEC. The inclusion of the PEC, though easier, does not change the result significantly as compared to the CC, while the SEC highly increases the complexity and the computational effort, at the cost of better predictions of muscle response. A good balance in multiple muscle systems can be made by inclusion of the SEC in the muscle model, for muscles with longer and stiffer tendons.