Modeling of Muscle Wrapping Phenomenon: A Geometric Method Based on Tangency Constraints

Abstract

Musculoskeletal simulations have become an essential tool for surgical planning and clinical evaluation of movement. Accurate simulations of muscle-tendon paths in musculoskeletal models promote better predictions of muscle forces and, therefore, joint torques. A precise description of a muscle-tendon unit behavior should consider the wrapping phenomenon around bones. State-of-the-art muscle wrapping methods do not guarantee continuity along the path, which may result in unrealistic solutions. Moreover, many use the well-known Newton-Raphson method, discarding other methods to solve the problem. This article presents a muscle wrapping formulation based on tangency constraints that, unlike other methods, guarantees <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {C}^{1}$</tex-math></inline-formula> continuity along the path over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> parametric surfaces. We first test our formulation over one cylinder to compare the numerical efficiency considering the Newton-Raphson and two Quasi-Newton (Broyden–Fletcher–Goldfarb–Shanno and Broyden) methods under two cases: 1) wrapping state all the time, and 2) from/to a wrapping to/from a non-wrapping state. Results show that all yields well-posed solutions according to the tangency constraints, and that Broyden's method is the fastest to converge. Finally, a simulation of muscle wrapping over two cylinders was performed using Broyden's method considering the transition from/to a wrapping to/from a non-wrapping state, resulting also in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {C}^{1}$</tex-math></inline-formula> muscle paths.