Gradient‐based optimization with B‐splines on sparse grids for solving forward‐dynamics simulations of three‐dimensional, continuum‐mechanical musculoskeletal system models

Abstract

Investigating the interplay between muscular activity and motion is the basis to improve our understanding of healthy or diseased musculoskeletal systems. To be able to analyze the musculoskeletal systems, computational models are used. Albeit some severe modeling assumptions, almost all existing musculoskeletal system simulations appeal to multibody simulation frameworks. Although continuum-mechanical musculoskeletal system models can compensate for some of these limitations, they are essentially not considered because of their computational complexity and cost. The proposed framework is the first activation-driven musculoskeletal system model, in which the exerted skeletal muscle forces are computed using 3-dimensional, continuum-mechanical skeletal muscle models and in which muscle activations are determined based on a constraint optimization problem. Numerical feasibility is achieved by computing sparse grid surrogates with hierarchical B-splines, and adaptive sparse grid refinement further reduces the computational effort. The choice of B-splines allows the use of all existing gradient-based optimization techniques without further numerical approximation. This paper demonstrates that the resulting surrogates have low relative errors (less than 0.76%) and can be used within forward simulations that are subject to constraint optimization. To demonstrate this, we set up several different test scenarios in which an upper limb model consisting of the elbow joint, the biceps and triceps brachii, and an external load is subjected to different optimization criteria. Even though this novel method has only been demonstrated for a 2-muscle system, it can easily be extended to musculoskeletal systems with 3 or more muscles.

Multiscale modeling of the neuromuscular system: Coupling neurophysiology and skeletal muscle mechanics

“…The only model predicting the motion of an agonist-antagonist musculoskeletal system (here a two-muscle-one-degree-of-freedom upper limb model) was proposed by Röhrle, Sprenger, and Schmitt (2017). This model was also the basis for developing a novel surrogate approach to achieve forward and inverse simulations in real time (Valentin, Sprenger, Pflüger, & Röhrle, 2018).…”

Solving High-Dimensional Dynamic Portfolio Choice Models with Hierarchical B-Splines on Sparse Grids

“…To the left and top, the one-dimensional not-a-knot B-spline functions are plotted. The resulting grid is shown on the right applications (Pflüger 2010;Valentin and Pflüger 2016;Valentin et al 2018). For p = 1 and homogeneous boundary conditions, the L 2 interpolation error on the full grid of level n is given by O(2 −2n ) and the number of grid points (i.e., the required function evaluations) is O(2 nd ) .…”

A Systematic Review and Meta-Analysis on the Longitudinal Effects of Unilateral Knee Extension Exercise on Muscle Strength

“…Using models to reveal the underlying principles of training do not necessarily exist on the musculoskeletal system level yet. However, such models can be integrated into multi-muscle forward-dynamics models (e.g., Röhrle et al, 2017;Valentin et al, 2018), in neuromuscular system models (e.g., Röhrle et al, 2019) and linked to homogenization techniques aiming to describe macroscopic muscle behavior based on microscopic constituents (e.g., Bleiler et al, 2019). Such models can then be used to investigate the musculoskeletal system in more detail.…”