Maria Eleni Athanasiadou

Maria Eleni Athanasiadou

Master's Thesis

A physics informed neural network for muscle mechanics


Dr.-Ing. Eva Dorschky, Prof. Dr. Anne Koelewijn


06 / 2023 – 12 / 2023


Muscle function and the respective force production are inherent to the structure of a muscle which is ultimately composed of sarcomeres, the basic contractile unit of a muscle. Sarcomeres are most commonly represented as a structure consisting of two proteins; myosin and actin. This structure has been originally modeled by Huxley [1]. However, in recent years the theoretical foundation of this model has been updated and a third protein, titin, has been included [7]. In practice, myosin and actin form cross-bridges leading a muscle to contract, shortening its length. According to the change of muscle length, two important relationships have been established, which describe muscle mechanics. The first is the force-length relationship which illustrates how a decrease in muscle length leads to an increase in muscle force, eventually achieving a maximum force at the optimal muscle length. However, a further decrease in muscle length leads to a decrease in muscle force [2]. The second relationship is the force-velocity relationship which suggests that as the shortening velocity of a muscle increases, the force produced decreases super-linearly [3]. In order to produce these relationships, the relevant data is recorded. Muscle forces are often recorded by using tendon buckles and measuring the force on the tendon that is attached to the muscle in question. To measure muscle length on the other hand, sonomicrometry or Diffusion Tensor Imaging (DTI) can be used. One of the most common phenomenological models, which, unlike Huxley’s model, incorporates these muscle mechanics is the Hill model. It is most commonly comprised of a contractile element (CE) representing the muscle, on which the aforementioned relationships apply [4], a parallel elastic element (PEE) representing the connective tissues of a muscle and a series elastic element (SEE) representing the tendon and aponeurosis connecting the muscle to the bone [5].
Despite the wide use of Hill-type muscle models, they continue to offer a rough approximation of the actual muscle behavior that is recorded during experiments. For example, the optimal muscle length changes depending on how much a muscle is stimulated. Hill-type models cannot incorporate this fluctuation of the optimal muscle length. Additionally, research suggests that the length-dependent dynamics of an activated muscle differ significantly from what the force-length relationship suggests. Specifically, muscles cannot remain fully activated while their length changes slowly since such an effort caused muscle fatigue and damage [6]. Apart from issues with fully incorporating muscle behavior, Hill-type models also tend to trade numerical stability for low computational loads, often leading to computational failures [6]. Considering these shortcomings of Hill-type models we suggest the alternative approach of using machine learning and data of muscle behavior to produce a neural network (NN) capable of replicating all muscle mechanics, as a result of its training. In previously conducted research, such a NN was able to outperform a Hill-type model with respect to muscle force prediction accuracy. Even though a NN trained on the available data was able to predict muscle force accurately, it did not reproduce the force-length-velocity relationship as it is known to exist. This result means that, while such a network is able to predict muscle force, it cannot be trusted to give accurate predictions that are far outside the range of the training data.
Therefore, we would like to investigate if we can increase robustness of such networks by combining data with physical knowledge using physics inspired neural networks (PINNs). In this thesis, we will first investigate the muscle mechanics literature and identify different muscle models that could be included in the training process. We will also define different ways to include these muscle models into the training process, and compare which models and which inclusion method will lead to a network that accurately predicts muscle forces but also is able to replicate muscle mechanics, such as the sliding filament theory.


[1] A. F. Huxley, “Muscle structure and theories of contraction,”Progress in Biophysics and Biophysical Chemistry, vol. 7, pp. 255–318, 1957.
[2] A. M. Gordon, A. F. Huxley, and F. J. Julian, “The variation in isometric tension with sarcomere length in vertebrate muscle fibres,”The Journal of Physiology, vol. 184, no. 1, pp. 170–192, 1966.
[3] A. V. Hill, “The Heat of Shortening and the Dynamic Constants of Muscle,”Proceedings of the Royal Society of London. Series B – Biological Sciences, vol. 126, no. 843, pp. 136–195, 1938.
[4] F. E. Zajac, “Muscle and tendon: properties, models, scaling, and application to biomechanics and motor control,”Critical reviews in biomedical engineering, vol. 17, no. 4, pp. 359–411, 1989.
[5] R. H. Miller, “Hill-based Muscle Modeling,”Handbook of Human Motion, pp. 1–23, 2018.
[6] S.-H. Yeo, J. Verheul, W. Herzog, and S. Sueda, “Numerical Instability of Hill-type muscle models,”Journal of The Royal Society Interface, vol. 20, no. 199, 2023
[7] W. Herzog, “The role of titin in eccentric muscle contraction,”Journal of Experimental Biology, vol. 217, no. 16, pp. 2825–2833, 2014.