## Overview The complex interaction of muscles with their surrounding anatomy and environment plays a vital role in the many activities that are required for animals to live and survive. Skeletal muscle composes a large portion of that musculo-skeletal system, and is controlled by the central nervous system in a conscious or unconscious manner. For humans in particular, the construction of- and control mechanisms behind skeletal muscle allows us to accomplish complex tasks ranging from those that are physically exerting to those that are delicate and require great dexterity. As an introduction into the biomechanics of the human muscular system, we combine a well known load-activation pattern taken from well established literature on the topic (both in the fields of human physiology and the computational simulation thereof) with an idealised model of a part of the human anatomy that most can easily identify with, namely the biceps brachii. ### An idealised model of the human biceps brachii To tackle this problem, we do not deviate particularly far from the approach that is comprehensively documented in `step-8`. The primary differences between this code-gallery example and the tutorial is the alteration of the geometry and boundary conditions that accompany it, as well as the extension of the constitutive law to include the transversely isotropic active muscle model. We thus present both the theory and associated nomenclature (which is mimicked in the code itself) in the [accompanying document.](./doc/theory/theory-linear_elastic_active_muscle_model.pdf) There you can observe the additional contributions to both the left- and right-hand sides of the linear system due to the integration of the rate-dependent, linearised muscle model. Although such linear model models are valid only under conditions of small deformations, in this scenario we will (with a healthy dose of skepticism) make a very coarse-grained first approximation of the muscles behaviour in the large deformation regime. The basic problem configuration, including a depiction of the underlying muscle microstructural orientation, is (loosely) summarised in the following image. ![Problem geometry](./doc/geometry/geometry.png) Note that the driver for the deformation of the muscle tissue are the applied traction alone when the muscle is in a passive state. However, during active contraction, as governed by the prescribed input neural signal, the muscle works against the applied traction. This condition, where the traction applied to a muscle is constant during periods of muscle activation, is known as isotonic contraction. More specifically, since overall the muscle shortens during contraction we are in fact modelling concentric contraction of the biceps. As for the specific geometry of the problem, we consider an idealised human biceps with a length of `250mm`, insertion and origin diameter of `20mm` and a diameter of `80mm` at its mid-point. We assume that there exists a single muscle fibre family orientated axially. The orientation of the underlying muscle fibres is, however, not parallel, but rather follows the curvature of the macroscopic anatomy. The longitudinal profile of the muscle is generated using a trignometric function, as opposed to being extracted from medical images. The benefit to doing so is that the geometry can be (parametrically) created in `deal.II` itself and the associated microstructural orientation can be directly linked to the user-defined geometry. ## Requirements * Version `8.5` or greater of `deal.II` There are no other requirements with regards to the third-party packages that `deal.II` can link to. ## Compiling and running Similar to the example programs, run ``` cmake -DDEAL_II_DIR=/path/to/deal.II . ``` in this directory to configure the problem. You can switch between debug and release mode by calling either ``` make debug ``` or ``` make release ``` The problem may then be run with ``` make run ``` Some simulation parameters may be changed by adjusting the `parameters.prm` file. Notably, its possible to switch between the model of the biceps and the reduced geometry used to reproduce the linearised counterpart of the isotonic contraction numerical experiments conducted by Martins. ## Reference for this work If you use this program as a basis for your own work, please consider citing it in your list of references. The initial version of this work was contributed to the deal.II project by the authors listed in the following citation: * J-P. V. Pelteret and T. Hamann, The deal.II code gallery: Linear Elastic Active Skeletal Muscle Mode, 2017. DOI: [10.5281/zenodo.437608](http://doi.org/10.5281/zenodo.437608) ### Acknowledgements The support of this work by the European Research Council (ERC) through the Advanced Grant 289049 MOCOPOLY is gratefully acknowledged by the first author. ## Recommended Literature * Kajee, Y. and Pelteret, J-P. V. and Reddy, B. D. (2013), The biomechanics of the human tongue. International Journal for Numerical Methods in Biomedical Engineering 29 , 4, 492-514. DOI: [10.1002/cnm.2531](http://doi.org/10.1002/cnm.2531); * J-P. V. Pelteret, A computational neuromuscular model of the human upper airway with application to the study of obstructive sleep apnoea. PhD Thesis, University of Cape Town, 2013. [http://hdl.handle.net/11427/9519](http://hdl.handle.net/11427/9519); * Martins, J. A. C. and Pires, E. B. and Salvado, R. and Dinis, P. B. (1998), A numerical model of passive and active behaviour of skeletal muscles. Computer Methods in Applied Mechanics and Engineering 151 , 419-433. DOI: [10.1016/S0045-7825(97)00162-X](http://doi.org/10.1016/S0045-7825(97)00162-X); * Martins, J. A. C. and Pato, M. P. M. and Pires, E. B. (2006), A finite element model of skeletal muscles. Virtual and Physical Prototyping 1 , 159-170. DOI: [10.1080/17452750601040626](http://doi.org/10.1080/17452750601040626); * Pandy, M. G. and Zajac, F. E. and Sim, E. and Levine, W. S. (1990), An optimal control model for maximum-height human jumping. Journal of Biomechanics 23 , 1185-1198. DOI: [10.1016/0021-9290(90)90376-E](http://doi.org/10.1016/0021-9290(90)90376-E); * T.J.R. Hughes (2000), The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover. ISBN: [978-0486411811](http://store.doverpublications.com/0486411818.html) ## Results The displacement of the central point on the insertion surface (i.e. the traction boundary) is plotted against the simulation time when the muscle is made to undergo concentric contraction. Within the first second, when the muscle is completely passive, the displacement increases linearly due to the applied pressure that ramps to a maximum over this duration. This response is not entirely unsurprising for this geometrically symmetric, linear elastic body. When the muscle is activated, it shortens considerably until during the `1s` for which the neural signal is applied. The activation level increases exponentially until is saturates near the `2s` mark. At this point the neural signal is removed and the muscle starts to relax. The contractile level decreases exponentially and the muscle is nearly completely relaxed by the end of the simulation. ![Problem geometry](./doc/results/results.png) As a supplement to the above, the following animation shows the concentric contraction (under the assumption that it experiences no additional gravitational loading is present). All of the highlights that are discussed above can be observed in the gross displacement of the body, as well as the activation level that is visualised through the depiction of the underlying microstructure directions. This also shows how the muscle's cross-section influences the shortening along the length of the muscle. ![Problem geometry](./doc/results/biceps-no_gravity-concentric_contraction.gif) ### Influence of gravity Just for fun, we can repeat the above experiment with a fraction of the full gravitational loading applied in the transverse direction. We apply only a fraction of the full load because the muscle is not sufficiently well constrained and does not see the support of its surrounding anatomy. The loading condition is thus somewhat unphysical and, due to the lack of constraint, the application of the full load results in excessive deformation. Here we see the fully passive muscle with partial gravitational loading and a full traction load ![Displaced solution](./doc/results/biceps-gravity-traction-passive.png) and its counterpart solution when in the active stage. ![Displaced solution](./doc/results/biceps-gravity-traction-active.png) The asymmetry of the solution is clearly observed, although the length change that it exhibits curing the concentric contraction cycle remains somewhat similar to that demonstrated before.