Keywords: musculoskeletal system, optimisation, equilibrium, simulations, muscle modelling.

Computational modelling of load-carrying structures is a conventional tool in mechanical, structural and robotic engineering.These modelling possibilities are more or less general in their representation of the behaviour of the structures and components, when subjected to load. Many such tools have also been used for bio-mechanical problems. In order to allow for these extremely complex simulation models, significant simplifications of the real behaviour of the biological systems are often necessary. These simplifications thereby often represent the human body parts as rigid, undeformable links, connected by perfect hinges and affected by very simplified muscle models - 'rubber-band' or, at best, 'Hill-type', [1]. The bio-mechanical models created can be subject to active or passive forces, and predict with some accuracy the expected response of the object modelled, at least for rather common situations of, primarily, repetitive movements at reasonable loading levels.

The musculoskeletal system is, from the computational viewpoint, normally a highly redundant force system for most loading cases, [2]. The indeterminacy in force distribution is related to the high number of alternative components over most joints. The redundancy allows very complex movements under neural control, but causes computational difficulties when simulating the behaviour.

The exact functioning of the control system is not fully known, but a common approach is to assume that the forces are distributed between the muscles following some optimisation rules: minimum forces, nominal efforts, activation, energy consumption, or maximum smoothness. A special aspect of this is the distribution of forces between synergistic muscles, which is often seen as a static or dynamic optimisation problem. Several methods and criteria for static optimisation can easily be set in a common algorithm, [3].

Such approaches demand that the muscular forces can be prescribed to certain values, at a certain time. Although these values might also be used in a dynamic simulation of motion, this is a rather static view of the muscular force production, where the force is computed from the current length and lengthening velocity. For simulations of optimally controlled motions of musculoskeletal systems, [4], considerably more physiologically relevant descriptions would increase the predictive capacity [5].

This paper discusses the numerical simulation of musculoskeletal systems from two viewpoints. First, the modelling of muscle mechanics is under intense debate. The general and accurate numerical representation of physiological muscle knowledge is complicated, as in full simulations the models must be able to describe the relation between force and muscle length for arbitrary movements. The effects from an interaction between the muscular forces and the partly externally induced movements are only to some degree understood and implemented in numerical models. These effects might be of critical importance for the results in many impact-type situations.

The second important area of study is related to dynamic simulations of the body, or parts of it.In addition to the handling of complex anatomical geometry, another important area is the choice of optimal movement strategies. The criteria for optimality are, however, not obvious, and probably not universal for all situations. An interesting question is related to the similarities and differences obtained with different criteria for optimality, and also the possibilities for introducing kinematic and strength restrictions in the search for optimal movement.

The paper is presented as follows. Firstly, a discussion on the treatment of redundant force systems, where the choice of alternative force patterns can be based on some optimality criterion: both static and dynamic viewpoints are considered. Secondly, a few aspects of the numerical modelling of muscles is considered, where the focus is set on aspects of muscle physiology which are important for general simulations of musculoskeletal systems. Finally, three illustrative numerical experiments are described and discussed.

1

A. Hill. "The heat of shortening and the dynamic constants of muscle". Proc. Royal Soc. London B, 126,136-195, 1938. doi:10.1098/rspb.1938.0050

2

A. Eriksson and A. G. Tibert. "Redundant and force-differentiated systems in engineering and nature". Comput. Methods Appl. Mech. Engrg., 2005. (in press). doi:10.1016/j.cma.2005.11.007

3

S. Heintz, E. Gutierrez-Farewik, and A. Eriksson. "Evaluation of load-sharing and load capacity in force-limited muscle systems". Comp. Meth. Biomech. Biomed. Eng., 2006. (submitted).

4

A. Eriksson. "Analysis methodology based on temporal FEM for bio-mechanical simulations". In M. Ursino, C. A. Brebbia, G. Pontrelli, and E. Magosso, editors, Modelling in Medicine and Biology VI, Southampton, 2005. WIT Press.

5

A. J. van Soest and G. P. van Galen. "Coordination of multi-joint movements: An introduction to emerging views". Human Movement Science, 14,391-400, 1995. doi:10.1016/0167-9457(95)00022-4

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