Modelling the pressure and force equilibrium in unipennate muscles with in-line tendons

Several of the models proposed in the literature of unipennate muscles, which have two tendinous sheets and in-line tendons, cannot meet the criterion of mechanical stability. Based on the theory of Van Leeuwen & Spoor (Phil. Trans. R. Soc. Lond. B 336, 275-292 (1992)), we discuss how mechanically stable solutions for (planar) unipennate architectures could be obtained. A mathematical model is proposed in which the muscle architecture is generated numerically using the principles of mechanical stability and assuming that all muscle fibres shorten by the same relative amount. The tendinous sheets are attached tangentially to their respective tendons, as predicted from their low bending stiffness. The curvature, however, is discontinuous at the junction because of the sudden absence of muscle fibres from aponeurosis to tendon. In two of the muscle shapes generated, the sheets adjacent to the tendon show a region of negative curvature connected to a region of positive curvature. A sheet with a concave outer side is defined to have a negative curvature. In another example, two negative curvature regions are present with a positive region in-between. We show also a generated shape with a negative curvature of the sheets over their whole length. A good resemblance was found between the unipennate medial gastrocnemius muscle of the cat and a simulated architecture. The pressure distribution has also been calculated. With all muscle fibres exerting the same tensile stress of 200 kPa, a high pressure region is present in the centre of the muscle belly, half-way along its length. The highest pressures are predicted for muscles with long tendinous sheets, large attachment angles, and strongly curved fibres. Maximum pressures (2.40, 9.54, 10.47, and 7.57 kPa for the four discussed examples, and 15.05 kPa for the simulated gastrocnemius muscle) were at the lower side of the range as predicted previously for bipennate muscles and the unipennate medial gastrocnemius muscle of man (Van Leeuwen & Spoor 1992).