EXISTENCE OF SOLUTIONS OF PARTIALLY DEGENERATE VISCO-ELASTIC PROBLEMS, AND APPLICATIONS TO MODELLING MUSCULAR CONTRACTION AND CARDIAC ELECTRO-MECHANICAL ACTIVITY PRAS PATHMANATHAN, CHRISTOPH ORTNER AND DAVID KAY Abstract. Muscle cells generate tension in response to electrical activation. This response is well-known to depend on both the stretch and stretch-rate of the cell. Models of cellular tension development have been proposed which include these effects, and have in recent years been embedded in models of the nonlinear passive mechanical response of tissue, in order to simulate muscular contraction, especially in cardiac tissue. However, it has been observed that such models are no longer purely elastic. Instead they are visco-elastic, but only in the muscle fibre direction. These models can be considered to be partially degenerate visco-elastic. The mathematical properties of such systems are unclear, and in fact preliminary observations [Pathmanathan et al., QJMAM, 2010] raised questions about their well-posedness. This paper lays a mathematical foundation for the use of partially degenerate visco-elastic models in muscular modelling. We define an abstract linear problem, for which we prove an existence theorem. This theorem allows jumps at initial time of particular spatial derivatives of the solution; we show that such jumps are always possible in general, but derive constraints on the initial condition that guarantee continuous solutions. Overall, the linear partially degenerate visco-elastic case is fully characterised, and the implication of these results for nonlinear muscular models (cardiac electro-mechanical models in particular) is discussed. 1. Introduction. Computational simulation of biological processes can be used to provide insight into mechanisms underlying biological phenomena, and are a use- ful tool for generating hypotheses. One mature field in computational physiology is cardiac modelling, in which numerical simulations have been performed for over half a century. The vast majority of such simulations are of cardiac electrical activity. In contrast, computing the mechanical deformation of the heart, and in particular the coupled problem of computing the deformation induced by electrical activation, is a challenging problem that has received much less attention in comparison, although reasonably sophisticated models and whole-organ simulations have been developed over the past fifteen years. Cardiac electro-mechanical models are normally comprised of several components, as illustrated in Figure 1.1. Firstly, there is a component governing tissue electrical activity, normally the so-called monodomain or bidomain models. To describe the deformation, a continuum mechanics theory is required, and due to the large de- formations undergone in muscular contraction, nonlinear elasticity (finite elasticity) is generally used. In addition, a coupling model, known as the contraction model, which describes cellular tension development (or cellular length-change) in response to electrical activation, is used to connect the electrical and mechanical models. There are two ways in which the contraction model can couple to the mechanical model. In the so-called active-stress approach [13, 9, 22, 15, 20], the contraction model determines cellular tension, which is converted to a stress and added to the passive stress, to obtain the total stress. In other words, the total stress is assumed to break down additively into a passive response and an active stress. In the active-strain approach [5, 12], the contraction model determines a new natural length at the cell level, and this affects the definition of strain. Active-stress is more physiologically intuitive, which is the reason why it has been far more commonly used in cardiac modelling. However, active-strain is much more safe mathematically: there are various open questions in the mathematical validity of active stress models, as is discussed in [17]. The focus of this paper is on one of these questions. 1