Some mathematical problems in cardiac mechanics
Abstract: Cardiac mechanics is a very attractive subject for mathematical modelling because of the variety of coupled bio-physical mechanisms that regulate (often self-regulate) the dynamics. Solid mechanics, fluid dynamics, electrochemical signalling: one challenge in this respect is to capture the overall complexity. I will instead focus on a couple of specific biomechanical questions that eventually lead to interesting mathematical problems. The first issue is the correct mathematical statement of cardiac mechanical activity in the equations of balance of forces: as the heart is a pump that autonomously undergoes contraction without external load, I will discuss suitable mathematical conditions that apply in modelling its active stress. Secondly, I will enforce the importance of reduced order models in mechanobiology, showing at what extent a simple elastic rod can be successful in addressing questions that arise from the clinical practice. Again, new nontrivial mathematical questions can be addressed in terms of perturbation methods.