Mathematical analysis of cardiac electromechanics with physiological ionic model

Modeling the cardiac electromechanical function: A mathematical journey

“…Concerning the active strain setting, instead, in [7], the authors proved existence of weak solutions and uniqueness of regular solutions for a simplified electromechanical model, where the nonlinear elastic model is linearized and where the FitzHugh-Nagumo ionic model [54] is employed. In [15] these results have been generalized to the more realistic Beeler-Reuter [14] and Luo-Rudy [52] ionic models. The latter results are based on the Faedo-Galerkin method and compactness arguments.…”

“…Partial results do exist concerning the electrophysiology subproblem (3.2) with suitable boundary conditions [34,93], as well as for the passive mechanics problem (5.1) under suitable boundary conditions [12,27]. See [7,15,60] for existence results on the coupled electromechanical model under simplifying assumptions. This may raise questions about the actual solvability of problem (7.1)-(7.5) and suggests a cautious attitude to mathematicians.…”

Unfolding homogenization method applied to physiological and phenomenological bidomain models in electrocardiology

“…The proof of the theorem follows closely the steps done in the case above of more general FitzHugh-Nagumo ionic function type. Using approximation systems and applying a Faedo-Galerkin method in space, one can obtain the existence of a weak solution for the approximation systems (similarly to section 4) then by a passage to the limit, the existence for the microscopic problem is obtained based on some technical results and a series of a priori estimates that are listed in the sequel but their detailed proofs are available in [10]. We also refer to [42] where a fixed point approach was used.…”

“…Based on the previous Lemmata, and proceeding in a similar way as in Section 5, one can easily obtain the following estimates on the solutions to the microscopic problem that are required for the passage to the limit as ε → 0 (the detailed derivation can be found in [10]). Lemma 7.6.…”

Simplification of weakly nonlinear systems and analysis of cardiac activity using them

“…The author of this work states that nonlinear ODE systems are useful for describing the action potentials of individual cardiac muscle cells, complex PDE systems for propagating a wave of electrical excitation through cardiac tissue or the whole heart. For example, in [2] we find a mathematical analysis of a coupled elliptic-parabolic system modeling the interaction between the propagation of electric potential coupled with general physiological ionic models and subsequent deformation of the cardiac tissue. The existence of weak solutions to the underlying coupled electromechanical bidomain model is proved numerically.…”