Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology

Abstract

We study the well-posedness of the bidomain model that is commonly used to simulate electrophysiological wave propagation in the heart. We base our analysis on a formulation of the bidomain model as a system of coupled parabolic and elliptic PDEs for two potentials and ODEs representing the ionic activity. We first reformulate the parabolic and elliptic PDEs into a single parabolic PDE by the introduction of a bidomain operator. We properly define and analyze this operator, basically a non-differential and non-local operator. We then present a proof of existence, uniqueness and regularity of a local solution in time through a semigroup approach, but that applies to fairly general ionic models. The bidomain model is next reformulated as a parabolic variational problem, through the introduction of a bidomain bilinear form. A proof of existence and uniqueness of a global solution in time is obtained using a compactness argument, this time for an ionic model reading as a single ODE but including polynomial nonlinearities. Finally, the hypothesis behind the existence of that global solution are verified for three commonly used ionic models, namely the FitzHugh-Nagumo, Aliev-Panfilov and MacCulloch models.

Optimal control of the bidomain system (III): Existence of minimizers and first-order optimality conditions

“…analogously to [4], page 478, Section 5.3. As a consequence of the imbedding theorem [7], page 286, Theorem 2,…”

“…In problem (P), the extracellular excitation I e acts as control, which is allowed to be applied on the subdomain Ω con only. 4 The pointwise constraint within the description (1.9) of C is included due to the obvious fact that one cannot apply arbitrary large electrical stimulations to living tissue without damaging it. In mathematical terms, this restriction is necessary in order to establish a stability estimate for the bidomain system (Thm.…”

“…Parabolic-elliptic form of the bidomain system; strong and weak solutions It is well-known that the bidomain system (1.1)-(1.6) can be equivalently stated in parabolic-elliptic form, cf. [4], page 459, and [12], page 4, (2.1)-(2.9). In its weak formulation, the system reads as follows:…”

“…Approximate solutions for the reduced adjoint system. By [4], page 464, Theorem 6, the bidomain bilinear form A( · , · ) gives rise to an orthonormal basis of eigenfunctions { ψ i } within the space W 1,2 (Ω), which are related to eigenvalues 0 = λ 0 < λ 1 λ 2 . .…”

Optimized Schwarz coupling of Bidomain and Monodomain models in electrocardiology

“…The transmembrane potential u is uniquely determined, while the intra and extracellular potentials u i and u e are determined up to the same function of time, whose value is usually obtained by imposing that u e has zero mean on Ω. For well-posedness analysis of the Bidomain problem associated with different ionic models see [2,3,8,38]. In what follows we will rely on a non-symmetric formulation (2.8) (see e.g.…”