Unfolding homogenization method applied to physiological and phenomenological bidomain models in electrocardiology
Introduction
The heart is the muscular organ that contracts to pump blood throughout the body. Its contraction is initiated by an electrical signal called action potential. At a microscopic level, the cardiac tissue is a complex structure composed of elongated connected cells (cardiomyocytes) that have a cylindrical shape and that are aligned in preferential directions forming fibers. Cardiomyocytes are encapsulated in a dynamic cell membrane (the sarcolemma) that separates the interior of the cell from the surrounding medium and maintains a potential difference (the transmembrane potential) between the two media due to the different concentrations of various ionic species on both sides. The elongated cardiomyocytes are endowed with special end-to-end connections (the gap junctions) that form the long fiber structure of the muscle, as well as with lateral junctions that permit the connection between the intracellular spaces of the elongated fibers. Since those connections have a low resistance, the cardiac tissue can be viewed as a single intracellular connected domain, separated from the extracellular domain by the surface of the cell membrane [1]. Moreover, the sarcolemma consists of a phospholipid bilayer in which are embedded ionic channels that ensure the flow of ionic currents from the extra- to intracellular space or vice versa. As a consequence of this transfer of ionic species between the two-spaces (intra- and extracellular spaces) a current flows across the cell membrane (transmembrane current). The capacitive, diffusive and conductive effects contribute to this current flux across the membrane [1], [2], [3].
From a physical point of view, the cardiac tissue can be viewed as partitioned into two ohmic conducting volumes (intra- and extracellular spaces). The intra- and extracellular domains act as volume conductors and can be described by a quasi-static approximation of elliptic equations in both spaces. These equations are complemented by a dynamical boundary equation at the interface of the two regions. It is worth mentioning in the sequel that the approximation of the ionic current flow is based on Ohm’s law and charge conservation and that these equations depend (at the microscopic level) on a small parameter () whose order of magnitude is the ratio of the two macro- and microscopic space scales.
In this paper we derive a macroscopic bidomain model of cardiac electrophysiology based on a microscopic bidomain model, using a rigorous homogenization method. Indeed, the microscopic model is unsuitable for numerical computations due to the complexity of the underlying geometry, which highlights the importance of the rigorous derivation of the macroscopic model while taking into account the properties of the physiological and microscopic structure. Classically, homogenization has been done by means of the multiple-scale method which permits to formally obtain the homogenized problem based on a formal asymptotic expansion [4], [5]. There are now various mathematical methods related to this theory: the oscillating test functions method due to L. Tartar in [6], the two-scale convergence method introduced by G. Nguetseng in [7], and further developed by G. Allaire in [8] (see also [9]) and recently the periodic unfolding method introduced by D. Cioranescu, A. Damlamian and G. Griso for the study of classical periodic homogenization in the case of fixed domains and adapted to homogenization in domains with holes in [10]. The idea of the unfolding operator was used in [11], [12], [13] under the name of periodic modulation or dilation operator. The name “unfolding operator” was then introduced in [10] and deeply studied in [14], [15]. The interest of the unfolding method comes, on one hand, from the fact that it only deals with functions and classical notions of convergence in spaces and it does not necessitate the use of a special class of test functions. On the other hand, the unfolding operator maps functions defined on oscillating domains into functions defined on fixed domains. Hence, the proof of homogenization results becomes quite simple.
Regarding the asymptotic behavior of a microscopic-level modeling problem for the bioelectric activity of the heart, there is the work by M. Pennachio, G. Savaré, and P. Franzone that rigorously studies the derivation of the bidomain model in the framework of -convergence theory presented in [16]. Recently, the two-scale method has been used in [17], [18] to obtain the homogenized macroscopic model using different ionic models and assumptions on the conductivity matrices. In [17], the authors derive a macroscopic bidomain model using simplified ionic models whereas in [18], the authors use the FitzHugh–Nagumo ionic model. In the present work, we treat a generalized class of ionic models including the FitzHugh–Nagumo model along with physiological models involving ionic concentrations that appear as arguments of a logarithmic function and that must be shown to be bounded away from 0. We further note that in [17], [18], the cardiac domain was assumed to be a cube in . Regarding the mathematical analysis of the microscopic model, we point out that in [19], the author used Schauder’s fixed point theorem and in [20], the authors used a variational approach to establish the well-posedness of the microscopic problem under different initial and boundary conditions. In the present work, we prove the existence of solution of the microscopic problem by a constructive method based on the Faedo–Galerkin approach without the restrictive assumption, usually found in the literature, on the conductivity matrices to have the same basis of eigenvectors or to be diagonal matrices (see for instance [21] where the authors prove the existence of a local in time strong solution of the bidomain equations after introducing the so-called bidomain operator). It is worth to mention that our approach is innovative and cannot be found in the literature in the context of existence of solutions to the microscopic bidomain model. The convergence of solutions of a sequence of microscopic problems to the solution of the macroscopic problem is established in properly chosen function spaces. We use the unfolding method in perforated domains [10], [14], for sequences of functions bounded in , or in on a micro-periodic domain. The difficulty of the homogenization problem for the bidomain equations is due, on one hand, to the degenerate structure of the equations, in combination with the highly oscillating underlying geometry. As a consequence, standard parabolic a priori estimates are not immediately available [20]. On the other hand, the (nonlinear) dynamics of the cellular model take place on the cell membrane which is a wildly oscillating surface. Hence, an ambiguity arises in defining a proper notion of “strong convergence” of functions in this context. However, some kind of strong convergence is required to pass to the limit in the nonlinear equations. For this reason, we also use the boundary unfolding operator along with a Kolmogorov–Riesz compactness argument [22], [23]. We stress that we do not restrict our study to the homogenization method of the bidomain model with nonlinear ionic function of FitzHugh–Nagumo type but also with physiological ionic function of Luo–Rudy type. Moreover, the approach presented herein can be extended to electropermeabilization models. We cite for instance [24] where a dynamical homogenization scheme is obtained from a physiological cell model and [25] where a conductivity dependent macroscopic tissue model is for the first time derived from first principles.
Note that thanks to homogenization, the resulting macroscopic bidomain model describes averaged intra and extracellular potential by a nonlinear anisotropic reaction–diffusion system. The cardiac tissue is then considered (at the macroscopic level) as the superposition of two anisotropic continuous media: the intra- and extracellular spaces, coexisting along with the cell membrane, at each point of the tissue. The most substantial mathematical description of the bidomain model is found in the review paper by Henriquez [26], which presents a formal definition of the model from its origins in the core conductor model, and outlines many of the approximations that can be made under certain assumptions.
The plan of this paper is outlined as follows. The microscopic problem and the main assumptions used for homogenization are presented in Section 2 and the main result is stated. In Section 3, existence of weak solutions to the microscopic problem is proved based on a Faedo–Galerkin approach, a priori estimates and a compactness argument. In Section 4, some estimates on the solutions of the microscopic problem are obtained and the microscopic problem is formulated using the unfolding operator. The passage to the limit using compactness and the unfolding method are established in Section 5. Then in Section 6, the macroscopic bidomain equations are recuperated from the limit equations obtained in Section 5 and the cell problem is decoupled. Finally, in Section 7, a microscopic bidomain model with physiological ionic model is homogenized to obtain the corresponding macroscopic model.
Section snippets
The microscopic bidomain model
We first list in the following paragraphs the assumptions used in sections 3 Existence of solutions to the microscopic model, 4 Convergence of solutions to the macroscopic problem, 5 “Unfolding” compactness, 6 Macroscopic bidomain model (proof of.
Assumptions on the domain. For our model we assume that (the cardiac tissue) is a bounded open subset of with smooth boundary . The cardiac tissue is composed of two connected regions, the intracellular and the extracellular . These two
Existence of solutions to the microscopic model
This section is devoted to proving existence of solutions to the microscopic bidomain model for fixed . The existence proof is based on the Faedo–Galerkin method, a priori estimates, and the compactness method.
We start with a weak formulation of the microscopic model.
Definition 3.1 Weak Formulation A solution of problem , (2), (3) is a four tuple such that , , , , , and satisfying the
Convergence of solutions to the macroscopic problem
This section consists in preparing the ground for the passage to the limit as . First, some a priori estimates are obtained on the solutions of the microscopic problem. Then, the unfolding operator for perforated domains and the boundary unfolding operator are introduced and some of their properties are recalled. Finally, the microscopic problem is written in an equivalent formulation, the so called “unfolded” formulation, making use of the unfolding operators.
“Unfolding” compactness
In this section, we establish the passage to the limit in (50), (51). First, note that by estimates (44)–(47) obtained above one has Also, by regularity of the test functions and , there holds and Now, consider and in and in such that and test Eq. (50) with functions where (see for e.g. [15]). Since and
Macroscopic bidomain model (proof of Theorem 2.2)
The next step is to obtain the weak formulation of the bidomain equations and the cell problem. So one needs to formulate the limit problem in terms of and alone and hence find an expression of and in terms of , respectively. First, to determine the cell problem, set in (65) and to 0, to get which corresponds to the classical cell problem obtained in Section 2 and it can be shown that the function can be written in terms of
Unfolding homogenization to physiological models
In this section, we extend the homogenization results obtained in the previous sections to physiological ionic models. So the ordinary differential equation (1d) is replaced by a system of ODEs for the gating variables , and the concentration variable .
The kinetics of a general physiological ionic model may be represented by the functions , and that satisfy assumptions (A.1)–(A.3), stated below. It can be verified that those assumptions are satisfied by several gating and
Acknowledgments
We would like to thank the anonymous referees for their constructive and thorough reports. Their suggestions significantly improved our original manuscript.
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