On a Transmission Problem Related to Models of Electrocardiology

To begin with, we consider a basic example related to models of electrocardiology. As known from clinical practice, see, e.g., [1], [2], electrical activity of cardiac cells is crucial for pumping function of heart, which is the result of rhythmical cycles of contraction-relaxation of the cardiac tissue. Anomalies of electrical activity often cause heart diseases, which makes these investigations, in particular, development of adequate mathematical models, very relevant nowadays.

Let us illustrate this by one model of electrocardiology [1, 2, 5]. Denote by $\Omega_{B}$ and $\Omega_{H}$ three-dimensional domains with piecewise smooth boundaries with $\partial\Omega_{B}$ and $\partial\Omega_{H}$ corresponding to a body and a heart (see Fig. 1). Then the domain $\Omega=\Omega_{B}\setminus\Omega_{H}$ with the boundary $\partial\Omega=\partial\Omega_{B}\cup\partial\Omega_{H}$ corresponds to the body without heart.

Usually, in standard models one assumes that the cardiac tissue can be divided into two parts – intracellular and extracellular parts separated by a membrane – to which the electric potential $u_{i}$ and $u_{e}$, respectively, is assigned. Regarding the cardiac tissue as a continuous medium we think of the potentials as defined in each point of $\Omega_{H}$ and satisfying the equation

where $M_{i}$ and $M_{e}$ are known tensor matrices that characterize intracellular and extracellular parts, and $\nabla$ is the gradient operator in $\mathbb{R}^{3}$.

One often considers the case when $M_{i}$ and $M_{e}$ are positively defined matrices with constant coefficients with entry values defined by conductivity of the cardiac tissue. For simplicity of the further analysis one assumes that these matrices are proportional

Based on equation (1) one considers two models of heart activity. In one model it is assumed that the heart is isolated and one considers the problem

where $M_{b}$ is the tensor matrix characterizing conductivity of the body, $\nu$ is the vector field of unit outward normal vectors to the boundary of the domain under the consideration and $u_{b}$ is the electric potential of the body.

In the second model one takes the body into account, and from the electrodynamics of stationary currents it follows that the electric potential of the body $u_{b}$ in the domain $\Omega$ is defined by the equations

A feature of the model is the fact that one is more interested not in potentials $u_{i}$ and $u_{e}$ separately but in their difference $v=u_{i}-u_{e}$ in $\Omega_{H}$ or at least on its boundary.

Since matrices $M_{i}$ and $M_{e}$ are positively defined and not degenerate, the problems (2), (3) can be studied in the framework of the theory of boundary (maybe ill-posed) problems for elliptic formally self-adjoint equations, see [1, 2, 5]. Moreover, notice that the problems above may be regarded as transmission problems for elliptic equations with discontinuous coefficients describing solutions in different domains of a continuum with the help of additional conditions on separating surfaces, see, for example, [6], [7].

Until now we have not used any functional spaces in the problems description, in the next section we give a precise formulation of a more general problem and specify functional classes for its solution.

Let $\theta$ be a measurable set in ${\mathbb{R}}^{n}$, $n\geq 2$. Denote by $L^{2}(\theta)$ a Lebesgue space of complex-valued functions on $\theta$ with the scalar product

If $D$ is a domain in ${\mathbb{R}}^{n}$ with a piecewise smooth boundary $\partial D$, then for $s\in\mathbb{N}$ we denote by $H^{s}(D)$ the standard Sobolev space with the scalar product

It is well-known that this scale extends for all $s>0$. Let now $H^{s}(D)$ for $s\in{\mathbb{R}}_{+}\setminus{\mathbb{Z}}_{+}$ be the standard Sobolev-Slobodeckij spaces. Denote by $H^{s}_{0}(D)$ the closure of the subspace $C^{\infty}_{\mathrm{comp}}(D)$ in $H^{s}(D)$, where $C^{\infty}_{\mathrm{comp}}(D)$ is the linear space of functions with compact supports in $D$.

The space of $k$-vectors $u=(u_{1},...,u_{k})$ whose components lie in $H^{s}(D)$ equipped with the scalar product

we shall denote by $[H^{s}(D)]^{k}$.

Further on, we shall consider linear matrix operators

where $p\in\mathbb{N}$ is the order of operator $A$, $\alpha\in{\mathbb{Z}}^{n}_{+}$, and $A_{\alpha}$ are $(l\times k)$-matrices with constant coefficients. By a formal adjoint of $A$ we call the differential operator

Let now $A_{e}$, $A_{i}$, and $A_{b}$ be linear differential operators of the first order with constant coefficients on $\overline{D}_{m}$, i.e.

where $m\in\{e,i,b\}$, $D_{e}\equiv D_{i}\equiv\Omega_{H}$, $D_{b}\equiv\Omega$.

Further on, we assume that principal symbols of operators $A_{m}$ are injective in the corresponding domains.

Denote by $A_{m}^{*}$ a formal adjoint of $A_{m}$ and consider a generalized Laplacian $A_{m}^{*}A_{m}$.

Under assumptions made above, the operator $A_{m}^{*}A_{m}$ is a strongly elliptic $(k\times k)$-matrix second order operator, i.e. it is elliptic and there exists a positive constant $c$ such that

The operator $A_{m}^{*}A_{m}$ is also formally self-adjoint, i.e.

in particular, the operator $A_{m}^{*}A_{m}$ is (formally) positively defined

Let, as before, $\nu$ be the outward normal vector operator on the boundary of the domain of the operator $A_{m}$. Introduce the conormal derivatives

associated with these operators via Green’s formula:

Assume that bounded domains $\Omega_{H}$, $\Omega$, and $\Omega_{b}$ have twice smooth boundaries and consider the following problem (5)-(6): find vector-functions $u_{i}$, $u_{e}$ from ${\left[H^{2}(\Omega_{H})\right]}^{k}$ and a vector-function $u_{b}$ from ${\left[H^{2}(\Omega)\right]}^{k}$ such that

where the equality on the boundary is in the sense of traces, and the equality in the domains is in the sense of distributions. In this case we can assume that traces of functions and their conormal derivatives are well-defined.

It is obvious that the problem (5-6) is a generalization of the problem (2-3). Note also that it incorporates several classical boundary problems.

On of the simplest additional conditions mentioned above leads to using of an ill-posed Cauchy problem. More precisely, it implies measuring the potential $u_{b}$ on the boundary of ‘the body’:

where $f$ is a given vector-function from ${\left[H^{3/2}(\Omega)\right]}^{k}$.

Unfortunately, as known very well, the problem (6), (11) is nothing else but an ill-posed problem for an elliptic operator $A_{b}^{*}A_{b}$. Let us see what the addition of the property (11) gives in a more general problem than those in cardiology.

Denote by $N(\Omega)$ the set of solutions to the problem (5), (6), (11) under the condition $f=0$. Let $S_{A_{e}}(\Omega_{H})$ be the space of generalized solutions of the equation $A_{e}h=0$ в $\Omega_{H}$. Since the operator $A_{e}$ has an injective symbol and its coefficients are real analytic, the Petrovsky theorem yields that the elements of the space $S_{A_{e}}(\Omega_{H})$ are real analytic vector-functions in $\Omega_{H}$.

In particular, it follows from Lemma 3.1 that the zero space of the problem (5) coincides with the space $S_{A_{e}}(\Omega_{H})\cap{\left[H^{2}(\Omega_{H})\right]}^{k}$.

Denote by $\ker A_{e}$ the kernel of a continuous linear operator $A_{e}:\,[H^{2}(\Omega_{H})]^{k}\rightarrow[H^{1}(\Omega_{H})]^{l}$ and consider several examples. In fact, $\ker A_{e}=S_{A_{e}}(\Omega_{H})\cap{\left[H^{2}(\Omega_{H})\right]}^{k}$.

It is clear that the operator $A_{e}$ should be chosen in a way that its kernel is at least finite dimensional.

Under assumptions of Theorem 3.1 the rest of the scheme of solving the problem (5), (6), (11) differs little from the standard one, see [5]. Namely, first we introduce a function $h(x)$ such that $h(x)={\lambda}^{2}u_{i}+u_{e}$, where $x\in\Omega_{H}$. From the conditions on the boundaries in (5) and the fact that $\nu_{A_{i}}=\lambda\nu_{A_{e}}$ we get that

Thus, we can rewrite the original problem (5), (6), (11) in new notation: knowing a vector $f\in[H^{3/2}(\partial\Omega_{H})]^{k}$, find vectors $h\in[H^{2}(\partial\Omega_{H})]^{k}$ and $u_{b}\in[H^{2}(\partial\Omega)]^{k}$ such that

The original problem splits into two – (20) and (21). The problem (21), as noticed above, is an ill-posed Cauchy problem for an elliptic operator $A_{b}^{*}A_{b}$. It is known that if a solution to this problem exists it is unique. The problem (20) is a Neumann problem for an elliptic operator $A_{e}^{*}A_{e}$. Unfortunately, in general the Neumann problem may also be ill-posed. For it to be Fredholm, the so called Shapiro-Lopatinsky conditions must be placed [13, Chapter 1, §3, condition II for $q=0$], [14] on the pair $(A_{e}^{*}A_{e},\nu_{A_{e}})$. In particular, they guarantee that the space $S_{A_{e}}(\Omega_{H})\cap[H^{2}(\Omega_{H})]^{k}$ is finite dimensional.

More precisely, let us consider the following Neumann problem: for a given vector $h_{0}\in[H^{1/2}(\partial\Omega_{H})]^{k}$ find a vector $h\in[H^{2}(\partial\Omega_{H})]^{k}$ such that

and formulate conditions for solvability.

Thus, under hypothesis of Theorem 3.2 for solvability of the Neumann problem (20) it is necessary and sufficient that for the vector $h_{0}=-\nu_{A_{b}}u_{b}$ the condition (23) is fulfilled. This can be achieved if we place additional conditions on relations between the operators $A_{e}$ and $A_{b}$. Namely, as we have seen above, it is quite natural to assume that

Denote by $S_{A_{b}}(\Omega)$ the zero space of solutions to the problem (21) in the domain $\Omega$.

The condition that the spaces $S_{A_{e}}(\Omega_{H})\cap[H^{2}(\Omega_{H})]^{k}$ and $S_{A_{b}}(\Omega)\cap[H^{2}(\Omega)]^{k}$ coincide seems to be rather strong, especially since these are spaces of solutions to different differential equations in different domains. Nevertheless, provided (25) holds, such a coincidence is possible if the operator $A_{e}$ is so much overdetermined that the space of its solutions in any domain is finite dimensional and coincides with the space of solutions in ${\mathbb{R}}^{n}$; the typical examples are the so-called stationary holonomic systems. Let us illustrate this by the following examples.