Existencia de solución en un modelo de actividad eléctrica de tipo monodominio para un ventrículo Existence of global solutions in a model of electrical activity of the monodomain type for a ventricle

Resumen Introducción: Se formula un modelo de monodominio de actividad eléctrica en un ventrículo aislado. Este modelo se escribe como una EDP de tipo reacción difusión acoplada a una EDO, se utiliza el modelo de Rogers-Mculloch para representar la actividad eléctrica a través de la membrana celular. Método: Se proponen definiciones de solución débil y fuerte respectivamente para el problema de Cauchy variacional asociado al modelo de monodominio. Se propone una sucesión de soluciones aproximadas de tipo Faedo-Galerkin. Resultados: Se demuestra que la sucesión de soluciones aproximadas converge a una solución débil según la definición que se propone. Finalmente, se obtiene que la solución débil es también una solución fuerte. Conclusión: El modelo de monodominio de actividad eléctrica en un ventrículo aislado que se propone tiene solución débil en un sentido apropiado. Además, esta solución débil también es una solución fuerte.


Introduction
The bidomain model represents an active myocardium on a macroscopic scale by relating membrane ionic current, membrane potential, and extracellular potential (Henriquez 1993).
Created in 1969 (Schmidt 1969), (Clerc 1976) and first developed formally in 1978 (Tung 1978), (Miller 1978, I), the bidomain model was initially used to derive forward models, which compute extracellular and body-surface potentials from given membrane potentials (Miller 1978, I), (Gulrajani 1983), (Miller 1978, II) and (Gulrajani 1998).Later, the bidomain model was used to link multiple membrane models together to form a bidomain reaction-diffusion (R-D) model (Barr 1984), (Roth 1991), which simulates propagating activation based on no other premises than those of the membrane model, those of the bidomain model, and Maxwell's equations.Other mathematical derivations of the macroscopic bidomain type models directly from the microscopic properties of tissue and using asymptotic and homogenization methods along with basic physical principles are presented in (Neu 1993), (Ambrosio 2000) and (Pennacchio 2005).
Monodomain R-D models, conceived as a simplification of the R-D bidomain models, with advantages both for mathematical analysis and computation, were actually developed before the first bidomain R-D models, and few papers have compared monodomain with bidomain results.Those that did, have shown small differences (Vigmond 2002), and monodomain simulations have provided realistic results (Leon 1991), (Hren 1997), (Huiskamp 1998), (Bernus 2002), (Trudel 2004) and (Berenfeld 1996).In (Potse 2006) has been investigated the impact of the monodomain assumption on simulated propagation in an isolated human heart, by comparing results with a bidomain model.They have shown that differences between the two models were extremely small, even if extracellular potentials were influenced considerably by fluid-filled cavities.All properties of the membrane potentials and extracellular potentials simulated by the bidomain model have been accurately reproduced by the monodomain model with a small difference in propagation velocity between both models, even in abnormal cases with the Na conductivity (Bernus 2002) reduced to 1=10 of its normal value, and have arrived at the same conclusions.The difference between the results that may be obtained with one or another model are small enough to be ignored for most applications, with the exception of simulations involving applied external currents or in the presence adjacent fluid on within, although these effects seem to be ignorable on the scale of a human heart.A formal derivation of the monodomain equation as we present here can be found in (Sundnes 2006).There are few references in the literature dealing with the proof of the well-posedness of the bidomain model.The most important seem to be Colli-Franzone and Savarés paper (Colli 2002), Veneroni's technical report (Veneroni 2009) and Y. Bourgault, Y. Coudière and C. Pierre's paper (Bourgault 2009).In (Colli 2002), global existence in time and uniqueness for the solution of the bidomain model is proven, although their approach applies to particular cases of ionic models, typically of the form (, ) = () +  and (, ) =  + , where  ∈  1 (ℝ) satisfies inf ℝ  ′ > −∞.In practice a common ionic model reading this form is the cubic-like FitzHugh-Nagumo model (Fitzhugh 1961), which, although it is important for qualitatively understanding of the action potential propagation, its applicability to myocardial excitable cells is limited (Keener 1998), (Panfilov 1997).However, from the results of (Colli 2002) is not possible to conclude the existence of solution for other simple two variable ionic models widely used in the literature for modelling myocardial cells, such as the Aliev-Panfilov (Aliev 1996) and MacCulloch (Rogers 1994) models.In (Veneroni 2009), Colli-Franzone and Savarés results have been extended to more general and more realistic ionic models, namely those taking the form of the Luo and Rudy I model (Luo 1991), this result still does not include the Aliev-Panfilov and MacCulloch models.In reference (Bourgault 2009), global in time weak solutions are obtained for ionic models reading as a single ODE with polynomial nonlinearities.These ionic models include the FitzHugh-Nagumo model (Fitzhugh 1961) and simple models more adapted to myocardial cells, such as the Aliev-Panfilov (Aliev 1996) and Rogers-MacCulloch (Rogers 1994) models.
In this paper, we give a definition of weak solution of the variational Cauchy problem and, from this one, we give a definition of strong solution.We aim to obtain the existence of a global weak solution for a monodomain R-D model when applied to a ventricle isolated from the torso in absence of blood on within, which is activated through the endocardium by a Purkinje current and for simpler ionic models reading as a single ODE with polynomial nonlinearities.Also, it is proved that this weak solution is strong in the sense of the given definition.We will consider a bounded subset Ω ∈ ℝ 3 simulating an isolated ventricle surrounded by an insulating medium.The boundary Ω of the spatial region is formed by two disjoint components; the component Γ 0 imulating the epicardium and the component Γ 1 simulating the endocardium.The way Ω is electrically stimulated is by means Purkinje fibers, which directly stimulate only the inner wall Γ 1 then the excitable nature of the tissue allows this stimulus to propagate by Ω.We will assume that the ventricle is isolated from the heart and torso, that is to say that Γ 0 is in contact with an electrically insulating medium.We will use the monodomain model and the    It is convenient to establish some notations that we will follow throughout this work.For convenience, we will denote  =  1 (Ω) and  =  2 (Ω) since we will make constant use of these spaces.It is important to note that in the context of this work the following inclusions are fulfilled for 2 ≤  ≤ 6 Note that only  is identified with its dual space.In particular, we will consider  = 4 from here on.As usual, ′ denotes a positive number such that Let  be a Banach space of integrable functions over Ω , we define the subspace Which is a Banach space with the norm induced by .For any  ∈ , we denote This paper is organized as follows.The spaces   (0, ; ) are the functional setting we will work in, so in section 2.1 the definition of this spaces along with some important facts about them are presented.In section 2.2 some preliminary results are established, mainly related to the diffusion term ∇(∇) and with the model for the ionic current  and .In section 2.3 we state the definition of weak and strong solution, and enunciate some results that allow us to find a relation between them.The existence will be shown in sections 3.1 and 4.1.
It is necessary to give a definition of the derivative of an element of   (0, ; ), for this we will consider the space of distributions on [0, ] with values in  , see (Lions 1969, 7).
Definition 1.We define ′(0, ; ), the space of distributions on [0, ] with values in , as where (0, ) is the set of infinitely differentiable functions of compact support in (0, ).
If  ∈ ′(0, ; ) we can define its derivative in the sense of distributions as For the chain of inclusions ( 9) and the fact that the immersion  →  is compact we can enunciate the following result, which is a particular case of a classic compactness result, see (Lions 1969, th. 5.1, p.58).

Definition of weak and strong solution
This section establishes the definition of the solution that will be obtained in section 3.1 for the model ( 1)-( 5) of a ventricle.Also, we define strong solution and give a result of selectivity of the weak solution.It will be necessary to consider the weak formulation both in time and space.In order to give a bit of context to this definition we will start by considering the variational formulation in the spatial variable of the original model, in this way it will be natural to introduce a succession of approximate solutions through a discretization of the space in which we will look for the solution.This procedure is known as the Faedo-Galerkin method.
In addition, the functions  and  satisfy where equality is considered in ′(0, ).
If, furthermore, given  0 ,  0 in , ,  are weak solutions that satisfy then we call ,  a weak solution of variational Cauchy problem associated to ( 1)-( 5).
Remark 1.The derivatives that appear in the first terms of the equations ( 29) and ( 30) refer to derivatives in the sense of distributions, that is, for  ∈ (0, ) we have Now, we can give a definition of strong solution for the variational formulation.Suppose that, ,  are weak solutions, in the sense of definition 4, and furthermore,  ∈  1,2, ′ (0, ;  ′ , ) and  ∈  1,2,2 (0, ; , ), then the equation ( 29) means that thus, by proposition 1, it has which implies that From the above it follows that, which holds in ′.In a similar for it is possible to prove that is fulfilled in .

Existence of global solution
The main result of this section is the following theorem.
The demonstration is developed in the following two subsections,  a sequence of approximate solutions   ,   is defined,  then, it is verified that the approximate solutions converge to a function that satisfies the definition 4.

Existence of approximate solutions
The next lemma states that the approximate solutions   ,   are defined for all  > 0, other important estimates are also established to demonstrate later that the succession of approximate solutions converges to a solution.The following norms will be used.
If (  ,   ) is not a global solution, this is   < 1, then it is not bounded in [ 0,   ).
Suppose that (  ,   ) is a maximal solution of ( 26 On the other hand, by Young's inequality we have for all  > 0 the following then, by taking  =  we get the following inequality that will be useful a little later. From (42) it follows immediately that Then, integrating with respect to  over the interval [0,   ) on both sides of the previous inequality we get Recall now that, there exist a constant  > 0, such that ‖  (0)‖  ≤  y ‖  (0)‖  ≤ , also we have that Ω is bounded.Then, from the previous inequality and from Gronwall's inequality it follows that there is a constant  1 that depends only on , , , ,  0 ,  0 , Ω, ,  ̂ and   , such that As a consequence we have that (  ,   ) is bounded in any finite interval of time, this is   = +∞.For  > 0 fixed we have shown (33).
In order to get (34) we begin by integrating ( 42 On the other hand, from the estimates (33), ( 34) and by lemma 6 The next thing will be to obtain a bound for the projection operator   .We begin by highlighting that, as   ( ′ ) ⊂   ⊂ , the restriction of   to V can be considered as an The previous inequality shows that the family of operators   is uniformly bounded in ′, Then, the following inequalities are met Inequality ( 35) is obtained from the previous inequalities and ( 43).We will proceed similarly to obtain the estimate for ′  .From ( 27) it follows that and therefore where we take the operator   restricted to the orthogonal projection  | , so ‖  ‖ ℒ(,) ≤ 1.

Convergence of approximate solutions
In the previous section it was shown that the approximate solutions proposed in (25) exist and are defined for all  > 0. In this section we will use the a priori estimates ( 33) -( 36) to show that, there exist subsequences of the approximate solutions (  ,   ) that converge, in a suitable form, to a weak solution according to the definition 4. Furthermore, we prove that this weak solutions is also a strong solution.
Proof.Let us take  ∈ ,  ∈ (0, ), and note that, by taking limit in the above equality we obtain Thus, we have obtained (50).Also, by the weak converge of   ′ , we get and, due to the uniqueness the weak limit that is In a similar form are proved the affirmations for .

Conclusion
By the three previous corollaries it is concluded that the functions  and  satisfy for all  ≥ 1 the following where equality is considered in ′(0, ).Then, because functions   ,  ≥ 0 are dense in , it follows that  and  satisfy the equations ( 29)-( 30) in the definition of weak solution 4.
In other words, we have proved that if the systems of Faedo-Galerkin ( 26)-( 27) are considered with uniformly bounded initial conditions the corresponding solutions,   ,   , have subsequences that converge, in a suitable form, to a weak solution of the considered problem.
Note that, in the case that the Cauchy problem be considered for the variational formulation, that is, initial conditions  0 ,  0 be given the systems of Faedo-Galerkin ( 26)-( 27) have initial conditions  0 ,  0 which are the projections of  0 ,  0 in the subspaces   , for each  = 0, 1, …, and are uniformly bounded.In fact, thus, by applying the results previously exposed we obtain the existence of weak solution of the variational Cauchy problem.
When we consider  0 and  0 as the orthogonal projections in  of  0 and  0 respectively, we obtain that (0) =  0 and (0) =  0 .

Rogers-
McCulloch model for ion currents through the cell membrane, in this way and for the above considerations this model can be written as one parabolic PDE with boundary conditions, coupled to a ODE, and some initial data: The unknowns are the scalar functions (, ) and (, ) which are the membrane potential and an auxiliary variable without physiological interpretation called the recovery variable, respectvely.We denote by  the unit normal to Ω out of Ω.The anisotropic properties of the tissue are included in the model by the conductivity tensor ().The functions (, ) and (, ) crrespond to the flow of ions through the cell membrane.The function : (0, +∞) → ℝ represents the electrical activation of the endocardium by means of Purkinje fibers.The function : Ω → ℝ represents the activation spatial density.Because we consider that Ω is surrounded by an insulating medium, there is no current flowing out of Ω, this is expressed in the boundary condition (3).The specific assumptions we will make about (1) -(5) are as follows:(h1) Ω has Lipschitz boundary Ω.

(
h2) () is a symmetric matrix, function of the spatial variable  ∈ Ω, with coefficients in  ∞ (Ω) and such that there are positive constants m and M such that Is met for almost all  ∈ Ω. (h3)  ∈  ∞ (0, +∞).