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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 f(u,w) = k(u) + αw and g(u,w) = βu + γw, where k ∈ C1(ℝ) satisfies infℝ k' > −∞. 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 ∈ ℝ3 simulating an isolated ventricle surrounded by an insulating medium. The boundary 𝜕Ω of the spatial region is formed by two disjoint components; the component Γ0 imulating the epicardium and the component Γ1 simulating the endocardium. The way Ω is electrically stimulated is by means Purkinje fibers, which directly stimulate only the inner wall Γ1 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 Γ0 is in contact with an electrically insulating medium. We will use the monodomain model and the 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 u(t,x) and w(t,x) which are the membrane potential and an auxiliary variable without physiological interpretation called the recovery variable, respectvely. We denote by η the unit normal to ∂Ω out of Ω. The anisotropic properties of the tissue are included in the model by the conductivity tensor σ(x). The functions f(u,w) and g(u,w) crrespond to the flow of ions through the cell membrane. The function s:(0,+∞) → ℝ represents the electrical activation of the endocardium by means of Purkinje fibers. The function φ: 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) σ(x) is a symmetric matrix, function of the spatial variable x ∈ Ω, with coefficients in L∞(Ω) and such that there are positive constants m and M such that
Is met for almost all x ∈ Ω.
(h3) s ∈ L∞(0,+∞).
(h4) φ ∈ L2(Γ1).
(h5) f(u,w) y g(u,w) y stands for Rogers-McCulloch ionic model,
(h6) u0, w0 ∈ L2(Ω).
It is convenient to establish some notations that we will follow throughout this work. For convenience, we will denote V = H1(Ω) and H = L2(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 ≤ p ≤ 6
Note that only H is identified with its dual space. In particular, we will consider p = 4 from here on. As usual, p′ denotes a positive number such that
Let X be a Banach space of integrable functions over Ω, we define the subspace
Which is a Banach space with the norm induced by X. For any u ∈ X, we denote
Thus [u] ∈ X/ℝ.
This paper is organized as follows. The spaces Lq(0,T;X) 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 ∇(σ∇u) and with the model for the ionic current f and g. 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.
Method
Lq(0,T;X) spaces
Let X be a Banach space, we denote by Lq(0,T;X) the space of the functions t → f(t) of [0,T] → X that are measurable with values in X such that
with this norm Lq(0,T;X) is complete. Observe that
where QT[0,T] × Ω.
It is necessary to give a definition of the derivative of an element of Lq(0,T;X), for this we will consider the space of distributions on [0,T] with values in X, see (Lions 1969, 7).
Definition 1. We define D'(0,T;X), the space of distributions on [0,T] with values in X, as
where D(0,T) is the set of infinitely differentiable functions of compact support in (0,T).
If f ∈ D'(0,T;X) we can define its derivative in the sense of distributions as
If f ∈ D'(0,T;X) it corresponds a distribution
In this way, we can define the derivative in the sense of distributions of a function f ∈ Lq(0,T;X) as
Theorem 1. Let QT a bounded open in ℝ × ℝN fn and f functions in Lq(QT), 1 < q < ∞, such that
for a certain constant C > 0, then,
Proof. (Lions 1969, lema 1.3, p. 12).
For the chain of inclusions (9) and the fact that the immersion V →H 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).
Theorem 2. We define for T finite and 0 < qi < ∞,i = 0,1,
endowed with the norm
Proposition 1.
Let with
where (·,·) represents the scalar product in H, and 〈ũ,ν〉
𝑉′×𝑉
represents the evaluation of functional ũ in u. That is, u is the distributional derivative of u, and is the only function
From now on, we write 〈∙,∙〉 instead of 〈∙,∙〉 𝑉′×𝑉 .
Theorem 3. If f ∈ Lq and ∂tf ∈ Lq(0,T;X) (1 ≤ q ≤ ∞),then, f is continuous
almost everywhere from (0,T) to X
Proof. (Lions 1969, lema 1.2, p. 7).
Preliminaries
Definition 2. For all u, v ∈ V × V we define the bilinear form
Proposition 2. The bilinear form 𝑎 (⋅,⋅) is symmetric, continuous and coercitive in V,
with α,M > 0. There is a growing sequence 0 = λ0 < ⋯ < λi < ⋯ ∈ ℝ and there is an orthonormal basis of H formed by eigenvectors {ψi}i∈ℕ such that, ψi ∈ V y
Proof. The symmetry of a(⋅,⋅) is immediate consequence of the symmetry of σ. By (h2). we have that σ is uniformly elliptic and symmetric, then satisfies the following inequality
then, integrating over Ω and adding
which shows (17), the continuity of a(⋅,⋅)is also a consequence of (6). The existence of egenvalues and eigenvectors is obtained by a classical result, see (Raviart 1992, thm 6.2-1 y rem. 6.2-2, p. 137-138), taking into account that λ0 = 0 because the bilinear form a(⋅,⋅) is canceled only for constant functions.
It is important to note that the properties of the bilinear form a(⋅,⋅)allow to introduce an operator in a natural way.
Definition 3. By the previous lemma, the hypotheses of the Lax-Milgram theorem for the bilinear form a(⋅,⋅) are fulfilled and therefore there is an operator A:V → V' injective and continuous with continuous inverse such that
If v is a function defined on Ω we denote its trace to the boundary ∂Ω also as v, its meaning will always be clear from the context.
Proposition 3. If φ ∈ L2(Γ1) then for v ∈ V the function
defines a linear and continuous functional. This is, we have
We will denote
with
Proposition 4. For p = 4, there are constants ci ≥ 0, i = 1,…,6, such that for all u ∈ ℝ the following inequalities hold.
Proof. Due to Young’s inequality the following estimates are met
Then,
Proposition 5. For =4, there are λ > 0, μ, c ≥ 0 such that for all(u,w) ∈ ℝ we have
Proof. By direct calculation from (20) we have
On the other hand, from Young’s inequality we have
Then,
To continue, it is necessary to extract a common term from the coefficients corresponding to |u|2 and |w|2, for this we can write
To conclude it is necessary to verify that θ, β and ρ can be chosen so that
which is fulfilled for
obviously, we can find a ρ small enough to meet such conditions. We have μ = γ, λ > 0 arbitrary,
Proposition 6. Let u ∈ Lp (Ω) and w ∈ H, Then f(u,w) ∈ Lp′ (Ω) and g(u,w) ∈ H. In addition, the following inequalities are met
where Ai ≥ 0, i = 0,...,3 y Bi ≥ 0, i = 0,...,3 are constants that depend only on ci, i = 1,...,6 and p.
Proof. Let (u,w) ∈ ℝ2, by proposition 4 we have
with B1 =c5,B2 = c6 y B3 = |g2|. On the other hand, by Young’s inequality, with
then, because
then, once more by Young’s inequality
If (u,v) ∈ Lp(Ω) × H, by direct calculation and taking into account that (p − 1)p′ = p,βp′ = 2 we have
In a similar way
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.
We will denote as Vm the linear space generated by {ψ0,ψ1,…,ψm}, where the functions ψi,i = 0,…,m, are eigenfunction of the bilinear form a(⋅,⋅) as established in the proposition 2. Note that Vm ⊂ V. For each m, we consider the variational problem restricted to the space Vm, that is, instead of v andz we take ψi,i = 0,…,m, and approximate u(t) and w(t) by um(t) and um(t) respectively, with
By means of these substitutions we obtain from (22)-(24) the following system
for i = 0,…,m.
Definition 4. (Weak Solution). Let τ > 0 and the functions u ∶ t ∈ [0,τ) ↦ u(t) ∈ H, w ∶ t ∈ [0,τ) ↦ w(t) ∈ H. We say that (u,w) is a weak solution of the varitional formulationof the problem (1)-(4) if for any T ∈ (0,τ),
u ∶ [0,T] ↦ and w ∶ [0,T] ↦ H are continuous.
For almost all t ∈ (0,τ), we have u(t) ∈ V, also u ∈ Lp (QT) ∩ L2 (0,T;V) and w ∈ L2 (QT), with QT = (0,T) × Ω.
In addition, the functions u and w satisfy
where equality is considered in D′(0,T).
If, furthermore, given u0 in H, u, w0 in H, u,w in, are weak solutions that satisfy
then we call u,w 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 ϕ ∈ D(0,T) we have
Now, we can give a definition of strong solution for the variational formulation. Suppose that, u,w are weak solutions, in the sense of definition 4, and furthermore, u ∈ W1,2,p′(0,T;V′,V) and w ∈ W1,2,2(0,T;H,H), then the equation (29) means that
thus, by proposition 1, it has
which implies that
From the above it follows that,
which holds in V′ In a similar for it is possible to prove that
is fulfilled in H.
Definition 5. (Strong Solution). Let be u ∈ W1,2,p′ (0,T;V,V′) and w ∈ W1,2,2(0,T,H,H) we call u,w strong solutions of the variational formulation problem (1)-(4), if they satisfy the equation (31)-(32) in V′ and H, respectively.
If, besides,
for u0, w0 given, we say that u,w are strong solutions of variational Cauchy problem associated to (1)-(5).
Results
Existence of global solution
The main result of this section is the following theorem.
Theorem 4. (Existence of weak solution). Under the hypotheses (h1)-(h5) plus
(h6’) the sequences um0, wm0 are bounded in H,
the system (1)- (4) has a weak solution (u,w) in the sense of the definition 4 with τ = +∞.
The demonstration is developed in the following two subsections,
Existence of approximate solutions
The next lemma states that the approximate solutions um, wm are defined for all t > 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.
Lemma 1. The Cauchy problem (26) - (28) has solution for all t > 0. In addition, there are
constants Ci > 0,i = 1,…,4, such that for all T > 0. The following estimates are met a priori
where
Proof. Note that the integrals in (26) and (27) are well defined, in deed, as um (t) ∈ V ⸦ Lp (Ω) and wm(t) ∈ H we have from proposition 6 that f(um(t),wm (t)) ∈ Lp′(Ω) ⊂ V′ and, g(um,wm) ∈ H, then because ψi ∈ V ⊂ Lp(Ω) and ψi ∈ H we have
The terms in (26) and (27) are continuous as functions of uim(t) and wim(t), then the initial value problem formed by (26) - (27) with initial conditions (28) has a unique maximal solution defined for t ∈ [0,tm) with uim and wim in C1, for each initial condition u0m, w0m, (by Cauchy-Peano theorem).
If (um,wm) is not a global solution, this is tm < 1, then it is not bounded in [ 0,tm). Suppose that (um,wm) is a maximal solution of (26)-(28). Multiplying (26) by λuim, (27) by wim and adding on i = 0,…,m we get
Note that for being {ψi} an orthonormal set we have
Then, by the previous observations, adding (37) and (38) we have for all t ∈ [ 0,tm)
On the other hand, note that for being a(⋅,⋅) coercitive, see (17), we have
Also, from proposition 5, by integrating both sides of (21) on Ω we get
Then, adding (40) and (41) we get
Adding
Then, reorganizing terms and adding
On the other hand, by Young’s inequality we have for all θ > 0 the following
then, by taking θ = λα we get the following inequality that will be useful a little later.
From (42) it follows immediately that
Then, integrating with respect to t over the interval [0,tm) on both sides of the previous inequality we get
Recall now that, there exist a constant c > 0, such that ‖um(0)‖H ≤ c y ‖wm(0)‖H ≤ c, y, also we have that Ω is bounded. Then, from the previous inequality and from Gronwall’s inequality it follows that there is a constant C1 that depends only on
As a consequence we have that (um,wm) is bounded in any finite interval of time, this is. tm = +∞. For T > 0 fixed we have shown (33).
In order to get (34) we begin by integrating (42) in the interval [0,T]
with
with k2 = k1 + (α + μ)C1T. Therefore, we have shown inequality (34) with
Integrating (33) on [0,T] we also get a bound for wm in L2(QT).
Now we will obtain the estimates for u’m and w’m. Consider the projection operator Pm ∶ V′ → V′ defined by
because u’m(t) ∈ Vm ⊂ V′, f(um(t),wm(t)) ∈ Lp′(QT) and v ∈ V ⊂ Lp(QT). Thus, from (26) it follows that
and then
where A is the weak operator defined in (19). For the continuity of A and the estimate (34) we have for all T > 0
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 Pm. We begin by highlighting that, as Pm(V′) ⊂ Vm ⊂ V, the restriction of Pm to V can be considered as an operator from V on V defined by
Therefore, for all u ϵ V we have
The previous inequality shows that the family of operators Pm is uniformly bounded in V′,
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 w′m. From (27) it follows that
and therefore
where we take the operator Pm restricted to the orthogonal projection Pm|H, so ‖Pm‖ℒ(H,H) ≤ 1. Then, for T > 0 fixed, from (33), (34) and by proposition 6, we have (36)
Convergence of approximate solutions
In the previous section it was shown that the approximate solutions proposed in (25) exist and are defined for all t > 0. In this section we will use the a priori estimates (33) - (36) to show that, there exist subsequences of the approximate solutions (um,wm) 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.
Lemma 2. There are subsequences, which for convenience are also denoted as um, u'm, wm and w'm such that
and
Proof. Evidently Lp(QT) ∩ L2(0,T;V) is a reflexive space since Lp(QT) is reflexive, see (Brezis 2011, prop. 3.20, p. 60). By inequality (34), um is a bounded sequence in Lp(QT) ∩ L2(0,T;V), then it has a subsequence that converge weakly, see (Brezis 2011, thm. 3.18, p. 69). So (44) has been proved. By a similar argument we obtain (45)-(47).
Note that, because 2 ≥ p′, we have Lp'(QT) + L2(0,T; V′) ⊂ Lp'(0,T; V′). By lemma 1 we know that u’m is bounded in Lp' (0,T; V′) while um is bounded in L2 (0,T; V) and then um is a bounded sequence in W1,2,p'(0,T; V,V′), see theorem 2. Then, by the compact immersion of W1,2,p'(0,T;V,V′) in L2(QT), there is a subsequence that converge in L2(Qt).
Corollary 1. The subquences um,wm satisfy
and also, it has that
in D′(0,T). That is,u ∈ W1,2,p' ′(0,T; V,V′), and w ∈ W1,2,2 (0,T;H,H).
Proof. Let us take v ∈ V, ϕ ∈ D(0,T), and note that,
by taking limit in the above equality we obtain
Thus, we have obtained (50). Also, by the weak converge of u'm, we get
and, due to the uniqueness the weak limit
that is
In a similar form are proved the affirmations for w.
Corollary 2. For ψi ,i ≥ 0 and the bilinear form a(⋅,⋅) defined in (15) we have
Proof. Because a(⋅,⋅) is a continuous bilinear form, the map
is a continuous linear functional on Lp(QT) ∩ L2(0,T;V), and then the result follows immediately from the fact that um converges to u weakly in Lp(QT) ∩ L2(0,T;V).
Corollary 3. For f and g defined in (7)-(8) and for all ψi,i ≥ 0, we have
Proof. Given that um → u, and wm → w, in L2(QT), it obtains
and by the continuity of f,
Also,
And
Using an argument of dominated convergence type, see (Lions 1969), we can affirm that f(um,wm), converges to f(u,w), and g(um,wm), converges to g(u,w), weakly in Lp'(QT), and L2(QT), respectively, that is, for all ζ ∈ Lp(QT) and η ∈ L2(QT), it has
taking ζ = ϕv, η = ϕℎ with ϕ ∈ D'(0,T), v ∈ V and ℎ ∈ H, it has the result.
Conclusion
By the three previous corollaries it is concluded that the functions u and w satisfy for all i ≥ 1 the the following
where equality is considered in D′(0,T). Then, because functions ψi,i ≥ 0 are dense in V, it follows that u and w satisfy the equations (29)-(30) in the definition of weak solution 4.
For other hand, by corollary 1, these weak solutions u,w belong to W1,2,p'(0,T;V,V') and W1,2,2(0,T;H,H), thus they are strong solutions , too.
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, um,wm, 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 u0,w0 be given the systems of Faedo-Galerkin (26)-(27) have initial conditions u0m,w0m which are the projections of u0,w0 in the subspaces, Vm, for each m = 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.
Continuity
From the previous section we have that u ∈ W1,2,p′(0,T; V,V′) ⊂ W1,2,2(0,T; V′ V′), and w ∈ W1,2,2(0,T;H,H). Then, by theorem (3) it follows that the functions u: t ∈ [0,T] → u(t) ∈ V′ and w:t ∈ [0,T] → w(t) ∈ H are continuous. Regarding u, it only shows that u, it is weakly continuous in V.
By corollary 1 it follows that
where equality is considered in D′(0,T). Then, from (52), we have
so that the function
When we consider um0 and wm0 as the orthogonal projections in H of u0 and w0 respectively, we obtain that u(0) = u0 and w(0) = w0.
