Abstract

We consider a mathematical model to describe the growth of a vascular tumor including tumor cells, macrophages, and blood vessels. The resulting system of equations is reduced to a strongly coupled nonlinear parabolic system of degenerate type. Assuming the initial data are far enough from 0, we prove existence of a global weak solution with finite entropy to the problem by using an approximation procedure and a time discrete scheme.

1. Introduction

The theory of mixtures is used to develop a mathematical model that governs the interactions of macrophages, tumor cells, and blood vessels within a vascular tumor, focusing on the ability of macrophages to both lyse tumor cells and stimulate angiogenesis. In recent years, a variety of macroscopic continuum models have been derived by [1–6]. In their wake, we present hereafter a simplified model.

The vascular tumor is viewed as a mixture of three constituents: tumor cells, tumor-associated macrophages (abbreviated by TAMs), and blood vessels. We denote their respective volume fractions by , and , and we assume that the mixture is saturated, so we take We suppose that the tumor undergoes one dimensional and one side growth, parallel to the -axis, by occupying the region . Each phase is associated with velocity, pressure, and spatial stress denoted, respectively, by for the tumor cells, for the TAMs and for the blood vessels. Formulating conservation of mass for the three volume fractions, under the assumption that each phase has the same constant density, we get where the indices and are set for partial derivatives and and are the rates of production related to each phase, satisfying . We suppose that the volume fraction of tumor cells increase by proliferation and decrease by apoptosis, necrosis or lysis, the volume fraction of TAMs increase by proliferation and by influx from capillaries and decrease by natural death or after lysing tumor cells. Thus, we write where and are nonnegative constants. Assuming that the momentum is conserved and the motions of cells and blood vessels are so slow that inertial terms can be neglected, we can write where represent the momentum supply related to each phase, is a positive constant and is assumed to be a common pressure. When neglecting viscous effect, the partial stress tensors are given by where represent the pressures due to cell-cell interactions exerted on tumor cells, macrophages and blood vessels respectively. and are nonnegative constants and is constant.

The following initial and boundary conditions are considered: We impose the no flux boundary condition at the free boundary that we suppose moving at the same velocity as the tumor cells, so Adding the three continuity equations (1.2) and the three momentum equations (1.5), we get using (1.13) The last equality and (1.9) imply Using (1.6), (1.9), and the first relation in (1.15), the first equation of (1.5) reduces to give either , which we reject because it can be only transient, or which together with (1.16) gives Using (1.10) and the fact that is constant, (1.18) can be rewritten as follows: Similarly, the second equation of (1.5) simplifies into Thus, substituting the relations (1.19)-(1.20) into (1.2), the equations of and become Having regard to the saturation condition (1.1), we omit the equation of . The resulting problem (1.21) is strongly coupled with full diffusion matrix which is generally not positive definite. To simplify it, we reduce the number of biological parameters by setting In this case, (1.13), (1.14), (1.19) and (1.20) reduce the boundary condition at to Without loss of generality, we set and for technical reasons, to get the maximum principle (see Lemma 3.7), we need to take in (1.4) and In summary, denoting and , the problem (1.21) simplifies into with and given by The system (1.25) is complemented with the boundary and initial conditions and has to be solved in

In recent years, cross-diffusion systems have drawn a great deal attention. For example in [7], the global existence was established, as well as the existence of a global attractor in a case of triangular positive definite diffusion matrix. In [8], the well-posedness and the properties of steady states for a degenerate parabolic system with triangular positive (semi) definite matrix, modeling the chemotaxis movement of cells, were investigated. In [9, 10], the existence of global weak solution was shown for a nonlinear problem with full diffusion matrix. The proof was based on a symmetrization of the problem via an exponential transformation of variables, backward Euler approximation of the time derivative and an entropy functional. Here, we use analogous arguments, but in our case, after the transformation of variables the resulting matrix is not positive definite. To overcome this difficulty, we approximate by positive definite matrices which tend towards as , if the condition is satisfied. This needs to prove that the set is time invariant.

Throughout this paper, we use the following notations: let be positive real numbers, we will denote by all the positive constants which are independent of . We set and the positive part of the real number . We write and for partial derivatives of a real-valued function . Moreover we will use the Sobolev space equipped with the norm of and we denote as usual, by the dual of . In the case of vectorial functions, we designate the corresponding Lebesgue and Sobolev spaces, respectively, by . Finally we set for and for .

The remainder of this paper is organized as follows. In Section 2, we introduce the weak formulation of the problem and state our main existence result in Theorem 2.2. In proving this theorem, we define and solve in Section 3 an auxiliary problem which will be useful further. Then, in Section 4, we formulate a semidiscrete version in time of the problem, using a backward Euler approximation, combined with a perturbation of the diffusion matrix. This leads to a recursive sequence of elliptic problems depending on the small parameter . Performing the limit as , with the help of Aubin compactness lemma and the Sobolev embedding , we get a weak solution to our problem. Finally, an appendix is devoted to the proof of a technical lemma.

2. Main Result

We set the following assumptions:

(H1) (H2) The matrix is not positive even if , so the problem (1.25)–(1.27) has no classical solution in general. A weak solution is defined as follows.

Definition 2.1. Let (H1)-(H2) be satisfied ant let is said to be a weak solution of problem (1.25)–(1.27) on if
(1) with ,(2) a.e. in , a.e. in (3) , for all , where is the dual product between and .

Our main result is the following.

Theorem 2.2. Assume (H1)-(H2) are satisfied. Then for every , there exists (at least) a weak solution on to the system (1.25)–(1.27), satisfying the entropy inequality where depends on being positive functions defined on by

The proof of this existence result is based on the entropy inequality (2.1), which is formally obtained by testing (1.25) with , and integrating by parts (see Section 3.3 for details).

This estimate suggests to use the change of unknown , which transforms the problems (1.25)–(1.27) into the following one with and the new diffusion matrix is and takes the form The matrix resulting of this transformation is still not positive definite. Nevertheless, in the case which is under interest, is positive definite. Moreover, this change of variables leads to nonnegative solutions, without using maximum principle, since and .

3. Auxiliary Problems

We will use a time discretization scheme to study (1.25)–(1.27). In order to prove global existence for the resulting stationary problem, it may be useful to introduce an artificial perturbation of the diffusion matrix of type , where and is the identity matrix (see the proof of Proposition 3.3 below and Lemma 3.6). Nevertheless, the choice of the parameter is technical and cannot be done independently of the time discretization parameter . Here, we take , this choice being dictated by the sake of coherency of the discretization scheme proposed in Section 4. Indeed, in the case where is independent of , this procedure is seriously compromised. More details on this question are given in Remark 3.8, at the end of this section. In summary, we need to solve the following problem: where is a small parameter and is a fixed function. Before giving the existence result for this problem, let us define the solutions we deal with.

Definition 3.1. is said to be a weak solution of problem (3.1) if and if for every , it holds

We have the following result.

Theorem 3.2. Let satisfy assumption (H2), such that a.e. in and . Then for all , there exists a weak solution of the problem (3.1). Moreover, and it holds where and are defined by (2.2).

For the proof, according to the change of unknown introduced in Section 2, we will consider the following stationary problem where is the matrix defined by is given by and is a diagonal matrix with The vector field is defined by its components Clearly if , then when and . In addition, we have for that where depends only on We will prove the following result.

Proposition 3.3. Assume that is such that a.e. in . Then for all , there exists a weak solution to problem (3.4).

The proof of this result relies on the Leray-Schauder fixed point theorem, so we start by studying the linear problems associated with (3.4).

3.1. Linear Problems Associated with (3.4)

In the sequel, we let once for all and fixed. Let be given; we consider the following linear problem: find satisfying We have the following result.

Lemma 3.4. For every , problem (3.10) has a unique solution

Proof. We will apply the Lax-Milgram lemma. We set , so (3.10) goes over into the following equivalent problem Next, we define a bilinear form on and a linear form on by setting The continuity of and follows from the boundedness of and . For the coerciveness of , it is sufficient to prove that the matrix defined by is positive definite. Let us compute for and . We consider first the case where so that The elementary inequality and Young inequality lead to so In the case , we have From the inequality we deduce that So, thanks to Young inequality we get Hence, we infer that for all , so Lax-Milgram lemma implies the existence of a unique solution of problem (3.11). Consequently, is the unique solution of (3.10).

3.2. Proof of Proposition 3.3

Lemma 3.4 and the embedding allow us to define the map by setting the solution of (3.10). We will establish, using the theorem of Leray Schauder, that has a fixed point in , so is a solution of the nonlinear problem (3.4).

First, we prove that is continuous. Let be a sequence in such that strongly in as and let . We use the test function in (3.10), estimate (3.22), and Poincaré inequality to get where is independent of . So taking into account (3.9), we get, thanks to Young inequality, Thus, is bounded in and from the compactness of the embedding we deduce that there exists a subsequence of , still denoted by and a function such that This implies the weak convergence in , hence there exists a subsequence of which converges towards . Moreover thanks to the uniqueness result for the system (3.10), we see that all the sequence converges to which ends the proof of continuity of .

The compactness of follows from the compactness of the embedding into and (3.24). Finally, let us check that the sets are uniformly bounded with respect to . Observe that and if the equation is equivalent to and for all The remainder of the proof is a direct consequence of the following lemma.

Lemma 3.5. Under the assumptions of Proposition 3.3, there exists a positive constant independent of such that if satisfies (3.26), then it holds

For the proof, we need the following technical result which will be checked in the appendix.

Lemma 3.6. For all , one has where is the matrix given in (3.5).

Proof of Lemma 3.5. Testing the equation of (3.26) with leads to The left-hand side is estimated using (3.28). We write ; the convexity of leads to where for , for all . Using the elementary inequality , valid for all , we get with , for all . Combining (3.30) and (3.31), we find We recall that a.e. in for , and since the functions are continuous on , we deduce that the right-hand side of (3.32) is uniformly bounded with respect to Now we infer from (3.9) and Poincaré inequality that The result follows by combining all these inequalities.

3.3. End of Proof of Theorem 3.2

Let be the solution of (3.4) provided by Proposition 3.3. Recalling that and , we see that satisfies the following problem: where the matrix is given by with Note that if then and .

We will focus on the and estimates satisfied by the function . We begin with the following bounds.

Lemma 3.7. Let the hypotheses of Theorem 3.2 hold, and let satisfy problem (3.34). One has

Proof. We write the equation satisfied by and test it with . We get using (1.24) that so . Consequently, , and hence .

As a consequence of Lemma 3.7, we easily see that is a solution to problem (3.1) in the sense of Definition 3.1. Moreover, since then . In order to check the entropy inequality (3.3), we test (3.1) with to get Then, from (3.15) and Young inequality, we get so, inserting (3.39) into (3.38) and using the fact that , we see that Using the boundedness of and the fact that the function is bounded in , we obtain , then the convexity of the functions and leads to which ends the proof.

Remark 3.8. All the results of this section remain valid if one consider, instead of (3.1), the following problem: where and is independent of . In particular, the solution to (3.42) satisfies the bounds and the entropy inequality given in Theorem 3.2, with the constant independent of (and ). Thus performing the limit as in (3.42), we get a function solving the problem which corresponds to the “natural” time discretization of our problem (1.25)–(1.27). However, from there, the situation becomes complicated because the obtained solution to problem (3.43) has its components which are only nonnegative and we no longer have (in fact, we cannot even take the of ). Therefore, the time discretization scheme based on (3.43) cannot be solved.

4. Proof of Theorem 2.2

4.1. The Time Discretization Scheme

Let assumptions (H1)-(H2) hold and let . We will use the backward Euler approximation of time derivative . We divide the time interval into subintervals of the same length . Then, we define recursively , as the weak solution of (3.1) provided by Theorem 3.2 corresponding to the data , that is, being the initial condition of problem (1.25)–(1.27). Let be the piecewise constant in time interpolation on of and , respectively, that is, and let be the function defined on by With these notations, we can rewrite (4.1) as Now, we set for

4.2. Uniform Estimates with Respect to

Lemma 4.1. Let and defined by (4.2). One has and there exists a positive constant independent of such that

Proof. We apply the results of Theorem 3.2, so the first part is immediate, then (3.3) leads to Summing these inequalities from to , for , we get
Therefore, which can be written as This means that Coming back to (4.9), we deduce that and we get the result.

We have also the following estimate.

Lemma 4.2. There exists a positive constant independent of such that

Proof. We use (4.4) and the results of Lemma 4.1 to deduce that is uniformly bounded. Now, since on , then thanks to Lemma 4.1 we deduce that is uniformly bounded. Finally, to check (4.15), we have from (4.3) that for . This leads to the result by using (4.14).

4.3. Passing to the Limit as : End of Proof of Theorem 2.2

Using (4.14), we deduce the existence of a function such that as at least for some subsequence, Then Aubin compactness lemma and the compactness of the embedding of into lead to the strong convergence Moreover, by Lemma 4.1, we infer the existence of a function in and a subsequence of such that and according to (4.15) and (4.16), we derive that . Moreover, we have the strong convergence Indeed, where is independent of . Using (4.7), (4.14), (4.15), and (4.17), it is straightforward to deduce that strongly in , and hence a.e. in . Consequently, satisfies Finally, since , the initial condition is satisfied and thus is a weak solution of (1.25)–(1.27) in the sense of Definition 2.1.

Remark 4.3. The result proved in this work remains valid if we replace the condition by one of these conditions: Indeed, in these cases, direct calculations show that (3.15) becomes

Appendix

Proof of Lemma 3.6

Inequality (3.28) is equivalent to say that, for all We write denoting the successive terms of the equality with and . We split the proof into two cases.

Case 1. We suppose that , so that where and denote, respectively, the successive terms of the first and second sums in (A.4). To handle , we use (3.15) to see that Next, to estimate the other terms, we use the inequality , together with , valid for and . We get successively that which concludes the proof in the first case.

Case 2. We suppose that , and we set where Then, , and we set . Here again, the enumeration respects the ordering of the terms in each sum. First, we get from (3.19) that Next, to estimate , we distinguish two situations. If , then and Otherwise, we have , and Notice that can be estimated in the same way. Now for , since , we see that To estimate , we consider first the case , so that we have then if , the inequality leads to can be estimated along the same lines as , and we get the result.