Renormalized and Entropy Solutions of Tumor Growth Model with Nonlinear Acid Production

This paper establishes the existence of renormalized and entropy solutions for a system of nonlinear reaction-diffusion equations which describes the tumor growth along with acidification and interaction. Under the assumptions of L data and no growth conditions with zero Dirichlet boundary conditions, we prove the existence of renormalized and entropy solutions for the considered mathematical model.


Introduction
Acid-mediated tumor invasion model confines a mechanism linking altered glucose metabolism with the ability of tumor cells to form invasive cancers. Glucose metabolism and increased glucose uptake observed in the majority of clinical cancers which are critical for development of the invasive phenotype. Tumor cells are resistance to acid induced toxicity which survive and proliferate in low pH micro environments, invade the damaged adjacent normal tissues. First, acidification of the tumor micro environment is shown by Gatenby et al. [15] and Martin et al. [17] to increase invasiveness and metastasis of cancer cells using mathematical model. Acidification induced by the result of glycolysis both in the presence of oxygen through Warburg effect and intermittent hypoxia which produce toxicity in the surrounding normal tissue provides the empty space for tumor cell proliferation and invasion. Gatenby and Gawlinski [14] used the acid-mediated invasion hypothesis in a reaction-diffusion framework which plays an important role in tumor progression. The mathematical representation of a reaction-diffusion system at the tissue scale in which many mathematical models have been developed to explore the relationships between tumor invasion, tissue acidity and cellular metabolism and energy requirements, see, for example, Bertuzzi et al. [6], Bianchini and Fasano [7], Ganesan and Lingeshwaran [13], McGillen et al. [19], Smallbone et al. [26], Tao et al. [27] and Venkatasubramanian et al. [28].
Acidification factor plays a major role in the development of tumor growth models. Therefore, in this work, we consider a PDE model for acid-mediated tumor growth extended from [14] with nonlinear acidification term to capture a wider range of tumor behaviors. Here, we also have included the interacting phenomena, that is, as tumor cell density increases with time, metabolism which produces H + ions leads to destruction of normal cells surrounding the tumor and thus a reduction in normal cell density. Thus, the nonlinear reactiondiffusion mathematical model is given by The mathematical model consists of three unknown variables: normal cell density u 1 (x, t), tumor cell density u 2 (x, t) and acid concentration u 3 (x, t). The homogeneous Dirichlet boundary condition means that the model (1.1) is selfcontained and has no population on the boundary ∂Ω. The density dependent diffusion coefficients for the normal cells and tumor cells are given by σ 1 (u 2 ) and σ 2 (u 1 ) respectively. Further, the excess H + ions diffuse chemically with constant diffusion rate with σ 3 . The normal cells and tumor cells obey the logistic growth ) with growth rates r 1 and r 2 and carrying capacities K 1 and K 2 . The competition relationship between the normal cells and tumor cells with the rates a 1 and a 2 are given by µ 1 (u 1 ) = a1u1 K1 and µ 2 (u 2 ) = a2u2 K2 . The interaction of healthy cells with the excess H + ions is given by π(u 1 , u 3 ) = α 1 u 1 u 3 which leads to a death rate proportional to the concentration of H + ions and denote the constant of proportionality by α 1 . Acidification caused at a rate r 3 proportional to the tumor cell density and an uptake term with constant proportionality α 3 is included to measure the mechanisms for increasing extracellular pH. Moreover, f (x, t), g(x, t) and h(x, t) denote the source terms of the respective equations. Further, we have assumed that all the coefficients are positive and σ i (·) ∈ C 2 (R), i = 1, 2.
The notion of renormalized solution is introduced by DiPerna and Lions for Boltzmann equations in [11]. Further, the same framework has extended for elliptic equations, parabolic equations and conservation laws, for example, see [5,8,9,10,18,22,23] and also see the references therein. Furthermore, a new class of study, equivalence between renormalized and entropy solutions studied for parabolic equations by many researchers, for example, see [1,3,12]. Moreover, in the literature, considerable amount of works are available for the existence and uniqueness of biological models using various mathematical techniques, for example, see, [2,4,24,25] and the references therein. As far as, acid-mediated cancer invasion model is concerned, only few papers available in the literature. Local and global existence of solutions of the model governed by acid-mediated tumor invasion established in [20]. Acid-mediated invasion model for tumor-stromal interactions under no flux boundary condition is concerned in [16] and the global existence and uniqueness proved using contraction mapping principle. A mathematical model focusing on the effect of heat shock proteins on the tumor cell migration is proposed and the local existence of a unique positive solution is obtained in [21]. On the otherhand, existence of renormalized and entropy solutions for the system of parabolic equations concerned, only few papers available in the literature, see, [3,4,24,25]. Therefore, in contrast to the above mentioned papers, in this work, the main novel point is to establish the existence of renormalized and entropy solutions of the model governed by the acid-mediated tumor growth under no growth conditions and integrable data.
The paper is organized as follows. In Section 2, we state the main theorem, that is, existence of renormalized solution of the parabolic system (1.1). Then, we introduce the regularized system of (1.1) and establish the existence of weak solutions of the regularized system. Further, we prove the existence of renormalized solution of (1.1) using the lemmas established in that section. Finally, in Section 3, we prove that renormalized solution of (1.1) is also an entropy solution.

Renormalized solutions for cancer invasion system
In this section, first we define the renormalized solution for the given parabolic system (1.1). After stating the main result of the work, we introduce regularized system for (1.1) and then we establish the existence of weak solutions of the regularized problem using the Faedo-Galerkin approximation method. Furthermore, we state and prove the certain lemmas which are useful to prove the existence of renormalized solutions of (1.1).
Further, we introduce the truncation function at the height k, Moreover, throughout this work, we use a generic constant C instead of different constants. For ε > 0, let us introduce the following approximations of the data: (H6) u ε i,0 ∈ L 2 (Ω), and u ε i,0 → u i,0 , i = 1, 2, 3 a.e. in Ω and strongly in L 1 (Ω) as ε tends to zero.
Consider the following approximation problem of the system (1.1) for ε > 0: with initial and boundary conditions ). Lemma 1. Under the hypotheses (H1−H3) and (H6−H7) the approximation system (2.1) admits unique weak solution Proof. To use the Faedo-Galerkin approximation method, let us consider an appropriate spectral problem, see [4], in which the corresponding eigenfunctions e l (x) form an orthogonal basis in H 1 0 (Ω) and orthonormal basis in L 2 (Ω). Our aim is to identify the finite dimensional approximation solutions for the system (2.1) as sequences {u ε i,n }, i = 1, 2, 3 defined for n ≥ 1, t ≥ 0 and x ∈ Ω by with the initial conditions Further, it should be remarked that the above form of solutions should satisfy the required boundary conditions. Next, we have to determine the set of coefficients {c i,n,l } n l=1 , i = 1, 2, 3 such that, for m = 1, 2, . . . , n (2.2) Now (2.2) can be rewritten in the following form: where the constant C(R, n) > 0 depends only on R and n. Similarly it is easy to obtain that where the constant C(R, n) > 0 depends only on R, n.
Definition 2. We define the Lipschitz continuous function in the following form Obviously the function Θ n (z) satisfies Θ n (z) L ∞ (R) ≤ 1, for any n ≥ 1 and Θ n (z) → 0, for any n ≥ 1 and also Θ n (z) → 0, for any z when n → ∞. Proof. Treating Θ n (u ε 1 ) as a test function and multiplying the first equation of (2.1) by Θ n (u ε 1 ) and integrating over Q × (0, t), we have for all t in (0, T ) and ε < 1 n+1 . For any subsequences u ε i (still denoted by u ε i ), Lemma 2 and (2.12) confirm that, as ε → 0 for any k > 0 and n ≥ 1. Using (H2), we can show that for i = 1, 2, 3 for η i,k ∈ L 2 (Q T ). From (2.5)-(2.8) and Lemma 3, we have From the boundedness of the solution triple (u ε 1 , u ε 2 , u ε 3 ), Lemma 2 and use Young's inequality, we get where C k is a constant independent of ε. Taking lim inf as ε tends to 0 in the above estimate and using the results (2.13) and Lemma 2, we have From the definition ofT k (u i ), i = 1, 2, 3, we deduce that for almost all t ∈ (0, T ) and (2.15) shows that u ∈ L ∞ (0, T ; L 1 (Ω)).
Lemma 5. Let k ≥ 0 be fixed and S be an increasing C ∞ (R) function such that S(z) = z for |z| ≤ k and supp S be compact. Then Proof. The proof is as similar as of Lemma 1 in [9].
Lemma 6. For i = 1, 2, 3, and η i,k which is defined in (2.14), the subsequence of u ε i (still denoted by u ε i ) satisfies, for i = 1, 2, 3, Proof. Let us introduce S n to be a sequence of increasing C ∞ (R) functions such that S n (z) = z, for |z| ≤ n, Multiply the first equation of (2.1) by S n (u ε 1 ) to get By similar procedure, we obtain that, for i = 1, 2, 3 ∂ ∂t S n (u ε i ) ∈ L 1 (Q T ) + L 2 (0, T ; H −1 (Ω)).

Entropy solutions for cancer invasion system
In this section, we have established the second main result of the paper, that is, the renormalized solution is also an entropy solution.