ANALYSIS OF A MODEL FOR TUMOR GROWTH AND LACTATE EXCHANGES IN A GLIOMA

. Our aim in this paper is to study a mathematical model for tumor growth and lactate exchanges in a glioma. We prove the existence of nonneg- ative (i.e. biologically relevant) solutions and, under proper assumptions, the uniqueness of the solution. We also state the permanence of the tumor when necrosis is not taken into account in the model and obtain linear stability results. We end the paper with numerical simulations.

u and the lactate concentrations in the intracellular and in the capillary domains, respectively, ϕ and ψ.
The resulting problem is given by a reaction-diffusion equation for u coupled with two parabolic nonlinear equations for the lactates, involving in particular cotransport terms. More precisely, it reads ∂ t u − div(D∇u) = a(ϕ, ψ) u(γ − u) − ug(ϕ, ψ), Dealing with a biomedical system, our first concern is to show that a biologically relevant solution originates from any biologically meaningful initial datum: therefore, considering density and concentrations, the solutions are expected to stay nonnegative if they depart from biological data. This is a crucial issue also on account of the possibly singular co-transport terms for the lactate concentrations. Then, we prove the existence of a weak solution on any finite time interval by the Galerkin method and further a priori estimates, which requires a careful management of the nonlinearities. Since the tumor proliferation rate is in general non-constant and it actually depends on the lactate concentrations, the uniqueness of the solution is not obvious. Indeed, this will be proved under suitable assumptions.
The boundedness of the solutions on any finite time interval is another desirable feature: such a task can be accomplished provided that the initial data are smooth enough. We conclude with some forecasts: we see that the illness is not extinguished when we neglect necrosis. Also, assuming a constant balance sheet for lactate, as far as production, consumption and export are concerned, under suitable conditions we observe two stationary solutions sharing the same fixed lactate concentrations, whose linear stability can be investigated. While the most favorable occurrence, corresponding to the absence of tumor is unstable, the other one is stable provided that suitable assumptions hold true.
In our model, the equation for the tumor cell density is a reaction-diffusion equation in which the proliferation rate a depends on both lactate concentrations: for fixed concentrations, the tumor cells increase their density till they reach their maximal capacity γ > 0. We also take into account the tumor cells necrosis whose rate is represented by the function g(ϕ, ψ). The equations governing the evolution of extracellular and capillary lactates are both diffusion equations (the terms −α∆ϕ and −β∆ψ correspond to random motions) and contain nonlinear co-transport terms through the brain-blood boundary, where k is the maximum transport rate between the blood and the cell, while K > 0 and K > 0 are the Michaelis-Menten constants for the intracellular and the capillary lactate, respectively. Concerning the extracellular lactate, we also keep into account a forcing term J, depending on time, the tumor cell density and the extracellular lactate seen as regulatory terms. The function J represents the balance sheet of the production, consumption or export of lactate by the cell. In the equation for capillary lactate, the model also accounts for the volume separating these two compartments, ε > 0, and the blood flow contribution both from arterial and venous lactate; in particular, L > 0 corresponds to the arterial lactate concentration and F to the cerebral blood flow.
The sole lactate kinetics has been investigated in [3,4,7]. In particular, wellposedness, bounds on the solutions and stability of the unique spatially homogeneous equilibrium have been obtained.
We also mention related models, also including diffusion and tumor growth ( [5]) and different phenotypes of the tumor cells ( [6]). These works account for glucoselactate dynamics (see also [8] in which diffusion is neglected).
The plan of the paper is as follows. In the next Section, we make precise our assumptions and introduce the proper functional setting, as well as the main tools that we employ. In Section 3, we prove the global in time existence of a weak solution, first addressing a modified problem where the symport terms are tamed so to be no longer singular. This further system is useful, since, for positive initial data, its solutions and those to the original problem coincide. Hence, having shown the local in time existence of a weak solution to the nonsingular problem by the Galerkin method, we are then able to prove that such a solution exists globally, owing to a priori estimates which are uniform in time. In order to tackle the uniqueness issue, we first devise suitable L ∞ − upper bounds on the solutions when the initial data are smooth enough and we neglect necrosis. These, or the reduction to two-space dimensions, allow us to properly handle the tumor proliferation rate when it is not constant, so that the uniqueness of the solution can be shown in three different cases. Finally, in the last section, we make some remarks on the asymptotic behavior of the solutions. Namely, without necrosis, permanence of the tumor is shown. Another case when we can infer something on the longtime behavior is when J is constant. Under suitable assumptions, two stationary solutions exist and are linearly unstable and stable, respectively.
2. Mathematical setting. We consider the following initial and boundary value problem: in Ω T = Ω × (0, T ), where Ω is a bounded and regular domain of R N with N = 1, 2, 3, unless otherwise stated. We recall that ε, α, β, γ, k, K, K , F, L are positive constants related with the biological mechanisms in the glioma, as described in the introduction.
We set H = L 2 (Ω) with inner product denoted by (·, ·) and corresponding norm · . We also set V = H 1 (Ω) equipped with the norm u 2 V = u 2 + ∇u 2 , and V its dual space, the symbol ·, · standing for the corresponding duality pairing. The proper phase space for biologically meaningful solutions is We recall the classical Gagliardo-Nirenberg inequalities: and, depending on the space dimension N , Throughout the paper, the letters c and C T denote generic positive constants which may vary from line to line. In particular, C T depends on the time interval [0, T ].
Eventually, we will be able to show that the weak solution to (1) issuing from initial data in H + coincides with the corresponding weak solution to (4), owing to the following.
3.1. Local existence of a weak solution to (4). In order to prove the local existence of a weak solution to (4), we introduce the Galerkin approximation of the problem. Thus, let 0 = λ 1 < λ 2 ≤ · · · be the eigenvalues of the minus Laplace operator associated with Neumann boundary conditions and e 1 , e 2 , · · · be associated eigenvectors such that {e j } ∞ j=1 forms an orthonormal basis in H which is also orthogonal in V . We set V n = Span{e 1 , . . . , e n } and denote by P n the corresponding projection.
Remark 1. The initial datum (P n u 0 , P n ϕ 0 , P n ψ 0 ) of the n−th approximated problem does not necessarily preserve the positivity of (u 0 , ϕ 0 , ψ 0 ). Therefore the maximum principle stated in Lemma 3.1 does not apply to (u n , ϕ n , ψ n ) and we have to deal with the modified system.
Thanks to the Cauchy Lipschitz theorem, for any n ∈ N there exists a unique solution (u n , ϕ n , ψ n ) ∈ C 1 ([0, t n ), V n ) to (6). Now in order to pass to the limit, we choose v = u n in the first equation of (6).
owing to (H1)-(H2), and having exploited (2) and the Young inequality. Here and throughout this proof, the constant c is independent of n but it is allowed to depend on T . Therefore, denoting by Λ(t) = u n (t) 2 , the assumption (H4) on D gives d dt for some c still independent of n. Arguing as in [2,Section 4], that is, by comparison with the solution to a suitable Cauchy problem for ODEs, we deduce that for some 0 < τ ≤ t n and c both independent of n. Integrating the above differential inequality over (0, τ ), we deduce that so that, by comparison, ∂ t u n L 2 (0,τ ;V ) ≤ c. Now, testing the second equation of (6) by ϕ n yields: Since s K +|s| ≤ 1 and, by (H3), noting that 0 ≤ J(·, ·, ·) ≤ J 2 , we get so that, as above The third component ψ n is analogously controlled, that is, we test the third equation of (6) by ψ n , obtaining Arguing as above, this leads to: and a further application of Gronwall's lemma followed by an integration in time yields Thus we conclude that, up to a subsequence, Moreover, since the previous a priori bounds and the interpolation inequality (2) lead to the uniform boundedness of u n in L 3 (Ω × (0, τ )), we have In order to prove that (u, ϕ, ψ) is indeed a weak solution to (4), we want to pass to the limit in the nonlinear terms in (6). Let w ∈ V and φ ∈ C ∞ 0 (0, τ ) be arbitrary and t ≤ τ . We only discuss the difficult terms, namely, The first term on the right-hand side is managed, thanks to (H1), the Hölder inequality and (7), as In order to show that also the second contribution tends to zero, we preliminarily observe that by (H1) on account of the boundedness and the Lipschitz continuity of a. Therefore having exploited (7). It is easier to tackle The symport nonlinear terms are easily treated thanks to the Lipschitz continuity of the map s µ+|s| for any µ > 0. For the reader's convenience we just show the first one, We can now prove the global in time existence of the weak solutions to (1) issuing from nonnegative initial data. Proof. Owing to Lemma 3.1 we learn that u(t), ϕ(t), ψ(t) ≥ 0 whenever defined. Therefore, multiplying the first equation of (1) by u, thanks to (H1),(H2) and (H4), we get : hence the dissipative estimate and the integral inequality with C T depending on T . The product of the second equation of (1) by ϕ yields: Since ψ K +ψ ≤ 1, and J(t, ϕ, u) ≤ J 2 due to (H3), we get 1 2 showing that no dissipation occurs for ϕ as Multiplying the third equation of (1) by ψ, we obtain: leading to: Thus we conclude that 1 2 4. L ∞ -upper bounds on the solutions. Lemma 4.1. Denoting by (u, ϕ, ψ) any solution departing from (u 0 , ϕ 0 , ψ 0 ) ∈ H + with u 0 ∈ L ∞ (Ω), we have u(t) L ∞ ≤ max{γ, u 0 L ∞ } for almost any t ≥ 0.
If instead u 0 L ∞ > γ, arguing as in [2], we compare u with the solution to the Cauchy problem y = a 1 Φ(y), For this, there exists a unique solution λ(t) for t ≥ 0 with γ < λ(t) ≤ u 0 L ∞ , for any t ≥ 0. Therefore, w = λ − u solves In particular, so that, taking the product by −w − , we obtain where Ω − = {x ∈ Ω : λ(t) ≤ u(x, t)}. Hence, by definition of Ω − , we have u ≥ λ and, since Φ is monotone decreasing on (γ, +∞), As a consequence, 1 2 and we can conclude that w − = 0, hence u ≤ λ, which, in particular, entails For more regular initial data, we can devise L ∞ -upper bounds on finite time intervals for both lactate concentrations, namely, be arbitrary. Then, for any given T > 0, there exists C T > 0 such that Proof. We multiply the first equation of (1) by −div(D∇u). This yields We handle the first term in the right-hand side employing (3) for N ≤ 3 and recalling (8), The other terms can be handled in the same way. Therefore, on account of d ∇u 2 ≤ (D∇u, ∇u). By Gronwall's lemma, we conclude that We next multiply the second equation of (1) by −∆ϕ. It yields: By (H3), we find 1 2 and we conclude that ∇ϕ(t) 2 ≤ C T and t 0 ∆ϕ(s) 2 ds ≤ C T ∀t ∈ [0, T ]. Now, the product of the third equation of (1) by −∆ψ yields 1 2 Again, we conclude that 1 2 d dt (ε ∇ψ 2 ) + β 2 ∆ψ 2 + F ∇ψ 2 ≤ c, so that, by Gronwall's lemma, . We then multiply the second equation of (1) by ∂ t ϕ, getting Owing to (H3), We rewrite the third equation of (1) in the following equivalent form: and differentiate this equation with respect to time. This yields: Multiplying this equation by ∂ t ψ, we obtain: Since the right hand side can be handled in the following way we obtain: A further application of Gronwall's lemma, on account of (9), yields To prove that ϕ(t) L ∞ ≤ C T , first notice that J(t, ϕ, u)+ kψ K +ψ ≤ J 2 +k =: C 2 , where C 2 > 0 is an absolute constant. Consider then the solution z to It is easy to check that z ≥ 0. Next, we prove that z(t) Taking the product of the first equation of (11) by z, we obtain . We now test (11) with −∆z, obtaining 1 2 Hence we get ∇z(t) ≤ C T , ∀t ∈ [0, T ]. We then differentiate (11) in time, and multiply the resulting equation by ∂ t z. We obtain: Then we deduce from the first equation of (11) that and, by Agmon's inequality, In order to compare ϕ with z, let w = ϕ − z. We can see that w solves Multiplying the equation in (12) by w + and using the fact that , we obtain: We conclude that w + (t) 2 ≤ w + (0) 2 = 0, which implies that ϕ(t) ≤ z(t) ≤ C T a.e. in Ω × [0, T ].
Multiplying the second equation in (13) by ϕ and recalling the definition of J i , we have on account of the Lipschitz continuity of J stated in (H3). Next the product of the third equation in (13) by ψ gives We take the product in H of the first equation in (13) by u, so that, recalling the definition of a i and g i , we deduce that The assumptions on a and g easily yield owing to (H1)-(H2), and on account of the Lipschitz continuity. Therefore, adding together the previous inequalities, we are left to consider d dt u 2 + ϕ 2 + ε ψ 2 + d ∇u 2 + 2α ∇ϕ 2 + 2β ∇ψ 2 ≤ c(1 + u 1 2 V )( u 2 + ϕ 2 + ε ψ 2 ) + 2|I 1 |. In the first case, we take a constant, so that I 1 vanishes and we can conclude thanks to (8).
In the second one, we may take advantage of (3) for N ≤ 2, so that Since this yields the uniqueness easily follows from (8).
In case u 0 ∈ L ∞ (Ω), then we know from Lemma 4.1 that We finally conclude that hence the uniqueness due to (8).
6. Permanence of the illness and linear stability. As far as the longterm dynamics is concerned, we will be able to prove that, when the function g vanishes, then there is permanence of the illness. In turn, when J is constant then, provided suitable conditions hold true, there exist two stationary states whose linear stability can be investigated: the former one is unstable, while the latter is stable.
Then v is solution to the equation: Multiplying this equation by v, we obtain, since a(ϕ, ψ) > 0 and u ≥ 0, meaning that extinction cannot occur.
(I) In order to prove that (0,φ,ψ) is unstable, we prove that the solution originating from a constant in space initial datum does not approach zero as time progresses. Fixũ 0 > 0 and consider the ODE This ODE can be explicitly solved, yielding (II) Now, we deal with the stability of (ū,φ,ψ). We can easily prove thatφ,ψ andũ are positive. So we just observe that, under our assumptions, the product of (15) byũ in H gives 1 2 yielding ũ(t) ≤ u 0 , ∀t ≥ 0.
Remark 2. When g = 0 and J is constant, the stability analysis complies with the aforementioned permanence of the disease for solutions departing from a biologically meaningful initial datum.
7. Numerical simulations. In the numerical simulations presented below, Ω is an ellipse parametrized by x = 6 cos θ, y = 8 sin θ, θ ∈ [0, 2π]. The tumor is initialized as u 0 (x, y) = 0.1e −10(x 2 +(y−2.5) 2 ) and is centered at the point (0; 2.5). We assume, as far as the lactate concentrations are concerned, that, far from the tumor area, the initial concentrations ϕ 0 and ψ 0 are close to the equilibrium valuesφ andψ and, in the tumor area, the initial lactate concentrations correspond to a spike (see Figure 1 and Figure 5).
7.1. Grompertz growth of the brain lactate concentration.
We can see in Figure 2 that the tumor spreads and the concentration increases with respect to time, up to its equilibrium value. Moreover the lactate concentrations in the tumor area tend to return to their equilibrium values.  7.2. Lactate decrease of the brain tumor. We now take initial lactate concentrations corresponding to patient 5 in [4], i.e. ϕ0 = 1.817mM, ψ0 = 2.291mM, J(t, ϕ, u) = 0.007mM.d −1 , the other parameters being unchanged. This particular value of J leads toφ ∼ 0.915,ψ ∼ 0.557 (henceφ +ψ ∼ 1.472) andū = 1.136. We can see in Figure 6 that the tumor spreads and the concentration increases with respect to time, up to its equilibrium value. Meanwhile the lactate concentrations inside the tumor decrease, tending to return to their equilibrium values.