A Glioblastoma PDE-ODE model including chemotaxis and vasculature

ESAIM: M2AN 56 (2022) 407–431 ESAIM: Mathematical Modelling and Numerical Analysis https://doi.org/10.1051/m2an/2022012 www.esaim-m2an.org A GLIOBLASTOMA PDE-ODE MODEL INCLUDING CHEMOTAXIS AND VASCULATURE Antonio Fern´ andez-Romero, Francisco Guill´ en-Gonz´ alez and Antonio Su´ arez* Abstract. In this work we analyse a PDE-ODE problem modelling the evolution of a Glioblastoma, which includes chemotaxis term directed to vasculature. First, we obtain some a priori estimates for the (possible) solutions of the model. In particular, under some conditions on the parameters, we obtain that the system does not develop blow-up at finite time. In addition, we design a fully discrete finite element scheme for the model which preserves some pointwise estimates of the continuous problem. Later, we make an adimensional study in order to reduce the number of parameters. Finally, we detect the main parameters determining different width of the ring formed by proliferative and necrotic cells and different regular/irregular behaviour of the tumor surface. Mathematics Subject Classification. 35A01, 35B40, 35M10, 35Q92, 47J35, 92B05. Received September 29, 2021. Accepted January 26, 2022. 1. Introduction Among the group of brain tumors, the Glioblastoma (GBM) is the most aggressive form with a survival of a little more than one year [26]. Moreover, GBM differs from many solid tumors in the sense that they grow infiltratively into the brain tissue, there exists an important presence of necrosis and they produce a high proliferation tumor cells. For all these reasons, GBM is one of the cancer types with more interest in the mathematical oncology community (see [1,4,32] and references therein). Some studies about the morphology of GBM are based in the magnetic resonance images (MRI) in order to obtain results related to prognosis and survival (see [23,27–29]). Specifically, Molab1group classifies the GBM depending on the width of the tumor ring and/or the tumor surface regularity (see [27,28] respectively). The study of [27] concludes that tumors with slim ring have better prognostic, specifically 7 months of more survival than tumors with thick ring. In [28], the survival of patients in relation to the surface growth, regular or irregular, of the GBM, show that tumors with a regular surface have better prognostic, more than 5 moths of survival, than tumor with irregular surface. In [39], the authors use the Fisher–Kolmogorov equation to reproduce the infiltrative characteristic of the GBM. However, more complex mathematical models are also built to simulate phenomena such that the tumor ring and the regularity surface of the GBM. One model appears in [30] where the tumor ring is studied by Keywords and phrases. Glioblastoma, chemotaxis, PDE-ODE system, numerical scheme. Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Universidad de Sevilla, Sevilla, Spain. *Corresponding author: [email protected] 1http://matematicas.uclm.es/molab/ c ○The authors. Published by EDP Sciences, SMAI 2022 This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 408 A. FERN´ ANDEZ-ROMERO ET AL. a PDE-ODE system of two equations (proliferative tumor and necrosis). In [11,12], the authors present a PDE-ODE system with three equations (proliferative tumor, necrosis and vasculature) which is able to capture different behaviours of tumor ring and regularity surface of the GBM via a nonlinear diffusion tumor increasing with vasculature. In this paper, we present a PDE-ODE system, also with three equations (tumor, necrosis and vasculature) and we study the biological behaviours of the GBM such as the tumor ring volume, studied in [27,30], and the regularity surface considered in [28]. Unlike the system considered in [11,12], we have included a chemotaxis term. This term has been already introduced to model the movement of some populations towards a higher concentration of the chemical substance or another living organism, see for instance the reviews given in [5,10, 15,16,31] and the references herein. Specifically, in this paper, we have included the chemotaxis term modelling the movement of tumor to vasculature. Some previous chemotactic PDE-ODE models have been extensively studied in the literature, see for instance [6,33–35] where the authors model the cells movement with a parabolic-ODE system. Specifically, in [35] a system of PDEs is considered using a probabilistic framework of reinforced random walks. The authors analyse various combinations of taxis and local dynamics giving examples of aggregation, blow-up and collapse. Later, in [33], some analytical and numerical results which support the numerical observations of [35] are presented using a similar model than in [35]. Moreover, in [3,6] a model of tumor inducing angiogenesis is proposed consisting of a equation with chemotaxis and haptotaxis term, and two nonlinear ODEs. Finally, in [34] a stochastic system related to bacteria and particles of chemical substances is discussed where the position of each particle is described by a equation of a chemotaxis system. Several works such as [19,36–38] have shown existence results for systems of three differential equations modelling cancer invasion. In [36] the global existence and boundedness of solution for a parabolic-parabolic-ODE system with nonlinear density-dependent chemotaxis and haptotaxis and logistic source is deduced. Furthermore, in [37], the authors have proved global existence of solutions for a parabolic-elliptic-ODE system with chemotaxis, haptotaxis and logistic growth. The study of existence of solutions for the chemotaxis and haptotaxis model with nonlinear diffusion is presented in [38]. The global existence of solution and its asymptotic behaviour are studied in [19] for a parabolic-parabolic-ODE system modelling the cells invasion process. Recently, a PDE-ODE model with chemotaxis is studied in [18] obtaining asymptotic stability results using a proper transformation and energy estimates. Another PDE-ODE with chemotaxis problem is considered in [24], see also [25], modelling the evolution of biological species and they obtain analytical results concerning the bifurcation of constant steady states and global existence of solutions for a range of initial data. In [14] a parabolic-ODE problem is analysed, and it is shown that, under several conditions, any stationary solution is locally stable. In this paper, we investigate the following parabolic PDE-ODE system in (0, 𝑇𝑓)×Ω (Ω ⊆R3is a bounded and regular domain and 𝑇𝑓>0 corresponds to the final time) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝜕𝑇 𝜕𝑡 −𝜈∆𝑇 ⏟ ⏞ Diffusion +𝜅∇ · (𝑇∇Φ) ⏟ ⏞ Chemotaxis =𝑓1(𝑇, 𝑁, Φ) 𝜕𝑁 𝜕𝑡 =𝑓2(𝑇, Φ) 𝜕Φ 𝜕𝑡 =𝑓3(𝑇, 𝑁, Φ) (1.1) endowed with non-flux boundary condition on the boundary 𝜕Ω (−𝜈∇𝑇+𝜅 𝑇 ∇Φ) ·𝑛= 0 (1.2) where 𝑛is the outward unit normal vector to 𝜕Ω and initial conditions at time 𝑡= 0: 𝑇(0,·) = 𝑇0, 𝑁 (0,·) = 𝑁0,Φ (0,·)=Φ0in Ω.(1.3) A GLIOBLASTOMA PDE-ODE MODEL 409 Table 1. Parameters. Variable Description Value 𝜈Speed diffusion cm2 s 𝜅Speed chemotaxis cm2 s·density 𝜌Tumor proliferation rate day−1 𝛼Hypoxic death rate day−1 𝛾Vasculature proliferation rate day−1 𝛿Vasculature destruction by tumor day−1 𝐾Carrying capacity cell/cm3 Here, 𝑇(𝑡, 𝑥), 𝑁(𝑡, 𝑥) and Φ(𝑡, 𝑥) represent the tumor and necrotic densities and the vasculature concentration at the point 𝑥∈Ω and time 𝑡 > 0, respectively. The nonlinear reactions functions 𝑓𝑖:R3→Rfor 𝑖= 1,2,3 have the following form ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝑓1(𝑇, 𝑁, Φ) := 𝜌 𝑃 (Φ, 𝑇)𝑇(︂1−𝑇+𝑁+ Φ 𝐾)︂ ⏟ ⏞ Tumor growth −𝛼 𝑆(Φ, 𝑇)𝑇 ⏟ ⏞ Hypoxia 𝑓2(𝑇, 𝑁, Φ) := 𝛼 𝑆(Φ, 𝑇)𝑇+𝛿 𝑄 (Φ, 𝑇) Φ 𝑓3(𝑇, 𝑁, Φ) := 𝛾 𝑅 (Φ, 𝑇 ) Φ (︂1−𝑇+𝑁+ Φ 𝐾)︂ ⏟ ⏞ Vasculature growth −𝛿 𝑄 (Φ, 𝑇) Φ ⏟ ⏞ Vascular destruction by the tumor . (1.4) The parameters in (1.1) have the following description [17,21,22] (Tab. 1). The functions 𝑃(Φ, 𝑇), 𝑆(Φ, 𝑇), 𝑅(Φ, 𝑇) and 𝑄(Φ, 𝑇) appearing in (1.4) are adimensional factors with the following biological meaning: (1) The tumor growth cells need space and a well amount of nutrients to grow. If this amount of nutrients per cell is suitable, the proliferation of tumor cells will occur. Hence, we introduce the tumor proliferation factor 𝑃(Φ, 𝑇) in 𝑓1as a volume fraction of the vasculature. (2) We consider the hypoxia as a decreasing term due to lack of vasculature. Hence, low vasculature produces more tumor destruction. Therefore, the factor 𝑆(Φ, 𝑇) must be a volume fraction of the lack of vasculature. (3) The vasculature growth factor 𝑅(Φ, 𝑇) will depend on the amount of tumor and the vasculature does not grow without tumor. Thus, 𝑅(Φ, 𝑇) will be a volume fraction of tumor. (4) The destruction of vasculature will increase with tumor and there will not be vascular destruction without tumor. In consequence, 𝑄(Φ, 𝑇) will be a volume fraction of tumor. Thus, these factor functions 𝑃(Φ, 𝑇 ), 𝑆(Φ, 𝑇), 𝑅(Φ, 𝑇) and 𝑄(Φ, 𝑇) must satisfy the following modelling conditions: 0≤𝑃(Φ, 𝑇), 𝑆 (Φ, 𝑇), 𝑄 (Φ, 𝑇), 𝑅 (Φ, 𝑇)≤1∀(𝑇, Φ) ∈R2,(1.5) and, 𝑃(Φ, 𝑇) = 0 for Φ = 0 and 𝑃(Φ, 𝑇) increases if Φ increases, (1.6) 𝑆(Φ, 𝑇) increases if Φ decreases, (1.7) 410 A. FERN´ ANDEZ-ROMERO ET AL. 𝑅(Φ, 𝑇) = 0 for 𝑇= 0 and 𝑅(Φ, 𝑇) increases if 𝑇increases (at least for 𝑇≤𝐾), (1.8) 𝑄(Φ, 𝑇) = 0 for 𝑇= 0 and 𝑄(Φ, 𝑇) increases if 𝑇increases. (1.9) We assume along the paper the following assumptions on the initial data 0≤𝑇0(𝑥), 𝑁0(𝑥),Φ0(𝑥)≤𝐾, 𝑎.𝑒. 𝑥 ∈Ω.(1.10) In order to obtain some estimates of the solutions of (1.1)–(1.3) (see (2.1)), we define the following truncated system of (1.1): ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝜕𝑇 𝜕𝑡 −𝜈∆𝑇+𝜅∇ · (𝑇+∇Φ) = 𝑓1(︀𝑇+, 𝑁+,Φ𝐾 +)︀ 𝜕𝑁 𝜕𝑡 =𝑓2(𝑇+,Φ+) 𝜕Φ 𝜕𝑡 =𝑓3(︀𝑇+, 𝑁+,Φ𝐾 +)︀ (1.11) subject to (1.2) and (1.3). We have denoted Φ𝐾 += min {𝐾, max {0,Φ}} and 𝑇+= max {0, 𝑇}and the same for 𝑁+and Φ+. The main contributions of this work are the following: (1) Theorem 1.1 (A priori estimates). (a) Any regular enough solution (𝑇, 𝑁, Φ) of the truncated problem (1.11)–(1.3)satisfies: 0≤Φ≤𝐾, 𝑇 ≥0 and 𝑁≥0, 𝑎.𝑒. in (0, 𝑇𝑓)×Ω and 𝑇, 𝑁 are bounded in 𝐿∞(︀0, 𝑇𝑓;𝐿1(Ω))︀. (b) Assuming that there exists a constant 𝐶1>0such that 𝐶1𝑃(Φ, 𝑇)≥𝑅(Φ, 𝑇 ) Φ,∀0≤Φ≤𝐾, and 𝑇≥0 (1.12) and 𝜌≥𝜅 𝜈𝛾 𝐶1,(1.13) then 𝑇, 𝑁 are bounded in 𝐿∞(0, 𝑇𝑓;𝐿∞(Ω)) . (c) Assuming additionally that there exist constants 𝐶𝑖>0for 𝑖= 2,3,4such that for all 0≤Φ≤𝐾and 𝑇≥0,𝜕(𝑅(Φ, 𝑇) Φ) 𝜕Φ,𝜕(𝑅(Φ, 𝑇) Φ) 𝜕 𝑇 ≤𝐶2,(1.14) 𝜕(𝑄(Φ, 𝑇) Φ) 𝜕Φ,𝜕(𝑄(Φ, 𝑇) Φ) 𝜕 𝑇 ≤𝐶3(1.15) and 𝜕(𝑆(Φ, 𝑇)𝑇) 𝜕Φ,𝜕(𝑆(Φ, 𝑇)𝑇) 𝜕 𝑇 ≤𝐶4,(1.16) then ∇𝑁, ∇Φ are bounded in 𝐿∞(︀0, 𝑇𝑓;𝐿2(Ω))︀, and ∇𝑇is bounded in 𝐿2(︀0, 𝑇𝑓;𝐿2(Ω))︀. A GLIOBLASTOMA PDE-ODE MODEL 411 By Theorem 1.1(a), for any (𝑇, 𝑁, Φ) solution of (1.11), we deduce that 𝑇+=𝑇,𝑁+=𝑁and Φ𝐾 += Φ and then, 𝑓𝑖(︀𝑇+, 𝑁+,Φ𝐾 +)︀=𝑓𝑖(𝑇, 𝑁, Φ) for 𝑖= 1,3 and 𝑓2(︀𝑇+,Φ𝐾 +)︀=𝑓2(𝑇, Φ). Hence, we obtain the following crucial corollary: Corollary 1.2. If (𝑇, 𝑁, Φ) is a solution of the truncated problem (1.11), then (𝑇, 𝑁, Φ) is also a solution of (1.1)–(1.3)and (𝑇, 𝑁, Φ) satisfies the estimates of Theorem 1.1. The existence of solutions of problem (1.11) is out of the scope of this paper. It is an interesting open problem that could be treated in a forthcoming paper. (2) In Section 3, we design a Finite Element numerical scheme, computing (︀Φℎ 𝑘, 𝑇ℎ 𝑘, 𝑁ℎ 𝑘)︀as an approximation of (Φ(𝑡𝑘,·), 𝑇(𝑡𝑘,·), 𝑁(𝑡𝑘,·)) where 𝑡𝑘is a partition of the time interval (0, 𝑇𝑓) and ℎis the mesh size. To build the scheme, we will use the change of variable in the PDE equation with chemotaxis, 𝑇=𝑒𝜅 𝜈Φ𝑢, similar to the used in [8,9,20], in order to obtain an equivalent system with diffusion for the new variable 𝑢. Theorem 1.3 (Discrete version of Theorem 1.1(a)).Scheme (3.3)–(3.9)has a unique solution satisfying the first pointwise estimates of Theorem 1.1(a), these are: 0≤Φ𝑘 ℎ≤𝐾, 𝑇 𝑘 ℎ≥0 and 𝑁𝑘 ℎ≥0,in Ω.(1.17) The design of a numerical scheme preserving the whole estimates of Theorem 1.1, and not only the estimates (1.17), remains as an open problem. (3) A parametric study through numerical simulations is made in order to detect different behaviours for the ring width and the regularity of the surface of the tumor. The outline of the paper is as follows. In Section 2, we prove Theorem 1.1. In Section 3we build a numerical scheme which preserves the a priori estimates of the continuous model given in Theorem 1.1(a). Later, in Section 4, we show a possible example of the dimensionless reaction functions of the system satisfying the hypotheses given in (1.6)–(1.9) and (1.12)–(1.16) and we make an adimensionalization of the model. Section 5 is dedicated to show, by means of some numerical simulations, the different behaviour of the ring width-volume and the regularity surface with respect to the dimensionless parameters. Finally, the more technical part of the proof of Theorem 1.1(b), obtained via an Alikakos’ argument, is given in an Appendix A. 2. A priori estimates of the solutions of (1.3)–(1.11) 2.1. Proof of Theorem 1.1(a) Lemma 2.1. Any solution (𝑇, 𝑁, Φ) of the truncated problem (1.11)satisfy the following pointwise estimates: 0≤Φ≤𝐾, 𝑇 ≥0 and 𝑁≥0, 𝑎.𝑒. in (0, 𝑇𝑓)×Ω.(2.1) Proof. Let (𝑇, 𝑁, Φ) be a solution of (1.11). Since one can rewrite 𝑓1(𝑇+, 𝑁+,Φ𝐾 +) = 𝑇+ 𝑓1(𝑇+, 𝑁+,Φ𝐾 +), multiplying the first equation of (1.11) by 𝑇−= min {𝑇, 0}and integrating in Ω, we get 1 2 d d𝑡∫︁Ω (𝑇−)2d𝑥+𝜈∫︁Ω | ∇𝑇−|2=∫︁Ω 𝑇−𝑇+ 𝑓1(︀𝑇+, 𝑁+,Φ𝐾 +)︀d𝑥= 0, 𝑎.𝑒. in (0, 𝑇𝑓). Hence, since 𝑇−(0, 𝑥) = 0, then 𝑇−(𝑡, 𝑥)=0a.e. (𝑡, 𝑥)∈(0, 𝑇𝑓)×Ω. We repeat the same argument for the other two equations of (1.11) using now that Φ−𝑓3(︀𝑇+, 𝑁+,Φ𝐾 +)︀= 0 and 𝑁−𝑓2(𝑇+,Φ+)≤0. To obtain the upper bound Φ ≤𝐾, we multiply the third equation of (1.11) by (Φ −𝐾)+= max {0,Φ−𝐾} and integrate in Ω, 412 A. FERN´ ANDEZ-ROMERO ET AL. 1 2 d d𝑡∫︁Ω(︀(Φ −𝐾)+)︀2d𝑥=∫︁Ω 𝑓3(︀𝑇+, 𝑁+,Φ𝐾 +)︀(Φ −𝐾)+d𝑥, 𝑎.𝑒. in (0, 𝑇𝑓). Since 𝑓3(𝑇+, 𝑁+,Φ𝐾 +)≤𝛾Φ𝐾 +(1 −Φ𝐾 + 𝐾), then 𝑓3(𝑇+, 𝑁+,Φ𝐾 +)(Φ −𝐾)+≤0. As (Φ (0, 𝑥)−𝐾)+= 0, then (Φ (𝑡, 𝑥)−𝐾)+= 0 a.e. (𝑡, 𝑥)∈(0, 𝑇𝑓)×Ω.  Lemma 2.2. Any solution of (𝑇, 𝑁, Φ) satisfies the estimates: ‖𝑇‖𝐿∞(0,𝑇𝑓;𝐿1(Ω)) +‖√︀𝑃(Φ, 𝑇)𝑇‖𝐿2(0,𝑇𝑓;𝐿2(Ω)) ≤𝐶(𝜌, 𝐾, |Ω|, 𝑇𝑓),(2.2) ‖𝑁‖𝐿∞(0,𝑇𝑓;𝐿1(Ω)) ≤𝐶(𝜌, 𝛼, 𝛿, 𝐾, |Ω|, 𝑇𝑓).(2.3) Proof. Let (𝑇, 𝑁, Φ) be a solution of (1.11). Integrating in Ω the first equation of (1.11) and using that 𝑃(Φ, 𝑇), 𝑆 (Φ, 𝑇)≥0, we obtain that d d𝑡∫︁Ω 𝑇d𝑥=∫︁Ω 𝜌 𝑃 (Φ, 𝑇)𝑇d𝑥−∫︁Ω 𝜌 𝑃 (Φ, 𝑇)𝑇2 𝐾d𝑥−∫︁Ω 𝜌 𝑃 (Φ, 𝑇)𝑇𝑁+ Φ 𝐾 ⏟ ⏞ ≥0 d𝑥 −∫︁Ω 𝛼 𝑆 (Φ, 𝑇)𝑇 ⏟ ⏞ ≥0 d𝑥≤∫︁Ω 𝜌 𝑃 (Φ, 𝑇)𝑇d𝑥−1 𝐾∫︁Ω 𝜌 𝑃 (Φ, 𝑇)𝑇2d𝑥. Thus, d d𝑡∫︁Ω 𝑇d𝑥+1 𝐾∫︁Ω 𝜌 𝑃 (Φ, 𝑇)𝑇2d𝑥≤∫︁Ω 𝜌 𝑃 (Φ, 𝑇)𝑇d𝑥. Rewriting 𝑃(Φ, 𝑇)𝑇=√︀𝑃(Φ, 𝑇 )√︀𝑃(Φ, 𝑇)𝑇and applying Young’s inequality for the right side, we get, d d𝑡∫︁Ω 𝑇d𝑥+1 𝐾∫︁Ω 𝜌 𝑃 (Φ, 𝑇)𝑇2d𝑥≤𝜌(︂1 2𝐾∫︁Ω 𝑃(Φ, 𝑇)𝑇2d𝑥+𝐾 2∫︁Ω 𝑃(Φ, 𝑇))︂. Hence, using that 𝑃(Φ, 𝑇)≤1, we conclude that d d𝑡∫︁Ω 𝑇d𝑥+𝜌 2𝐾∫︁Ω 𝑃(Φ, 𝑇)𝑇2d𝑥≤𝜌 𝐾 2|Ω|. Integrating in (0, 𝑡) for 0 < 𝑡 ≤𝑇𝑓, we obtain that ‖𝑇(𝑡, ·)‖𝐿1(Ω) +𝜌 2𝐾∫︁𝑡 0∫︁Ω 𝑃(Φ, 𝑇)𝑇2d𝑥d𝑡≤𝑇𝑓 𝜌 𝐾 2|Ω|,∀𝑡∈(0, 𝑇𝑓) whence we deduce (2.2). To prove (2.3), we integrate the second equation of (1.11) in Ω ×(0, 𝑡), with 0 < 𝑡 ≤𝑇𝑓, ‖𝑁(𝑡, ·)‖𝐿1(Ω) ≤𝛼∫︁𝑡 0∫︁Ω 𝑇d𝑥d𝑡+𝛿∫︁𝑡 0∫︁Ω Φ d𝑥d𝑡 where we have used (1.5). Thus, using that Φ ≤𝐾and the bound obtained for 𝑇in (2.2), we get (2.3).  A GLIOBLASTOMA PDE-ODE MODEL 413 2.2. Proof of Theorem 1.1(b) In order to obtain the 𝐿∞estimate for 𝑇, firstly we make a change of variable such that we rewrite the diffusion term and chemotaxis term as an unique diffusion term depending on the new variable. In fact, we consider: 𝑤= log (𝑇)−𝜒Φ⇔𝑇=𝑒𝑤𝑒𝜒Φ=𝑒𝜒Φ𝑢(2.4) with 𝑢=𝑒𝑤and 𝜒=𝜅 𝜈. Thus, the first equation of (1.1) changes to (︀𝑒𝜒Φ𝑢)︀𝑡−𝜈∇ · (︀𝑒𝜒Φ∇𝑢)︀=𝑓1(︀𝑒𝜒Φ𝑢, 𝑁, Φ)︀(2.5) and the boundary condition (1.2) to ∇𝑢·𝑛= 0.(2.6) Lemma 2.3 (Proof of Theorem 1.1(b)).Assume (1.12)and (1.13). Then, given any solution (𝑇, 𝑁, Φ) of (1.11), it holds that 𝑢is bounded in 𝐿∞(0, 𝑇𝑓;𝐿∞(Ω)) and ∇𝑢is bounded in 𝐿2(︀0, 𝑇𝑓;𝐿2(Ω))︀. Moreover, 𝑇 and 𝑁are bounded in 𝐿∞(0, 𝑇𝑓;𝐿∞(Ω)). Proof. To obtain the 𝐿∞estimates for 𝑇and 𝑁, taking into account the 𝐿∞estimates for Φ, it suffices that 𝑢be 𝐿∞. The proof of 𝑢is based in 𝐿𝑝estimates with an Alikakos’ argument. Let (𝑇, 𝑁, Φ) be a solution of (1.11). We multiply (2.5) by 𝑢𝑝−1(for any 𝑝≥2), and analyse term by term: – Time derivative term: (︀𝑒𝜒Φ𝑢)︀𝑡𝑢𝑝−1=𝜒Φ𝑡𝑒𝜒Φ𝑢𝑝+1 𝑝𝑒𝜒Φ(𝑢𝑝)𝑡(2.7) and the second term of the right side of (2.7) can be expressed as 1 𝑝𝑒𝜒Φ(𝑢𝑝)𝑡=1 𝑝(︀𝑒𝜒Φ𝑢𝑝)︀𝑡−𝜒 𝑝𝑒𝜒Φ𝑢𝑝Φ𝑡.(2.8) Hence, from (2.7) and (2.8), (︀𝑒𝜒Φ𝑢)︀𝑡𝑢𝑝−1=1 𝑝(︀𝑒𝜒Φ𝑢𝑝)︀𝑡+𝑝−1 𝑝𝜒Φ𝑡𝑒𝜒Φ𝑢𝑝.(2.9) – Nonlinear diffusion term: −𝜈∇ · (︀𝑒𝜒Φ∇𝑢)︀𝑢𝑝−1=−𝜈∇ · (︀𝑒𝜒Φ(∇𝑢)𝑢𝑝−1)︀+𝜈 𝑒𝜒Φ(𝑝−1) 𝑢𝑝−2| ∇ 𝑢|2 =−𝜈∇ · (︀𝑒𝜒Φ(∇𝑢)𝑢𝑝−1)︀+𝜈 𝑒𝜒Φ(𝑝−1) 4 𝑝2| ∇(𝑢𝑝/2)|2.(2.10) – Reaction term: 𝑓1(︀𝑒𝜒Φ𝑢, 𝑁, Φ)︀𝑢𝑝−1=𝜌 𝑃 (Φ, 𝑇 )𝑒𝜒Φ𝑢𝑝(︂1−𝑒𝜒Φ𝑢+𝑁+ Φ 𝐾)︂−𝛼 𝑆 (Φ, 𝑇)𝑒𝜒Φ𝑢𝑝.(2.11) Rewriting in (2.9) the function Φ𝑡as 𝑓3(︀𝑒𝜒Φ𝑢, 𝑁, Φ)︀and adding (2.9)–(2.11), we get: 1 𝑝(︀𝑒𝜒Φ𝑢𝑝)︀𝑡−𝜈∇ · (︀𝑒𝜒Φ(∇𝑢)𝑢𝑝−1)︀+𝜈 𝑒𝜒Φ(𝑝−1) 4 𝑝2| ∇(𝑢𝑝/2)|2+𝛼 𝑆 (Φ, 𝑇)𝑒𝜒Φ𝑢𝑝 +(︂𝜌 𝑃 (Φ, 𝑇)−(︂𝑝−1 𝑝)︂𝜒 𝛾 𝑅 (Φ, 𝑇) Φ)︂𝑒𝜒Φ𝑢𝑝(︂𝑒𝜒Φ𝑢+𝑁+ Φ 𝐾)︂ 414 A. FERN´ ANDEZ-ROMERO ET AL. =(︂𝜌 𝑃 (Φ, 𝑇)−(︂𝑝−1 𝑝)︂𝜒 𝛾 𝑅 (Φ, 𝑇) Φ)︂𝑒𝜒Φ𝑢𝑝+𝜒 𝑝𝑒𝜒Φ𝑢𝑝𝛿 𝑄 (Φ, 𝑇) Φ.(2.12) Due to hypothesis (1.13) and (1.12), it is easy to see in (2.12) that, 𝜌 𝑃 (Φ, 𝑇)−(︂𝑝−1 𝑝)︂𝜒 𝛾 𝑅 (Φ, 𝑇) Φ ≥0. Using now that 0 ≤Φ≤𝐾, (1.5) and (1.13) we obtain that 1 𝑝(︀𝑒𝜒Φ𝑢𝑝)︀𝑡−𝜈∇ · (︀𝑒𝜒Φ(∇𝑢)𝑢𝑝−1)︀+𝜈 𝑒𝜒Φ(𝑝−1) 4 𝑝2| ∇(𝑢𝑝/2)|2+𝛼 𝑆 (Φ, 𝑇)𝑒𝜒Φ𝑢𝑝 ≤𝐶 𝑒𝜒Φ𝑢𝑝(2.13) with 𝐶 > 0. Integrating (2.13) in Ω, it holds that 1 𝑝 d d𝑡∫︁Ω 𝑒𝜒Φ𝑢𝑝d𝑥+𝜈(𝑝−1) 4 𝑝2∫︁Ω 𝑒𝜒Φ| ∇(𝑢𝑝/2)|2d𝑥+𝛼∫︁Ω 𝑆(Φ, 𝑇)𝑒𝜒Φ𝑢𝑝d𝑥 ≤𝐶∫︁Ω 𝑒𝜒Φ𝑢𝑝d𝑥(2.14) with 𝐶 > 0 independent of 𝑝(along the proof, we will denote by 𝐶different constants independent of 𝑝). Using the auxiliary variable 𝑤=𝑢𝑝/2, we can rewrite (2.14) as follows 1 𝑝 d d𝑡‖𝑒𝜒Φ 2𝑤‖2 𝐿2(Ω) + 4 𝜈(𝑝−1) 𝑝2‖𝑒𝜒Φ 2∇𝑤‖2 𝐿2(Ω) ≤𝐶‖𝑒𝜒Φ 2𝑤‖2 𝐿2(Ω).(2.15) Thus, applying Gronwall’s lemma, we deduce for 𝑝= 2 that ∇𝑢is bounded in 𝐿2(︀0, 𝑇𝑓;𝐿2(Ω))︀. Now, using the following equivalent norms with constants independent of 𝑝 ‖𝑧‖2 𝐿2(Ω) ≤ ‖𝑒𝜒Φ 2𝑧‖2 𝐿2(Ω) ≤𝑒𝜒 𝐾‖𝑧‖2 𝐿2(Ω),(2.16) multiplying (2.15) by 𝑝and using that 𝑝−1 𝑝≥1 2for any 𝑝≥2, we obtain that d d𝑡‖𝑒𝜒Φ 2𝑤‖2 𝐿2(Ω) + 2 𝜈‖∇𝑤‖2 𝐿2(Ω) ≤𝐶 𝑝 ‖𝑤‖2 𝐿2(Ω).(2.17) We are going to apply the following Gagliardo-Nirenberg interpolation inequality ([13], Thm. 10.1) ‖𝑤‖2 𝐿2(Ω) ≤𝜀‖∇ 𝑤‖2 𝐿2(Ω) +𝐶(︂1 𝜀)︂𝑛/2 ‖𝑤‖2 𝐿1(Ω) (2.18) with 𝜀 > 0 and 𝑛the dimension of Ω (in this case 𝑛= 3). Applying (2.18) for 𝜀=𝜈 𝐶 𝑝 in the right hand side of (2.17), we deduce that d d𝑡‖𝑒𝜒Φ 2𝑤‖2 𝐿2(Ω) +𝜈‖∇𝑤‖2 𝐿2(Ω) ≤𝐶 𝑝2‖𝑤‖2 𝐿1(Ω).(2.19) Using (2.18) in (2.19) but now for 𝜀=𝜈, it holds that d d𝑡‖𝑒𝜒Φ 2𝑤‖2 𝐿2(Ω) +‖𝑤‖2 𝐿2(Ω) ≤𝐶(︀𝑝2+ 1)︀‖𝑤‖2 𝐿1(Ω).(2.20) A GLIOBLASTOMA PDE-ODE MODEL 415 Finally, due to (2.16), we can deduce that d d𝑡‖𝑒𝜒Φ 2𝑤‖2 𝐿2(Ω) +𝐶1‖𝑒𝜒Φ 2𝑤‖2 𝐿2(Ω) ≤𝐶(︀𝑝2+ 1)︀‖𝑤‖2 𝐿1(Ω) (2.21) where 𝐶1=𝑒−𝜒 𝐾. Hence, we obtain that max 𝑡∈(0,𝑇𝑓)‖𝑢‖𝑝 𝐿𝑝(Ω) ≤ ‖𝑒𝜒Φ 2𝑤(𝑡)‖2 𝐿2(Ω) ≤𝑒−𝐶1𝑡𝐶‖𝑢0‖𝑝 𝐿∞(Ω) +𝐶(︀𝑝2+ 1)︀𝑒−𝐶1𝑡∫︁𝑡 0 𝑒𝐶1𝑠(︂∫︁Ω 𝑢𝑝/2d𝑥)︂2 𝑑𝑠 ≤𝐶‖𝑢0‖𝑝 𝐿∞(Ω) +𝐶(︀𝑝2+ 1)︀max 𝑡∈(0,𝑇𝑓)‖𝑢‖𝑝 𝐿𝑝/2(Ω) ≤𝐶max {︁(︀𝑝2+ 1)︀max 𝑡∈(0,𝑇𝑓)‖𝑢‖𝑝 𝐿𝑝/2(Ω),‖𝑢0‖𝑝 𝐿∞(Ω)}︁.(2.22) Following a similar argument to used by Alikakos in [2] (see Appendix A), from (2.22) we can obtain that 𝑢is bounded in 𝐿∞(0, 𝑇𝑓;𝐿∞(Ω)) . As consequence, 𝑇is bounded in 𝐿∞(0, 𝑇𝑓;𝐿∞(Ω)). Since 𝑁𝑡=𝑓2(𝑇, Φ) and 𝑇and Φ are bounded in 𝐿∞(0, 𝑇𝑓;𝐿∞(Ω)) we obtain that 𝑁is bounded in 𝐿∞(0, 𝑇𝑓;𝐿∞(Ω)).  2.3. Proof of Theorem 1.1(c) Let (𝑇, 𝑁, Φ) be a solution of (1.11). Taking gradient in the second and third equation of (1.11), (∇Φ)𝑡=𝛾[︂(︂𝜕(𝑅(Φ, 𝑇) Φ) 𝜕Φ∇Φ + 𝜕(𝑅(Φ, 𝑇 ) Φ) 𝜕 𝑇 ∇𝑇)︂(︂1−𝑇+𝑁+ Φ 𝐾)︂ −𝑅(Φ, 𝑇) Φ 𝐾(∇𝑇+∇𝑁+∇Φ)]︂−𝛿(︂𝜕(𝑄(Φ, 𝑇) Φ) 𝜕Φ∇Φ +𝜕(𝑄(Φ, 𝑇) Φ) 𝜕 𝑇 ∇𝑇)︂,(2.23) (∇𝑁)𝑡=𝛼(︂𝜕(𝑆(Φ, 𝑇)𝑇) 𝜕Φ∇Φ + 𝜕(𝑆(Φ, 𝑇 )𝑇) 𝜕 𝑇 ∇𝑇)︂ +𝛿(︂𝜕(𝑄(Φ, 𝑇) Φ) 𝜕Φ∇Φ + 𝜕(𝑄(Φ, 𝑇 ) Φ) 𝜕 𝑇 ∇𝑇)︂.(2.24) Using the change of variable 𝑇=𝑒𝜒Φ𝑢as in Lemma 2.3, we deduce that ∇𝑇=𝜒 𝑒𝜒Φ𝑢∇Φ + 𝑒𝜒Φ∇𝑢=𝜒 𝑇 ∇Φ + 𝑒𝜒Φ∇𝑢(2.25) and we know from Lemma 2.3 that ∇𝑢is bounded in 𝐿2(︀0, 𝑇𝑓;𝐿2(Ω))︀. Taking into account that 𝑇and Φ are bounded in 𝐿∞(0, 𝑇𝑓;𝐿∞(Ω)), it holds that |∇ 𝑇| ≤ 𝐶(|∇ Φ|+|∇ 𝑢|). Thus, rewriting (2.23) and (2.24) in terms of ∇𝑢, multiplying (2.23) and (2.24) by ∇Φ and ∇𝑁respectively and integrating in Ω, we deduce 1 2 d d𝑡‖∇ Φ‖2 𝐿2(Ω) ≤𝐶1‖∇ Φ‖2 𝐿2(Ω) +𝐶2∫︁Ω |∇ 𝑢| |∇ Φ|d𝑥+𝐶3∫︁Ω |∇ 𝑁| |∇ Φ|d𝑥, (2.26) 422 A. FERN´ ANDEZ-ROMERO ET AL. Table 2. Dimensionless parameters. Dimensionless parameter 𝜅*𝛼*𝛾*𝛿* Original parameter 𝐾𝜅 𝜈 𝛼 𝜌 𝛾 𝜌 𝛿 𝜌 Finally, the adimensionalizated system is the following: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝜕𝑇 𝜕𝑡 −∆𝑇+𝜅∇ · (𝑇∇Φ) = 𝑃(Φ, 𝑇)𝑇(1 −(𝑇+𝑁+ Φ)) −𝛼 𝑆 (Φ, 𝑇)𝑇, 𝜕𝑁 𝜕𝑡 =𝛼 𝑆 (Φ, 𝑇)𝑇+𝛿 𝑄 (Φ, 𝑇) Φ, 𝜕Φ 𝜕𝑡 =𝛾 𝑅 (Φ, 𝑇) Φ (1 −(𝑇+𝑁+ Φ)) −𝛿 𝑄 (Φ, 𝑇) Φ. (4.8) 5. Numerical simulations In this section, we will show some numerical simulations in order to detect which parameters of (4.8) are more important in the behaviour of the ring width between necrosis and tumor and the regular or irregular growth of the surface of a GBM. For the numerical simulations we will use the uncoupled and linear fully discrete scheme defined in (3.3)– (3.5) by means of an Implicit-Explicit (IMEX) Finite Difference in time approximation and 𝑃1continuous finite element with “mass-lumping” in space. We will use the computational domain, Ω = (−9,9) ×(−9,9), the final time, 𝑇𝑓= 500, the structured triangulation, {𝒯ℎ}ℎ>0of Ω such that Ω = ⋃︀𝒦∈𝒯ℎ𝒦, partitioning the edges of 𝜕Ω into 45 subintervals, corresponding with the mesh size ℎ= 0.4 and the time step, d𝑡= 10−3. We consider along the work necrosis zero initially and initial tumor given by Figure 5. For the vasculature, we will take different initial conditions depending on the kind of tumor growth considered. 5.1. Ring width Here, we present some numerical simulations according to the tumor-ring. Based on the study [27], we know that tumors with a thick tumor ring have the worst prognosis as we can see in the Figure 1. In order to measure different rings, we will compare the density of tumor with respect to the density of tumor and necrosis. In every simulation, we will change the value of one parameter and testing how the tumor growth changes. Since the subjects of study are tumor and necrosis, we change the parameters of the tumor and necrosis equations, these are, 𝜅and 𝛼. Then, in all the simulations the value of 𝛾and 𝛿are fixed (see Tab. 3). For 𝜅and 𝛼, we will take either 𝜅= 5 and 𝛼∈[10,100] or 𝜅∈[1,10] and 𝛼= 45 (see Tab. 4). Moreover, we take the initial vasculature defined uniformly in space. 5.1.1. Tumor ring quotient We will start studying the ratio between proliferative tumor density, ∫︁Ω 𝑇d𝑥and total tumor density, ∫︁Ω (𝑇+𝑁) d𝑥and we consider the different values of 𝜅and 𝛼given in Table 4. In fact, we compute the A GLIOBLASTOMA PDE-ODE MODEL 423 Figure 1. Survival versus the ring width of GBM. Table 3. Fixed value parameters. Variable 𝛾 𝛿 Value 0.255 2.55 Table 4. Variable value parameters. Variable (fixed value) 𝜅(5) 𝛼(45) Ranges [1,10] [10,100] following “ring quotient” (RQ) coefficient: 0≤RQ = ∫︁Ω 𝑇d𝑥 ∫︁Ω (𝑇+𝑁) d𝑥 ≤1.(5.1) Thus, if RQ is near to zero, there exists a high density of necrosis (which implies slim tumor ring) whereas if RQ is close to one, there is not enough necrosis in comparison with proliferative tumor density (which means thick tumor ring). We can see in Figures 2a and 2b, how the model captures two kinds of tumor ring changing the parameter 𝛼 and the tumor rings for different 𝜅do not change. This means that a change of the rate of tumor destruction for hypoxia produces much difference in the tumor rings. Hence, the best configurations to obtain a slim (resp. thick) ring would be choose a big (resp. small) 𝛼. 5.1.2. Density tumor growth In Figure 3, we compute the total tumor ∫︁Ω (𝑇+𝑁) d𝑥for the values of 𝜅and 𝛼given in Table 4. We can see in Figure 3how the variation in the parameters 𝜅and 𝛼produces changes in the total tumor density. In fact, the total tumor decreases with respect to 𝜅and 𝛼. Therefore, we conclude that 𝛼is the most important parameter in order to change the tumor ring and both 𝜅and 𝛼have relevance to change the total density in the tumor growth. 424 A. FERN´ ANDEZ-ROMERO ET AL. Figure 2. RQ versus time for 𝜅and 𝛼. (a) RQ versus time for 𝜅. (b) RQ versus time for 𝛼. Figure 3. ∫︁Ω (𝑇+𝑁) d𝑥versus time for 𝜅and 𝛼. (a) ∫︁Ω (𝑇+𝑁) d𝑥versus time for 𝜅. (b) ∫︁Ω (𝑇+𝑁) d𝑥versus time for 𝛼. 5.2. Regularity surface In this case, we will test if our model (4.8) can develop different regularities for the tumor surfaces. Now, we will base our results on the study published in [28] where appears the following survival curve. From Figure 4, the authors conclude that tumors with a regular surface have better prognosis than tumors with irregular surface. Along this section, we simulate the tumor growth with the initial tumor defined in Figure 5, necrosis zero and the vasculature distributed in different zones as in Figure 6. Thus, the question is if the chemotaxis term (of tumor going to the vasculature) implies tumor growth with regular or irregular surface. We remember that the chemotaxis term in (4.8) is defined by 𝜅∇ · (𝑇∇Φ) with 𝜅 > 0. Now, we want to detect which parameter is more relevant changing the regularity of the tumor surface, showing some simulations in which we move the value of one of them and observe how the tumor changes. For A GLIOBLASTOMA PDE-ODE MODEL 425 Figure 4. Survival versus the regularity surface of GBM. Figure 5. Initial tumor. Figure 6. Initial vasculature. 426 A. FERN´ ANDEZ-ROMERO ET AL. Table 5. Variable value parameters. Variable (fixed value) 𝜅(5) 𝛼(45) 𝛾(0.255) 𝛿(2.55) Ranges [1,10] [10,100] [0.01,0.5] [0.1,5] this, it is important the interaction between tumor and vasculature. Then, we will move the parameters which appear in tumor and vasculature equations, 𝜅,𝛼,𝛾and 𝛿. For these parameters we take the values of Table 5(each parameter will change its value in the range, jointly the other parameters take fixed values). 5.2.1. Regularity surface quotient The pictures of Figure 7show the quotient between the area occupied by the total tumor (tumor and necrosis) and the area of ratio the smallest circle containing the tumor. Thus, we present these computations for the different values of 𝜅,𝛼,𝛾and 𝛿chosen in Table 5. In fact, we compute the following “surface quotient” (SQ) coefficient: 0≤SQ = ∫︁Ω (𝑇+𝑁)min d𝑥 𝜋·(Rmax)2≤1 (5.2) where (𝑇+𝑁)min and Rmax are defined as follows: (𝑇+𝑁)min =⎧ ⎨ ⎩ 1 if 𝑇+𝑁≥0.001, 0 otherwise. (5.3) Rmax = max {ratio of the subdomain where (𝑇+𝑁)min = 1}.(5.4) Thus, we will deduce that if SQ is near to zero, tumor will have an irregular surface whereas if SQ is close to one, tumor will have a regular surface. Remark 5.1. By the size of mesh considered, at the beginning of the pictures given in Figure 7, the value of SQ is larger than 1 and it is observed oscillations in the graphs of SQ. Indeed, if we consider a mesh size smaller, these initial values of SQ and the oscillations can be reduced. In order to check this, we show an example of SQ versus time for different 𝜅considering a mesh size smaller. However, we think that it is not necessary the use of a mesh size smaller since we obtain the same behaviour (in average) for the mesh considered initially and with this mesh, we reduce the computational time. We see in Figure 7, how our model differentiates two kinds of tumor growth changing the parameter 𝜅, see Figure 7a, and with lower variation for 𝛼, see Figure 7b. On the other hand, we do not appreciate changes in the variation of parameters 𝛾and 𝛿for the irregularity of tumor growth as we see in Figures 7c and 7d. 5.2.2. Area Once we have identified that the more important parameters for the regularity surface are firstly 𝜅and later 𝛼, we measure the area of total tumor for these parameters as in Table 5(Fig. 8). We see in Figure 9how the largest area corresponds to the smallest 𝛼= 10 and the smallest area holds for the highest 𝛼= 100. In the case of variation of 𝜅, Figure 9a, a similar influence in the total tumor area for 𝜅= 1 and 𝜅= 10 is observed. Thus, we have obtained a higher variation of total area for the different values of 𝛼than for 𝜅, see Figure 9. Nevertheless, in the simulation of the “surface quotient” (SQ), we obtained more variation between the different values of 𝜅that for the different values of 𝛼, see Figure 7. Hence, the factor which modifies this change is Rmax, defined by (5.4). In fact, Rmax will change more with the variation of 𝜅than for the variation of 𝛼. A GLIOBLASTOMA PDE-ODE MODEL 427 Figure 7. SQ versus time for 𝜅1,𝛼,𝛾and 𝛿. (a) SQ versus time for 𝜅. (b) SQ versus time for 𝛼. (c) SQ versus time for 𝛾. (d) SQ versus time for 𝛿. Figure 8. SQ versus time for 𝜅using a mesh size smaller. 428 A. FERN´ ANDEZ-ROMERO ET AL. Figure 9. Area of total tumor versus time for 𝜅and 𝛼. (a) Area of total tumor versus time for 𝜅. (b) Area of total tumor time for 𝛼. Figure 10. Irregular tumor growth for 𝜅= 10. (a) 𝑡= 50. (b) 𝑡= 100. (c) 𝑡= 150. (d) 𝑡= 200. (e) 𝑡= 250. 5.2.3. Tumor growth Here, we examine the tumor growth for 𝜅= 10 in five times step in order to see the variation in space of tumor (see Figure 10). For this growth, the rest of parameters take the fixed values showed in Table 5. We observe an irregular tumor growth for 𝜅= 10 when time increases. These results are in concordance with Figure 9a, where we observed a great irregularity for 𝜅= 10 and with Figure 9a, where the area of the tumor for 𝜅= 10 is increasing. Finally, we conclude that 𝜅is the more relevant parameters in the irregular surface of tumor and 𝛼is the most important parameter for total area in the tumor growth. 5.3. Discussion Summarizing the results obtained with respect to the ring width and the regularity surface for the chemotactic and dimensionless system (4.8) related to GBM growth model, we deduce that this model can capture these two properties varying some parameters. Moreover, we have proved that the parameters more relevant according to the tumor growth are 𝜅and 𝛼. For the tumor ring, where the vasculature is uniformly distributed, the results show that the hypoxia parameter 𝛼is the most relevant coefficient as we can observe in Figures 2a and 2b. In the case of regularity surface, where the vasculature is non-uniformly distributed, the parameter which produces more irregularity in the tumor surface is the chemotaxis parameter 𝜅, see Figures 7a–7d. A GLIOBLASTOMA PDE-ODE MODEL 429 Finally, after the reduction of our model (1.1) from 7 initial parameters to 2 (𝜅and 𝛼) which capture the different behaviour of tumor growth, we conclude that hypoxia coefficient 𝛼is the main parameter for the tumor ring and area of tumor and 𝜅is the most influential parameter for the regularity surface. Appendix A. In this appendix, we will prove an Alikakos’ recursive 𝐿∞estimate. Following the proof of Lemma 2.3, we obtain in (2.22) that max 𝑡∈(0,𝑇𝑓)‖𝑢‖𝑝 𝐿𝑝(Ω) ≤ 𝐶max {︁(︀𝑝2+ 1)︀max 𝑡∈(0,𝑇𝑓)‖𝑢‖𝑝 𝐿𝑝/2(Ω),‖𝑢0‖𝑝 𝐿∞(Ω)}︁.(A.1) In [2], the authors obtained an estimate starting from an estimate like (A.1) but with power 𝑝instead of 𝑝2. Taking in (A.1)𝑝= 2𝑘for all 𝑘≥1, it holds that, max 𝑡∈(0,𝑇𝑓)∫︁Ω 𝑢2𝑘d𝑥≤𝐶max {︁(︀22𝑘+ 1)︀max 𝑡∈(0,𝑇𝑓)(︂∫︁Ω 𝑢2𝑘−1d𝑥)︂2 ,‖𝑢0‖2𝑘 𝐿∞(Ω)}︁ ≤𝐶 𝐶2max {︃(︀22𝑘+ 1)︀[︃max {︃(︁22 (𝑘−1) + 1)︁max 𝑡∈(0,𝑇𝑓)(︂∫︁Ω 𝑢2𝑘−2d𝑥)︂2 , ‖𝑢0‖2𝑘−1 𝐿∞(Ω)}︃]︃2 ,‖𝑢0‖2𝑘 𝐿∞(Ω)⎫ ⎬ ⎭ ≤𝐶 𝐶2𝐶22max {︃(︀22𝑘+ 1)︀(︁22 (𝑘−1) + 1)︁2(︂max 𝑡∈(0,𝑇𝑓)∫︁Ω 𝑢2𝑘−3d𝑥)︂22 , (︀22𝑘+ 1)︀‖𝑢0‖2𝑘 𝐿∞(Ω)}︁ ≤𝐶 𝐶2𝐶22𝐶23max{︃(︀22𝑘+ 1)︀(︁22 (𝑘−1) + 1)︁2(︁22 (𝑘−2) + 1)︁23 max 𝑡∈(0,𝑇𝑓)(︂∫︁Ω 𝑢2𝑘−3d𝑥)︂23 , (︀22𝑘+ 1)︀(︁22 (𝑘−1) + 1)︁2‖𝑢0‖2𝑘 𝐿∞(Ω)}︂≤. . . ≤ ≤(︀𝐶(︀22𝑘+ 1)︀)︀(︁𝐶(︁22 (𝑘−1) + 1)︁)︁2(︁𝐶(︁22 (𝑘−2) + 1)︁)︁22 . . . (︀𝐶(︀22+ 1)︀)︀2𝑘−1 𝐾2𝑘,(A.2) where  𝐾is the constant that dominates ‖𝑢‖𝐿1(Ω) for all time, since 𝑢∈𝐿∞(︀0, 𝑇𝑓;𝐿1(Ω))︀(using Lem. 2.2, taking into account that ‖𝑢0‖𝐿∞(Ω) and the hypothesis (1.10)). Thus, from (A.2) max 𝑡∈(0,𝑇𝑓)∫︁Ω 𝑢2𝑘d𝑥≤(︀𝑎22𝑘)︀(︁𝑎22(𝑘−1))︁2(︁𝑎22(𝑘−2))︁22(︁𝑎22(𝑘−3))︁23 . . . (︀𝑎22)︀2𝑘−1 𝐾2𝑘,(A.3) for a certain 𝑎≥3𝐶since 𝐶(︀22(𝑘−𝑗)+ 1)︀≤𝑎22𝑘if 𝑎≥3𝐶for all 𝑗= 0, . . . , 𝑘 −1. Thus, we can express (A.3) as max 𝑡∈(0,𝑇𝑓)∫︁Ω 𝑢2𝑘d𝑥≤𝑎∑︀𝑘−1 𝑗=0 2𝑗22∑︀𝑘−1 𝑗=0 (𝑘−𝑗)2𝑗 𝐾2𝑘=𝑎2𝑘−12(−𝑘−6+2𝑘+1) 𝐾2𝑘.(A.4) Taking the limit as 𝑘→+∞of the 1/2𝑘power of both sides of (A.4) we obtain max 𝑡∈(0,𝑇𝑓)‖𝑢‖𝐿∞(Ω) ≤lim 𝑘→+∞(︃𝑎2𝑘−1 2𝑘2(−𝑘−6+2𝑘+1) 2𝑘 𝐾)︃=𝑎22 𝐾. (A.5) 430 A. FERN´ ANDEZ-ROMERO ET AL. Hence, 𝑢∈𝐿∞(0,+∞;𝐿∞(Ω)) . Acknowledgements. The authors were supported by PGC2018-098308-B-I00 (MCI/AEI/FEDER, UE). References [1] J.C.L. Alfonso, K. Talkenberger, M. Seifert, B. Klink, A. Hawkins-Daarud, K.R. Swanson, H. Hatzikirou and A. Deutsch, The biology and mathematical modelling of glioma invasion: A review. J. R. Soc. Interface 14 (2017) 20170490. [2] N.D. Alikakos, An application of the invariance principle to reaction-diffusion equations. J. Differ. Equ. 33 (1979) 201–225. [3] A.R.A. Anderson and M.A.J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull. Math. Biol. 60 (1998) 857–899. [4] A. Baldock, R. Rockne, A. Boone, M. Neal, C. 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