Analysis of a tumor model as a multicomponent deformable porous medium

We propose a diffuse interface model to describe tumor as a multicomponent deformable porous medium. We include mechanical effects in the model by coupling the mass balance equations for the tumor species and the nutrient dynamics to a mechanical equilibrium equation with phase-dependent elasticity coefficients. The resulting PDE system couples two Cahn–Hilliard type equations for the tumor phase and the healthy phase with a PDE linking the evolution of the interstitial fluid to the pressure of the system, a reaction-diffusion type equation for the nutrient proportion, and a quasistatic momentum balance. We prove here that the corresponding initial-boundary value problem has a solution in appropriate function spaces. Introduction Tumor growth is nowadays one of the most active area of scientific research, especially due to the impact on the quality of life for cancer patients. Starting with the seminal work of Burton [10] and Greenspan [35], many mathematical models have been proposed to describe the complex biological and chemical processes that occur in tumor growth, with the aim of better understanding and ultimately controlling the behavior of cancer cells. In recent years, there has been a growing interest in the mathematical modelling of cancer, see for example [1, 2, 5, 9, 16, 20, 22]. Mathematical models for tumor growth may have different analytical features: in the present work, we are focusing on the subclass of continuum models, namely diffuse interface models. There are various ways to model the interaction between the tumor and the surrounding host tissue. A classical approach is to represent the interfaces between the tumor and healthy tissues as idealized surfaces of zero thickness, leading to a sharp interface description that differentiates the tumor and the surrounding host tissue cell-bycell. These sharp interface models are often difficult to analyze mathematically, and may fail when the interface undergoes a topological change. Metastasis, which is the spreading of cancer to other parts of the body, is one important example of a change of topology. In such an event, the interface can no longer be represented as a mathematical surface, and thus the sharp interface models do no longer properly describe the reality. On the other hand, diffuse interface models consider the interface between the tumor and the healthy tissues as a layer of non-infinitesimal thickness in which tumor and healthy cells can coexist. The main advantage of this approach is that the mathematical description is less sensitive to topological changes. This is the reason why recent efforts in the mathematical modeling of tumor growth have been mostly focused on diffuse interface models, see for example [15, 16, 21, 30, 33, 36, 43, 50], and their numerical simulations demonstrating complex changes in tumor morphologies due to mechanical stresses and interactions with chemical species such as nutrients or toxic agents. Regarding the recent literature on the mathematical analysis of diffuse interface models for tumor growth, we can further refer to [11, 12, 13, 18, 24, 25, 27, 29] as mathematical references for Cahn–Hilliard-type models and [6, 28, 37, 41] for models also including a transport effect described by Darcy’s law. DOI 10.20347/WIAS.PREPRINT.2842 Berlin 2021 P. Krejčí, E. Rocca, J. Sprekels 2 A further class of diffuse interface models that also include chemotaxis and transport effects has been subsequently introduced (cf. [30, 33]); moreover, in some cases the sharp interface limits of such models have been investigated generally by using formal asymptotic methods (cf. [42, 45]). Including mechanics in the model is clearly an important issue that has been discussed in several modeling papers, but has been very poorly studied analytically. Hence, the main aim of this paper is to find a compromise between the applications and the rigorous analysis of the resulting PDE system: we would like to introduce here an application-significant model which is tractable also analytically. Regarding the existing literature on this subject, we can quote the paper [46], where, using multiphase porous media mechanics, the authors represented a growing tumor as a multiphase medium containing an extracellular matrix, tumor and host cells, and interstitial liquid. Numerical simulations were also performed that characterize the process of cancer growth in terms of the initial tumor-to-healthy cell density ratio, nutrient concentration, mechanical strain, cell adhesion, and geometry. However, referring to [47] for more details on this topic, we mention here that many models in the literature are based on the assumption that the tumor mass presents a particular geometry, the so-called spheroid, and in that case the models mainly focus on the evolution of the external radius of the spheroid. The resulting mathematical problem is an integro-differential free boundary problem, which has been proved to have solutions (cf. [8, 23]) and to predict the evolution of the system. Variants of this approach have been then considered, e.g., in [17] differentiating between viable cells and the necrotic core. Further extensions of the model introduced in [47] can be found in [44]. Very recently, in [32], a new model for tumor growth dynamics including mechanical effects has been introduced in order to generalize the previous works [38, 39] with the goal to take into account cell-cell adhesion effects with the help of a Ginzburg–Landau type energy. In their model an equation of Cahn– Hilliard type is then coupled to the system of linear elasticity and a reaction-diffusion equation for a nutrient concentration, and several questions regarding well-posedness and regularity of solutions have been investigated. In this paper, following the approach of [47], we introduce a diffuse interface multicomponent model for tumor growth, where we include mechanics in the model, assuming that the tumor is a porous medium. In [47], the tumor is regarded as a mixture of various interacting components (cells and extracellular material) whose evolution is ruled by coupled mass and momentum balances. The cells usually are subdivided into subpopulations of proliferating, quiescent and necrotic cells (cf., e. g., [15, 16]), and the interactions between species are determined by the availability of some nutrients. Here, we restrict ourselves to the case where we distinguish only healthy and tumor cells, even if we could, without affecting the analysis, treat the case where we differentiate also between necrotic and proliferating tumor cells. Hence, we represent the tumor as a porous medium consisting of three phases: healthy tissue φ1 , tumor tissue φ2 , and interstitial fluid φ0 satisfying proper mass balance equations including mass source terms depending on the nutrient variable % . The nutrient satisfies a reaction-diffusion equation nonlinearly coupled with the tumor and healthy tissue phases by a coefficient characterizing the different consumption rates of the nutrient by the different cell types. We couple the phases and nutrient dynamics with a mechanical equilibrium equation. This relation is further coupled with the phase dynamics through the elasticity modulus depending on the proportion between healthy and tumor phases. We refer to [19] for a mathematical model of a multicomponent flow in deformable porous media from which we take inspiration. The mass balance relations are derived from a free energy functional which, in the domain Ω where the evolution takes place, can be written as F = ∫ Ω ( F̂ (p) + |∇φ1| 2 + |∇φ2| 2 + (ψ + g)(φ1, φ2) + |%| 2 ) dx , where p denotes the fluid pressure and F̂ is a suitable nonnegative function of the pressure. The DOI 10.20347/WIAS.PREPRINT.2842 Berlin 2021 Analysis of a tumor model as a multicomponent deformable porous medium 3 sum ψ + g represents the interaction potential between tumor and healthy phases, with dominant component ψ which is convex with bounded domain, while g is its smooth nonconvex perturbation, which is typically of double-well character. The quantity % represents the mass content of the nutrient. Notice that the gradient terms in the free energy are due to the modeling assumption that the interface between healthy and tumor phases is diffuse (we take the parameters in front of the gradients equal to 1 here for simplicity, but, in practice, they determine the thickness of the interface and have to be chosen properly). The quantities φ0, φ1, φ2 are relative mass contents, so that only their nonnegative values are meaningful. We also assume that all the other substances present in the system are of negligible mass, that is, the identity φ0 + φ1 + φ2 = 1 is to be satisfied as part of the problem. Hence, we choose the domain of ψ to be included in the set Θ := {(φ1, φ2) ∈ R : φ1 ≥ 0, φ2 ≥ 0, φ1 + φ2 ≤ 1} . Classically, ψ can be taken as the indicator function of Θ or a logarithmic type potential (cf. [26]). Under proper assumptions on the data, we prove the existence of weak solutions for the resulting PDE system, which we will introduce in the next Section 1, coupled with suitable initial and conditions. The PDEs consist of two Cahn–Hilliard type equations for the tumor phase and the healthy phase with a PDE linking the evolution of the interstitial fluid to the pressure of the system, a reaction-diffusion type equation for the nutrient proportion and the momentum balance. The technique of the proof is based on a regularization of the system, where, in particular, the nonsmooth potential ψ is replaced by its Yosida approximation ψε . Then, we prove existence of the approximated problem by means of a Faedo–Galerkin scheme, and we pass to the limit by proving suitable uniform (in ε ) a priori estimates and applying monotonicity and compactness arguments. A key point in the estimates consists in proving that the mean value of the phases belong to the interior of the domain Θ of ψ , which in turns leads to the es