Tumor growth in complex, evolving microenvironmental geometries: A diffuse domain approach

Abstract

We develop a mathematical model of tumor growth in complex, dynamic microenvironments with active, deformable membranes. Using a diffuse domain approach, the complex domain is captured implicitly using an auxiliary function and the governing equations are appropriately modified, extended and solved in a larger, regular domain. The diffuse domain method enables us to develop an efficient numerical implementation that does not depend on the space dimension or the microenvironmental geometry. We model homotypic cell-cell adhesion and heterotypic cell-basement membrane (BM) adhesion with the latter being implemented via a membrane energy that models cell-BM interactions. We incorporate simple models of elastic forces and the degradation of the BM and ECM by tumor-secreted matrix degrading enzymes. We investigate tumor progression and BM response as a function of cell-BM adhesion and the stiffness of the BM. We find tumor sizes tend to be positively correlated with cell-BM adhesion since increasing cell-BM adhesion results in thinner, more elongated tumors. Prior to invasion of the tumor into the stroma, we find a negative correlation between tumor size and BM stiffness as the elastic restoring forces tend to inhibit tumor growth. In order to model tumor invasion of the stroma, we find it necessary to downregulate cell-BM adhesiveness, which is consistent with experimental observations. A stiff BM promotes invasiveness because at early stages the opening in the BM created by MDE degradation from tumor cells tends to be narrower when the BM is stiffer. This requires invading cells to squeeze through the narrow opening and thus promotes fragmentation that then leads to enhanced growth and invasion. In three dimensions, the opening in the BM was found to increase in size even when the BM is stiff because of pressure induced by growing tumor clusters. A larger opening in the BM can increase the potential for further invasiveness by increasing the possibility that additional tumor cells could invade the stroma.

A computational framework for the personalized clinical treatment of glioblastoma multiforme

“…by considering more complex mechanical constitutive equations of the cancer cell phase. In particular, we aim to consider the elastic deformation of the brain during tumor growth and after its surgical removal, following the numerical strategy in [16]. This refinement will allow describing the structural changes within the brain tissues and the availability of nutrients depending on the spatial distribution of the blood vessels.…”

A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis

“…which resemble the system of equations studied in [8,9,20,54,56]. Note in particular that only µ T is needed to drive the evolution of ϕ i , 2 ≤ i ≤ L. However, the mathematical treatment of these types of models is difficult due to the fact that the equation for ϕ i is now a transport equation with a high order source term div (M (ϕ i )∇µ T ), and the natural energy identity of the model does not appear to yield useful a priori estimates for ϕ i .…”

“…In [8,9,20,45,54,56] a mass-averaged velocity is used instead of the volume-averaged velocity considered in our present approach and also in [49]. Meanwhile, in [37,38] the velocities of the cell components are assumed to be negligible.…”

“…Aside from mitosis proportional to the local density of nutrients, and constant apoptosis for the tumour cells, certain sink terms for one cell type become source terms for another, for example the term Aϕ 2 in (1.8b). It is commonly assumed that the host cells are homeostatic [8,9,20,56,54], and so the source term for the host cells is zero. In [37,38], where quiescent cells are also considered, a two-sided exchange between the proliferating cells and the quiescent cells, and a one-sided exchange from quiescent cells to necrotic cells based on local nutrient concentration are included.…”

Model of vascular desmoplastic multispecies tumor growth

“…The square gradient theory can be used in this approach to describe the smooth transition within a thin interfacial region. The gradient contributes to the Helmholtz free energy, from which the component velocities, pressures, and diffusive terms are derived (Chen and Lowengrub, 2014; Wise et al, 2008). Continuum single- or multi-phase models that consider the effects of cell-cell and/or cell-ECM adhesion include among others (Ambrosi and Preziosi, 2009; Bearer et al, 2009; Chatelain Clément et al, 2011; Escher and Matioc, 2013; Frieboes et al, 2007; Frieboes et al, 2013; Kuusela and Alt, 2009), while in (Arduino and Preziosi, 2015; Gerisch and Chaplain, 2008; Preziosi and Tosin, 2009b; Psiuk-Maksymowicz, 2013; Sciume et al, 2014a; Sciume et al, 2014b; Wu et al, 2013), the ECM is represented as one of the key components of the tumoral tissue.…”