Mechanics of growing tumors: impact of modeling assumptions and boundary conditions on reliability of numerical results

A new computational model based on porous media mechanics has been recently developed for prediction of tumor growth [1]. The tumor mass is modeled as a four-phase system consisting of a solid phase, the extracellular matrix (ECM), and three immiscible fluid phases: the interstitial fluid (IF); the tumor cells (TC) and the healthy cells (HC) (TC and HC are modeled as adhesive fluids). Being the tumor growth strongly influenced by nutrients availability, the diffusion of oxygen coming from the nearby existing vessels is also considered. The mathematical model – governed by mass balance equations of phases and species and by the linear momentum balance equation of the solid scaffold (ECM) – has been discretized in space by finite elements and in the time domain by finite differences, and implemented in Cast3M (FE code of the French Atomic Agency). When in 2011 we started working on this model we introduced two simplifying assumptions: i) a unique pressure was considered for both cell populations (pTC = pHC) and ii) the ECM was assumed rigid. Then, the introduction of relevant constitutive relationships for the pressure difference among each pair of fluid phases (these are based on relative wettability of fluids and fluid–fluid interfacial tensions, see [2]) has allowed for relaxation of the first hypothesis and for a more realistic modeling of cell adhesion and invasion [3]. More recently also the second hypothesis has been relaxed and ECM deformability and its impact on tumor growth can be properly taken into account [4]. Our final aim is to develop a numerical tool which can be a complement for in vivo and in vitro experimental tests and help scientists in better understanding physical interaction between tumor and its surrounding. Hence, we will present our recent efforts to extensively validate the model, the impact of modeling assumptions and boundary conditions on reliability of numerical results, and some perspectives of enhancement of the model.