A stochastic model featuring acid-induced gaps during tumor progression

Abstract

In this paper we propose a phenomenological model for the formation of an interstitial gap between the tumor and the stroma. The gap is mainly filled with acid produced by the progressing edge of the tumor front. Our setting extends existing models for acid-induced tumor invasion models to incorporate several features of local invasion like formation of gaps, spikes, buds, islands, and cavities. These behaviors are obtained mainly due to the random dynamics at the intracellular level, the go-or-grow-or-recede dynamics on the population scale, together with the nonlinear coupling between the microscopic (intracellular) and macroscopic (population) levels. The wellposedness of the model is proved using the semigroup technique and 1D and 2D numerical simulations are performed to illustrate model predictions and draw conclusions based on the observed behavior.

Mathematical Analysis and Numerical Simulations for the Cancer Tissue Invasion Model

“…Another approach analyzing the e¤ect of chemotaxis and haptotaxis has been made in [8]. On the other hand, in some cases models may contain stochastic e¤ects like the ones in the studies [9,10,11] considering the stochasticity in pH dynamics.…”

Global existence for a go-or-grow multiscale model for tumor invasion with therapy

“…Another relevant feature of tumor migration is its multiscality: the macroscopic behavior of the whole cell population is conditioned by processes taking place on the individual cell level and on the subcellular scale and influences, in turn, these processes. Apart from discrete or hybrid settings (see e.g., [3] and the references therein), several continuum models connecting the subcellular and the population scales or also accounting for the mesoscopic individual level dynamics of cells have been recently proposed and analyzed e.g., in [14,28,39,40] and [20,26,5], respectively. Newer multiscale models also accounting for the tumor heterogeneity in the sense mentioned above (go-orgrow dichotomy) were proposed in [6,17,43] and in the context of acid-mediated tumor invasion (active vs. quiescent cells) in [29].…”

On a coupled SDE-PDE system modeling acid-mediated tumor invasion

“…Recently we considered in [20] a two-scale model with nonlocal sample dependence describing the proton dynamics in a tumor, where the intracellular one is governed by an SDE that is coupled to a reaction-diffusion equation for the macroscopic concentration of extracellular protons. The models in [15,14] have a multiscale character, as well; they couple random ODEs with PDEs of reaction-(cross)diffusion-taxis type and show the relevance of stochasticity in explaining transiently observed phenomena like hypocellular gaps between the tumor and the surrounding normal tissue, further infiltrative growth patterns, or tumor aggressivity depending on cell phenotype switching. In this work we consider a model connecting the subcellular scale (dynamics of intracellular protons, described by an SDE) with the macroscopic one (tumor cell density and extracellular proton concentration, each described by a reaction-diffusion PDE -the one for tumor cells also including pH-taxis).…”

“…Another well studied approach -still within the context of stochastic evolution equations in Hilbert spaces-relies on the theory of semigroups (see e.g., [3]) and requires, too, rather strong smoothness assumptions about the involved operators; in particular, it is not clear how to apply it for SDE-PDE systems including nonlinear diffusion and taxis effects. In [14] we applied such method to a larger system involving nonlinear diffusion and pH-taxis, however coupling PDEs with an ODE and a random ODE; the proof was quite involved, but ensured global well-posedness under the conditions imposed on the coefficients. When only weaker assumptions can be made about the functions and coefficients of the system to be studied in this or related stochastic settings then -as mentioned-an approach relying on compactness and a priori estimates (as in the deterministic framework) would be desirable, since it allows to approximate the highly complex problem at hand with a less complicated one.…”