A Bistable Field Model of Cancer Dynamics

Abstract

Cancer spread is a dynamical process occurring not only in time but also in space which, for solid tumors at least, can be modeled quantitatively by reaction and diffusion equations with a bistable behavior: tumor cell colonization happens in a portion of tissue and propagates, but in some cases the process is stopped. Such a cancer proliferation/extintion dynamics is obtained in many mathematical models as a limit of complicated interacting biological fields. In this article we present a very basic model of cancer proliferation adopting thebistableequation for a single tumor cell dynamics. The reaction-diffusion theory is numerically and analytically studied and then extended in order to take into account dispersal effects in cancer progression in analogy with ecological models based on the porous medium equation. Possible implications of this approach for explanation and prediction of tumor development on the lines of existing studies on brain cancer progression are discussed. The potential role of continuum models in connecting the two predominant interpretative theories about cancer, once formalized in appropriate mathematical terms, is discussed.

A gradient Discretisation Method For Anisotropic Reaction Diffusion Models with applications to the dynamics of brain tumours

“…The forward Euler method is utilized for the time-discretization, incorporating the use of time step δt. In accordance with [5], we utilize the initial data, incorporating several distorted Gaussian functions that are focused on a range of points chosen as Returning to the reaction model (1.1), anisotropic diffusion A has a specific symmetric and positive definite matrix and is regarded as being in direct proportion to the measured water diffusion tensor (DT ) harvested by DTI data [32,33]. DT details water molecule diffusion employing a Gaussian model.…”

“…With the proliferation term F (c) that details a population of tumor cells' growth, the majority of researchers have regarded it as a quadratic function that may be exponential: F (c) = ρc with constant proliferation rate ρ, illustrating that cellular division is obedient to a cycle, or logistic: F (c) = ρc(1 − c) with a decrease in ρ, the proliferation parameter, for zones of high cellular density. Nevertheless, in this research, following [5], cell growth is modeled with a cubic behavior so that the influence of the threshold of cancer cell density can be captured, something that is ignored when a quadratic function is employed. The threshold is a local driver of normal tissue for cancer cell regimes or will not permit cancer cells to colonize tissues, keeping it free of tumors, and this is an important factor to consider.…”

“…The threshold is a local driver of normal tissue for cancer cell regimes or will not permit cancer cells to colonize tissues, keeping it free of tumors, and this is an important factor to consider. Thus we consider a reaction term of [5,38]:…”

A stochastic mathematical model to study the autoimmune progression towards type 1 diabetes

“…This step would be useful to realistically simulate glucose, insulin, glucagon and somatostatin dynamics by possibly including other communication pathways (i.e. direct internalization [44]) together with the here adopted gap junctions, on the lines of previous mathematical studies performed by some of the authors in other different biomedical scenarios [45][46][47][48][49][50][51][52].…”

From Syndemic Lesson after COVID-19 Pandemic to a “Systemic Clinical Risk Management” Proposal in the Perspective of the Ethics of Job Well Done

“…“Reduction” is also useful and necessary for medicine that wants to be scientific and increasingly effective, but requires in order being conducted correctly, specific logical expedients, such as the use of non-linear logic [ 30 ], systemic correlations [ 31 ] and multidimensional observation. The application of these devices bears results at different levels of Public Health intervention in the area of patient safety and quality of care.…”