The Minimal, Phase-Transition Model for the Cell-Number Maintenance by the Hyperplasia-Extended Homeorhesis

Abstract

Oncogenic hyperplasia is the first and inevitable stage of formation of a (solid) tumor. This stage is also the core of many other proliferative diseases. The present work proposes the first minimal model that combines homeorhesis with oncogenic hyperplasia where the latter is regarded as a genotoxically activated homeorhetic dysfunction. This dysfunction is specified as the transitions of the fluid of cells from a fluid, homeorhetic state to a solid, hyperplastic-tumor state, and back. The key part of the model is a nonlinear reaction-diffusion equation (RDE) where the biochemical-reaction rate is generalized to the one in the wellknown Schlögl physical theory of the non-equilibrium phase transitions. A rigorous analysis of the stability and qualitative aspects of the model, where possible, are presented in detail. This is related to the spatially homogeneous case, i.e. when the above RDE is reduced to a nonlinear ordinary differential equation. The mentioned genotoxic activation is treated as a prevention of the quiescent G0-stage of the cell cycle implemented with the threshold mechanism that employs the critical concentration of the cellular fluid and the nonquiescentcell-duplication time. The continuous tumor morphogeny is described as a time-space-dependent cellular-fluid concentration. There are no sharp boundaries (i.e. no concentration jumps exist) between the domains of the homeorhesis-and tumor-cell populations. No presumption on the shape of a tumor is used. To estimate a tumor in specific quantities, the model provides the time-dependent tumor locus, volume, and boundary that also points out the tumor shape and size. The above features are indispensable in the quantitative development of antiproliferative drugs or therapies and strategies to prevent oncogenic hyperplasia in cancer and other proliferative diseases. The work proposes an analytical-numerical method for solving the aforementioned RDE. A few topics for future research are suggested.

Propagation of Acoustic Waves Caused by the Accelerations of Vibrating Hand-Held Tools in Viscoelastic Soft Tissues of Human Hands and a Mechanobiological Picture for the Related Injuries

“…Journal of Applied Mathematics and Physics this respect, the mentioned populations are similar to other long-lasting lesions such as scars or other infiltrates in chronic inflammation, and lesions caused by homeorhetic dysfunctions. The infiltrates and the notion of a homeorhetic dysfunction are discussed, for example, in [38] and a number of the references therein.…”

Homeorhesis-based modelling and fast numerical analysis for oncogenic hyperplasia under radiotherapy

“…Let m be the so-called scaled concentration of the cells in a population (e.g., [7,8,6], see also (2) below). Note that m > 0 because of feature (i).…”

“…Note that m > 0 because of feature (i). Let also m c be the creode value of m. The simplest space-time model for morphogeny is an RDE (e.g., see [6,Section 10] for a recent discussion). Assume that concentration m is described with an RDE where the diffusion coefficient D of a cell is space-time-independent.…”

“…The advantage of description (1) is that it does not presume to model creode concentration m c : the latter can be obtained from the corresponding experimental data (e.g., see [17] for examples of these data). Function Φ is derived independently of the creode modelling in [6] within the PhasTraM approach. The PhasTraM form of (1) is discussed in Section 4.…”

“…As it is well known (e.g., [3,4,5]), oncogenic hyperplasia is a genotoxically activated homeorhetic dysfunction. The minimal mathematical model that takes into account the listed features was developed in [6] (see also [7,8]). The purpose of the present work is three-fold:…”