Reaction-diffusion model for the growth of avascular tumor

Ferreira, Silvio C.; Martins, Marcelo L.; Vilela, M.J.

doi:10.1103/physreve.65.021907

Cited by 201 publications

(211 citation statements)

References 21 publications

Supporting

Mentioning

196

Contrasting

Unclassified

Order By: Relevance

“…The tumour morphologies obtained with this model suggest that the shape of the tumour very much depends on its growth conditions, and are similar to those observed in the models of Ferreira et al (2002), Anderson (2005) and Gerlee and Anderson (2007a). For normal oxygen concentrations we find tumours with a smooth circular morphology, while in low oxygen concentration we observe tumours with invasive fingered shape.…”

Section: Discussionsupporting

confidence: 84%

“…The first work using an individual based cellular automata model in cancer modelling was done by Düchting and Vogelsaenger (1984), who used it to investigate the effects of radiotherapy. More recent work in this direction has investigated the impact of the micro-environment on tumour growth (Ferreira et al, 2002;Anderson, 2005;Anderson et al, 2006). Other studies have focused on how the cell processes information from the environment and how this influences tumour growth dynamics (Gerlee and Anderson, 2007a;Mansury et al, 2006Mansury et al, , 2002.…”

Section: Previous Workmentioning

confidence: 99%

“…The decay rates of the metabolites are known to be considerably smaller than the respective cellular consumption rates and for simplicity we therefore disregard these in the equations. This allows us to develop a minimal model of the chemical fields, similar to those in the models of Patel et al (2001) and Ferreira et al (2002). The time evolution of the oxygen (3), glucose (4) and hydrogen ion (5) fields are therefore governed by the following set of partial differential equations,…”

Section: Chemical Fieldsmentioning

confidence: 99%

See 2 more Smart Citations

A hybrid cellular automaton model of clonal evolution in cancer: The emergence of the glycolytic phenotype

Gerlee

Anderson

2008

Journal of Theoretical Biology

119

113

View full text Add to dashboard Cite

We present a cellular automaton model of clonal evolution in cancer aimed at investigating the emergence of the glycolytic phenotype. In the model each cell is equipped with a micro-environment response network that determines the behaviour or phenotype of the cell based on the local environment. The response network is modelled using a feed-forward neural network, which is subject to mutations when the cells divide. This implies that cells might react differently to the environment and when space and nutrients are limited only the fittest cells will survive. With this model we have investigated the impact of the environment on the growth dynamics of the tumour. In particular we have analysed the influence of the tissue oxygen concentration and extra-cellular matrix density on the dynamics of the model. We found that the environment influences both the growth and evolutionary dynamics of the tumour. For low oxygen concentration we observe tumours with a fingered morphology, while increasing the matrix density gives rise to more compact tumours with wider fingers. The distribution of phenotypes in the tumour is also affected, and we observe that the glycolytic phenotype is most likely to emerge in a poorly oxygenated tissue with a high matrix density. Our results suggest that it is the combined effect of the oxygen concentration and matrix density that creates an environment where the glycolytic phenotype has a growth advantage and consequently is most likely to appear.

show abstract

Section: Discussionsupporting

confidence: 84%

Section: Previous Workmentioning

confidence: 99%

Section: Chemical Fieldsmentioning

confidence: 99%

See 1 more Smart Citation

A hybrid cellular automaton model of clonal evolution in cancer: The emergence of the glycolytic phenotype

Gerlee

Anderson

2008

Journal of Theoretical Biology

119

113

View full text Add to dashboard Cite

show abstract

“…According to the above recipe, one takes into account the corresponding diffusion term (cf., Burton, 1966;De Angelis and Preziosi, 2000;Pettet et al, 2001;Akabani et al, 2002;Ferreira et al, 2002;Kar et al, 2002;Jiang et al, 2002). This extends ODE (7.29) to the following reaction-diffusion equation (RDE) , , , (10.1) where is the diffusion parameter of the cellular fluid (e.g., see (10.4) below) and column is the Hamilton differential expression in the entries of vector , i.e.…”

Section: The Diffusion Generalization Of the Core Model To Morphogenymentioning

confidence: 90%

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

2006

View full text Add to dashboard Cite

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.

show abstract

“…We also provide simulations of diffusion processes in two dimensions in order to corroborate our results. [3,4] and tumour growth [5]. Of particular interest to mathematical biologists in recent years have been the effects of noise on those systems which form patterns via diffusion-driven instability [6].…”

Section: Introductionmentioning

confidence: 99%

Importance of the Voronoi domain partition for position-jump reaction-diffusion processes on nonuniform rectilinear lattices

Yates

Baker

2013

Phys. Rev. E

View full text Add to dashboard Cite

Position-jump processes are used for the mathematical modelling of spatially extended chemical and biological systems with increasing frequency. A large subset of the literature concerning such processes is concerned with modelling the effect of stochasticity on reaction-diffusion systems. Traditionally, computational domains have been divided into regular voxels. Molecules are assumed well-mixed within each of these voxels and are allowed to react with other molecules within the same voxel or to jump to neighbouring voxels with predefined transition rates. In this report, we investigate theoretically the propriety of the Voronoi domain partition as an appropriate method to partition domains for position-jump models in higher dimensions. We also provide simulations of diffusion processes in two dimensions in order to corroborate our results.

show abstract

Reaction-diffusion model for the growth of avascular tumor

Cited by 201 publications

References 21 publications

A hybrid cellular automaton model of clonal evolution in cancer: The emergence of the glycolytic phenotype

A hybrid cellular automaton model of clonal evolution in cancer: The emergence of the glycolytic phenotype

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

Importance of the Voronoi domain partition for position-jump reaction-diffusion processes on nonuniform rectilinear lattices

Contact Info

Product

Resources

About