Numerical solutions for a model of tissue invasion and migration of tumour cells

The Dynamics of a Fractional-Order Mathematical Model of Cancer Tumor Disease

This article explores the application of the reduced differential transform method (RDTM) for the computational solutions of two fractional-order cancer tumor models in the Caputo sense: the model based on cancer chemotherapeutic effects which explain the relation between chemotherapeutic drugs, tumor cells, normal cells, and immune cells using a fractional partial differential equations, and the model that describes the different cases of killing rate K of cancer cells (the killing percentage of cancer cells K (I) is dependent on the number of cells, (II) is a function of time only, and (III) is a function of space only). The solutions are presented using Mathematica software as a convergent power series with elegantly computed terms using the suggested technique. The proposed method gives new series form results for various values of gamma. To clarify the complexity of the models, we plot the two- and three-dimensional and contour graphics of the obtained solutions at varied values of fractional-order gamma and the selected system parameters. The solutions are analyzed with fractional and reduced differential transform methods to obtain an idea of invariance regarding the computed solution of the designed mathematical model. The obtained results demonstrate the efficiency and preciseness of the proposed method to achieve a better understanding of chemotherapy effects. It is observed that chemotherapy drugs boost immunity against the specific cancer by decreasing the number of tumor cells, and the killing rate K of cancerous cells depend on the cells concentration.

An Unconditional Positivity-Preserving Difference Scheme for Models of Cancer Migration and Invasion

In this paper, we consider models of cancer migration and invasion, which consist of two nonlinear parabolic equations (one of the convection–diffusion reaction type and the other of the diffusion–reaction type) and an additional nonlinear ordinary differential equation. The unknowns represent concentrations or densities that cannot be negative. Widely used approximations, such as difference schemes, can produce negative solutions because of truncation errors and can become unstable. We propose a new difference scheme that guarantees the positivity of the numerical solution for arbitrary mesh step sizes. It has explicit and fast performance even for nonlinear reaction terms that consist of sums of positive and negative functions. The numerical examples illustrate the simplicity and efficiency of the method. A numerical simulation of a model of cancer migration is also discussed.

The influence of soluble fragments of extracellular matrix (ECM) on tumor growth and morphology

A major challenge in matrix-metalloproteinase (MMP) target validation and MMP-inhibitor-drug development for anti-cancer clinical trials is to better understand their complex roles (often competing with each other) in tumor progression. While there is extensive research on the growth-promoting effects of MMPs, the growth-inhibiting effects of MMPs has not been investigated thoroughly. So we develop a continuum model of tumor growth and invasion including chemotaxis and haptotaxis in order to examine the complex interaction between the tumor and its host microenvironment and to explore the inhibiting influence of the gradients of soluble fragments of extracellular matrix (ECM) density on tumor growth and morphology. Previously, it was shown both computationally (in one spatial dimension) and experimentally that the chemotactic pull due to soluble ECM gradients is anti-invasive, contrary to the traditional view of the role of chemotaxis in malignant invasion [1]. With two-dimensional numerical simulation and using a level set based tumor-host interface capturing method, we examine the effects of chemotaxis on the progression and morphology of a tumor growing in nutrient-rich and nutrient-poor microenvironments which was not investigated before. In particular we examine how the geometry of the growing tumor is affected when placed in different environments. We also investigate the effects of varying ECM degradation rate, the production rate of matrix degrading enzymes (MDE), and the conversion of ECM into soluble ECM. We find that chemotaxis due to ECM-fragment gradients strongly influences tumor growth and morphology, and that the instabilities caused by tumor cell proliferation and haptotactic movements can be prevented if chemotaxis is sufficiently strong. The influence of chemotaxis and the above factors on tumor growth and morphology are found to be more prominent in nutrient-poor environments than in nutrient-rich environments. So we extend our investigations of these antinvasive chemotactic influences by examining the effects of cell-cell and cell-ECM adhesion and low proliferation rate for tumors growing in low-nutrient environments. We find that as the extent of chemotaxis increases, the effects of adhesion on tumor growth and shape become negligible. Under conditions of low cell mitosis, chemotaxis may cause the tumor to shrink, as the extent of chemotaxis increases. Both stable and unstable tumor shrinkage are predicted by our model. Unexpectedly, in some cases chemotaxis may contribute toward developing instability where haptotaxis alone induces stable growth.

A novel method for simulating the extracellular matrix in models of tumour growth

A novel hybrid continuum-discrete model to simulate tumour growth on a cellular scale is proposed. The lattice-based spatiotemporal model consists of reaction-diffusion equations that describe interactions between cancer cells and their microenvironment. The fundamental ingredients that are typically considered are the nutrient concentration, the extracellular matrix (ECM), and matrix degrading enzymes (MDEs). The in vivo processes are very complex and occur on different levels. This in turn leads to huge computational costs. The main contribution of the present work is therefore to describe the processes on the basis of simplified mathematical approaches, which, at the same time, depict realistic results to understand the biological processes. In this work, we discuss if we have to simulate the MDE or if the degraded matrix can be estimated directly with respect to the cancer cell distribution. Additionally, we compare the results for modelling tumour growth using the common and our simplified approach, thereby demonstrating the advantages of the proposed method. Therefore, we introduce variations of the positioning of the nutrient delivering blood vessels and use different initializations of the ECM. We conclude that the novel method, which does not explicitly model the matrix degrading enzymes, provides means for a straightforward and fast implementation for modelling tumour growth.

Integrated cellular and gene interaction modelling of pattern formation

Cellular behaviour depends on and also modifies protein concentration and activity. An integrated cellular and gene interaction model is proposed to reveal this relationship. In this model, protein activity varies spatiotemporally with cellular location, gene interaction, and diffusion. In the meanwhile, cellular behaviour can vary spatially, driven by cell-cell signalling and inhomogeneous protein distribution across cells. This model integrates two components. The first component adopts a variation of the reaction-diffusion mechanism at the gene expression level. The second component is a lattice cellular model based on the Differential Adhesion Hypothesis (DAH) for cell sorting at the cellular level. Cell sorting and tumour invasion were simulated to illustrate the model. This model approximates cellular pattern formation more closely than existing models based on cell density.