There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

This paper proposes a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the classical Keller–Segel model and its subsequent modifications, which, in several cases, have been developed to obtain models that prevent the non-physical blow up of solutions. The presentation is organized in three parts. The first part focuses on a survey of some sample models, namely the original model and some of its developments, such as flux limited models, or models derived according to similar concepts. The second part is devoted to the qualitative analysis of analytic problems, such as the existence of solutions, blow-up and asymptotic behavior. The third part deals with the derivation of macroscopic models from the underlying description, delivered by means of kinetic theory methods. This approach leads to the derivation of classical models as well as that of new models, which might deserve attention as far as the related analytic problems are concerned. Finally, an overview of the entire contents leads to suggestions for future research activities.

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

https://iopscience.iop.org/article/10.1088/0951-7715/23/1/R01/pdf

Nonlinear modelling of cancer: Bridging the gap between cells and tumours

This review will outline a number of illustrative mathematical models describing the growth of avascular tumors. The aim of the review is to provide a relatively comprehensive list of existing models in this area and discuss several representative models in greater detail. In the latter part of the review, some possible future avenues of mathematical modeling of avascular tumor development are outlined together with a list of key questions.

/pdf/mathematical-models-of-avascular-tumor-growth-1k9m7fkgnk.pdf

Mathematical Models of Avascular Tumor Growth

Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations.

This chapter shows how the mathematical tools presented in Chapter 3 can be developed to derive mathematical models in social sciences and to understand how interactions at the microscopic scale can lead to the collective social behaviors.

Modeling Social Behavioral Dynamics

Methods of mathematical kinetic theory have been recently developed to describe the collective behavior of large populations of interacting individuals such that their microscopic state is identified not only by a mechanical variable (typically position and velocity), but also by a biological state (or sociobiological state) related to their organized, somehow intelligent, behavior. The interest in this type of mathematical approach is documented in the collection of surveys edited in [1], in the review papers [2], [3], and in the book [4].

Methods and tools of mathematical kinetic theory towards modelling complex biological systems

In this paper, we propose a numerical scheme to solve the kinetic model for chemotaxis phenomena. Formally, this scheme is shown to be uniformly stable with respect to the small parameter, consistent with the fluid-diffusion limit (Keller-Segel model). Our approach is based on the micro-macro decomposition which leads to an equivalent formulation of the kinetic model that couples a kinetic equation with macroscopic ones. This method is validated with various test cases and compared to other standard methods.

An asymptotic preserving scheme for kinetic models for chemotaxis phenomena

This paper deals with the derivation of mathematical models at the macroscale of biological tissues corresponding to angiogenesis phenomena. The derivation is obtained by mathematical description delivered at the microscale of cells using a kinetic theory approach. A classical Chapman–Enskog expansion properly truncated is used to obtain the desired result. It is shown that the approach is general enough to describe a broad variety of different angiogenesis models corresponding to well-defined assumptions on interactions at the cellular scale.

Abdelghani Bellouquid

Papers

Modeling Social Behavioral Dynamics

Methods and tools of mathematical kinetic theory towards modelling complex biological systems

An asymptotic preserving scheme for kinetic models for chemotaxis phenomena

On the derivation of angiogenesis tissue models: From the micro-scale to the macro-scale