Abdelghani Bellouquid

MATHEMATICAL TOPICS ON THE MODELLING COMPLEX MULTICELLULAR SYSTEMS AND TUMOR IMMUNE CELLS COMPETITION

This paper deals with a critical analysis and some developments related to the mathematical literature on multiscale modelling of multicellular systems involving tumor immune cells competition at the cellular level. The analysis is focused on the development of mathematical methods of the classical kinetic theory to model the above physical system and to recover macroscopic equation from the microscopic description. Various hints are given toward research perspectives, with special attention on the modelling of the interplay of microscopic (at the cellular level) biological and mechanical variables on the overall evolution of the system. Indeed the final aim of this paper consists of organizing the various contributions available in the literature into a mathematical framework suitable to generate a mathematical theory for complex biological systems.

MULTICELLULAR BIOLOGICAL GROWING SYSTEMS: HYPERBOLIC LIMITS TOWARDS MACROSCOPIC DESCRIPTION

This paper deals with the analysis of the asymptotic limit towards the derivation of hyperbolic macroscopic equations for a class of equations modeling complex multicellular systems. Cellular interactions generate both modification of biological functions and proliferating destructive events related to growth of tumor cells in competition with the immune system. The asymptotic analysis refers to the hyperbolic limit to show how the macroscopic tissue behavior can be described by linear and nonlinear hyperbolic systems which seem the most natural in this context.

From a multiscale derivation of nonlinear cross-diffusion models to Keller–Segel models in a Navier–Stokes fluid

This paper deals with a micro–macro derivation of a variety of cross-diffusion models for a large system of active particles. Some of the models at the macroscopic scale can be viewed as developmen...

Asymptotic analysis of a nonlinear integro-differential system modeling the immune response

This paper deals with the qualitative analysis of a nonlinear model of immune response with special attention to the actions applied by cytokine signals to activate the last one, in the framework of the mathematical kinetic theory for active particles. The content focuses on the early stage of the competition before the onset of proliferation. The analysis gives evidence how parameters and initial conditions influence the asymptotic behaviors of solutions.

From systems biology to analytic problems: Comment on the paper “On the interplay between mathematics and biology, hallmarks toward a new systems biology” by N. Bellomo et al.

The new systems approach in biological sciences proposed in [1] is a constructive reply to the hints of the scientific community [2] and opens new research paths toward developments of new mathematical methods to model the living matter. This comment focuses on some analytic and modeling issues which are somewhat related to my personal scientific knowhow. I look forward to the answer to them as a critical analysis on the topics presented in the following might contribute to research activity in the field and, specifically, to the complex interplay between mathematics and biology. Therefore, after some introductory comment, I pose, for each issue, a specific question.

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

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.

Identication Parameters in a Biological Model of Immune Competition : Global Optimization and Kriging Method

In this work, we study the problem of identifying parameters in the model of immune competition developed in ([1], [2]). Specically, we use the approach of the inverse problem which will allow the identication of parameters from measurements of densities of two populations of cells in the proliferation case. The reformulation of the given nonlinear identica- tion problem was considered as a parametric optimization problem using the Least Square criterion. In this work, a design procedure for global ro- bust optimization is developed using Kriging [4] and global optimization approaches [3]. Robustness is determined by the Kriging model to reduce the number of real functional calculations of Least Square criterion. The technical of the global optimization methods is adopted to determine the global robust optimum of a surrogate model. References [1] A. Bellouquid, M. Delitala, Modelling Complex Biological Systems: Akinetic Theory Approch, Birkuser, Boston, Basel, Berlin (2006). [2] N. Bellomo, A. Bellouquid and M. Delitala, Mathematical Topics onthe Modelling Complex Multicellular Systems and Tumor Immune CellsCompetition, Mathematical Models and Methods in Applied SciencesVol. 14, No. 11 , 1683-1733 (2004). [3] Donald R.Jones, Matthias Schonlau and William J.Welch EcientGlobal Optimization of Expensive Black-Box Functions Journal ofGlobal Optimization 13:455-492 (1998). [4] L. Afraites, J. Hazart, P. Schiavone, Application of the Kriging methodto the reconstruction of ellipsometric signature, Microelectronic Engi-neering, Vol. 86, issue 4-6, pp. 1033-1035 (2009).

Asymptotic Preserving Scheme from Kinetic to Macroscopic Scale for Multicellular Growing Systems

In this work, we develop a numerical method to solve a model for Kinetic Theory of Active Particles (in brief KTAP) which is able to capture amacroscopic models of biological system of two populations cells.The asymptotic preserving (AP) schemes are based on the micro/macrodecomposition technique, which applies to general collision operators.We also present several numerical tests to illustrate the efficiency of ourapproach.

On the Derivation of Biological Tissue Models from Kinetic Models of Multicellular Growing Systems

This paper deals with the derivation of macroscopic equations for a class of equations modelling complex multicellular systems delivered by the kinetic theory for active particles. The analysis is focused on growing cancer tissues. A critical analysis is proposed to enlighten the technical difficulties generated by dealing with living tissues and to focus the strategy to overcome them by new mathematical approaches.