Jean-Pierre Françoise

Mathematical modeling of energy metabolism and hemodynamics of WHO grade II gliomas using in vivo MR data.

Therapeutic management of low-grade gliomas (LGG) is a challenge because they have undergone anaplastic transformation with variable delay. Today, only progressive volume growth on successive MRI allows an in vivo monitoring of this evolution. On the other hand, multinuclear spectroscopy and perfusion available during MRI may also provide assessment of metabolic changes underlying morphological modifications. To overcome this drawback, we developed a mathematical model of the metabolism and the hemodynamic of gliomas, based on a physiological model previously published, and including the MR parameters. This allows us to suggest that some specific profiles of metabolic and hemodynamic changes would be good indicators of potential anaplastic transformation.

Mathematical Modeling of Metabolism and Hemodynamics

We provide a mathematical study of a model of energy metabolism and hemodynamics of glioma allowing a better understanding of metabolic modifications leading to anaplastic transformation from low grade glioma.This mathematical analysis allows ultimately to unveil the solution to a viability problem which seems quite pertinent for applications to medecine.

Une approche au problème du centre-foyer de Poincaré

Cherkas transform yields a version of the Poincare center-focus problem for the trigonometric Abel equations. In this Note, we consider the same problem for polynomial Abel equations. We show the existence of an integrating factor whose coefficients display a linear recurrency relation. We define the Bautin ideal and the Bautin index for these types of recurrency relations and we compute them in several cases. We introduce a tangential version of the centre problem and we solve it using the classical moment theorem. The moment theorem seems to be one of the key ingredient for solving the general case.

A differential geometry approach for biomedical image processing.

We show in this paper how simple considerations about bio-arrays images lead to a peak segmentation allowing the genes activity analysis. Bio-arrays images have a particular structure and the aim of the paper is to present a mathematical method allowing their automatic processing. The differential geometry approach used here can be also employed for other types of images presenting grey level peaks corresponding to a functional activity or to a chemical concentration. The mathematical method is based on elementary techniques of differential geometry and dynamical systems theory and provides a simple efficient algorithm when the peaks to segment are isolated.

On a Minimal Model for Hemodynamics and Metabolism of Lactate: Application to Low Grade Glioma and Therapeutic Strategies

WHO II low grade glioma evolves inevitably to anaplastic transformation. Magnetic resonance imaging is a good non-invasive way to watch it, by hemodynamic and metabolic modifications, thanks to multinuclear spectroscopy 1H/31P. In this work we study a multi-scale minimal model of hemodynamics and metabolism applied to the study of gliomas. This mathematical analysis leads us to a fast-slow system. The control of the position of the stationary point brings to the concept of domain of viability. Starting from this system, the equations bring to light the parameters that push glioma cells out of their domain of viability. Four fundamental factors are highlighted. The first two are cerebral blood flow and the rate of lactate transport through monocarboxylate transporters, which must be reduced in order to push glioma out of its domain of viability. Another factor is the intra arterial lactate, which must be increased. The last factor is pH, indeed a decrease of intra cellular pH could interfere with glioma growth. These reflections suggest that these four parameters could lead to new therapeutic strategies for the management of low grade gliomas.

From biological and clinical experiments to mathematical models.

The theory of dynamical systems and of nonlinear partial differential evolution equations and their applications in biology, physiology and medicine are fostered with the following papers comprising the present issue. They form a focused issue, which has its roots in an international congress on the

Extension of Bautin theory to any dimension

Bautin made some years ago a decisive contribution to the algebraic approach of the perturbation theory of periodic orbits of plane polynomial vector fields. This article presents first steps of a general framework in which a generalization of Bautin’s ideas to any dimension could be developed. The main result is the generalization of the algorithm of the successive derivatives of return mappings for 2-dimensional systems to any dimension in this framework.

Reduced Models for Unidirectional Block Conduction and Their Geometrical Setting

This article revisits a reduced model of cardiac electro-physiology which was proposed to understand the genesis of unidirectional block pathology and of ectopic foci. We underline some specificities of the model from the viewpoint of dynamical systems and bifurcation theory. We point out that essentially the same properties are shared by a simpler system more accessible to analysis. With this simpler system, it becomes possible to give a new presentation of the phenomenon in a phase plane with time moving slow manifolds. This presentation can be of interest both for cardiac electro-physiologists and for mathematicians.