Section: Research Program

Growth of biological tissues

Project-team positioning

The originality of our work is the derivation, analysis and numerical simulations of mathematical model for growing cells and tissues. This includes mechanical effects (growth induces a modification of the mechanical stresses) and biological effects (growth is potentially influenced by the mechanical forces).

This leads to innovative models, adapted to specific biological problems (e.g. suture formation, cell polarisation), but which share similar features. We perform linear stability analysis, and look for pattern formation issues (at least instability of the homogeneous state).

The biophysical literature of such models is large. We refer to the groups of Ben Amar (ENS Paris), Boudaoud (ENS de Lyon), Mahadevan (Harvard), etc.

Our team combines strong expertise in reaction-diffusion equations (V. Calvez) and mechanical models (P. Vigneaux). We develop linear stability analysis on evolving domains (due to growth) for coupled biomechanical systems.

Another direction of work is the mathematical analysis of classical tumor growth models. These continuous mechanics models are very close to classical equations like Euler or Navier Stokes equations in fluid mechanics. However they bring there own difficulties: Darcy law, multispecies equations, non newtonian dynamics (Bingham flows). Part of our work consist in deriving existence results and designing acute numerical schemes for these equations.

Recent results

We have worked on several biological issues. Cell polarisation is the main one. We first analyzed a nonlinear model proposed by theoretical physicists and biologists to describe spontaneous polarisation of the budding yeast S. cerevisae. The model assumes a dynamical transport of molecules in the cytoplasm. It is analogous to the Keller-Segel model for cell chemotaxis, except for the source of the transport flux. We developped nonlinear analysis and entropy methods to investigate pattern formation (Calvez et al 2012). We are currently validating the model on experimental data. The analysis of polarization of a single cell is a preliminary step before the study of mating in a population of yeast cells. In the mating phase, secretion of pheromones induces a dialogue between cells of opposite types.

We also derive realistic models for the growth of the fission yeast S. pombe. We proposed two models which couple growth and geometry of the cell. We aim to tackle the issue of pattern formation, and more specifically the instability of the spherical shape, leading to a rod shape. The mechanical coupling involves the distribution of microtubules in the cytoplasm, which bring material to the cell wall.

In parallel, we have built a realistic biomechanical models for the onset of instability in the growth of cranial sutures. The basic assumption is that mechanics influences the local orientation of fibers in the tissue. Then cells move preferentially in the direction of fibers, so that growth of the suture interface is coupled to the mechanics. On the other hand, the geometry of the interface has a strong impact on the distribution of mechanical stresses. We were able to perform the full linear stability analysis of this complex model, and derive analytical conditions for the instability of the planar interface. We also performed 2D numerical simulations using FreeFEM++.

Over the evaluation period, Paul Vigneaux developped expertise in modelling and design of new numerical schemes for complex fluid models of the viscoplastic type. Associated materials are involved in a broad range of applications ranging from chemical industry to geophysical and biological materials. In the context of NUMED, this expertise is linked to the development of complex constitutive laws for cancer cell tissue. During the period, NUMED used mixed compressible/incompressible fluid model for tumor growth and viscoelastic fluid model. Viscoplastic is one of the other types of complex fluid model which is usable in the field. Mathematically, it involves variational inequalities and the need for specific numerical methods.

More classically, Séverine Enault and Emmanuel Grenier studied the coupling between transport equation and Darcy law in multi population models and obtained in some case existence of weak solutions for all time and in other case blow up in finite time. They in particular underline the link with Euler equations for incompressible fluids. It turns out that these equations are also used in petrology. As a by product they proved the well known Arp's law of exploitation of mature petroleum fields.


  • V. Calvez (ENS de Lyon, Inria NUMED), Th. Lepoutre (Inria DRACULA), N. Meunier, (Univ. Paris 5), N. Muller (Univ. Paris 5), P. Vigneaux (ENS de Lyon, Inria NUMED): mathematical analysis of cell polarisation, numerical simulations

  • V. Calvez (ENS de Lyon, Inria NUMED), N. Meunier, (Univ. Paris 5), M. Piel, (Institut Curie, Paris), R. Voituriez (Univ. Paris 6): biomechanical modeling of the growth of S. pombe

  • D. Bresch (Univ. Chambéry), V. Calvez (ENS de Lyon, Inria NUMED), R.H. Khonsari (King's College London, CHU Nantes), J. Olivier (Univ. Aix-Marseille), P. Vigneaux (ENS de Lyon, Inria NUMED): modeling, analysis and simulations of suture formation.

  • Didier Bresch (Univ Chambéry), Benoit Desjardins(Moma group): petrology.

ANR JCJC project "MODPOL", Mathematical models for cell polarization, led by Vincent Calvez (ENS de Lyon, CNRS, Inria NUMED).