Section: New Results

Axis 2: Bio-physical modeling for personalized therapies

Electropermeabilization

Non-Linear Steady-State Electrical Current Modeling for the Electropermeabilization of Biological Tissue [8]

Team participants: Clair Poignard, Michael Leguebe

Other participants: Marie Breton, Lluis M. Mir (Vectorology and Anticancer Therapies), Francois Buret, Riccardo Scorretti, Damien Voyer, Laurent Krähenbühl (Ampère Laboratory (Lyon) participants), Ronan Perrussel (LAPLACE - Laboratoire Plasma et Conversion d'Energie, Toulouse)

We propose a non-linear steady-state model of irreversible electropermeabilization in a biological tissue. The non-linear problem is solved using a modified fixed point iteration. The unknown parameters are experimentally estimated from the observation of the necrosis on a potato tissue for different applied voltages. A variability study of the parameters involved in the model is performed.

A second-order Cartesian method for the simulation of electropermeabilization cell models [12]

Team participants: Clair Poignard, Michael Leguèbe

Other participant: Lizl Weynans (Memphis team, Inria)

In this work, we present a new finite differences method to simulate electropermeabilization models, like the model of Neu and Krassowska or the recent model of Kavian et al. These models are based on the evolution of the electric potential in a cell embedded in a conducting medium. The main feature lies in the transmission of the voltage potential across the cell membrane: the jump of the potential is proportional to the normal flux thanks to the well-known Kirchoff law. An adapted scheme is thus necessary to accurately simulate the voltage potential in the whole cell, notably at the membrane separating the cell from the outer medium. We present a second-order finite differences scheme in the spirit of the method introduced by Cisternino and Weynans for elliptic problems with immersed interfaces. This is a Cartesian grid method based on the accurate discretization of the fluxes at the interface, through the use of additional interface unknowns. The main novelty of our present work lies in the fact that the jump of the potential is proportional to the flux, and therefore is not explicitly known. The original use of interface unknowns makes it possible to discretize the transmission conditions with enough accuracy to obtain a second-order spatial convergence. We prove the second-order spatial convergence in the stationary linear one-dimensional case, and the first-order temporal convergence for the dynamical non-linear model in one dimension. We then perform numerical experiments in two dimensions that corroborate these results.

Cell membrane permeabilization by 12-ns electric pulses: Not a purely dielectric, but a charge-dependent phenomenon [15]

Team participants: Clair Poignard, Michael Leguèbe

Other participants: Aude Silve (KIT - Karlsruhe Institute of Technology), Isabelle Leray, Lluis M. Mir (Université Paris Sud)

Electric pulses of a few nanoseconds in duration can induce reversible permeabilization of cell membrane and cell death. Whether these effects are caused by ionic or purely dielectric phenomena is still discussed. We address this question by studying the impact of conductivity of the pulsing buffer on the effect of pulses of 12 ns and 3.2 MV/m on the DC-3F mammalian cell line. When pulses were applied in a high-conductivity medium (1.5 S/m), cells experienced both reversible electropermeabilization and cell death. On the contrary, no effect was observed in the low-conductivity medium (0.1 S/m). Possible artifacts due to differences in viscosity, temperature increase or electrochemical reactions were excluded. The influence of conductivity reported here suggests that charges still play a role, even for 12-ns pulses. All theoretical models agree with this experimental observation, since all suggest that only high-conductivity medium can induce a transmembrane voltage high enough to induce pore creation, in turn. However, most models fail to describe why pulse accumulation is experimentally required to observe biological effects. They mostly show no increase of permeabilization with accumulation of pulses. Currently, only one model properly describes pulse accumulation by modeling diffusion of the altered membrane regions.

Cell protrusion

Free boundary problem for cell protrusion formations: theoretical and numerical aspects [20]

Team participants: Olivier Gallinato, Clair Poignard

Other participants: Masahito Ohta (Tokyo University of Sciences), Takashi Suzuki (Osaka University)

In this work, we derive a free boundary problem for cell protrusion formation in which the cell membrane is precisely described thanks to a level-set function, whose motion is due to specific signalling pathways. The model consists in Laplace equation with Dirichlet condition inside the cell coupled to Laplace equation with Neumann condition in the outer domain. The motion of the interface is due the gradient of the inner quantity. We prove the well-posedness of our free boundary problem under a sign condition on the datum similarly to the Taylor criterion in water waves. We also propose an accurate numerical scheme to solve the problem and we exhibit the main biological features that can be accounted for by the model. Even though simplistic from the modeling point of view, we claim that this work provides the theoretical and numerical grounds for single cell migration modeling. In particular, specific chemical reactions that occurred at the cell membrane could be precisely described in forthcoming works.