In this talk, I will present the use of the Lifshitz-Slyozov equations for modeling a population of adipose cells. These cells are present in adipose tissue and are responsible for the storage of energy in the form of lipids. Their distribution in size (radius of cells or quantity of lipids inside the cell) has a singular shape : it is bimodal. The aim of my PhD thesis is to model this bimodal shape and provide new insights on the modeling of adipose cells. The modeling assumptions lead us to a set of equations resembling the classical Lifshitz-Slyozov equations. These equations are the coupling of an advection equation and a non-local constraint, where the constraint acts on the velocity of the transport. The main differences between the Lifshitz-Slyozov equations and our model is that the constraint term in the velocity is non-linear in our case and that we provided null-flux boundary conditions which leads to the conservation of the mass. This allows us to circumvent some technicalities in the classical proofs. Additionally, I will present an extension to this model with the goal of obtaining smoother stationary solutions. This extension is the addition of a diffusive term to the advection equation. Using the theory on the convergence of the Becker-D¨oring model toward the Lifshitz-Slyozov model, one can choose heuristically the form of such a diffusive term. I will introduce a new proof for showing the convergence using tails of distributions and a probabilistic proof for the convergence toward the extended model. I will conclude by showing some numerical results using a well-balanced scheme for the Lifshitz-Slyozov model.
References
[1] Soula, H. A., et al. ”Modelling adipocytes size distribution.” Journal of theoretical biology 332 (2013): 89-95.
[2] Lifshitz, Ilya M., and Vitaly V. Slyozov. ”The kinetics of precipitation from supersaturated solid solutions.” Journal of physics and chemistry of solids 19.1-2 (1961): 35-50.
[3] Becker, Richard, and Werner D¨oring. ”Kinetische behandlung der keimbildung in ¨ubers¨attigten d¨ampfen.” Annalen der physik 416.8 (1935): 719-752.
[4] Goudon, Thierry, and Laurent Monasse. ”Fokker-Planck Approach of Ostwald Ripening: Simulation of a Modified Lifshitz–Slyozov–Wagner System with a Diffusive Correction.” SIAM Journal on Scientific Computing 42.1 (2020): B157-B184.