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v3.2.9
Adipose cells or adipocytes are the specialised cells composing the adipose tissue in a variety of species. Their role is the storage of energy in the form of a lipid droplet inside their membrane. Based on the amount of lipid they contain, one can consider the distribution of adipocyte per amount of lipid and observe a peculiar feature : the resulting distribution is bimodal, thus having two local maxima. The aim of this talk is to introduce a model built from the work in [1] that is able to reproduce this bimodal feature.
Considering that cells are spheres and that our system is closed (no loss of lipids) the following set of equation can be derived :
∂ t f (t, x) + ∂ x (v(x, L(t))f (t, x)) = 0, (t, x) ∈ R 2+
L(t) + R + xf (t, x)dx = λ, t ∈ R +
Where f : (t, x) ∈ R 2+ → R + is the distribution of adipocyte per amount of lipid x and L : t ∈ R + → R + is
the amount of lipids available in the medium. The transport speed v(x, L) takes the form
a(x)M (L) − b(x), where a is the lipid inflow rate and b the lipid outflow rate. The function M represents the dependency of the inflow rate to the outside resource, i.e. the lipids in the medium. The value λ is the total amount of lipids in the system and is assumed to be constant in time.
This coupling of a transport equation and a conservation equation of this form is related to the Lifshitz-Slyozov model introduced in [2]. We will introduce a related set of equations for the adipose cells : the Becker-Döring equations [3] and prove the classical result of convergence from one to another.
We will also present some extensions to the model using a transport-diffusion equation arising from the previous convergence result and a rewriting of the model using stochastic processes. In both cases, we shall describe stationary solutions. Finally, we will present some numerical results using a standard finite volume scheme introduced in [4] for the transport and transport-diffusion models.
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 Döring. ”Kinetische behandlung der keimbildung in übersättigten dämpfen.” 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.