Homogenization of advection-diffusion and solid diffusion in poroelastic media for modelling transport of soluble factors in biological tissues
Résumé
This work has been motivated by the study of solid tumor growth and more
specifically by the examination of the most frequent pediatric primary bone
tumors (osteosarcoma), which are characterised in their early stage by the for-
mation of non-mineralised bone tissue, called osteoid [1]. As the tumor evolves
in time, mineralisation of this growing tissue can take place. The purpose of
this work thus is to investigate the advection-diffusion of soluble factor such
as calcium and phosphate [2] in the context of tumor mineralisation, how this
mineralisation of bone tissue may become a barrier to treatment in the context
of drug transport [3] but also how proteins and growth factors can play a role in
tumor growth [4]. For this purpose, the method of asymptotic homogenization
is used so as to derive the macroscopic models for describing solute transport
in poroelastic media, with an advective-diffusive regime in the fluid-saturated
pores and diffusion in the solid phase, by starting from the description on the
pore scale. The fluid/solid equations lead to Biot's model of poroelasticity.
Then, homogenization of the transport equations leads to three macroscopic
models that relate to three orders of magnitude of the diffusivity ratio: a model
in which the solid diffusion only in uences the accumulation term; ii) a model
with memory effects; iii) a model without solid diffusion. Initially expressed by
means of orders of magnitude of the diffusivity ratio, the domains of validity
of each of these three models can be expressed in terms of relative orders of
magnitude of two characteristic times. The three models contain a solute-solid
interaction term, due to the advection regime and they are coupled to the poroe-
laticity model via the advection term.
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