Tamtam Proceedings - lamsin

A topology Nash game 27While still viewing the tumor+extracellular matrix+existing vessel as an overall system,we assume that the tumor and the host tissue are acting in a competition framework,each competitor willing to fulfill a given objective. We make two assumptions on therespective objectives of the tumor and of the host tissue. The first one is that angiogenesisprovides the tumor with an optimal drainage mechanism. Natural circulatorynetworks, including human vascular network are known to have arterial branchings optimalstructure with respect to the maximal-drainage objective ; An evidence is providedby Schreiner [8] "Arterial branchings closely fullfil several ’bifurcation rules’ which aredeemed to optimize blood flow. The question is whether these local criteria in conjunctionwith a general optimization principle can explain the overall structure of an arterialtree.” Schreiner concludes in the cited study that "The comparison between the modeland real coronary arterial trees shows good agreement regarding structural appearance,morphometric parameters, and pressure profiles." The second assumption we make is thatthe host tissue is willing to keep its structural integrity as a way to fight against the tumorgrowth. Indeed, this assumption seems to gain audience among biologists. Based on invitro studies, Helmlinger et al. [9] demonstrated that solid stress inhibits tumor growthin vitro, regardless of host species, tissue of origin, or differentiation state, which madeRoose et al. [7] suggest that “the host tissue provides resistance to tumor growth“.2.1. A porous media model for the tumorThe extracellular matrix as well as the tumoral vasculature are seen as a porous medium,which occupies a volume Ω ⊂ R N (N = 2 or 3), with a variable porosity denoted by ρ, which lies between the matrix porosity ρ M and blood vessel porosity ρ V . The simplesteffective model for porous media is the following, also known as the D’Arcy Law, wherethe physical variable is pressure p :⎧⎪⎨⎪⎩−div (ρ∇p) = Q in Ωρ ∂p∂n= ρg over Γ V∂p(1)∂n= 0 over Γ Np = 0 over Γ TThe right-hand side Q represents a residual source of nutrients by diffusion throughthe host tissue, it is assumed to be negligible compared to the inward blood flow g. Itshould be noticed that we do not take into account what happens inside the tumor itself,considering only its boundary Γ T as an outlet.Obviously, the pressure field depends on the porosity distribution.As mentioned before, we postulate that angiogenesis provides the tumor with an optimaldrainage mechanism, i.e. with a porosity such that the tumor optimal blood networkminimizes the averaged pressure drop.TAMTAM –Tunis– 2005