Tamtam Proceedings - lamsin

28 HabbalThe pressure drop denoted by L 1 (ρ; p) is given by the formula :∫ ∫L 1 (ρ; p) = Qp dx + ρgp ds2.2. A structural model for the extracellular matrixΩNow, one may also consider the host surrounding tissue as a continuum medium, letsay a linear isotropic, nonhomogeneous, elastic material. This model is of course a coarseapproximation of the actual mechanical behavior of the living tissues, which is rather ofvisco-elastic nature [6]. This medium is composed of healthy and degraded tissues. Thedegradation could be due to established vascularization or to an early enzyme’s action,like as the MMPs family.The elasticity tensor E lies then (in a certain sense) between thedegraded material tensor E D , and the original -sane- extracellular matrix tensor E M .Conforming to the linear elasticity classical equilibrium equations, the displacementvector u = (u j ) solves⎧⎪ ⎨⎪ ⎩−div (Eɛ(u)) = b in ΩΓ Vu = 0 over Γ VEɛ(u).n = 0 over Γ NEɛ(u).n = t over Γ T(2)The strain tensor denoted by ɛ(u) is defined with obvious notations asɛ(u) ij = 1 ( ∂ui+ ∂u )j2 ∂x j ∂x iThe mechanical stress tensor is given by σ(u) = Eɛ(u).The body forces -such as selfweight- are denoted by b, and the normal tension whichmodels the stress induced by the tumor growth is denoted by t. The tissue is assumedto be clamped to the mother vessel Γ V . A related model can be found in [1] where theauthors study the stress induced during avascular tumor growth.The displacement vector u depends on the Elasticity tensor E. The latter itself dependson the interaction between activators and inhibitors of tissue degradation.As said in the introductory section, we assume that the host tissue is willing to keepits integrity, by using all available factors it could control (one example is inhibitors ofMMPs). In continuum mechanics, it is usual to consider that such goal is achieved bymaximizing the stiffness, or equivalently, minimizing the compliance :∫ ∫L 2 (E; u) = b.u dx + t.u dsΩΓ TTAMTAM –Tunis– 2005