The use of expandable megaprostheses for reconstruction of the distal femur in pediatric patients after bone tumor resection

Expandable prostheses are becoming increasingly popular in the reconstruction of children with bone sarcomas of the lower limb. Since the introduction of effective chemotherapy in the treatment of these pathologies, in the 70s, there has been need for new limb salvage techniques. In children, limb salvage of the lower limbs is particularly challenging, not in the last place, because of the loss of growth potential. Therefore, expandable prostheses have been developed. However, the first experiences with these implants were not very successful. High complication rates and unpredictable outcomes raised major concerns on this innovative type of reconstruction. The rarity of the indication is one of the main reasons why there has been a relatively slow learning curve and implant development regarding this type of prosthesis. This PhD thesis, gives an overview of the introduction, the development, the current standards, and the future perspectives of expandable prostheses for the reconstruction of the distal femur in children.

Incompressible limit and well-posedness of PDE models of tissue growth

Both compressible and incompressible porous medium models are used in the literature to describe the mechanical aspects of living tissues. Using a stiff pressure law, it is possible to build a link between these two different representations. In the incompressible limit, compressible models generate free boundary problems where saturation holds in the moving domain. Our work aims at investigating the stiff pressure limit of reaction-advection-porous medium equations motivated by tumor development. Our first study concerns the analysis and numerical simulation of a model including the effect of nutrients. A coupled system of equations describes the cell density and the nutrient concentration and the derivation of the pressure equation in the stiff limit was an open problem for which the strong compactness of the pressure gradient is needed. To establish it, we use two new ideas: an L3-version of the celebrated Aronson-Bénilan estimate, and a sharp uniform L4-bound on the pressure gradient. We further investigate the sharpness of this bound through a finite difference upwind scheme, which we prove to be stable and asymptotic preserving. Our second study is centered around porous medium equations including convective effects. We are able to extend the techniques developed for the nutrient case, hence finding the complementarity relation on the limit pressure. Moreover, we provide an estimate of the convergence rate at the incompressible limit. Finally, we study a multi-species system. In particular, we account for phenotypic heterogeneity, including a structured variable into the problem. In this case, a cross-(degenerate)-diffusion system describes the evolution of the phenotypic distributions. Adapting methods recently developed in the context of two-species systems, we prove existence of weak solutions and we pass to the incompressible limit. Furthermore, we prove new regularity results on the total pressure, which is related to the total density by a power law of state.