With the recent developments of new technologies in biomedical engineering and medicine, the need for new mathematical and numerical methdologies to aid these developments has never been greater. In particular, the design of next generation drug-eluting stents for the treatment of coronary artery disease, and the design of an implantable bioartificial pancreas for the treatment of Type 1 diabetes, hinge on the development of mathematical and computational techniques for solving nonlinear moving boundary problems. In this talk we present recent developments and open problems in the mathematical theory of nonlinear moving boundary problems describing the interaction between viscous, incompressible fluids, and elastic and poroelastic structures. Applications of the mathematical results to coronary angioplasty with stenting, and to bioartificial pancreas design will be shown.

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