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Shape design of a polymer microstructure for bones Dondl, Patrick


We consider a shape optimization problem related to the design of polymer scaffolds for bone tissue engineering. Globally, bone loss due to trauma, osteoporosis, or osteosarcoma comprises a major reason for disability. To this day, autograft, i.e., a graft of bone tissue from a different place in the same body, remains the gold standard for large scale bone loss. This is despite major issues, for example donor site morbidity and limited availability. An ideal scaffold to be implanted in place of lost bone tissue must satisfy a number of different criteria, apart from the requirement of biocompatibility. It should be bioresorbable, so that no foreign objects remain after the regeneration time. In particular, however, during the regeneration time, it should provide adequate mechanical stability, while not preventing osteogenesis. Inspired by this biomechanical challenge, we consider the following shape optimization problem. In a periodic setting, by means of homogenization, one can obtain the effective elastic modulus for a given structure of a linearly elastic material occupying a set $E\subset \Omega=[0,1]^3$ under a certain loading condition. The objective is to find a set $E_\text{opt}$ (the occupied scaffold volume in a unit cell) such that the minimum of the effective elastic modulus for $E_\text{opt}$ and of the effective elastic modulus of the complement of the set $E^c_\text{opt}$ (i.e., of the regenerated bone tissue) is maximized. Joint work with Martin Rumpf and Stefan Simon (both Bonn).

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