Order Reduction of a Nonlinear Convection-Dominated Model of Heart Valve Tissue Maturation

The mechanical strength of tissue-engineered heart valves is defined by complex dynamics of the corresponding production process. During the so-called maturation process, the tissue develops in a living implant consisting of patient-own components such as cells and proteins, whose concentrations are dependent on space and time. In this work, a distributed-parametric model of the maturation process is described and a corresponding model order reduction based on the POD-Galerkin method is implemented and evaluated. Due to occurring tissue contraction represented by convectional terms, classic projection-based model reductions pose challenges. Hence, the model reduction is implemented in a Lagrangian frame of reference whereas the spatial grid points move according to the moving steep gradients of the solution. The results indicate an accurate and efficient low-order approximation of the model solution. Corresponding root-mean-square errors vary between 0.017 % and 0.066 %. Consequently, common methods such as control design and parameter estimation can be applied to the resulting reduced order model instead of the expensive high-dimensional approximation.