Physics-based manifold learning in scaffolds for tissue engineering: Application to inverse problems

In the field of bone regeneration, insertion of scaffolds favours bone formation by triggering the differentiation of mesenchymal cells into osteoblasts. The presence of Calcium ions ( Ca 2+ ) in the interstitial fluid across scaffolds is thought to play a relevant role in the process. In particular, the Ca 2+ patterns can be used as an indicator of where to expect bone formation. In this work, we analyse the inverse problem for these distribution patterns, using an advection-diffusion nonlinear model for the concentration of Ca 2+ . That is, given a set of observables which are related to the amount of expected bone formation, we aim at determining the values of the parameters that best fit the data. The problem is solved in a realistic 3D-printed structured scaffold for two uncertain parameters: the amplitude of the velocity of the interstitial fluid and the ionic release rate from the scaffold. The minimization in the inverse problem requires multiple evaluations of the nonlinear model. The computational cost is alleviated by the combination of standard Proper Orthogonal Decomposition (POD), to reduce the number of degrees of freedom, with an adhoc hyper-reduction strategy, which avoids the assembly of a full-order system at every iteration of the Newton’s method. The proposed hyper-reduction method is formulated using the Principal Component Analysis (PCA) decomposition of suitable training sets, devised from the weak form of the problem. In the numerical tests, the hyper-reduced formulation leads to accurate results with a significant reduction of the computational demands with respect to standard POD.