An Inverse Finite Element u/p-Formulation to Predict the Unloaded State of In Vivo Biological Soft Tissues

Physically realistic patient-specific biomechanical modelling is of paramount importance for many medical applications, where the geometry of tissues or organs is usually constructed from in vivo images. However, it is common for such biological structures to correspond to a deformed state due to being under external loadings. This necessitates the determination of the stress distribution of the known deformed state through an inverse analysis approach. To achieve this, we propose here a generalised finite element displacement/pressure (u/p)-formulation for evaluating the unloaded configuration of in vivo biological soft tissues that exhibit quasi-incompressible behaviour under finite deformations. Validity and applicability of the proposed numerical framework to practical inverse analysis problems in biomechanics is demonstrated through various numerical examples. The corresponding simulations utilise in vivo measurements of patient-specific geometries derived from different medical imaging modalities, and include recovery of the pressure-free configuration of human aortas and the gravity-free shape of the female breast.