Identification of residual stresses in multi-layered arterial wall tissues using a variational framework

Abstract

In the past decades a considerable amount of literature has been published addressing the study of the mechanical behavior of arterial walls. Ex-vivo experimentation made possible the development of constitutive models and the characterization of material parameters contributing to the understanding of the mechanobiological response of vascular tissues. Moreover, the existence of residual stresses in configurations free of load was revealed, and its impact in the general stress state of the tissue was quantified. In recent years, data assimilation techniques have seen a rapid development in cardiovascular modeling field, primarily focusing on the estimation of material parameters for arterial wall segments using information provided by medical imaging as well as by in-vitro settings. However, concerning the estimation of residual stresses, this research field is in its early stages, and much work is still required for the full functional characterization of arterial tissues. In this context, a conceptual variational framework for the development of residual stress estimation tools is proposed. Particularly, a variational formulation for the characterization of residual deformations and the associated stresses in arterial walls, based on full displacement field measurements of the vessel, is presented. Considering as known data the material parameters characterizing the behavior of the tissue and a set of arterial wall configurations at equilibrium with well defined pressure loads, we propose a cost functional that measures the mechanical imbalance caused by the lack of knowledge of residual stresses. In this manner, the characterization of residual stresses becomes a problem of minimizing such cost functional. Three numerical examples are presented highlighting the potential of the proposed approach. Among these examples, the characterization of residual stresses in a cylindrical geometry representing a three-layered aorta artery is performed.