A complementary energy approach accommodates scale differences in soft tissues

Abstract

The mechanics of biological entities, from single molecules to the whole organ, has been extensively analyzed during the last decades. At the smaller scales, statistical mechanics has fostered successful physical models of proteins and molecules, which have been later incorporated within constitutive models of rubber-like materials and biological tissues. At the macroscopic scale, the additive decomposition of energy functions i.e., a parallel arrangement of the tissue constituent, has been recurrently used to account for the internal heterogeneity of soft biological materials. However, it has not yet been possible to unite the mechanics at the tissue level with the actual response of the tissue components. Here, we exemplify our approach using cardiovascular tissue where the mechanical response at the tissue scale is in the range of kPa whereas the elastic modulus of collagen, the main component of the vascular tissue, is in the range of MPa GPa. In this work we develop a novel theoretical framework based on a complementary strain energy function that builds up on a full network model. The complementary strain energy function introduces naturally an additive decomposition of the deformation gradient for the tissue constituents, i.e an arrangement in series of the constituents. We demonstrate that the macroscopic response of the tissue can be reproduced by just introducing the underlying mechanical and structural features of the micro-constituents, improving in a fundamental manner previous attempts in the mechanical characterization of soft biological tissues. The proposed theoretical framework unveils a new direction in the mechanical modeling of soft tissues and biological networks.