Biorheology of soft tissues1

To understand the physiological function of vital organs we must know the mechanical properties of the tissues. Experimental determination of the mechanical properties of living tissues has many difficulties, such as the small size, large deformation, active contraction, damage due to dissection, i naccessibility or non-existence of a “natural” state, and the necessity of keeping the specimens alive. In this paper, major features of the rheology of soft tissues obtained in our laboratory are summarized, and a mathematical description is offered to serve as a starting point for the analysis of the function of the organs. Almost all published rheological data on soft tissues were obtained in one-dimensional condition—simple elongation of a slender cylindrical body, strip-biaxial or homogeneous—biaxial tension of a membrane. Recently we have collected data on two-dimensional testing of the skin, and torsion of the mesentery. From these we propose the following stress (σij) -strain (eij) relation for such tissues as the skin, the mesentery, and the muscle in the passive state, when subjected to loading and unloading at a constant rate σij=Cijkl′ekl+Cijkleklexpamn (emn−emn (0) ) +b (J2−J2 (0) ) δij+pδi where J2=16 (e11−e22) 2+ (e22−e33) 2+ (e33−e11) 2+e122+e232+e312 is the second strain invarient; x1, x2, x3 are axes of orthotropic symmetry, Cijkl, Cijkl′ are orthotropic tensors of rank 4 familiar in the classical theory of elasticity, amn and b are constants which differ in loading from unloading (defined by whether ∂ (amnemn+bJ2) /∂t is positive or negative), but are only slightly dependent on the strain rate. This equation does not apply to highly structured tissues such as blood vessels or the lung. amn emn (0) and J2 (0) are the largest values of these strain invariants for which the formulas are expected to be applicable. The indexes range over 1, 2, 3. For membranes in plane stress the indexes range over 1, 2 and the p term should be deleted. The summation convention over a repeated index is used.