Arteries: Mechanics, Mechanobiology, and the Need for a New Class of Models - Multiscale Modeling in Biomechanics and Mechanobiology - page 203

blood flow; internal tears or fissures can lead to life threatening blood clots whereas
dissections and transmural ruptures can lead to life-threatening bleeding. Second,
because vascular cells are mechanosensitive, understanding the mechanical stimuli
to which they are subjected is fundamental to understanding many of their biological
responses, including migration, proliferation, differentiation, and apoptosis. Indeed,
for this reason, the mechanics is fundamental to arterial development, homeostasis,
adaptation, disease progression, and response to injury [ 25 ].

9.3 Constitutive Relations—A Historical Perspective

As noted many years ago by Y.C. Fung, one of the most important needs in biome-
chanics is identification of nonlinear constitutive equations for tissues that experience
multiaxial loads in vivo (cf. [ 11 ]). Arteries are prototypical of such tissues.
Phenomenological Relations . The first class of stress-strain (constitutive)
relations developed for arteries can be classified as phenomenological, that is, rela-
tions that describe observed responses primarily in pressure-diameter and axial force-
length tests. Because of the ubiquitous nonlinear material behavior exhibited by
arteries over finite deformations, an important advance realized in the late 1960s
and the early 1970s was the use of the theory of finite elasticity to quantify arterial
behavior. Early contributors in this regard included H. Demiray, Y.C. Fung, I. Mirsky,
B. Simon, R. Vaishnav, and R. Vito, among others (cf. [ 22 ]). Briefly, in each case,
these investigators sought to identify a strain energy function W that depended on
an appropriate measure of the finite deformation, as, for example, the right Cauchy-
Green tensor C or the Green strain tensor E . That is, given a functional form for
W ( C )or W ( E ), one could then compute the associated stress via established, general
(e.g., based on the entropy inequality) constitutive relations, namely

det F F ∂

2 W

∂ C F T

t =

F T F and E where F is the deformation gradient tensor (with C .

One of the most popular phenomenological relations is that due to Fung, which can
be written as W

= = ( C − I )/ 2 )

e Q where Q is typically written as a quadratic function

of Green strain (note: by using a quadratic form, one can exploit material symmetry
arguments from linearized elasticity, which is recovered directly in the limit as the
strain becomes small). Although such relations can provide excellent estimates of the
transmural distribution of wall stress, and indeed were fundamental in the discovery
of the importance of residual stresses in arteries [ 7 ], values of the best-fit material
parameters (typically determined via nonlinear regression) do not have any physical
meaning, which limits the interpretive value when comparing relations as a function
of species, genotype, arterial location, age, adaptation, or disease status.
Structurally Motivated Relations . For this reason, there was further motivation to
pursue structurally motivated constitutive relations. Toward this end, there have been

= c ( − 1 )