A Model of Electromechanical Coupling in the Small Intestine - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 190

4.2 Modeling Passive Tension

Passive tension is an intrinsic material property and refers to the stiffness in the
material that resists deformation. In smooth muscle tissue, this stiffness is largely
due to inelastic collagen fibers, which prevent further deformation at high wall
tensions [ 26 ]. It is important to include the effects of passive tension in a
mechanics model of the intestine, because the ease with which a segment of the
intestine is actively distended depends on passive stress, which in turn influences
the flow and mixing within the intestinal lumen [ 26 ].
Specifying the passive behavior of smooth muscle tissue in response to stress
requires two important specifications. These are the compressibility constraint and
the constitutive behavior of the tissue. The intestinal tissue can be assumed to be
incompressible, as smooth muscles consist mainly of incompressible fluids. Pop-
ular constitutive laws used to model isotropic material behavior include the
Neo-Hookean and Mooney-Rivlin laws. However, the Mooney-Rivlin relation-
ship may not be appropriate to model the intestinal tissue because the layered
muscle structure, i.e., a hyperelastic and anisotropic constitutive law may be
required and large strains [ 6 ]. Another important aspect of constitutive law
modeling is viscoelasticity; although, viscoelasticity is challenging to model
because of the time-dependency. However, an important assumption can be made
in the case of smooth muscle tissue. Following a period of pre-stretching, after the
muscle undergoes repeated loading-unloading cycles, the stress-strain relationship
becomes relatively independent of strain rate; the response becomes constant and
predictable. Fung termed this approximation ''pseudoelasticity'' [ 24 ]. The exis-
tence of a unique stress-strain relationship means that the tissue has an associated
strain energy function, and can therefore be modeled as a hyperelastic material
[ 24 ]. In general, the stress can be related to the strain energy function (W) and the
strain (E) tensor of the hyperelastic material,

T ¼ o W ð E Þ
oE

ð 23 Þ

:

The general form of the strain energy function can be modeled using many dif-
ferent mathematical formulations, each with parameters that can be adjusted to fit
data from material tests and yield the constitutive equations for the tissue. One
such constitutive law is the Fung-type model, based on a previous biaxial study of
intestinal tissue by Bellini et al. [ 6 ],

W ¼ C
2

e Q 1

ð 24 Þ

where,

Q ¼ a 1 E 11 þ a 2 E 22 þ a 3 E 11 E 22 ; ð 25 Þ