A Model of Electromechanical Coupling in the Small Intestine - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 180

(a) (b)

x

z

y

(c)

ξ

3

ξ 1

Fig. 1 Anatomical models of the small intestine. a An anatomical model constructed from the
Visible Human Project using the cubic Hermite basis functions [ 37 ]. b A more structurally
detailed linear model with idealized representation of the inner circular (brown) and outer
longitudinal muscle layers (light brown). The green lines represent the orientations of the smooth
muscle fibers in each geometric element. c A end-view of the linear intestinal anatomical model

idealized intestinal model was represented using the Cartesian coordinates (x, y, z)
as the global coordinate system. Alternative coordinate systems such as the polar
cylindrical coordinate system could also be used in this case. A local coordinate
system (n 1 ; n 2 ; n 3 ) is usually used in the finite element method to represent the
geometry within each local element. The local geometric coordinates were inter-
polated over the element using linear Lagrange basis functions in the circumfer-
ential (n 1 ) and transmural (n 3 ) directions, and a cubic Hermite basis function in the
longitudinal (n 2 ) direction. This basis functions scheme represented a balance
between computational efficiency and accuracy, since the physical dimension in
the transmural and circumferential directions was smaller than the geometry in the
longitudinal direction.
The wall of the intestine anatomical model was further divided into two smooth
muscle layers of equal thickness, to represent the inner circular and outer longi-
tudinal muscle layers (Fig. 1 b). To represent the different alignment of the muscle
fibers in the longitudinal and circular directions, a fiber coordinate system was
prescribed at each node, and subsequently interpolated over the whole element, in
this case using a trilinear basis function scheme. Note the other anatomical layer of
the small intestine, e.g., the submucosal layer, was not explicitly defined in this
initial model, because they do not contribute significantly to the active mechanical
properties of the intestine.
Anisotropic material properties, such as electrical conductivities and the
parameters for the constitutive equations, were specified along material axes which
are referred to the fiber directions. These axes specify three directions: (i) 'fiber',
which was aligned with the direction of the fibers; (ii) ''sheet'', which was per-
pendicular to the muscle fiber, but parallel to the plane of the muscle layer; and