Adaptive Quasi-Linear Viscoelastic Modeling - Computational Modeling in Tissue Engineering - page 52

The Adaptive QLV model can be written in generalized form, as with the
Generalized Fung QLV model, by assuming different nonlinear behavior for dif-
ferent shape functions:

r ð t Þ¼ r o ð e ð t ÞÞ þ X

M

k i ð e ð t ÞÞ V ð e Þ ð t Þ

i

i ¼ 1

ð 9 Þ

ð t Þ¼ Z

t

g i ð t s Þ de ð s Þ
ds

V ð e Þ
i

ds ; i ¼ 1 ; 2 ; ... ; M

1

where r o ð e Þ is a pure function of strain representing the long-term elastic part of
the response. Each g i (t) could be any relaxation function such that g i (0) = 1 and
g i (?) = 0. g i (t) is termed a shape function rather than a reduced relaxation
function because the relaxation curves are a weighted summation of g i (t) func-
tions. Usually the shape functions are chosen to be exponential terms (i.e.,
g i ð t Þ¼ e t = s i ).
The main advantage of the Adaptive QLV model is simplicity of calibration.
The Adaptive QLV model involves a convolution of the relaxation function with
the strain function rather than with the elastic stress as in the Fung QLV model.
The strain function is prescribed, and can be chosen to be any function. In contrast,
the elastic stress depends on the mechanical characteristics of the tissue and needs
to be calibrated along with the relaxation function. Calibrating only one unknown
function (the relaxation function) through the convolution integral is far simpler
than calibrating two unknown functions through the convolution integral. For a
great many displacement inputs, the convolution integral (Eq. ( 9 )) of the Adaptive
QLV model can be found in closed analytical form. Some particularly useful
examples of these are described in the next section.

2.3.1 Representation of the Adaptive QLV Model with Maxwell Elements

When the shape functions g i (t) are represented by exponential terms, the Adaptive
QLV model can be represented in terms of parallel Maxwell elements, although
such elements need not exist physically [ 55 ]. Viewing the Adaptive QLV model in
this way allows for physical insight into the underlying principles, and is thus the
focus of this section. These elements involve a set of nonlinear springs and
dashpots where the spring stiffnesses and dashpot coefficients are functions of
overall tissue strain and not their individual strain (Fig. 2 ). For each element i:

(

þ V ð e Þ

V ð e Þ
i

s i ð e Þ ¼ e
r i ¼ k i ð e ð t ÞÞ V ð e Þ
i

i

ð 10 Þ

ð t Þ

where: