Adaptive Quasi-Linear Viscoelastic Modeling - Computational Modeling in Tissue Engineering - page 57

r o De ðÞ¼ R n t !1 ð Þ: ð 26 Þ

Next, the values of k i can be calibrated for each strain level tested. If the intent
is to model the viscoelastic properties of a tissue using some predetermined shape
functions g i , then for each relaxation test the values of k i are chosen to minimize I n
defined as:

I n ¼ Z

þ1

Þ 2 dt ð

R n ðÞ r n ðÞ n ¼ 1 ; 2 ; ... ; N :

ð 27 Þ

1

where r n (t) is the model prediction of the relaxation stress using Eq. ( 25 ). To
increase accuracy, the shape functions may instead be expressed in terms of
unknown parameters that can also be calibrated. For example, we may assume
M = 3 shape functions of the form g i ðÞ¼ exp ð t = s i Þ in which the three time
constants s 1 , s 2 and s 3 are unknown parameters that should be calibrated to
experimental data. Here, because the model uses the same shape functions to
model all relaxation tests, all of the relaxation curves should be calibrated at once.
The calibration procedure is then to determine all k i values and the parameters of
the shape functions (e.g., s 1 , s 2 and s 3 ) by minimizing I defined as:

Z 2

þ1

I ¼ X

N

R n ðÞ r n ðÞ
R n ðÞ

d : ð 28 Þ

n ¼ 1

1

With the values of the functions k i (e) calibrated at several strain levels De n ,an
appropriate interpolation method may be used to determine these functions for all
values of strain.

3.1.2 Generalized Fung QLV Model

Substituting for strain from Eq. ( 23 ) into the Fung QLV model (Eq. ( 2 )) or the
Generalized Fung QLV model (Eq. ( 7 )), the model prediction for stress in the nth
relaxation test is of the following form:

Z

t

r n ðÞ¼ r o e ð t ðÞþ X
i

g i t s ð Þ A i e ð s Þ ð Þ De n d ðÞ ds

ð 29 Þ

1

¼ r o De ðÞþ X
i

A i De ð De n g i ðÞ

In the Fung and Generalized Fung QLV models, the nonlinear functions are
determined for the tissue strain at the beginning of the test (time t = 0). However,
at the beginning of the test the strain is discontinuous. In deriving Eqs. ( 29 ), the
strain at time zero is assumed to be the strain at time 0 + (the final strain) rather than