Adaptive Quasi-Linear Viscoelastic Modeling - Computational Modeling in Tissue Engineering - page 62

of these functions. Sophisticated optimization algorithms may be used to reduce
the number of trials by identifying and subsequently choosing functions based on
previous trails, and indeed several functional choices of g(t) and r (e) (e) are com-
mon in the literature [ 52 , 53 , 65 , 66 ]. The optimization procedure requires com-
puting the convolution integrals of Eq. ( 41 ) for numerous choices of g(t) and
r (e) (e) functions, optimizing the prediction of Eq. ( 41 ) to find the parameters of
these functions. Additional optimization is required for choosing amongst the fits
to find optimum functions and parameters.

3.4 Incremental Ramp-and-Hold Protocol

Although a single, large amplitude ramp-and-hold test is a simple and fast protocol
for calibrating a QLV model to a biological or bio-artificial tissue, the fit that
results may be oversimplified. The drawback is that a single, large amplitude
ramp-and-hold test might not contain sufficient information to characterize the way
that the relaxation function evolves with tissue length: such a protocol is appro-
priate only if a tissue is well-modeled by a single reduced relaxation function that
is valid for all tissue lengths. The incremental ramp-and-hold protocol (Fig. 3 d) is
an

alternative that generates more information than either the single large

amplitude ramp-and-hold protocol or the incremental relaxation protocol.
The incremental ramp-and-hold protocol is similar to the incremental relaxation
protocol, except that the stepwise stretches are replaced by ramp stretches. The
strain function for the nth ramp-and-hold test can be written:

8
< ð n 1 Þ De ; t\0

Þ De þ D T e n ð t Þ¼ ð n 1 t ; 0\t\T ð 42 Þ

:

nDe ; t [ T

where T is the duration of ramp loading in each increment. We denote the
experimentally recorded stress during the nth ramp-loading phase (0 B t B T)as
P n (t), and that recorded during nth hold-relaxation phase (t C T) as H n (t).

3.4.1 Adaptive QLV Model

Substituting the strain from Eq. ( 42 ) into Eq. ( 9 ), the viscoelastic strains in the nth
relaxation test are:

8
<

R

t

Þ D T ds ¼ D T c i ð t Þ;

g i t s ð 0\t\T

V ð e Þ
ni

0

ðÞ¼ ð 43 Þ

R

T

:

g i t s ð Þ D T ds ¼ D T ð c i ð t Þ c i ð t T Þ Þ; t [ T

0