Adaptive Quasi-Linear Viscoelastic Modeling - Computational Modeling in Tissue Engineering - page 49

where r is the uniaxial stress and e is the corresponding strain. G(t) is the unit step
relaxation function, which is the recorded stress in response to a unit step elon-
gation of the tissue. Qualitatively, G(t-s) represents the diminishing effect of the
strain state at a time s before the current time, t, on the current stress, r(t).
The Fung QLV model incorporates nonlinearity into Eq. ( 1 ) by replacing strain
with a nonlinear function of strain. The stress state is then calculated by the
following convolution integral:

r ðÞ¼ Z

t

dr ð e Þ

ð e ð s Þ Þ

gt s ð Þ ds

ds

1

ð 2 Þ

¼ Z

t

dr ð e Þ ðÞ
de

de ðÞ
ds

gt s ð Þ ds

1

where g(t) is called the ''reduced'' relaxation function, which is the unit step
relaxation function normalized by its initial value (so that g(0) = 1), and r ðÞ ðÞ is
a function of strain called the ''elastic stress.'' In this model, the nonlinearity is
included within the elastic tangent stiffness term, dr ð e Þ = de.
The ''reduced'' relaxation function g(t) represents the shape of the normalized
unit step relaxation curve, which is the decay in isometric stress level following an
instantaneous stretch. For different tissues and depending on the shape of the
experimentally recorded reduced relaxation functions, g(t) may be an exponential,
polynomial, logarithmic or any other mathematical function. A common choice of
g(t) is the sum of several exponential terms:

g ðÞ¼ a o þ X

M

a i e t = s i ; ð 3 Þ

i ¼ 1

where each of the M exponential terms (each time constant s i and corresponding
amplitude a i ) can be associated with a series combination of a spring and a dashpot
(a Maxwell element), and a o can be associated with a spring.
Fung [ 39 ] also observed that many biological tissues are well modeled by a
reduced relaxation function with a continuous spectrum of time constants:
g ð t Þ¼ 1 þ R 1

0 S ð s Þ e t = s ds

1 þ R 1
0

; ð 4 Þ

S ð s Þ ds

where S(s) dictates the distribution of time constants. As did Neubert in his earlier
model of rubber viscoelasticity [ 62 ], Fung adopted the form S(s) = c/s, where c is
a constant; this provides g(t) with a uniform spectrum of time constants, as is
appropriate for many tissues that exhibit a logarithmic decay in isometric force
following a stretch [ 22 ].