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Proceedings of IMECE2002
ASME International Mechanical Engineering Congress & Exposition
November 17–22, 2002, New Orleans, Louisiana
IMECE2002-32605
A KELVIN-VOIGHT FRACTIONAL DERIVATIVE MODEL FOR VISCOELASTIC
CHARACTERIZATION OF LIVER TISSUE
Lawrence S. Taylor (1), Amy L. Lerner (2), Deborah J. Rubens (2,3), and Kevin J. Parker (1, 2, 3)
(1) Electrical and Computer Engineering
University of Rochester
Rochester, NY 14627
[email protected]
(2) Biomedical Engineering
University of Rochester
Rochester, NY 14627
(3) Department of Radiology
University of Rochester
Rochester, NY 14624
INTRODUCTION
There has been interest in the mechanical properties of the non-load
bearing soft body tissues (brain, liver, prostate etc.) in recent years.
The motivation comes from three areas: characterizing tissue response
to crash injuries [1], modeling for robotic surgical devices [2] and
elastographic diagnosis of disease processes using ultrasound [3].
dashpot is equal to the fractional derivative of order α of the strain.
Figure 1 shows the model. Eo refers to the spring elastic constant, η
refers to the dashpot parameter, σ is stress and α refers to the order of
the fractional derivative. The differential equation for the KVFD
model is:
Spring and dashpot models are useful in viscoelastic characterization
of materials because of their simplicity and ease of use. It is widely
believed that the three parameter standard linear solid (a KelvinVoight model in series with a spring) is the simplest of the springdashpot models that has both a creep and stress relaxation response
which resemble real materials. Caputo [4] introduced the fractional
calculus into the field of viscoelasticity in 1967 when he generalized
the Kelvin-Voight model by introducing a derivative of real order, the
so called fractional derivative, into the relation between stress and
strain in the viscous element. The creep compliance and stress
relaxation responses of the Kelvin-Voight fractional derivative
(KVFD) model are presented and it is shown that both functions are
realistic responses. These functions are used to curve fit experimental
liver data .
σ (t ) = E0ε (t ) + ηDα ε (t )
where ε is strain. In the standard Kelvin-Voight model α = 1, and the
stress relaxation is equal to a delta function at time zero with a
constant response after. No real material follows this function. When
0 < α < 1 in the KVFD model the stress relaxation has the form t-α ,
where t is time. The creep compliance, J(t), and stress relaxation, G(t),
functions, for the KVFD model are [5]:
  E0 t  α 
1 
J (t ) =
 
1 − Eα −
E0 
  η  
where the Eα , is the Mittag-Leffler function given by:
FRACTIONAL DERIVATIVE
The idea of the fractional derivative was first explored by
mathematicians in the 19th century, who started by recognizing that
differentiation and integration are inverse processes. The formula for
the fractional derivative of order α of f(t) is:
Dα f (t ) =
(n)
t f
1
(τ )dτ
∫
0
Γ (α − n) (t − τ )α +1− n
xn
n =1 Γ (αn + 1)
∞
Eα ( x ) = 1 + ∑
and :
n −1< α < n
KELVIN-VOIGHT MODEL
The KVFD model is a generalization of the Kelvin-Voight model
where the stress in the dashpot is equal to the first derivative with
respect to time of the strain. In the KVFD model the stress in the
G(t ) = E0 + η
1
(1 − α )t −α
Γ (2 − α )
Copyright  2002 by ASME
LIVER DATA
Adult bovine liver samples were tested using unconfined uniaxial
compression. Strain levels of 15% where used for the stress relaxation
tests. 0.3 N was applied for the creep testing. Fig. 2 shows typical
experimental data for the creep compliance of liver along with the
theoretical curve fit for the KVFD model. Fig. 3 shows a typical curve
for stress relaxation with the theoretical curve from the model.
DISCUSSION
The Kelvin-Voight fractional derivative model has not been widely
used in the biomechanics community. Like the standard linear solid
(SLS) it is a three parameter model. However, it differs from the SLS.
First, the creep compliance of the SLS has an instantaneous
discontinuous response a time zero while the KVFD has a gradual
continuous response. Second, the stress relaxation of the SLS model is
a decaying exponential while the KVFD model varies as t-α. Finally
the frequency response of the complex Young's modulus in the KVFD
model has the dependence ωα where ω, refers to radian frequency.
This function is monotonically increasing, which is not the case for the
SLS. This is of interest because measurements of the velocity of shear
wave propagation in liver at discrete values from 40 Hz [3] to 14 MHz
[6] indicate that shear velocity is a monotonically increasing function.
Fig 1. Kelvin-Voight fractional derivative
model
The strain levels in our testing were quite large so it is likely that the
data are non-linear. The curve fit for the creep compliance and the
stress relaxation required very different model parameters further
suggesting the data was non-linear.
SUMMARY AND CONCLUSIONS
We have applied the Kelvin-Voight fractional derivative viscoelastic
model to experimental data on bovine liver. We argue that this
model should be considered when using spring and dashpot type
viscoelastic models for modeling soft tissue viscoelasticity.
REFERENCES
1 . Thibault, K. L., and Margulies, S. S., 1998, "Age-dependent
material properties of the porcine cerebrum: effect on pediatric
inertial head injury criteria," J of Biomech, Vol. 31, 1119-1126.
2. Miller, K., et al., 1997 "Constitutive modeling of brain tissue:
experiment and theory," J of Biomech, Vol. 30, 1115-1121.
3 . Sanada, S., et al. 2000. "Clinical evaluation of sonoelasticity
measurement in liver using ultrasonic imaging of internal forced
low-frequency vibration," Ultrasound in Med. & Biol., Vol. 26,
1455-1460.
4 . Caputo, M., 1967, "Linear models of dissipation whose Q is
almost frequency independent-II," Geophys. J. R. Astr. Soc.,
Vol. 13, 529-539.
5. Koeller, R., 1984 "Applications of the fractional calculus to the
theory of viscoelasticity" Journal of Applied Mechanics, Vol. 51,
299-307
6. Sarvazyan, A. P., et al., 1995 "Biophysical bases of elasticity
imaging." In Acoustical Imaging. Vol 21. Proceedings of the 16th
International Symposium, 223-240. Plenum Press.
Fig. 2 Creep Compliance
Fig. 3. Stress relaxation of a bovine liver sample
2
Copyright  2002 by ASME