EXERCISE AND QUALITY OF LIFE
Research article
Volume 5, No. 1, 2013, 6167
UDC 517:616.75808
612.76
The application of fractional Zener model
on MCL
Nada Santrač and Jasmina Pavkov*,
Faculty of Medicine
University of Novi Sad, Serbia
nadasantrac@hotmail.com
Abstract
On the basis of recently published experimental results of Abramowitch et al. (2004), dealing
with viscoelastic properties of sham operated and healing MCL, we show that the fractional Zener
model of viscoelastic body seems to be very tractable tool for rheological description of different
states of biological tissues. In this paper medial collateral ligament of the knee will be described
by means of the method of Dankuc et al. 2010, who examined middle ear structures and rampand
hold stress relaxation experiments. Two different states of MCL were described by four different
constants representing modulus of elasticity, the order of fractional derivative and two relaxation
constants. Predictions of the model are in good agreement with the experimental results.
Keywords: medial collateral ligament biomechanics, rampandhold stress relaxation
Introduction
Injuries of knee ligaments are very common. Some estimations showed that the incidence
could be 2/1000 people every year in the general population and for those involved in sports it has
a much higher rate (Woo, Abramovitch, Kilger, & Liang, 2006).
About 90% of knee ligament injuries involve the anterior cruciate ligament (ACL) and the
medial collateral ligament (MCL) /Woo et al., 2006/. After rupture MCL can heal spontaneously.
However, laboratory studies have shown that healing MCL ligament substance has the mechanical
properties which remain considerably inferior to those of normal ligaments even after years of
remodeling (Abramowitch, Woo, Clineff, & Debski, 2004). After a few weeks of immobilization
of rabbit hind limbs decreases in the structural properties of the femurMCLtibia complex were
* Corresponding author. Faculty of Medicine, University of Novi Sad, Hajduk Veljkova 3, 21000 Novi Sad, Serbia,
email: jasmina.pavkov.ns@gmail.com
© 2013 Faculty of Sport and Physical Education, University of Novi Sad, Serbia
N. Santrač & J. Pavkov
observed. 9 weeks of immobilization required up to one year of remobilization to reverse those
changes and to return ligament properties to normal levels.
Water constitutes about 65 to 70% of ligament’s total weight. The major constituent which
is primarily responsible for ligament’s tensile strength is Type I collagen. Another fibrous protein
found in ligaments is elastin and its role is to increase flexibility. Other major components are
collagen Type III and Type V (Woo et al., 2006).
Ligaments are threedimensional anisotropic structures. Also, ligament substance have
properties of solid materials as well as fluid materials thus is denoted as viscoelastic material.The
complex interaction of collagen with elastin, proteoglycans, ground substance and water results in
the time and historydependent viscoelastic behaviors of ligaments (Woo et al., 2006; Grahovac
and Zigic, 2010).
There are such medical conditions which can change soft tissue remodeling due to
degenerative processes, infectious or mechanic lesions. Those conditions have huge impact to the
tissue function and adaptive response to the traumatic conditions (Dankuc, Kovincic, & Spasic,
2010).
Due the interaction between adaptation of the tissue and the mechanical condition within
tissue is very complex, for studying this interrelation mathematical models are needed. These models
are mainly studied within biomechanics. One of the first definitions of biomechanics  mechanics
applied to biology  should be replaced with the one that highlights development, extension and
application of mechanics for better understanding of physiology and pathophysiology, but also the
diagnosis and treatment of disease and injury (Dankuc et al., 2010).
Ligaments exhibit viscoelastic properties that depend on their state. Bodies that show
creep, stressrelaxation, and hysteresis like behavior in force displacement diagram and memory
effects are called viscoelastic bodies. Therefore viscoelastic properties of the ligaments can be
described by nonlinear models, the standard linear viscoelastic model (so called the Zener model),
or the standard fractional viscoelastic model called the fractional Zener model (see Spasic &
Charalambakis, 2002). By use of the lastone we shall describe viscoelastic properties of MCL.
Namely, the fractional Zener model is able to predict behavior of the viscoelastic material with
significant accuracy, including only four parameters, which is the main advantage when compared
to other possibilities, no matter how complex material is considered.
Figure 1 shows the system under consideration and the position of the ligament to be
modeled.
Figure 1. System under consideration
62
The application of fractional Zener model on MCL
We recognize the knee structure where PL stands for patellar ligament, MCL  medial
collateral ligament, MM  medial meniscus, ACL  anterior cruciate ligament, PCL  posterior
cruciate ligament, LM  lateral meniscus, LCL  lateral collateral ligament.
In what follows we shall put the results of the rampandhold strain type of stressrelaxation
experiments of Abramowitch et al. (2004), in the framework of fractional model suggested by
Dankuc et al. (2010).
The model (methods)
For a body in a deformed state, the Hooke law states that stress (force per unit area) and
strain can be related by
v=
E
f,
(1)
where σ stands for the stress at time t, E > 0 is the constant called modulus of elasticity,
and ε stands for the strain at time t. The standard viscoelastic body or the Zener model, states the
stressstrain relation in the following form:
^1h
^1
v+xv1
v
= f+E
1
xf
1
f
^{h}.
(2)
with xv1 as a constant known as stress relaxation time, E 1 is modulus of elasticity and xf1 denotes
^1h
d
strain relaxation time constant. Here
^: h
=
^: h
stands for the first derivative with respect
dt
to time t. The second law of thermodynamics and the stability conditions implies that in (2) the
following restrictions on the constants must be satisfied
E1 > 0,
xv1 > 0,
xf1 > xv1
(3)
In order to describe specific class of viscoelastic materials we introduce α−the derivative,
with 0 < α <1, of a function u(t) in the RiemannLiouville form, and replace the first derivative in
(2) by the one of order α, say:
a
t
d
^ah
d
1
u^phdp
a
u^th
=
u
/
#
a
,
dt
dt
C^1

ah
0
^t

ph
where Γ is the Euler gamma function. In such a way we obtain the fractional derivative type
generalization of (2), usually called a fractional Zener body, (see Bagely & Torvik, 1986):
^ah
^a
v + xvav
=Eaf+Eaxfaf
^{h},
(4)
where 0 < α < 1, and xva , xfa and Ea are constants. The dimension of xva and xfa is time to the
power of α.
The constitutive equation (4) describes uniaxial, isothermal deformation of the viscoelastic
body of negligible mass, together with fundamental restrictions on the coefficients of the model,
that follow from the ClausiusDuhem inequality (Bagely & Torvik, 1986; Atanackovic, 2002):
Ea > 0, xva > 0, xfa > xva .
(5)
63
N. Santrač & J. Pavkov
For examining the rampandhold strain then stress relaxation experiment, the preparation
should be done by applying the Laplace transform, see Dankuc et al. (2010), to the following strain
function
lt,for
0
#
t
#
t
k
,
f^th
=
*
y
(6)
f
0
=
lt
k
=
const.,for
t
$
t
k
where κ = const. stands for the initial strain rate, and t_{k}, represent the time instant when the
prescribed strain is achieved.
Following the lines of Dankuc et al. (2010.) one gets the following form for the stress in
the viscoelastic tissue after applying strain as in eq. (6):
Z
t
]
Ealt
+
nl
#
e
a^
p,mhdp,for
0
<
t
#
t
k
]
]
0
v^th=
[
y
(7)
t
t t
k
]
]Ealt
k
+
nl=
#
e
a
^p,mhdp

#
e
a
^p,mhdpG, for
t
>
t
k
0
9
\
where ea ^t; mh stands for the MittagLeffler function, and where λ and μ denotes 1/xva and
/
Ea ^xfa
xva  1h respectively. The deformation pattern given in form (6), (7) is known as the real
stimulus, since almost all real materials exhibit that type of behavior, see Figure
2.
Figure 2.The rampandhold strain than stress relaxation deformation pattern.
It should be noted that for the values of parameters satisfying (5) the expression (7) exhibits
behavior shown in Figure 2.
64
The application of fractional Zener model on MCL
In the following section we present it for two sets of recent experimental data corresponding
to sham operated and healing MCL. From the stress relaxation curves several points were chosen
and then equation (7) was forced to pass through those points. Fitting procedure was performed by
the Newton method.
Results
In Abramowitch et al. (2004) study the rampandhold stress relaxation test was performed
on the healing goat MCL and shamoperated controls. Each femurMCLtibia complex specimen
was elongated from 0 to 3mm at 10mm/min and held constant for a period of 60min. Deformation
f0 = 1.7% for sham operated specimens and f0 = 1.8% for heeling ones, while tk = 18.4 s for
both.
The values of the four unknown constants a , E, xfa and xva which are describing
viscoelastic properties of the medial collateral ligament are computed by use of the suggested
numerical procedure. E and v are given in MPa, and time t is given in seconds.We present obtained
results for these cases  sham operated MCL:
a = 0.301, E = 453.188, xfa = 1.211,
1 439 $10 ^{3}
xva =
and healing MCL:
a = 0.345, E = 128.466, xfa = 2.883,
xva =
0 107
The agreement between the model and experimental results for sham operated MCL is
shown in Figure 3, where plots are shown only for t !60, 300@s .
Figure 3. Measured stress relaxation experiment data for sham operated MCL (Abramowitch et al,
2004)  dashed line with marks, and predicted using fractional Zener model  solid line.
The agreement between the model and experimental results for healing MCL is shown in
Figure 4.
65
N. Santrač & J. Pavkov
Figure 4. Measured stress relaxation experiment data for healing MCL (Abramowitch et al., 2004)
 dashed line with marks, and predicted  solid line.
From Figures 3 and 4 it can be observed that the difference between the experimental data
and the model is negligible.
Discussion
In this paper the rheological descriptions of the sham operated and healing MCL by
means of fractional calculus were introduced. The rampandhold stress relaxation experiment
performed for both sham operated and healing MCL were related with the constitutive equation
called fractional Zener model of viscoelastic body. First the Laplace transform was applied to that
constitutive equation and the strain function corresponding to ramp and hold behaviour. Then the
inversion procedure was performed. Finally the obtained solution was forced to pass trough four
selected experimental points.
We found good agreement between experimental results and theoretical predictions obtained
by use of the Laplace transform method applied to the fractional Zener model of viscoelastic body.
The important characteristic of the proposed model is that it is capable of predicting
behavior of viscoelastic materials with significant accuracy. Also, it has only four parameters
and it is simple. Four parameters could be determined for normal and pathological conditions of
different intensity. Knowing the obtained constants for two different states of MCL we may predict
their behavior for different load which can be done in future work. The same procedure can be
applied for different states of other structures within a human body as well as for corresponding
cyclic or impact loading what can be used in proper design of rehabilitation treatments.
Aknowledgements
Authors appreciate helpful discussions with Professor Zigic and Professor Spasic.
66
The application of fractional Zener model on MCL
References
Abramowitch, S. D., Woo, S. LY., Clineff, T. D., & Debski, R. E. (2004). An evaluation of the
quasilinear viscoelastic properties of the healing medial collateral ligament in a goat
model. Annals of Biomedical Engineering, 32(3), 329334.
Atanackovic, T. M. (2002). A modified Zener model of viscoelastic body. Continuum Mechanics
and Thermodynamics, 14, 137148.
Bagely, R. L. & Torvik, P. J. (1986). On the fractional calculus model of viscoelastic behavior.
Journal of Rheology, 30, 133155.
Dankuc, D. V., Kovincic, N. I., & Spasic, D. T. (2010). A new model for middle ear structures with
fractional type dissipation pattern. In: Proceedings of FDA’10. The 4^{th} IFAC Workshop on
Fractional Differentiation and its Applications, Badajoz, Spain, October 1820th, 2010.
Article No FDA10_156. Laxenburg: by International Federation of Automatic Control.
Gorenflo, R., & Mainardi, F. (2000). Fractional calculus: Integral and differential equations of
fractional order. In A. Carpinteri & F. Mainardi (Eds), Fractals and Fractional Calculus in
Continuum Mechanics (pp. 223276). Wien and New York: SpringerVerlag.
Grahovac, N. M., & Zigic, M. M. (2010). Modeling of the hamstring muscle group by use of
fractional derivatives. Computers and Mathematics with Applications, 59(5), 16951700.
Spasic, D. T., & Charalambakis, N. C. (2002). Forced vibrations with fractional type of dissipation.
In Proceedings of the International Conference on Nonsmooth/Nonconvex Mechanics
with Applications in Engeneering (pp. 323330). Thessaloniki: Aristotle University of
Thessaloniki.
Woo, S. LY., Abramovitch, S. D., Kilger, R., & Liang, R. (2006). Biomechanics of knee ligaments:
injury, healing and repair. Journal of Biomechanics, 39, 120.
67