Mathematical Modeling of Uniaxial Mechanical Testing
of Biological Tissue
By
Guus Verhaar
Bachelor student Physics and Astrophysics at the UvA
Student number: 0520470
Scientific Abstract
A mathematical model to describe mechanical testing of biological tissue was
developed as an addition to a research done on uniaxial mechanical testing of collagen
gels. This model was made as a start for further research on hypertension and remodeling
of blood vessels. A generalized Maxwell model was used to model the viscoelastic
behavior of a collagen gel.
A Maxwell element is a composition of a spring and a dashpot acting in series.
The generalized Maxwell model consists of a single spring and an arbitrary number of
Maxwell elements. This research uses one, two and three Maxwell elements. Using three
Maxwell elements results in the best fit of the experimental data. Mathematical
description of the generalized Maxwell model results in a differential equation which is
solved numerically using a Matlab programming code.
Straining experiments done on collagen gels (Lagerburg, 2008) provide us with a
stress relaxation curve, from which we obtained three relaxation times, the results are
comparable with previous researches performed on the same subject.
Comparison of the generalized Maxwell model with the experimental data shows
that the model does not display the stress of a collagen gel correctly, especially during
straining of the collagen gel and the ration of the stress relaxation limit compared to the
peak stress, so it needs to be adjusted.
Popular Abstract
Collagen is a very important protein in animals; it is for example responsible for
the stiffness of blood vessels. Therefore it is important to monitor the reaction of collagen
when it is stretched. In this research a mathematical model is developed to simulate the
reaction of a collagen gel when a known strain is applied. It follows that the model does
not completely describe the stress behavior of a collagen gel, but that the model needs to
be adjusted.
Furthermore some experimental data is analyzed. The values we found for the
elasticity and relaxation time of the stress curve when the strain is released, stroke with
literature.
2
Contents
Introduction………………………………… 4
Theory……………………………………… 5
Materials and Methods……………………... 12
Results…………………………………….... 13
Discussion………………………………….. 18
Conclusion…………………………………. 21
Acknowledgements………………………… 22
References………………………………….. 22
Appendix: the programming code…………. 24
3
Introduction
Collagen is a very important protein in animals, especially in mammals. About 25
% of the total protein content in this type of animal consists of collagen, being therefore
the most abundant protein. In general collagen consists of three polypeptide strands that
are bounded in a left-handed triple helix. In total 28 types of collagen exist, type I being a
major stress-carrying protein and appearing in e.g. skin, bones, tendons, fascia and blood
vessels. In this last example our particular interest exists for collagen being an important
factor in the adaptation of the structure to local conditions such as pressure (Van Bavel,
2006).
Hypertension or high blood pressure is a state in which the blood pressure is
chronically raised and an important factor in the development of cardiovascular diseases.
An understanding of the adaptation of blood vessels to pressure is therefore of the highest
importance. Usually vessels respond to a change in pressure by vasoconstriction or
vasodilatation, accomplished by smooth muscle cells in the wall of the vessels (Bouman).
When this change is chronicle, vessels are remodeled. This remodeling can take place in
different ways (Van Bavel, 2006). The amount of wall material can decrease
(hypotrophy), increase (hypertrophy) or stay equal (eutrophy). In this last case, the wall
can expand outward or shrink inward.
In large vessels, hypertension causes hypertrophy to take place and the amount of
wall material is increased by the synthesis of matrix and cell proliferation (Bakker, 2004).
However, small vessels respond to hypertension by eutrophic inward remodeling,
existing wall material is rearranged around a smaller lumen. Resistance vessels (d < 200
µm) are such vessels and contribute to the vascular resistance by 70-80 %. For an
adequate perfusion of the organs the capacity of these vessels should be sufficiently high,
but due to inward remodeling the resistance is elevated.
Eutrophic remodeling is thought to take place as a result of two processes
(Bakker, 2006). Because of chronicle vasoconstriction, smooth muscle cells are
repositioning for a maximum force development. Furthermore, the matrix of mainly
collagen is remodeled, probably by the formation of cross-links between collagen fibers.
Transglutaminases are demonstrated to have an effect on cross-linking matrix elements,
but the relation to physical remodeling are still relatively unknown (Orban 2004, Bakker,
2005).
To obtain a better insight in the process of remodeling we will look to matrix
remodeling in vitro by using artificial matrices of collagen (Lagerburg, 2008), being the
most important element for remodeling of the vessel wall. Two ways exist to do this: in a
macroscopic gel compaction setup or a gel force setup.
In the macroscopic gel compaction setup a collagen gel is poured in a Petri dish
and can be seeded with smooth muscle cells. Now the gel will compact and the area of
the gel is monitored. An important feature of this type of experiment is that the gels are
mechanically unloaded and very little force is required to alter the structure of the gel.
This is why it is thought that only few cells or cross-linking is needed for gel compaction.
Another feature is that it is hard to say anything quantitatively about the forces that are
acting in the gel. The characteristic that can be measured is the compaction: the relative
surface area of the collagen gel. The macroscopic unloaded gels can also be studied by
following the interaction of individual cells with the collagen matrix microscopically
(Van den Akker, 2008).
4
Aiming to get more quantitative results about the forces acting in a collagen gel, a
special interest in this research is directed at the gel force setup. Over the last few years
different studies have been performed to measure the mechanical properties of collagen
matrices (Wagenseil, 2003; Wagenseil, 2004; Pryse, 2003; Krishnan, 2004; Cacou, 2000;
Thomopoulos, 2005; Nekouzadeh, 2007; Roeder, 2002; Sheu, 2001; Feng, 2003). In a
previous research at the institute of Medical Physics at the AMC, a system was developed
as well to measure quantitatively the mechanical properties of collagen gels (Sleutel,
2007). In this system a collagen gel is placed between two clamps of a myograph that can
strain the gel uniaxially and measure the reacting force of the gel.
This research is divided up into an experimental part and a mathematical,
theoretical part. In the experimental part collagen gels will be tested under various
conditions (Lagerburg, 2008). Incubation periods of one, four and seven days will be
compared and gels will be tested with and without smooth muscle cells. With these tests
we hope to expand the basis that was created in the former research at the institute in
which only few experiments could be performed (Sleutel, 2007).
This paper will cover the theoretical part of the research. It aims at
mathematically modeling the mechanical testing of biological tissues, in particular
collagen. To model the mechanical properties of a collagen gel a generalized Maxwell
model is used. By adding different elements step by step a generalized model is
developed. Following from this model an ordinary differential equation is formulated,
which is solved numerically by using a Matlab programming code. Furthermore the
results of the mathematical model are compared to the experimental data found by
Lagerburg, 2008. Specific aims of the research are to model mechanical testing of
biological tissue, particularly collagen gels, and to put a physical interpretation on the
experimental results we found.
With the right straining protocol and the corresponding results we hope to get
more information about the mechanical properties of collagen gels and the mechanism of
remodeling in these gels.
Theory
In order to create a mathematical model we must first consider the properties of
viscoelastic behavior. In Lagerburg the force on the collagen gels in measured so we
must first convert this to a stress. From that we can model the behavior of the entire
system by introducing different elements each accounting for elasticity and viscosity.
By measuring the force of a collagen gel using a myograph we can determine the
stress inside the collagen gel.
 
F
A
(1)
Where σ equals the stress, F the measured force and A the surface of the cross
section of the collagen gel perpendicular to the direction of the force. So by measuring
5
the force a collagen gel exerts on a myograph we can image the stress inside the gel. A
typical stress curve is shown in figure 1.
Fig 1: typical stress curve. By measuring the force of the collagen gel and its cross
section surface the stress could be determined. During increase of the stress a strain is
applied to the gel, after one minute the stress remains constant and the gel relaxates.
During the first minute of the straining protocol an increasing strain is applied, the
strain rate is constant and equals 20% per minute, during this minute the stress appears to
increase linearly with the strain. After one minute the strain remains constant at 20%, at
this moment the stress decreases exponentially to a stress-relaxation limit (see also
Lagerburg, 2008). This typical mechanical behavior of collagen is well-known as
viscoelastic behavior.
We can now distinguish three different conditions to which a theoretical model
must suffice to describe the viscoelastic behavior of a collagen gel properly:
1. When increasing strain is applied, the stress must increase (red square in figure 1);
2. When the strain reaches its maximum the stress should drop exponentially (green
square in figure 1);
3. The stress should not drop to the pre-straining level, but it should saturate to a
stress-relaxation limit (blue square in figure 1).
The easiest way to describe the behavior of a collagen gel would be to consider it
as a linear-elastic material satisfying Hooke’s Law (equation (2)).
6
  E 
(2)
Where σ is the stress of the spring in Pa, E the elastic modulus in Pa and ε the applied
strain. A simple model that describes such behavior would be an ordinary spring to which
a strain is applied.
Fig 2: conceptual model of elasticity, an ordinary spring to which a strain is applied.
Typical stress strain relations are shown in the graphs added. It shows that the stress is
linear to the strain, with linear modulus E.
This simple model would adequately describe the linear increase of the stress in
the first minute, after this minute the relaxation does not appear at all though (figure 2). It
is obvious that an adjustment must be made to satisfactory describe the process of
relaxation. In fact, the collagen gels consist of a large amount of collagen fibers all
interacting with each other, and thus generating resistance to stretch, in rheology this
phenomenon is called viscosity and in a model it is represented by a dashpot in series
with a spring (Roylance, 2001). This system is called a Maxwell element (figure 3).
Moreover under influence of tension the collagen fibers unravel which causes a decrease
in stress; this phenomenon is called creep and is represented in the model by placing a
dashpot parallel to a spring. The effect of creep is so thorough that the stress will
eventually drop to pre-straining level.
To model viscous behavior of a collagen gel, a dashpot with a known stress-strain
relation (equation (3)) is used.
d  
d
dt
(3)
7
Again ε is the strain, now σd is the stress of the dashpot and η is the viscosity of the
collagen gel.
Fig 3: a Maxwell element. A dashpot is placed in series with a spring to simulate viscous
behavior.
In order to describe the appropriate behavior of this system an equation that
accounts for both the strain of the dashpot as well as the strain of the spring is needed. In
a Maxwell element the following condition applies:
s d
E s  
d d
dt
(4)
If we now divide the total initial length of the system, L0, up into the initial length of the
dashpot section, Ld0, and the initial length of the spring section, Ls0, we can define a strain
for the dashpot section, εd, and a strain for the spring section, εs, given by:
d 
Ld  Ld 0
Ld 0
L  Ls 0
s  s
Ls 0
(5)
Where Ls and Ld are the current length of the spring- and dashpot section. We can now
also define a parameter α which has the following properties:
Ls 0    L0
Ld 0  (1   )  L0
(6)
Furthermore:
L  L0 (1   )
Ls  Ls 0 (1   s )   (1   s ) L0
And:
8
d 
Ld  Ld 0 L  Ls  ( L0  Ls 0 ) (1   ) L0   (1   s ) L0  L0  L0    s



Ld 0
L0  Ls 0
L0  L0
(1   )
d d
d 
1
 d


 s 
dt
(1   )  dt
dt 

d

1 d
s  s   s 
E E
dt
E dt
Inserting these results into equation (4) yields:
d 
 d

 s 
(1   )  dt
dt 

1 d
d
 (1   )  E      
 
E
E dt
dt
   d
d
 

 
(1   ) E dt
dt
E s 

(7)
From this differential equation we can predict the stress-strain relationship for a single
Maxwell element; a very rough estimate is shown in figure 4.
Fig 4: a rough estimate of the reaction of the stress of a Maxwell element when a strain is
applied.
The influence of the relaxation is already visible when the strain is applied. This
results in a non-linear increase of the stress. The non-linearity obviously depends on the
relaxation time, which depends on the viscosity and the elastic modulus. If the strain
9
remains constant at 20% the time derivative of ε becomes zero. Thus the differential
equation simplifies to:
  d
d

 
0
(1   ) E dt
dt
d
(1   ) E

  
dt
 

(8)
This ordinary differential equation has a solution:
  c  et / 
Where  
(9)
 
is the relaxation time of the system. In the continuation of
(1   )  E
this paper we will assume α = 0.5, so that τ = η / E.
Now we need to consider the stress-relaxation limit. We can account for this
phenomenon by placing a Maxwell element parallel to a single spring (figure 5).
Fig 5: standard linear solid model. A single spring is placed parallel to a Maxwell
element. This model accounts for both the relaxation curve and the stress-relaxation
limit.
The mechanical properties of this model are described as the sum over the two
stresses:
 t  1   2  E   
d
  d


dt (1   ) E dt
(10)
With: σt the total stress, σ1 the stress on the spring section, σ2 the stress on the
Maxwell element, E Young’s modulus or linear modulus of the system. This results in a
stress-strain relation that seems to satisfy every condition (figure 6).
10
Fig 6: reaction of the total strain to an applied strain in the standard linear solid model.
By adding multiple Maxwell elements we create a generalized Maxwell model
(Wagenseil, 2003; Nekouzadeh, 2007; Roylance, 2001) or a Maxwell-Weichert model
(figure 7). This method provides a range of relaxation times, every Maxwell element that
is included in the model stands for a different relaxation time (Wagenseil, 2003).
Hopefully this will make it possible to approximate viscoelastic behavior of collagen gel
better. Figure 8 shows the exponential decay of the stress in the collagen gel and the
difference between a first, second and third order fit. Especially at the beginning of the
relaxation curve a second or third time constant is necessary to model the viscoelastic
behavior of the collagen gel properly.
Fig 7: a generalized Maxwell model, by adding multiple Maxwell elements different time
scales to which relaxation applies are taken into account in a model.
Again, the mechanical properties are described by summation of the stresses of
every component:
11
k
 t   n
(11)
n 1
With k the total number of Maxwell elements in the generalized Maxwell model.
This results in k-1 differential equations in the shape of equation (6). These equations are
solved numerically by using a Matlab programming code.
For more detailed mathematical information see Roylance, 2001.
Materials and Methods
This section will cover for the two main subjects of this paper, namely the
determination of the relaxation times of the collagen gels and the creation of the
programming code, which will offer a solution for the mathematical problem.
Determining the relaxation times
To determine the relaxation time of the collagen gel several gels with different
incubation times (1, 4 and 7 days) were placed in a myograph and stretched by applying a
strain for one minute. After this minute the strain was fixed at 20% and the gel was left to
relax for about one hour, in which the stress in the gel had enough time to decrease to the
stress-relaxation limit. The obtained data was analyzed using the computer program
“OriginPro 7.5”. By fitting a third order exponential decay to the relaxation curves the
three relaxation times were acquired. The increase in stress during straining was not
included when fitting the relaxation curves. More information about the straining
protocol and acquiring the experimental data can be found in a paper covering the
experimental aspects of this research (Lagerburg, 2008).
Determining the viscosity and elasticity
In order to say anything qualitative about the viscosity and elastic modulus of the
collagen gels, we fitted a first order exponential decay through the relaxation curves of
every measurement of the first series done after an incubation time of seven days.
Assuming τ = η / E, we can determine the viscosity of the collagen gel, since we now
know the relaxation time τ and the linear modulus E.
The programming code
To simulate viscoelastic behavior a Matlab programming code was written. To keep
the overview picture the model was built from scratch by first looking only at the spring
model. Then a dashpot was introduced to the model to simulate the mechanics of a
Maxwell element. By combining a spring and several Maxwell elements the generalized
Maxwell model was simulated. In this code equation (2) and equation (7) were solved
numerically, this made it relatively easy to image the stress in equation (11). The final
programming code made it possible to simulate every model consisting of springs and
Maxwell elements by setting certain parameters to zero.
The system is based on the fact that every elastic modulus of every spring and also
the viscosity is known. It cannot determine the relaxation times itself; instead, it is more a
12
program to control if the theory could reproduce the experiments by using the relaxation
times found from the experimental data (Wagenseil, 2003).
Results
Relaxation times
The acquired data was analyzed using “OriginPro 7.5”. For every dataset the
relaxation curve was fitted with a third order exponential decay function in the form:
 (t )  A1* e t /   A2 * e t /   A3 * e t /   y0
1
2
3
(12)
Where A1, A2 and A3 are the amplitudes corresponding to the three different
relaxation times τ1, τ2 and τ3. A typical stress relaxation curve is shown in figure 8. To see
if incubation time of the gel and the seeding of cells on the collagen gel (Lagerburg,
2008) was of any influence on the mechanical properties of the gel, we determined the
strain modulus and the relaxation times. Determination of three different relaxation times
was performed by fitting a third order exponential decay to the relaxation tail of the stress
curve (figure 10). Moreover, the peak stress was determined to see if that was of any
influence. The results are shown in table 1.
Fig 8: exponential decay fitted with a first, second and third order exponential decay.
The experimental data was obtained during straining of collagen gels (Lagerburg, 2008).
13
τ1
τ2
τ3
Peak stress(Pa) Strain modulus (kPa)
M1
with cells (1 day)
with cells(4 days)
with cells (4 days)
without cells (4 days)
with cells 1 (7 days)
with cells 2 (7 days)
without cells 1(7 days)
without cells 2 (7 days)
9
7.8
15.8
14.6
14
7.6
11.8
14.6
75.4
57.2
216
185.8
169.2
57.1
110.3
179
429.8
695.7
1532.9
1211.7
1212.7
638.9
991.7
1430.1
1594
2856
10530
1382
2002
1824
2868
2237
----22.9
21.4
23.0
27.4
M2
with cells (1 day)
with cells (4 days)
with cells (7 days)
13.5 153.7
11.6 142.3
25.2 376
1482.9
1531.6
2482.7
1184
2004
3606
14.8
27.7
39.6
M3
without cell (4 days)
15.5 195.8 1563.9 2506
27738.4
without cell (7 days)
14.4 14.4
100.5
1331
19253
Table 1: results of relaxation time and strain modulus determination using “OriginPro
7.5” and fitting a third order exponential decay to the relaxation tail of the graph and
fitting a linear function through the linear part of the graph. M1, M2 and M3 stand for
gels of the first, second and third series (see also Lagerburg, 2008). Four values for the
strain modulus are missing because we did not trust the gels; they were damaged too
much after straining them to say anything sensible about the strain modulus.
Viscosity and elasticity
A first order exponential decay was fitted through the relaxation curves of every
measurement of the first series done after an incubation time of seven days. The viscosity
of the collagen gel was determined. Results are shown in table 2. The strain modulus was
obtained by fitting the linear part of the strain curve between the 45th and 55th second of
the straining protocol and creating a linear fit, a typical example is shown in figure 9.
Sample
τ (s)
E (kPa)
η (MPa.s)
with cells 1 (7 days)
626
23.1
14.5
with cells 2 (7 days)
434
21.7
9.4
without cells 1(7 days) 532
23.3
12.4
without cells 2 (7 days) 766
27.7
21.2
Table 2: results of viscosity determination, the viscosity is found by multiplying the
relaxation time τ with the linear modulus E. In literature values for E ~10 kPa are found
(Roeder, 2002) for gels with an incubation time of 1 day.
14
Fig 9: determination of the linear modulus of the generalized Maxwell model. The x-axis
shows the strain while the y-axis shows the stress. This shows only the first part of the
stress strain relation.
Fig 10: relaxation curve (black line) from experimental data and a third order
exponential decay fit (red line). This figure shows only the second part of the complete
stress strain relation.
15
The mathematical model and the programming code
The mathematical model we developed was based on the generalized Maxwell
model and it solves the differential equation derived in equation (7) numerically, the code
was written in Matlab. The construction of the programming code was chosen so that we
can account for three different Maxwell elements and one single spring (for the
programming code, see the appendix). For every Maxwell element a differential equation
was solved, using different viscosities to determine the relaxation times. Two different
linear moduli were introduced: one to account for the single spring element and thus
determining the stress relaxation limit, and one to account for the springs in the Maxwell
elements. The relaxation times used in the model were based on the experimental data
(table 1).
So by setting certain parameters to zero we can create stress-strain relations that
simulate the behavior of a collagen gel when it is stretched. Figure 11 shows the results
of the model when only one Maxwell element is used. In this case there is only a short
relaxation time and the stress drops to a stress-relaxation limit very rapidly.
Fig 11: simulation of a generalized Maxwell model consisting of a spring and only one
Maxwell element.
When we insert a second Maxwell element (figure 12), a long term relaxation
time appears, but still the stress still drops to a stress relaxation limit quite rapidly.
16
Fig 12: simulation of a generalized Maxwell model consisting of a spring and two
Maxwell elements.
To model the viscoelastic behavior of a collagen gel better we used three Maxwell
elements in the model. This provides us with a better simulation of the data acquired in
the experiments. The exponential decay tail is stretched a lot more because of the third
relaxation time which is quite long. Results of the three Maxwell element model are
shown in figure 13.
Fig 13: simulation of a generalized Maxwell model consisting of a spring and three
Maxwell elements.
If we now compare the experimental results with the simulation the model makes
we can see that there still are quite a few differences between the experimental stress
curve (figure 14) and the theoretical stress curve (figure 13), the question what these
differences are and what they mean will be discussed in the discussion.
17
Fig 14: experimental stress curve
Discussion
In this paper a mathematical model to simulate uniaxial mechanical testing of
biological tissue, in particular a collagen gel was developed. The model was built up from
a single spring to model linear elasticity, to a generalized Maxwell model to model linear
viscoelasticity. Specific interest went to the stress of the collagen gel when a strain is
applied. There are several imperfections in the model compared to experimental data;
especially during the straining of the collagen gel the theoretical model does not match
experimental data.
Firstly; during the first seconds of straining the stress in the collagen gel does not
increase linearly, instead the increase of stress is divided up into two regions before it
reaches the linear increase (Fratzl, 1997; Lagerburg, 2008; Roeder, 2002 figure 3)). In the
model the increase in stress starts directly and is proportional to the strain. But because
three different time scales to which relaxation takes place are inserted, relaxation already
shows during straining. This was not visible when the stress of the collagen gel was
measured. In the collagen gel, the collagen fibrils have to order first and the molecules of
which the collagen fibers have to be stretched first, this takes time and force so only after
a while, the linear part of the stress curve is reached. In the linear part the collagen fibers
glide along each other. The ordering of the collagen fibers during straining is disregarded
in the model. No solution for the relaxation during straining was found. A possibility
would be to leave the short relaxation time out of the simulation during straining, but this
resulted in another problem, namely that the decreases of stress after straining becomes to
idle.
Secondly the stress-relaxation limit in the experimental data is about 50% of the
peak stress, in the model this is only about 25%. This probably has to do with the same
problem as mentioned above, namely that the short relaxation time is of to much
influence during the straining and of to little influence during the stress relaxation. So it
seems we have to make adjustments to several parameters of the system. The single
parameter where they almost all come together is the relaxation time:
18

 
(1   )  E
(13)
By changing α, we can adjust the difference between the stress-relaxation limit and the
peak stress. This would result in different relaxation times though, which is a feature we
do not want to occur if the comparison to the experimental data must remain.
In literature relaxation times of τ1 = 1-10 s, τ2 = 10-100 s and τ3 > 1000 s are
found (Wagenseil, 2003). We find relaxation times ranging from τ1 = 7-17 s, τ2 = 50-200
s and τ3 = 400-1500 s, which is comparable.
Furthermore a third order Maxwell model is preferred above a first or second
order model because it accounts for a long relaxation curve, as well for a smaller drop in
stress during straining.
Despite the flaws mentioned, the model does accurately describe other
viscoelastic properties. When the strain rate is increased, the peak stress also increases,
because there is less time to relax. Moreover it describes the behavior of a generalized
Maxwell model properly; probably the collagen gel does not behave like a linear
viscoelastic material during stretching, which is of influence on the reaction of the stress
when the strain is released.
If the generalized Maxwell model does not describe the behavior of a collagen gel
totally, are there any other models that may describe the viscoelastic behavior better? A
more intuitive way would be to see if the relaxation curve is a straight line on a double
logarithmic scale. Instead of using equation (12) to fit the relaxation curve one would
rather use:
 (t )  a  t b
(14)
This results in the fit shown in figure 15. In this case a third order exponential fit matches
the acquired data better.
A more physical way to look at a change in model would be to consider the composition
of the collagen gel again. If we consider it to be a more glassy substance it is possible to
describe the behavior of the relaxation curve using a so called stretched exponential
(Berry, 1997; Abou, 2001). This is an equation of the form:
 (t )  c  e ( t /  )

(15)
In this way we can model the fact that a collagen gel is not homogeneous, but in fact
consists out of very small regions, which each have its own contribution to the relaxation
time. Results of such a fit are shown in figure 16.
19
Fig 15: experimental data fitted with a third order exponential decay function (red line,
r2 = 0.99) and an allometric function (equation (14), blue line, r2 = 0.95).
Fig 16: experimental data (black line) fitted with a stretched exponential (red line, r2
=0.9996).
20
The stretched exponential fit is by far the best fit and provides us with a physical
understanding.
The advantage of a generalized Maxwell model is that one can adjust the number
of Maxwell elements manually, this makes modeling easier than using a stretched
exponential or a power function (equation 14).
All in all a generalized Maxwell model can describe viscoelastic behavior of a
collagen gel adequately. And with a few adjustments it should be able to model the stress
of a collagen gel, also during strain.
Conclusion
We were able to create a model that simulates linear viscoelasticity. It was clear
that we needed at least three time constants to model the viscoelastic behavior we
measured during the experiments. Besides this, more research could be done to optimize
the model, especially during straining of the collagen gel. Also other models than the
generalized Maxwell models should be studied.
Our straining protocol was good enough to determine the linear modulus and the
relaxation times of the collagen gel. From these values we could determine the viscosity.
The values we found were comparable to values found in similar researches (Roeder,
2002; Wagenseil, 2003). To say more about experimental data, the method of
measurements should be optimized (see also Lagerburg, 2008). The main problem would
be to model the behavior of a collagen gel during the first minute of the straining
protocol. If one could solve this problem it might be possible to predict the viscosity of a
collagen gel instead of using the viscosity found by performing experiments.
21
Acknowledgements
Special thanks go to ir. Jeroen van den Akker for his willingness to help, his
patience when we broke yet another gel and his support during preparation for the
presentation. Also I would like to thank prof. dr. Ed van Bavel for involving us in his
research and his view on the acquired data, when we thought we could not get anything
out of it anymore. Also special thanks to Angela van Weert for helping us creating the
collagen gels and seeding them with smooth muscle cells, and all the other small but
important things she taught us when using equipment of the department. Thanks also to
dr. Dirk Faber for making it possible to complete at least one measurement of the
thickness of a collagen gel with the OCT-scan. Also thanks to Jeroen Goedkoop for being
our independent assessor, and his comments at our final presentation.
And of course thanks to everyone I forgot, but who made it possible to perform
our short research.
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23
Appendix
Programming code
%This program describes the viscoelastic behaviour of a collagen gel.
The
%chosen model is the so called Kelvin-Voigt model, which consists of a
%spring and several Maxwell elements. The total stress is calculated by
%adding all contributions of the different elements to each other.
%setting the time
dt=0.1;
t=0:dt:1000;
%boundary conditions
eps(1) = 0;
epsdot(1) = 0;
sigma1(1) = 0;
sigma2(1) = 0;
sigmadot(1) = 0;
%EXPERIMENTAL SETUP makes it possible to customize the simulation by
manual
%input of (experimental) parameters
prompt={'Enter the maximum strain','Enter the strainrate','Enter
Eta1','Enter Eta 2','Enter Eta 3','Enter Linear Modulus','Enter Youngs
Modulus'};
%name of the dialog box
name='Experimental setup';
%number of lines visible for your input
numlines=1;
%the default answer
defaultanswer={'0.2','0.00333','1160','14230','153160','100','50'};
%creates the dialog box. the user input is stored into a cell array
answer=inputdlg(prompt,name,numlines,defaultanswer);
%notice we use {} to extract the data from the cell array
maxstrain = str2num(answer{1});
strainrate = str2num(answer{2});
eta1 = str2num(answer{3});
eta2 = str2num(answer{4});
eta3 = str2num(answer{5});
E = str2num(answer{6});
Espring = str2num(answer{7});
%setting the correct matrix size of every used vector
eps = zeros(size(t));
sigma = zeros(size(t));
sigma1 = zeros(size(t));
sigma2 = zeros(size(t));
sigma3 = zeros(size(t));
sigmaspring = zeros(size(t));
sigmadot1 = zeros(size(t));
sigmadot2 = zeros(size(t));
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sigmadot3 = zeros(size(t));
epsdot = zeros(size(t));
%setting the relaxation time
alpha = 0.5;
tau1 = (alpha*eta1)/((1-alpha)*E);
tau2 = (alpha*eta2)/((1-alpha)*E);
tau3 = (alpha*eta3)/((1-alpha)*E);
%in this loop we calculate the stress by solving the ODE:
%sigma + A*sigmadot = eta * epsdot by first determining sigmadot from
the
%boundary conditions and then calculating sigma from the previous
values.
%The relaxation time is dependant on the value of Youngs modulus E and
the
%viscosities eta1, eta2 and eta3. The stress-relaxation limit is
inserted
%by the spring element sigmaspring
for i = 2:length(t);
eps(i) = eps(i-1)+ strainrate * dt;
if eps(i) > maxstrain;
eps(i) = maxstrain;
end;
epsdot(i) = (eps(i)-eps(i-1))/dt;
%the contribution to the stress of the first maxwell element, if
%tau equals zero, the contribution to the stress of the first
% Maxwell element is not included in the total stress.
if tau1 <= 0;
sigmadot1(i) = 0;
sigma1(i) = 0;
else
sigmadot1(i) = (- sigma1(i-1) + eta1 * epsdot(i) ) / tau1;
sigma1(i) = sigma1(i-1) + sigmadot1(i) * dt;
end;
%the contribution to the stress of the second maxwell element
if tau2 <= 0;
sigmadot2(i) = 0;
sigma2(i) = 0;
else
sigmadot2(i) = (- sigma2(i-1) + eta2 * epsdot(i) ) / tau1;
sigma2(i) = sigma2(i-1) + sigmadot1(i) * dt;
end;
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%the contribution to the stress of the third maxwell element
if tau3 <= 0;
sigmadot3(i) = 0;
sigma3(i) = 0;
else
sigmadot3(i) = (- sigma3(i-1) + eta3 * epsdot(i) ) / tau3;
sigma3(i) = sigma3(i-1) + sigmadot3(i) * dt;
end;
%the contribution to the stress of the spring element
sigmaspring(i) = Espring * eps(i);
%the total stress
sigma(i) = sigma1(i) +
sigma2(i) + sigma3(i) + sigmaspring(i);
end;
%plotting the stress and the strain in a subplot
subplot(211); plot(t,sigma,'r','LineWidth',2);
xlabel('time(s)','FontSize', 16); ylabel('stress (Pa)', 'FontSize',16);
title(['maximum strain = ' num2str(maxstrain), ',
strainrate = '
num2str(strainrate), ',
E = ' num2str(E),',
Espring = '
num2str(Espring),',
tau1 = ' num2str(tau1), ',
tau2 = '
num2str(tau2), ',
tau3 = ' num2str(tau3)],'FontSize',14)
subplot(212); plot(t,eps,'r','LineWidth',2);xlabel('time',
'FontSize',16); ylabel('strain', 'FontSize',16);
26