# Rheological Analysis and Evaluation of Measurement Techniques for the Curing
Polymethylmethacrylate Bone Cement in Vertebroplasty

Zubin Trivedi<EMAIL_ADDRESS>[  Jacek K. Wychowaniec
Dominic Gehweiler  Christoph M. Sprecher  Andreas Boger  [  Boyko Gueorguiev
Matteo D’Este  [  Tim Ricken  [  Oliver Röhrle  [

###### Abstract

Vertebroplasty is a minimally invasive surgical procedure used to treat
vertebral fractures, which conventionally involves injection of a
polymethylmethacrylate (PMMA) bone cement into the fractured vertebra. A
common risk associated with vertebroplasty is that of cement leaking out of
the vertebra during the injection. This could happen since the flow behaviour
of the bone cement is not conventional, but it varies with many factors like
curing time, shear rate, and temperature. Therefore, experiments to quantify
the cement’s flow properties are necessary to be able to understand and
properly handle the bone cement. In this study, we aimed to characterize the
behaviour of the PMMA bone cement in its curing stages, in order to obtain
parameters that govern the flow behaviour during injection flow. The
measurements were done using rotational and oscillatory rheometry as well as
using a custom-made injetor setup replicating a typical vertebroplasty
setting. Our results showed that the complex viscoelastic behaviour of the
bone cement, which evolves over a two-phase curing process, can be
significantly affected by deformations and temperature. We found that the
results from rotational tests were affected by wall slip and “ridge”-like
formations in the test sample, which we confirmed visually by microscopical
assessment. We also found that the Cox-Merz rule to be conditionally valid,
which impacts the use of oscillatory tests to obtain shear-thinning
characteristics for the bone cement.

###### keywords:

American Chemical Society, LaTeX

imsb-unistuttgart]Institute for Modelling and Simulation of Biomechanical
Systems, University of Stuttgart, Pfaffenwaldring 5a, 70569 Stuttgart These
authors contributed equally to this work These authors contributed equally to
this work hs-ansbach]Ansbach University of Applied Sciences, Residenzstraße 8,
91522 Ansbach, Germany ari-davos]AO Research Institute Davos,
Clavadelerstrasse 8, 7270 Davos, Switzerland isd-unistuttgart]Institute of
Structural Mechanics and Dynamics in Aerospace Engineering, University of
Stuttgart, Pfaffenwaldring 27, 70569 Stuttgart imsb-unistuttgart]Institute for
Modelling and Simulation of Biomechanical Systems, University of Stuttgart,
Pfaffenwaldring 5a, 70569 Stuttgart [simtech]Stuttgart Center for Simulation
Science (SC SimTech), Pfaffenwaldring 5a, 70569 Stuttgart IR,NMR,UV

##  1 Introduction

Vertebral fractures are a major source of suffering and disability, carrying
risk of back pain, immobility, and potentially severe post-complications that
impact the quality of life 1, 2. Up to 28% of patients with osteoporotic
fracture-related issues resulted in death by 2008 3. Unlike many other
musculoskeletal complications where bone grafting or alternative methods can
be used 4, the most commonly used procedure to treat vertebral fractures is
vertebroplasty 5, 6, 7. In this procedure, a rapidly curing bone cement is
injected into the porous cancellous part of the vertebra to serve as
mechanical support 8, 9, 10 and allow early mobilization, especially in the
elderly, where lack of mobility often causes a cascade of medical issues and
often death. A successful vertebroplasty can provide instant pain relief 11 to
the patient. On the other hand, improper injection could lead to cement
leaking outside the vertebra, which could cause serious complications like
pulmonary embolism and paralysis 12, 13. The bone cement is a material with
complex mechanical behaviour dependent on many factors, which makes it hard to
predict its behaviour during injection, especially for practitioners who rely
on intuition, skill, and sensory feedback. Therefore, characterization of the
curing bone cement using various measurement techniques is helpful to
understand its behaviour during injection in various conditions. Mathematical
constitutive models derived from the characterization data can quantify the
various dependencies, which can be used in analytical or computational models
to reliably predict the behaviour of the bone cement during injection 14, 15,
16, 17, 18, 19.

The bone cements typically consist of ceramic components and fast-curing,
minimally-toxic polymers that rapidly polymerize into, e.g. poly(methyl
methacrylate) (PMMA) 8, 20, 21, 22. The rapid polymerization (curing) of bone
cement is designed to provide the required structural support for the damaged
vertebra, whereas ceramic components like zirconium dioxide function as radio-
opacifiers that allow for easy visualization during the vertebroplasty
procedure 23. Hydroxyapatite, a calcium phosphate mineral naturally found in
bone, is also sometimes added to improve biocompatibility and to promote bone
growth around the cement. The curing typically relies on free radical
polymerization, the initiators react to form free radicals which cause the
methlyl methacrylate (MMA) monomer molecules to form PMMA polymer chains 24,
25. The polymerization is an exothermic process 24, meaning that it is further
accelerated by the heat released and the energy introduced, including
mechanical shear.

In general, it should be noted that PMMA bone cement is far from behaving like
an ideal fluid, rather it has a dough-like consistency while it is curing. The
behaviour of the bone cement can be categorized as viscoelastic, i.e. it is a
combination of solid-like and fluid-like behaviour. Moreover, the behaviour is
dependent on many factors, including time, flow conditions (e.g. shear rate),
and temperature. Due to this multifactorial dependency, the complete
characterization of the bone cement is a challenge. There are many previous
studies measuring the rheological properties of the bone cements. Different
studies use different brands of PMMA bone cement for their measurements.
Moreover, the set of conditions used to carry out the measurements are
different in each study. Not only the conditions, but also the methods used
for the measurements are different in each study, e.g. while some studies
report steady shear viscosity from rotational shear measurements, others
report complex viscosity from oscillatory measurements. The steady state and
complex viscosities can be compared only using the Cox-Merz rule 26, which
says that the complex viscosity measured must be equal to the steady shear
viscosity if the angular frequency is equal to the shear rate, respectively.
However, this rule is not valid for all kinds of materials. Nevertheless, its
validity for the PMMA bone cement is often assumed rather than being
explicitly investigated. The differences in some of these studies are shown in
Table 1. Due to these reasons, it is difficult to compare or compile the data
from these studies to understand the complete rheological behaviour of a bone
cement over the range of shear rates and temperatures expected during a
vertebroplasty.

Hence, the aim of this work was to understand the flow behaviour of the bone
cement during injection in its curing phase in conditions relevant to
vertebroplasty, taking Vertecem V+ as an example. The main objectives of this
work were as follows:

  1. 1.

To quantify various aspects affecting injection flow behaviour, viz.
viscosity, yield stress, loss factor, flow point, etc. from measurements

  2. 2.

To fit the experimental results to mathematical models that describe the
rheological behaviour, like the power law model, and obtain the model
parameters that can be used for prediction and simulation of injection flow
behaviour

  3. 3.

To compare the different methodologies used for rheological measurement, viz.
injection tests as well as rotational and oscillatory measurements on the
rheometer, namely their advantages and limitations

  4. 4.

To check the validity of the Cox-Merz rule for the PMMA bone cement

Authors | Cement | Methods | Measurements and conditions
---|---|---|---
Boger et al. 27 | Unnamed | Oscillatory | Time evolution, 1 Hz, various $T$
Deusser et al. 28 | Vertecem | Oscillatory | Time evolution, 1 Hz, 23 °C and 37 °C
Dunne and Orr 29 | Unnamed | Capillary | Time evolution
Farrar and Rose 30 | Various | Oscillatory | Time evolution, 0.05%, 5 Hz, various $T$
Kolmeder and Lion 31 | Osteopal | Rotational CP | Time evolution, various $\dot{\gamma}$
|  | Capillary | Time evolution, various $\dot{\gamma}$
Kolmeder et al. 32 | Unnamed | Oscillatory PP | Time evolution, 1 Hz, 0.1%, various $T$
Krause et al. 33 | Various | Rotational CP | Time evolution, 0.04 and 1 s-1
|  | Capillary | Shear rate dependence, 23 °C
Lepoutre et al. 17 111These studies assume the validity of the Cox-Merz rule. | Osteopal | Oscillatory PP | Time evolution, 50%, various $f$ and $T$
|  | Injector | Shear rate dependence, various $\dot{\gamma}$
Lewis and Carroll 34 | Orthoset | Oscillatory CuP | Time evolution, 1 Hz, 18 °C
Lian et al. 18 111These studies assume the validity of the Cox-Merz rule. | Simplex | Oscillatory PP | Frequency sweep, 0.2%
Nicholas et al. 35 | Various | Oscillatory PP | Time evolution, 5 Hz
Table 1: An overview of previous studies found in literature done to
characterize PMMA bone cement, the methods used, and the conditions used for
measurements

##  2 Experimental

###  2.1 Bone cement

The bone cement used for this study was purchased as non-sterile bulk material
from OSARTIS GmbH (Germany), which is equivalent of the bone cement
commercially sold by the name VertecemTM V+ by DePuy Synthes. The bone cement
is supplied as a powder consisting of 40% zirconium dioxide, 15%
hydroxyapatite, 44.6% poly(methyl methacrylate) (PMMA), and 0.4% benzoyl
peroxide; and a liquid consisting of 99.35% methyl methacrylate (MMA)
stabilized with 60 ppm hydroquinone, and 0.65% N,N-dimethyl-p-toluidine
(DMPT). Typical application of the bone cement uses the provided cement mixing
kit (DePuy Synthes), where the initiator and powder are mixed with MMA
monomers in 26 grams of powder to 10 mL of liquid ratio, upon which
immediately the user is required to push and pull the mixing chamber kit
handle from endpoint to endpoint for 20 seconds, at a steady pace of 1–2
strokes per second. The resultant mixture has a dough-like consistency which
can be filled into syringes and injected. The polymerization occurs through
the free radical polymerization method, wherein the activator DMPT reacts with
the initiator benzoyl peroxide to form free radicals. These free radicals
react with monomer MMA molecules that cause them to form polymer chains of
PMMA. PMMA is the main component of bone cement and is responsible for its
mechanical properties. The entire polymerization process takes about 20
minutes at room temperature.

###  2.2 Rheometer measurements

The rheological measurements were performed on MCR302 rheometer from Anton
Paar using a parallel plate (PP) setup. Disposable top and bottom plates were
used for the measurements. The diameter of the top plate was 25 mm. The bone
cement contains particles of size 5.2 ± 1.5 microns 36, hence the gap between
the two plates was kept at 1.5 mm to ensure sufficiently larger gap size
compared to the particle size 30 and avoid any particle-scale effects. The
temperature during the measurements was controlled using the Peltier module of
the rheometer, so that the measurements are not affected by the heat released
during curing. The temperature was fixed at 23 °C in all tests, unless
otherwise specified. The rheometer and the associated instrumentation was
switched on for at least one hour to equilibrate the temperature and all other
conditions before carrying out measurements.

The bone cement prepared using the typically provided mixing kit would produce
a large offset of unnecessary material for measurements. Hence, we established
a mixing process for our laboratory conditions that produced only sufficient
amount of bone cement for each measurement. To ensure that our laboratory
mixing process replicates the bone cement behaviour from the original mixing
process, we established a benchmark test as quality control. In the test, the
bone cement sample was subjected to oscillations at a constant torque of 3 mNm
and 1 Hz frequency at 23 °C, with a measuring point every 5 seconds. After
multiple trials, the successful and reproducible method consisted of the
following steps:

  1. 1.

2.6 grams of the PMMA powder were weighed in a 10 mL beaker.

  2. 2.

Separately, 10 mL of the MMA monomer liquid was prepared in a batch glass
vial.

  3. 3.

At time $t$ = 0 s, 1.0 mL MMA monomer liquid was dropped into the 10 mL beaker
with the powder using a positive displacement pipette, and a stopwatch for
measuring the time was started.

  4. 4.

The powder and the liquid were stirred using a non-reacting
polyetheretherketone (PEEK) stirring road for 20 seconds and counting about 20
rotations.

  5. 5.

After 20 s, part of sample was gently dropped onto rheometer bottom plate and
top plate was gently lowered down to avoid any disturbance.

  6. 6.

Silicone oil was then spread around the sample and the humidity control hood
was used to avoid sample evaporation. The test was then started and the time
on the stopwatch was recorded.

The test on the rheometer commenced once the bone cement was prepared and
placed between the parallel plates. The time since start of mixing to start of
test for all tests varied in the range of 180 $\pm$ 30 seconds. For the
characterization of the bone cement, various tests were done using the
rheometer, namely

  * •

Test Rh1a: Oscillatory test at 0.2% strain amplitude and 1 Hz frequency for 30
minutes: this test was done to observe the evolution of the bone cement
properties in time at low strain value.

  * •

Test Rh1b: Oscillatory test at 20% strain amplitude and 1 Hz frequency for 30
minutes: this test was done to observe the evolution of the bone cement
properties in time under continuous larger deformation.

  * •

Test Rh2a: Oscillatory test with 0.1% strain amplitude, 0.1 Hz (5 mins) – 20%
strain amplitude, 1 Hz (3 mins) – 0.1% strain amplitude, 0.1 Hz (22 mins).
This procedure was done to replicate a typical injection, in which the bone
cement is at rest at first after mixing, then it is deformed as it is filled
and injected through the syringe and cannula, and finally is at rest once it
is injected. The aim here was to investigate how the bone cement breaks under
deformation and recovers thereafter.

  * •

Test Rh2b: Oscillatory test with 0.1% strain amplitude, 0.1 Hz, 23 °C (5 mins)
– 20% strain amplitude, 1 Hz, 23 °C (3 mins) – 0.1% strain amplitude, 0.1 Hz,
37 °C (22 mins). This procedure was done to replicate a typical injection into
a human vertebra, hence the change in temperature in the last step from room
temperature to physiological body temperature.

  * •

Test Rh3: Rotational shear stress applied in steps of 0 Pa (1 min) - 100 Pa (1
min) - 0 Pa (1 min) - 500 Pa (1 min) - 0 Pa (1 min) - 2000 Pa (1 min) - 0 Pa
(1 min) to evaluate creep behaviour.

  * •

Test Rh4: Rotational shear strain applied in steps of 0% (2 min) - 1% (2 min)
- 0% (2 min) and 0% (2 min) - 100% (2 min) - 0% (2 min) to evaluate stress
relaxation behaviour.

  * •

Test Rh5: Amplitude sweep from 0.01% to 1000% strain amplitude, 1 Hz frequency
to obtain the yield stress and the crossover point of storage and loss moduli.

  * •

Test Rh6: Frequency sweeps from 0.1 Hz to 100 Hz at various strain amplitudes.
The frequency sweep yields the dependence of the complex shear modulus and the
complex viscosity on oscillation frequency. The strain amplitudes were chosen
a posteriori as per the different regions obtained from the amplitude sweep
test.

  * •

Test Rh7a: Rotational shear rate sweep, various tests were done to obtain flow
curves with the shear rate ranging over 2 or 3 orders of magnitude in each
test. In this way, steady-state viscosity values for shear rates from 10-4 s-1
to 103 s-1 were obtained.

  * •

Test Rh7b: Rotational shear rate sweep from 0.1 to 100 s-1 with 100-grit
sandpaper stuck to the top and bottom plates using double-sided tape. The test
was done to check if there was a difference in the result with and without
sandpaper, which would provide evidence for the occurrence of wall slip during
the test.

  * •

Test Rh8: Rotational tests with constant shear rate for 20 minutes. These
tests are done at shear rates 10-4, 10-3, 10-2, 10-1, 1, and 100 s-1. These
tests were done to evaluate the time-evolution of viscosity at various flow
rates.

###  2.3 Optical microscopy (tests Rh9 and Rh10)

Two additional tests were done on the rheometer to make bone cement samples
for investigation with optical microscopy. In the first test (test Rh9), bone
cement was prepared and simply left on the rheometer between the plates while
maintaining 1.5 mm gap for 45 minutes without any action. In the second test
(test Rh10), bone cement was prepared and subjected to 15 minutes of
rotational 100 s-1 shear rate, like in test Rh7b, and then 30 minutes of no
action on the rheometer. The two cured bone cement discs formed as a result of
these tests were investigated using an Axiotech microscope, which is a
material microscope for non-transparent samples, equipped with a 2.5x
objective. The images were captured using a digital camera Axiocam 105, and
processed using software Axiovision 4.9.1. The microscope, camera, and
processing software were provided by Carl Zeiss AG, Germany. Reflected light
bright field technique was used for the illumination.

###  2.4 Tests on the custom-made injector (Tests Inj1 – Inj5)

(a)

(b)

Figure 1: (a) Setup for the injection tests (b) Schematic of the simplified
geometry used for analytical calculations

The setup for the injection experiments is shown in Figure 1(a). The injector
consisted of a carriage which was driven by a stepper motor using a ball screw
with 5 mm feed per revolution. The stepper motor had a resolution of 1.8°
corresponding to 200 steps per revolution, amounting to a calculated carriage
resolution of 0.025 mm. The syringe was placed horizontally through a hole at
the other end of the device as shown in the Figure 1(a) and held in place with
a bolt. The cannula was then attached at the end of the syringe. The carriage
pushed the plunger at the desired rate which could be programmed using a
computer. At the point of pushing the plunger, a 200 N load cell was mounted
on the carriage to measure the forces applied to the plunger during injection.

To prepare the bone cement, a 10 mL syringe was taken and sealed at its open
end using a removable plug. The plunger of the syringe was removed and 3 mL of
the monomer liquid was poured in it using a pipette. 7.8 grams of the PMMA
powder were added directly into the syringe using a funnel and then mixed in
the syringe using a PEEK stirring rod for 20 seconds (20 rotations). After
mixing the components, the plunger was put back in the syringe and the plug at
the other end was removed. The mixed bone cement was then first transferred to
a 2 mL syringe until it was completely full, which was then used to fill up an
8-gauge cannula. The cannula had a side-opening and open front end, of which
the side-opening was sealed using a strong adhesive tape so the bone cement
could only come out of the front end when injected. The emptied volume of the
2 mL syringe was then again filled up using the remaining bone cement in the
10 mL syringe. At the beginning of the test, the carriage was set at the
‘unloaded’ position, which is far back to be able to mount the filled 2 mL
syringe easily. Once the syringe and the cannula were mounted, the carriage
was set to ‘loaded’ position, which was just behind the fully pulled back
plunger. The ‘unloaded’ and ‘loaded’ positions were programmed in the device
so it automatically arrived at them when commanded. At this point, the device
was given the command to inject with a specified speed according to the
required flow rate. The time from the start of mixing to the start of
injection was measured using a stopwatch. The tests, named Inj1 to Inj5, were
carried out with flow rates of 0.025, 0.05, 0.1, 0.2, and 0.4 mL/s
respectively. During the injection, the load cell measured the force required
for injection every 0.05 seconds. The time from mixing to start of injection
in all tests was between 230 and 245 seconds.

###  2.5 Analytical calculations

The measurements obtained from the injection tests were then used to obtain
the rheological parameters of the bone cement according to the power law
constitutive equation

$\mu=K\,\dot{\gamma}^{n-1},$ (1)

where $\mu$ is the viscosity, $\dot{\gamma}$ is the shear rate, and
consistency index $K$ and flow index $n$ are the material parameters. Assuming
such a fluid flows through a tube of radius $R$ and length $L$ with flow rate
$Q$, the shear stress $\tau$ at the wall relates to the pressure difference
$\Delta p$ as

$\tau=\frac{R\,\Delta p}{2\,L},$ (2) $\Delta
p=\frac{2\,L\,K}{R}\bigg{(}\frac{1+3\,n}{n}\frac{Q}{\pi\,R^{3}}\bigg{)}^{n}$
(3)

Furthermore, the shear rate in the tube can also be obtained using the
equation

$\dot{\gamma}(r)=\bigg{(}\frac{1+3\,n}{n}\frac{Q}{\pi\,R^{3}}\bigg{)}.$ (4)

Detailed derivations of these equations can be found in the works of 37, 38.

To apply these equations to our problem, the geometry of the apparatus was
simplified as shown in Figure 1(b). The syringe and the cannula were assumed
to be cylinders with the given dimensions. The coupling connecting the two
parts had the same inner diameter as the cannula, hence its length was
included in the cannula in the simplified geometry. The nozzle of the syringe
was short in length and was therefore ignored. The change of cross-section
from syringe to cannula would add to the pressure loss in reality, however
this was ignored in our study. We then used the Equation 3 to obtain the
pressure loss in our system. The pressure loss is then the sum of losses in
the syringe $\Delta p_{s}$ and the cannula $\Delta p_{c}$, given as

$\displaystyle\Delta p$ $\displaystyle=\Delta p_{s}+\Delta p_{c}$
$\displaystyle=\frac{2\,L_{s}\,K}{R_{s}}\bigg{(}\frac{1+3\,n}{n}\frac{Q}{\pi\,R_{s}^{3}}\bigg{)}^{n}+\frac{2\,L_{c}K}{R_{c}}\bigg{(}\frac{1+3\,n}{n}\frac{Q}{\pi\,R_{c}^{3}}\bigg{)}^{n},$
(5)

where $L_{s}=65$ mm, $R_{s}=3.2$ mm were the length and radius of the syringe
respectively. Similarly, $L_{c}=165$ mm and $R_{c}=1.7$ mm were the length and
radius of the cannula respectively. Given force equilibrium in the system, the
pressure loss $\Delta p$ obtained from Equation 5 is the pressure that would
be required to be created by the injection force $F$ applied externally on the
syringe plunger. Hence, the required injection force $F$ is then given as

$F=\Delta p\,(\pi\,R_{s}^{2}).$ (6)

##  3 Results and Discussion

###  3.1 Oscillations at constant strain and frequency (Tests Rh1a and Rh1b)

(a)

(b)

(c)

(d)

Figure 2: Evolution of complex modulus (solid blue line, left Y-axis) and loss
factor (solid red line, right Y-axis) as a function of time in oscillatory
tests. (a) Test Rh1a: complex modulus increases in two phases indicating the
two-phase curing process, dashed line shows linear regression for complex
modulus in first 20 minutes (b) Test Rh1b: complex modulus reduces initially
as shown in the zoomed part, then resumes increasing in two phases. Dashed
line shows linear regression for complex modulus from 10 to 20 minutes (c)
Test Rh2a: drop in modulus was observed upon increasing strain amplitude,
restoring the original amplitude resulted to recovery of modulus and
resumption of the two-phase curing (d) Test Rh2b: similar to Rh2a, but the
rapid curing phase was triggered as the temperature was increased to
physiological body temperature

At 0.2% strain amplitude and 23 °C, the complex modulus of the bone cement
increased with time due to the ongoing polymerization. The polymerization
occurred with a slow first phase, when the modulus increased gradually at a
rate of about 29 kPa/min (Figure 2(a)). This is the working phase when the
bone cement gradually hardens with time but maintains an injectable paste-like
consistency. The slow first phase was followed by a rapid curing phase after
about 20 minutes when the bone cement polymerized rapidly, and became solid in
the next 5-10 minutes. The complex modulus after solidification was above 25
MPa. Interestingly, the loss factor always remained below 1, implying that the
bone cement shows elastic prevalence, i.e. behaving more like an elastic solid
than a liquid, even at the early stages of curing. The loss factor was between
0.5 and 0.7 until 25 minutes and rapidly decreased thereafter (Figure 2(a)).

When the same test was carried out at higher 20% strain amplitude, the modulus
was about two orders of magnitude lower (Figure 2(b)). Zooming in on the graph
showed that initially, the modulus decreased with time. The decrease was an
exponential decay at first and a continued decrease afterwards. The
exponential decay is typical of viscoelastic stress relaxation behaviour,
while the continued decrease afterwards could be probably because of fatigue
resulting from repeated oscillations. After about 8 minutes, the modulus
started to increase again, which was probably due to new polymer chains
forming faster than the rate of breaking. The increase was, however, much
slower compared to that observed in the test with 0.2% strain. Interestingly,
the loss factor here was greater than 1, implying that the bone cement behaves
more like a fluid when it is being significantly deformed. The loss factor did
not go lower than 1 until after 25 minutes, when the rapid curing phase had
already begun.

The properties of the bone cement continuously evolve because of the curing
reaction. Hence, it is important to evaluate the so-called mutation time,
which is defined as the time required for change in a given property by a
factor equal to the Euler’s number $\mathfrak{e}$. Taking the complex modulus
as the property of interest, the bone cement has a mutation time of about 180
seconds and 700 seconds at 0.2% and 20% strain amplitude respectively. For
reliable measurements, it is important that the measurement times are much
less than mutation times to avoid the effect of the evolving properties in
measurements. This is ensured to be the case for most of our measurements. As
an exception, the measurement times are not negligible in the initial few
points in the frequency sweep but they are still much smaller than mutation
times.

These results show that although the bone cement behaviour is predominantly
solid when it is at rest or at very small deformations, while high
deformations cause the bone cement to behave more like a fluid. This indicates
that the bone cement is suitable for injections, despite this not being
apparent from the test Rh1a. Furthermore, the curing process is inhibited by
the deformations, as is evident from the much lower magnitude as well as a
delayed and lower rate of increase in the complex modulus during the entire
duration of the test.

###  3.2 Replication of injection (Tests Rh2a and Rh2b)

Figure 2(c) shows the results of the test Rh2a. During the initial stage of
the test when the strain amplitude and frequency were low, the complex modulus
increased gradually due to polymerization, slightly faster than at initial
time of Rh1a (Figure 2(a)) test due to the lower strain used. In the next
stage of the test, while it was subjected to higher strain and frequency, the
modulus immediately dropped to 1% of the original magnitude and the loss
factor jumped to a value higher than 1. This again showed the bone cement’s
transition to more fluid-like behaviour as soon as it is deformed during
polymerization. This was followed by an exponential decay of the modulus, as
was also observed in the previous test Rh1b (Figure 2(b)). When the bone
cement is brought to (almost) rest again in the final stage of the test (i.e.
at about 12 minutes time), there is an immediate but small recovery of about
8% in the modulus. The polymerization is resumed and the rapid curing phase
occurs at about similar time of 25 minutes, after which the bone cement
solidifies. We note however that properties of bone cement modulated in this
way (simulating more injection-like scenario), are different to those measured
at low deformations in test Rh1a (Figure 2(a)). For example at the end of the
test, the complex modulus is 2.5-fold lower and loss modulus is about 2-fold
higher due to structural disturbances during curing. Carrying out the same
test with the bone cement kept at 37 °C in the last stage of the test (Rh2b),
there is hardly any recovery, however the rapid curing phase of the bone
cement is triggered immediately. The bone cement is a fully cross-linked
polymer solid at 22 minutes, as evident from the plateaued complex modulus and
the loss factor value of 0.05.

The results show that the curing time is shortened when the bone cement is
exposed to higher temperature. Once inside the vertebra of the patient, the
bone cement starts curing rapidly. Vertebroplasty is often done in steps of
smaller injections, in which case it is important for practitioners to be
aware that the bone cement cures rapidly when exposed to the physiological
body temperature. If the subsequent injections are not done in a timely
manner, the viscosity of the cement inside the vertebra could get much higher
than the one being injected. This viscosity difference could result in
disadvantages like requiring higher injection force, unintuitive injection
force development over the course of the injection, and unstable flow patterns
inside the vertebra, as was highlighted in a previous work 19.

###  3.3 Step stress (Test Rh3) and step strain (Test Rh4)

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: (a) Test Rh3: Creep behaviour of bone cement upon applying stress in
consecutive steps. The steps are separated by dashed gray lines. (b) Test Rh4:
Stress relaxation behaviour of bone cement upon applying step strain. The
steps are separated by dashed gray lines. The shaded region in the following
graphs indicates the range of values, the dashed line inside the region
indicates the average. (c) Test Rh5: Complex modulus (blue) and loss factor
(red) during amplitude sweep. (d) Test Rh5: Complex viscosity (green) during
amplitude sweep (e) Test Rh6: Complex modulus (blue) and loss factor (red)
during frequency sweep (f) Test Rh6: Complex viscosity (green) during
frequency sweep.

Figure 3(a) shows the behaviour of the bone cement when subjected to step
stress. The resultant strain was observed to be exponentially decaying, which
is typical of viscoelastic behaviour, both when stress was applied and then
removed. Upon removal of the stress, the strain did not converge back to zero,
which indicated that there was permanent deformation in the sample. At very
high stress, i.e. at 2000 Pa, the bone cement showed almost purely viscous
behaviour as it deformed continuously, and there was no recovery after removal
of the stress.

Figure 3(b) shows the stress response of the bone cement to step strain. Two
tests were carried out with 1% and 100% step strain respectively. The
resultant stress was maximum at the start of the applied step strain, and then
decreased exponentially with time to a non-zero stress value, typical of
viscoelastic stress relaxation behaviour. As the applied strain was removed,
returning to zero strain state required stress in the opposite direction,
which again showed similar relaxation behaviour. The decay curve could be fit
to the relaxation equation

$\sigma(t)-\sigma_{\infty}=(\sigma_{0}-\sigma_{\infty})e^{-t/t_{\text{relax}}},$
(7)

where $t$ is the time since the start of step strain, $\sigma(t)$ is the
stress at time $t$, $\sigma_{0}$ is the stress at the start of relaxation,
$\sigma_{\infty}$ is the final stress the material relaxes to, and
$t_{\text{relax}}$ is the relaxation time. The relaxation times obtained from
curve fitting are shown in Figure 3(b). Compared to the case of 1% strain, the
relaxation times in the case of 100% step strain are lower, indicating that
the bone cement relaxes faster owing to lesser resistance to deformation. This
reinforces our previous observation that the bone cement has more fluid-like
behaviour while it is being deformed.

###  3.4 Amplitude sweep (Test Rh5)

Figures 3(c) and 3(d) show the range and mean of the complex modulus, loss
factor, and complex viscosity for three amplitude sweep tests on the bone
cement. The modulus grew initially, likely because the strain was too small to
inhibit the polymerization. At 0.02 - 0.04% strain, the modulus stopped
growing and started lowering afterwards. The stress at which the complex
modulus and the complex viscosity start to lower is the yield stress, which
was observed to be around 70-110 Pa for this bone cement. This is the stress
at which the bone cement starts to flow. Hence, this parameter is important to
understand the injectability of the bone cement. The given yield stress range
corresponds to injection force $F$ of about 0.7 N according to the equation
$F=(\pi R_{s}^{2})\frac{2L_{c}}{R_{c}}\sigma_{y}$ obtained from Equation 2,
which is relatively small and hence corroborates the use of bone cement in
clinical injections. However, it is important to note that this value holds
only for 295 $\pm$ 20 seconds time since mixing, and would increase the more
it is allowed to rest. In the 1-10% strain amplitude range, the loss factor
went above 1. This is called the crossover point, or the flow point, and
corresponded to about 500-1000 Pa stress ($\approx$ 3-7 N of injection force).
Finally, there was a sudden drop in the modulus near 200% strain or 2500-4000
Pa stress mark, which was likely because of insufficient friction between the
rheometer plates and the bone cement, causing the plates to slip. The evidence
for this phenomenon, also called wall slip, is provided in later sections.

###  3.5 Frequency sweep (Test Rh6)

Figures 3(e) and 3(f) show the range and mean of the complex modulus, loss
factor, and the complex viscosity over three frequency sweep tests on the bone
cement. We observed that the complex modulus kept increasing throughout the
duration of the test. On the other hand, the complex viscosity initially
increased slightly, then kept dropping as the angular frequency increased
beyond 1.3 rad/s. This angular frequency can be considered the inception point
of shear-thinning behaviour. The shear-thinning of the viscosity according to
power law, i.e. Equation 1, started at about 2.6 rad/s. Fitting the curve to
the power law equation yields the power law flow index $n=0.21$. Value of $n$
between 0 and 1 indicates shear-thinning, which is expected for the PMMA bone
cement. Measurements at higher angular frequency could not be executed due to
the rheometer limitation of maximum frequency of 100 Hz.

###  3.6 Shear rate dependence (Tests Rh7a, Rh7b, and Rh8)

(a)

(b)

(c)

Figure 4: (a) Flow curves from tests Rh6 (dashed red lines), Rh7a (solid black
lines) and Rh7b (dashed black lines) (b) Shear stress plotted against shear
rate from tests Rh7a (solid black lines) and Rh7b (dashed black lines) (c)
Viscosity evolution as a function of time at different shear rates from tests
Rh8. Values of $n$ are obtained by fitting the viscosity values at the
instances indicated by dashed lines to the power law equation (Equation 1).

Figure 4(a) shows the results from the various steady shear rate sweep tests
performed in rotational mode, also known as flow curves. The viscosity values
were not affected by whether the test was done over a shorter shear rate
ranges or all the way from 10-3 to 103 s-1. This implied that the duration of
the test did not affect the results, at least within the first 10 minutes of
mixing. The influence of curing was seen at very low shear rates in the order
of 10-4 s-1, where the viscosity increased. The shear-thinning is steady until
about 1 s-1. Upon curve-fitting, the flow index $n$ was obtained in the range
of 0.20-0.30, and the consistency index $K$ in the range of 700-1100 Pa sn.

As the shear rate went above 1 s-1, the viscosity drop was noticeably steeper,
until it started plateauing at shear rates greater than 20 s-1. The values of
$n$ were less than zero in this range, implying that the shear stress reduced
as the shear rate was increased, as seen in Figure 4(b). This could be a
result of structural breakdown of the material or a measurement artefact due
to the suspected wall slip effect. Performing the same tests using sandpaper
attached to both the top and bottom plates (see test Rh7b in ‘Experimental’
section), the results showed a slight improvement in the maximum shear stress
achieved (Figures 4(a) and 4(b)). This provided evidence that the drop in
shear stress indeed occurred due to wall slip. This was confirmed from visual
inspection through microscopy, as detailed in a later section.

Figure 4(c) shows results of tests Rh8, i.e. time evolution at various
constant shear rates. At the lowest shear rate 10-4 s-1, the bone cement
showed a linear increase in viscosity with time of about 480 kPa s/min. The
tests with higher shear rate showed lower magnitudes and slower increase with
time. At 1 s-1 shear rate, the viscosity kept decreasing for almost the entire
duration of the test. At 100 s-1 shear rate, the viscosity dropped almost
entirely in the early stages (first 30 seconds) of the test . Viscosity values
taken at various points in time for each of these tests were used to calculate
the flow index $n$ with time. For this, the viscosity for shear rate 10-4 s-1
was excluded since previous tests showed it was too low to induce shear-
thinning. The results showed that the flow index $n$ was 0.33 in the beginning
and 0.25 if the initial value from 100 s-1 is taken into calculation. These
values are in the same range as observed in tests Rh7a for shear rate less
than 1 s-1. However, the value of $n$ kept dropping with time and became
negative soon after the initial stages, as seen at 5 minutes mark in Figure
4(c). In fact, this was true even when the viscosity values from shear rates 1
and 100 s-1 were excluded from the calculation for $n$. This implied that the
viscosity values only at the beginning of the test gave plausible shear-
thinning characteristics. The same could be said for the consistency index,
which when interpreted as the viscosity at unit shear rate, gave $K=1083$ Pa
sn and kept decreasing with time.

###  3.7 Validity of the Cox-Merz rule for PMMA bone cement

Figure 5: Validation of the Cox-Merz rule

According to the Cox-Merz rule, for the same magnitude of shear rate and
angular frequency, the respectively measured steady shear viscosity and
complex viscosity must also be equal 26. To extend this rule to materials with
yield stress, Doraiswamy et al. 39 suggested multiplying the angular frequency
to the strain amplitude for the Cox-Merz rule to be true. To validate this for
the Vertecem V+ bone cement, the results from shear rate sweep and three
repetitions of frequency sweep measurements at various strain amplitudes are
superimposed in Figure 4(a). Note that, as we learned from the amplitude sweep
measurements shown in Figure 3(c), the 0.01% strain amplitude is within the
linear viscoelastic region, 0.03% is near the transition to non-linear region,
0.2% is in the non-linear region. The frequency sweep curves for 0.01% did not
coincide with those of steady shear viscosity, but they were closer for 0.03%.
The graphs coincided at 0.2%, which is in the non-linear viscoelastic range.
As a check, a single frequency sweep was carried out at 200% strain amplitude,
which is beyond the crossover point. The resulting curve seemed to extrapolate
the curves from the tests with 0.2% strain, and coincided with the rotational
shear rate graph until the aforementioned dip in viscosity due to wall slip.
This is surprising since the frequency sweep is generally done at a strain
amplitude that lies within the linear viscoelastic range. A possible
explanation is provided by Shafiei-Sabet et al. 40, that the PMMA structures
formed as a result of polymerization break under shear flow, but oscillatory
tests at low strains do not affect these structures. This means that the Cox-
Merz rule is not applicable to the Vertecem bone cement if the strain
amplitude is low, i.e. within the linear viscoelastic range, but it is only
valid for higher strain amplitudes in the non-linear range. This reveals the
conditional nature of the validity of the Cox-Merz rule for the Vertecem bone
cement.

###  3.8 Optical microscopy and visual inspection

Figure 6: Bone cement samples of 25 mm diameter and 1.5 mm thickness after
undergoing testing and curing on the parallel plate rheometer setup are shown
in (a), (b), (c), and (d). Microscopic images taken using 2.5x objective for
samples in (a) and (b) are shown in (e) and (f) respectively, giving a closer
look on the “ridge”-like formations on the sheared sample. Geometric analysis
of sample in (c), shown in (g), reveals the radius and shear rate beyond which
the ridges were formed

Examples of cured bone cement samples after tests on the rheometer are shown
in Figure 6. These samples were inspected microscopically to gain better
insight into the results. Figure 6a shows an unsheared sample simply left at
rest between the rheometer plates until it was cured, from test Rh9. Figure 6b
shows sample subjected to 100 s-1 for 15 minutes before letting it cure, from
test Rh10. We observed circular “ridge”-like formations formed in the sheared
sample as a result of continuous rotational deformation. The images from
optical microscopy, shown in Figure 6e and 6f, showed that the depth of the
ridges went almost through the entire thickness of the sample. The same ridges
were observed even in the sample sheared at 0.1 s-1 for 20 minutes (referred
to test Rh8, Figure 4(c)), as shown in Figure 6c. A closer look, shown in
Figure 6g, revealed that the sample looked normal in the center upto a radius
of about 13.66 mm, and the ridges were formed only beyond this radius. In a
parallel plate rheometer, the shear rate is not uniform across the sample,
hence the specified shear rate is actually the average shear rate along the
radius. Given that the shear rate at the periphery is 1.5 times the specified
shear rate, and the shear rate increases linearly with the radius, we could
calculate the critical shear rate as 0.082 s-1, beyond which the ridge
formation was observed.

Interestingly, there were no visible ridges in the sample exposed to
oscillatory deformation with 20% strain and 1 Hz for 30 minutes (refer test
Rh1b, Figure 2(b)), as shown in Figure 6d. This implies that rotational
shearing on the rheometer causes deformations in the bone cement that results
in the formation of deep circular ridges, likely stemming from the combination
of elastic-like instabilities and possibly Taylor-type vortex flows, as has
been observed for other types of materials 41, 42, 43. This could happen even
at low shear rates if the sample is sheared for sufficiently long time. The
ridges would not only cause a loss of contact with the top plate, but also a
lack of sufficient contact between neighbouring flow layers of the bone cement
needed for viscous resistance, which would result in underestimation of the
viscosity. This phenomenon limits the conditions in which the rheometer tests
can be done on the bone cement and thus makes standardization of tests very
difficult.

###  3.9 Tests on the custom-made injector (Tests Inj1 – Inj5)

(a)

(b)

Figure 7: (a) Injection force for various flow rates against time normalized
by respective injection times (b) The average force values for each flow rate
over a 5 second time period closest to the 245 seconds mark from the time of
mixing were used for regression analysis using Equations 5 and 6 to obtain
power law parameters $K$ and $n$.

The results of the injection experiments using the custom-made injector setup
at various flow rates are presented in Figure 7, which shows the measured
force against time normalized by the total duration of injection. As expected,
the results showed that the higher the flow rate, the more injection force is
required. For the two slowest flow rates, the force gradually increased
because the duration of injection was long enough and the shear rates were
small enough for the cement to cure while being injected. To obtain reliable
results, the average force values for each flow rate over a 5 second time
period closest to the 245 seconds mark from the time of mixing were used for
the regression analysis, as shown in Figure 7(b). This was done to avoid any
influence of curing time on the outcome of curve-fitting. The regression was
carried out using Equations 5 and 6 obtained from analytical calculations,
with power law parameters $K$ and $n$ as fitting parameters. The regression
analysis gave a good fit with $R^{2}=0.97$ yielding values $K=974$ Pa sn and
$n=0.30$. The value of $n$ is similar to those observed in test Rh7a for shear
rate less than 1 s-1 and in test Rh8 during the initial 30 seconds. Using
Equation 4, we could calculate the maximum shear rate, which were 10 s-1 and
165 s-1 for the lowest and the highest flow rates respectively. For shear
rates in this range, we observed decreasing viscosity with time in test Rh8
(Figure 4(c)). However, if this was truly the case, it would cause a decrease
in the injection force with time, which was not observed in our results
further evidence that the reducing viscosity seen in Figure 4(c) is due to the
ridges formed in the sample as a result of the rotational deformation.

###  3.10 Comparison of the methods

Test | Angular frequency * Strain amplitude/ |  $K$ (Pa sn) | $n$
---|---|---|---
| Shear rate / Flow rate |  |
Rh6, 0.01% strain | 6.28 $\times$ 10-5 to 0.0628 rad/s | 331 | 0.33
Rh6, 0.03% strain | 1.88 $\times$ 10-4 to 0.188 rad/s | 589 | 0.28
Rh6, 0.2% strain | 0.00126 to 1.26 rad/s | 1060 | 0.27
Rh7 | 0.001 to 1 s-1 | 700-1100 | 0.20-0.30
Rh8 | 0.001 to 1 s-1 | 1083 | 0.33
Inj1–Inj5 | 0.025 to 0.4 mL/s | 974 | 0.30
Table 2: Summary of power law parameters as obtained from various tests

The values of the parameters of the power law equation $K$ and $n$ are useful
for analyzing and simulating the injection flow behaviour of bone cement.
Usually, these values are extracted from experimentally measured flow curves
using rheometer tests like test Rh7a, or the oscillatory frequency sweep test
Rh6 if the Cox-Merz rule is known to be valid. They can also be extracted from
a capillary rheometer or an injector setup like in tests Inj1–5. The power law
parameters extracted from these tests are summarized in Table 2.

Analytical calculations show that the shear rate range during an actual
injection could go as high as 200 s-1. As previously discussed, the rotational
rheometer is not suitable at such high shear rates because of the wall sip and
ridge formation caused by rotational deformation. This could be avoided by
reducing the gap size between the rheometer plates, but that cannot be done in
this case since that the gap size must also be sufficiently large to ensure no
particle effects. For the same reason, we could not use our cone-plate setup
on the rheometer as it could only be used with a fixed gap of 49 microns,
which is not large enough for our application. Hence, only measurements for
shear rates below 1 s-1 can be reliably used, as the power law parameters thus
obtained are close to those obtained from the injection experiment. It must be
noted, however, that even for low shear rates, the ridge formation could occur
if the test duration was sufficiently long. It is important to exercise these
These cautions when performing rotational rheometer tests on PMMA bone
cements.

The flow curves can also be obtained from the oscillatory frequency sweep, but
as seen from Figure 5, it is important to ensure that the strain amplitude
used for the frequency sweep lies within the non-linear viscoelastic range.
Only then is the Cox-Merz rule valid and a feasible comparison to steady shear
flow (like during an injection) possible. This is a crucial finding since this
is in contrast to the conventional norm of performing the frequency sweep with
a strain amplitude within the linear viscoelastic region. Note that the
viscosity drop seen as a result of wall slip in the steady shear viscosity
from rotational measurements is not seen in the oscillatory frequency sweep
results. This, along with previously detailed visual and microscopical
inspection, confirmed that that problem of wall slip or ridge formation is
absent in oscillatory measurements. This demonstrates the advantage of
oscillatory tests over rotational tests, especially as the shear rates go
higher.

The setup of a capillary rheometer or the custom-made injector setup used in
this study is closest to an actual vertebroplasty setting. This kind of setup
circumvents the drawbacks of the rotational or oscillatory rheometry, and
yields mechanical parameters like effective or apparent viscosity and
injection force that is most representative of the real-world setting. It is
important to note that the flow model used and the analytical calculations
play an important role in the accuracy of the extracted rheological
parameters. The Herschel-Buckley model is arguably the more suited model since
the bone cement exhibited yield stress. However, the yield stress is low
compared to the shear stresses applied during the measurements in most cases,
hence the power law model was used in this study for simplicity. There needs
to be correction factors applied for flow entry, exit, change in tube
diameter, etc., but they are beyond the scope of this work. Hence, the
rheological parameters extracted from our setup are indicative numbers, and
are not expected to have the accuracy or sophistication of industry-standard
instruments.

##  4 Conclusion

From the various tests done, the curing PMMA bone cement was found to exhibit
viscoelastic behaviour with a two-phase curing process: a slow initial phase
followed by a rapid curing phase. The curing process is significantly affected
by deformations and surrounding temperature. Exposure to physiological body
temperature immediately triggered the rapid curing phase resulting in sudden
increase in viscosity. This could negatively affect the vertebroplasty
procedure if it is done in step-wise injections as a result of flow
instabilities caused by injection of a low-viscosity fluid in a high-viscosity
fluid environment. Properties like low yield stress and fluid-like behaviour
under deformation, as opposed to predominantly elastic contribution at rest,
make it suitable for injection.

While measuring the shear-thinning characteristics, rotational rheometry
underestimated the viscosity above 0.082 $s^{-1}$ shear rate. It was confirmed
from microscopical visual inspection that the dip in viscosity was caused due
to slippage between material and rheometer plates, and formation of circular
ridges within the sample material. The same was not observed in a sample
subjected to oscillatory rheometry and tests using our injector setup.

An important finding of the study is that Cox-Merz rule is valid only for
oscillatory frequency sweep tests done at strain amplitude in the non-linear
viscoelastic region, in contrast to the conventional norm of doing the test
within the linear viscoelastic region. This conditionally validity of the Cox-
Merz rule is important to take into consideration, especially when oscillatory
measurements are used to obtain the flow curves.

## Funding

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) – Project Number 327154368 – SFB 1313. We also acknowledge the
support by the Stuttgart Center for Simulation Science (SimTech). J.K.W.
acknowledges the European Union’s Horizon 2020 (H2020-MSCA-IF-2019) research
and innovation programme under the Marie Sklodowska-Curie grant agreement
893099 — ImmunoBioInks.

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