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journal of the mechanical behavior of biomedical materials 112 (2020) 104038
Available online 24 August 2020
1751-6161/© 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Contents lists available at ScienceDirect
Journal of the Mechanical Behavior of Biomedical Materials
journal homepage: www.elsevier.com/locate/jmbbm
Research paper
Hyperelastic and viscoelastic characterization of hepatic tissue under
uniaxial tension in time and frequency domain
Sarah-Jane Estermann a,b,c, Dieter H. Pahr a,c, Andreas Reisinger a,∗
aDepartment Anatomy and Biomechanics, Karl Landsteiner University of Health Sciences, Dr.-Karl-Dorrek-Straße 30, 3500 Krems an der Donau, Austria
bAustrian Center for Medical Innovation and Technology, Viktor Kaplan-Straße 2/1, 2700 Wiener Neustadt, Austria
cInstitute for Lightweight Design and Structural Biomechanics, TU-Wien, Getreidemarkt 9, 1060 Wien, Austria
ARTICLE INFO
Keywords:
Hepatic tissue
Tensile testing
Viscoelasticity
Hyperelasticity
Stress relaxation
Dynamic mechanical analysis
ABSTRACT
In order to create accurate anatomical models for medical training and research, mechanical properties of
biological tissues need to be studied. However, non-linear and viscoelastic behaviour of most soft biological
tissues complicates the evaluation of their mechanical properties. In the current study, a method for measuring
hyperelasticity and viscoelasticity of bovine and porcine hepatic parenchyma in tension is presented.
First, non-linear stress–stretch curves resulting from ramp loading and unloading, were interpreted based
on a hyperelastic framework, using a Veronda–Westmann strain energy function. Strain-specific elastic moduli,
such as initial stiffness 𝐸I, were thereupon defined in certain parts of the stress–stretch curves. Furthermore,
dissipated and stored energy density were calculated. Next, the viscoelastic nature of liver tissue was examined
with two different methods: stress relaxation and dynamic cyclic testing. Both tests yielded dissipated and
stored energy density, as well as loss tangent (tan 𝛿), storage modulus (𝐸′), and loss modulus (𝐸′′ ). In tension,
stress relaxation was experimentally more convenient than dynamic cyclic testing. Thus we considered whether
relaxation could be used for approximating the results of the cyclic tests.
Regarding the resulting elastic moduli, initial stiffness was similar for porcine and bovine liver (𝐸I∼
30 kPa), while porcine liver was stiffer for higher strains. Comparing stress relaxation with dynamic cyclic
testing, tan 𝛿of porcine and bovine liver was the same for both methods (t an 𝛿= 0.05 − 0.25 at 1 Hz). Storage
and loss moduli matched well for bovine, but not as well for porcine tissue.
In conclusion, the utilized Veronda–Westmann model was appropriate for representing the hyperelasticity
of liver tissue seen in ramp tests. Concerning viscoelasticity, both chosen testing methods – stress relaxation and
dynamic cyclic testing – yielded comparable results for 𝐸′,𝐸′′, and t an 𝛿, as long as elasticity non-linearities
were heeded.
The here presented method provides novel insight into the tensile viscoelastic properties of hepatic tissue,
and provides guidelines for convenient evaluation of soft tissue mechanical properties.
1. Introduction
Surgical training requires scenarios that are as life-like as possible.
However, before practising on real in vivo patients, teaching models
are necessary. The use of fresh human tissue is limited due to ethical
concerns, availability, and safety issues. Thus, models usually consist
of preserved human cadavers, animal organs or artificial materials.
Especially, the increasing trend towards minimally invasive meth-
ods in general surgery calls for improved laparoscopic training sys-
tems (Alli et al.,2017;Armijo et al.,2018;Chen et al.,2020). Chole-
cystectomy and hepatectomy, for instance, are common surgical pro-
cedures that are widely being performed laparoscopically and require
corresponding training systems which must include the liver (Alli et al.,
∗Corresponding author.
E-mail address: andreas.reisinger@kl.ac.at (A. Reisinger).
2017;Yoshida et al.,2019). In this context, porcine and bovine liver
models are deemed useful for laparoscopic training (Hildebrand et al.,
2007;Laird et al.,2011;Liu et al.,2018). Furthermore, liver models are
important for risk analysis in crash tests, due to the high susceptibility
of the liver to injury during vehicular crashes (Yoganandan et al.,
2000;Kemper et al.,2010). Another application for hepatic models is
in practising palpation to distinguish fibrotic from healthy tissue and
identifying tumours in open surgery approaches (Hata et al.,2011). It is
therefore important to accurately measure liver mechanical properties
in order to manufacture realistic models.
In general, characterizing mechanical properties of soft tissue can
be challenging due to potentially non-linear and viscoelastic material
https://doi.org/10.1016/j.jmbbm.2020.104038
Received 29 May 2020; Received in revised form 3 August 2020; Accepted 10 August 2020
Journal of the Mechanical Behavior of Biomedical Materials 112 (2020) 104038
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S.-J. Estermann et al.
behaviour, as well as the need for measuring small loads resulting from
the low stiffness (with the Young’s modulus typically ∼1 MPa (Akhtar
et al.,2011)).
Hepatic tissue is difficult to handle in experimental setups and has
predominantly been studied in compression (Tamura et al.,2002;Kiss
et al.,2004;Ocal et al.,2010;DeWall et al.,2012;Jing et al.,2016)
and shear (Liu and Bilston,2002;Nicolle et al.,2010;Wex et al.,2013;
Zhu et al.,2013;Capilnasiu et al.,2020), regarding elastic as well
as viscoelastic tissue properties. Tensile tests on hepatic parenchyma
have been conducted for evaluating failure properties (Brunon et al.,
2010;Kemper et al.,2010;Lu et al.,2014;Duong et al.,2015;Dun-
ford et al.,2018); thus, illuminating the effect of large strains (>
10% (Marchesseau et al.,2017)) on the tissue, as experienced in the
context of trauma. However, it is also interesting to study liver under
small strains relevant for tactile properties.
Ramp tests of hepatic tissue result in non-linear stress–strain curves
(Chen et al.,1996), suggesting the use of hyperelastic modelling (Chui
et al.,2004;Gao et al.,2010;Umale et al.,2013). However, a purely
hyperelastic approach neglects viscoelastic behaviour which plays an
important role in liver tissue (Liu and Bilston,2000). As a partial
solution to this problem, the idea of pseudoelasticity was introduced
by Fung (1993) for interpreting loading–unloading curves exhibiting
hysteresis (which is an indicator for viscoelasticity), utilizing a hyper-
elastic framework: Loading and unloading is thereby modelled sepa-
rately, yielding two distinct stets of hyperelastic parameters. Another
approach to analysing non-linear behaviour is to define certain strain
ranges, where elastic moduli can be calculated (Fung,1967). For
example, characteristic low-strain and high-strain elastic moduli were
determined for liver capsule (Hollenstein et al.,2006;Karimi and Sho-
jaei,2018) and for kidney capsule (Snedeker et al.,2005) in uniaxial
tension.
However, these methods are only descriptive of what is observed
but fail to describe the constitutive effect of viscoelasticity itself. To
study viscoelasticity in a more rigorous manner, the most common
methods are: testing at different strain rates, stress relaxation (under
constant strain), creep (under constant stress), and dynamic cyclic
testing (oscillating stress and strain) which is also known as dynamic
mechanical analysis (DMA). The results of these various testing meth-
ods can be compared with each other. For example, Bartolini et al.
(2018) compared viscoelastic properties from indentation at different
oscillatory frequencies with indentation at different strain rates for a
soft silicone polymer — thus connecting frequency domain and time
domain experimental data. Following this train of thought, viscoelastic
properties extracted from relaxation experiments could be compared
with the same type of properties found in dynamic cyclic tests.
Regarding stress relaxation of hepatic tissue, experiments have been
conducted under compression (Taylor et al.,2002), shear (Liu and Bil-
ston,2000,2002), tension (Chen et al.,2011) and indentation (Mattice
et al.,2006;Estermann et al.,2020). DMA has been done on liver tissue
in compressive and shear conditions (Kiss et al.,2004;Capilnasiu et al.,
2020). However, the authors are not aware of publications on liver
parenchyma exposed to tensile oscillatory strain for assessing tissue
viscoelastic properties. This may be due to experimental difficulties
connected to tensile testing of extremely soft materials; with DMA being
more complex than other testing methods, such as stress relaxation or
creep. Thus, the question arises whether simple relaxation data could
be utilized instead of DMA for hepatic tissue in tension. Viscoelastic
properties, which are usually extracted directly from dynamic cyclic
tests (e.g. loss tangent, storage modulus, and loss modulus), can be cal-
culated based on relaxation data via transformation from time domain
to frequency domain (Ocal et al.,2010).
The current study aims at a comprehensive characterization of
hepatic tensile mechanical properties, meaning that non-linear as well
as viscoelastic behaviour must be taken into account. For this reason,
different types of tests were conducted on porcine and bovine hepatic
parenchyma tissue. First, ramp loading and unloading was done, yield-
ing hysteresis and non-linear behaviour. These curves were interpreted
with a pseudoelastic model and elastic moduli, defined at specific strain
levels, were calculated. Furthermore, based on energy considerations,
the ratio of dissipated to stored energy density was evaluated.
Next, two different methods of viscoelastic characterization – stress
relaxation and DMA – were utilized for extracting loss tangent (tan 𝛿),
storage modulus (𝐸′), and loss modulus (𝐸′′). These properties stem-
ming from the two testing methods were finally compared with each
other, addressing the question whether a simplified viscoelastic eval-
uation via stress relaxation (assuming an ideal step-displacement) is
justified for hepatic tissue in tension. In addition to the above men-
tioned viscoelastic properties, dissipated and stored energy density
were calculated for relaxation and DMA, yielding results that are
specific for each testing method.
2. Materials and methods
2.1. Tensile test specimen
Hepatic samples were taken from eight porcine, and two bovine
livers, obtained from a local butchery about 24 h after slaughtering
and prepared for testing immediately upon arrival in the laboratory
(Fig. 1a and b). Between slaughtering and sample preparation, whole
livers were stored at ∼4 ◦C in sealed plastic bags. Dunford et al. (2018)
recommended storage of liver in large blocks as opposed to small cut
samples. Rectangular tensile test specimen were extracted with their
long axis orientated parallel to the diaphragmatic and the visceral
surface of the organs: First a rectangular block (measuring around
75 ×50 ×30 mm) was cut out of a relatively homogeneous region
of the liver (Fig. 1c). After removing the capsule, the block was placed
in a 5 mm thick 3D printed cutting guide and a long knife was used
for extracting thin rectangular slices by cutting parallel to the guide
(Fig. 1d). Next, a 75 ×20 mm stencil was placed on the liver slice
and samples were obtained by cutting around the stencil (Fig. 1e). The
resulting samples consisted of parenchyma tissue without the Glisson’s
capsule, free of large blood vessels and bile ducts to ensure relative
homogeneity (Fig. 1f). Sample thickness was chosen as ∼5 mm due to
the fact that samples of under 2.5 mm thickness exhibited dehydration
at the sand paper contact area when being clamped in the mechanical
testing machine, while significantly thicker samples were difficult to
mount and clamp. The total numbers of porcine samples was 𝑛= 36,
as was for the bovine samples whose total number was also 𝑛= 36. All
samples were kept well hydrated by submerging them in 0.9% saline
solution. Mechanical testing was conducted immediately after sample
preparation at room temperature (∼23 ◦C), thus the tissues were never
frozen.
2.2. Mechanical testing
Tensile testing was conducted on a ZwickiLine testing machine
(ZwickRoell GmbH & Co. KG, Ulm, Germany) which allows axial forces
𝐹up to 2.5 kN and a machine displacement 𝑢Mof up to 113 cm. In soft
tissue testing, the resulting forces are very small and thus an additional
load cell for measuring 𝐹, with a measuring range up to 100 N and
accuracy of 0.02% (S2M/100N, Hottinger Baldwin Messtechnik GmbH,
Darmstadt, Germany) was connected to the universal data acquisition
module (DAQ) QuantumX MX840B (Hottinger Baldwin Messtechnik
GmbH, Darmstadt, Germany). The load cell was gripped by the lower
machine clamp.
Custom-built tissue clamps enabled manual fine-tuning of the grip-
ping force with a screw. One tissue clamp was secured in the upper
machine clamp and the other one inserted above the external load cell.
The tissue clamping surface was covered with sand paper (grit P80) to
prevent sample slippage during tension.
Journal of the Mechanical Behavior of Biomedical Materials 112 (2020) 104038
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S.-J. Estermann et al.
Fig. 1. Sample preparation: (a) and (b) whole bovine and porcine organs; (c) block of hepatic tissue is placed in the 3D printed cutting guide; (d) cutting along the surface of
the cutting guide yields a thin tissue layer; (e) rectangular stencil is placed on the tissue layer; (f) sample that is cut from the tissue layer (75 ×20 ×5 mm).
Fig. 2. Overview of the mechanical test setup: (a) the complete setup with the machine displacement 𝑢M, upper and lower machine clamps, digital image correlation (DIC) system,
and data acquisition module (DAQ) — connected to the load cell and displacement sensor (LVDT) in (b); furthermore in (b), hepatic sample clamped with the tissue clamps, point
markers to measure the length 𝑙, the tensile force 𝐹, and the displacement of the LVDT 𝑢LVDT.
Journal of the Mechanical Behavior of Biomedical Materials 112 (2020) 104038
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S.-J. Estermann et al.
For the evaluation of the dynamic cyclic tests, extremely small phase
shifts between force and displacement of a few milliseconds need to be
measured, requiring perfectly synchronous measurements. Thus, a lin-
ear variable differential transformer (LVDT) position sensor (Hottinger
Baldwin Messtechnik GmbH, Darmstadt, Germany) was also connected
to the same DAQ, for measuring displacement 𝑢LVDT at exactly the same
time as the load cell measurement.
The LVDT displacement, however, does not only describe tissue
displacement, but also includes contributions stemming from machine
and setup stiffness as well as clamping of the sample. For this reason,
a specific measuring length 𝑙was defined with point markers on the
sample whose change in length was recorded optically by the digital
image correlation (DIC) system ARAMIS (GOM GmbH, Braunschweig,
Germany). To certify uniform displacement within the sample, three
marker pairs were arranged along the width of the sample and length
𝑙was reported as the average of the three distances. Actual tissue
strains were based on these DIC displacements, while the temporal
information, needed for the dynamic cyclic tests was extracted from
the LVDT measurement.
An overview of the mechanical test setup and the measured proper-
ties (force 𝐹, machine displacement 𝑢M, position sensor displacement
𝑢LVDT, and sample measuring length 𝑙) are depicted in Fig. 2.
A mounting procedure of the samples in the testing machine, similar
to the one described by Manoogian et al. (2009) and Kemper et al.
(2010), was followed: First the top tissue clamp was removed from the
testing machine and the sample was placed on the sand paper surface
of the clamp and aligned. Used sand paper was always replaced by a
new layer for each sample. Next, the sample was inserted in the testing
machine, attached to the top clamp with its bottom end hanging freely.
At this stage, each sample was allowed to hang under its own weight
(∼0.08 N) for approximately 3 min to ensure an equal small preloading
of the tissue. Finally the bottom clamp was carefully closed, the sticky
markers for DIC were applied, and sample cross-section was measured
with an analogue calliper at three locations along the measuring length.
The initial cross-section 𝐴0was taken as the average of these three
measurements. The initial length 𝑙0was defined based on a single DIC
image taken before starting mechanical loading.
Each sample was exposed to one of the following mechanical tests:
ramp test, stress relaxation, or dynamic cyclic test (Fig. 3). All 36
samples of the porcine and bovine tissue were divided among the three
testing methods, resulting in 12 samples per test and tissue type.
Ramp test. Upon starting extension of the samples, a trigger signal was
sent from the testing machine to the DIC system, prompting both mea-
surements to start. The cross-head of the testing machine was displaced
by 𝑢M= 10 mm (resulting in a maximum engineering tissue strain of
0.10 − 0.14) at a speed of 5 mm/min (corresponding to a strain rate of
∼0.001 s−1) and then moved back to its initial position at the same speed
(see Fig. 3a). Due to the highly strain rate dependent behaviour of soft
biological tissues, evaluation of elastic properties requires quasistatic
strain rates (Fung,1967;Miller and Chinzei,1997). A similar strain
rate was utilized as the rate reported for hepatic capsule in tension
by Brunon et al. (2010) who considers 0.001 s−1 to be quasistatic. Force
𝐹, from the 100 N load cell, and lengths 𝑙between the markers, from
DIC for calculating strains, were both recorded at a measuring rate of
10 Hz.
Stress relaxation. A cross-head displacement of 𝑢M= 5 mm (∼0.06 tissue
strain based on DIC measurement) was applied within ∼0.4s and held
for 300 s (see Fig. 3b). Force 𝐹and lengths 𝑙between the markers
were both recorded at a rate of 10 Hz by the 100 N load cell and DIC,
respectively.
Dynamic cyclic test. After capturing a single image with the DIC system
for defining the initial length between the markers 𝑙0, a cross-head
displacement of 𝑢M= 6 mm (∼0.08 tissue strain) was applied and
then held for 250 s to allow best possible relaxation. Next, sinusoidal
displacements of ±0.5mm (±0.006 tissue strain) were applied at 𝑓= 0.5,
𝑓= 1.0,𝑓= 1.5, and 𝑓= 2.0 Hz for 100 cycles at each frequency.
Low frequencies were chosen to avoid inertia effects (Nicolle and
Palierne,2010;Chatelin et al.,2011) and to represent tactile palpation
of the material (Estermann et al.,2020). Samples were allowed to relax
at 𝑢M= 6 mm for 250 s between each tested frequency to enable
comparison of the frequency-dependent tissue response. See Fig. 3c for
an overview of the testing procedure.
The chosen strain level ensured that samples were not compressed
but stayed in tension throughout the test. Lengths between markers 𝑙
were recorded by the DIC system (at 10 Hz for 𝑓= 0.5 Hz and at 20 Hz
for the other frequencies). Pretests showed that phase shifts between
stress and strain were expected to be ∼15 ms for liver. Thus, in order
to resolve such small phase shifts, measuring of force and displacement
must be perfectly synchronized. For this reason, 𝑢LVDT from the LVDT
was utilized alongside 𝐹from the load cell, both being controlled by the
same data acquisition module and software at a sampling frequency of
100 Hz. The LVDT position sensor yielded the actual occurrence time
of the displacement without delay, while DIC provided the accurate
displacement (and strain) amplitude.
2.3. Data analysis
Uniaxial tissue stretch 𝜆(in direction of the main sample axis) is
expressed as the ratio between deformed length 𝑙and the initial length
𝑙0as
𝜆=𝑙
𝑙0
=𝜀+ 1 ,(1)
with 𝜀being engineering strain. Cauchy (true) stress 𝜎T– assuming
incompressibility of hepatic tissue (Chui et al.,2004) – and engineering
stress 𝜎Ewere calculated based on the measured tensile force 𝐹and
initial cross-section 𝐴0as
𝜎T=𝐹
𝐴0
𝜆(2)
and
𝜎E=𝐹
𝐴0
.(3)
Ramp test. Hepatic parenchyma tissue is viscoelastic, exhibiting hys-
teresis even during extremely slow loading and unloading cycles. Thus,
a purely hyperelastic treatment of the material is not valid, as this
would completely ignore viscoelasticity. The so-called pseudoelastic
approach – introduced by Fung (1993) – allows the separate evalu-
ation of the loading and unloading branch with two distinct sets of
characteristic hyperelastic parameters.
In order to establish a stress–strain relation, in terms of a hy-
perelastic model, the strain energy function 𝛹is introduced. For an
isotropic material, 𝛹=𝛹(𝐼1, 𝐼2, 𝐼3)depends on the strain invariants
𝐼1,𝐼2, and 𝐼3. The strain invariants are calculated based on the finite
deformation applied to the material. In the following sections, liver
tissue is modelled as incompressible (𝜈= 0.5), as is often done for soft
tissues (Fung,1967;Miller and Chinzei,1997;Gao et al.,2010;Roan
and Vemaganti,2011). The Poisson’s ratio of 𝜈= 0.434±0.16 for hepatic
parenchyma in tension reported by Chui et al. (2004), furthermore
supports the assumption of incompressibility.
Thermodynamic, symmetry, and energy considerations enable the
choice of 𝛹(𝐼1, 𝐼2, 𝐼3). Based on the comparison of different hyperelastic
models presented in Appendix A, the strain energy function suggested
by Veronda and Westmann (1970) for feline skin was utilized:
𝛹=𝑐1e𝛽(𝐼1−3) − 1+𝑐2(𝐼2− 3) + 𝑔(𝐼3),(4)
Journal of the Mechanical Behavior of Biomedical Materials 112 (2020) 104038
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S.-J. Estermann et al.
Fig. 3. Overview of the mechanical testing protocols with 𝑢Mbeing the machine displacement and 𝑡the time: (a) ramp loading and unloading sequence, (b) holding phase for
relaxation, and (c) dynamic cyclic testing (100 cycles for each frequency).
with 𝑐1,𝑐2, and 𝛽being the model parameters, and 𝑔(𝐼3)being a
function of tissue compressibility. After inserting the strain invariants
𝐼1=𝜆2+2
𝜆,𝐼2= 2𝜆+1
𝜆2, and 𝐼3= 1 for uniaxial tension of
an incompressible material into Eq. (4) and further simplifying with
𝑔(𝐼3) = 𝑔(1) = 0,𝑐2= −𝑐1
𝛽
2, and 𝑐1=𝑐, the strain energy function
results in
𝛹=𝑐e𝛽(𝐼1−3) − 1−𝑐𝛽
2(𝐼2− 3) ,(5)
leaving the two material parameters 𝑐and 𝛽to be determined.
In case of incompressibility, the uniaxial Cauchy stress 𝜎Tcan
be expressed in terms of the strain invariants 𝐼1and 𝐼2, according
to Holzapfel (2000), as
𝜎T= 2𝜆2−1
𝜆𝜕 𝛹
𝜕𝐼1
+1
𝜆
𝜕𝛹
𝜕𝐼2,(6)
Using Eqs. (5) and (6), the stress of the Veronda–Westmann model
can thus be written as
𝜎VW = 2𝜆2−1
𝜆𝑐𝛽 e𝛽(𝐼1−3) −1
2𝜆.(7)
To now obtain material parameters for the loading and unloading part
of the tensile ramp experiments, Eq. (7) was fit to the experimental
true stress with non-linear least squares method, utilizing a Levenberg–
Marquardt algorithm (Marquardt,1963), for loading and unloading
curves separately. Thus, the resulting fitting parameters were 𝑐load and
𝛽load (for loading) as well as 𝑐unload and 𝛽unload (for unloading). The
coefficient of determination 𝑟2was calculated in order to evaluate
how well the model data corresponded to the experimental data. See
Fig. 4 for a typical experimental stress–stretch curve alongside the
corresponding hyperelastic model.
In addition to the above described hyperelastic approach, a second
type of data evaluation was applied, that is more descriptive of the
stress–stretch curve shape. For each loading–unloading sequence, three
characteristic elastic moduli were defined: 𝐸Ias the initial elasticity,
𝐸II for the highest stretch during loading, and 𝐸III for the first stretch
during unloading. 𝐸I,𝐸II, and 𝐸III were calculated based on the slope
of tangent of 𝜎VW for the corresponding stretch ranges (see Fig. 4).
Furthermore, the ramp tests were interpreted based on energy con-
siderations: For stretching the material, mechanical work is required
Fig. 4. Typical stress–stretch curve of a bovine ramp test with the measured true stress
𝜎T(dashed red line), the corresponding Veronda-Westmann curve fit 𝜎VW (solid black
line), and the calculated elastic moduli (𝐸I,𝐸II, and 𝐸III ). The engineering stress 𝜎E
(dashed grey line) is used to calculate the dissipated energy density 𝑊dis and stored
energy density 𝑊st.
in the loading phase, which is partly dissipated and partly stored as
elastic strain energy. During the following unloading phase, part of the
stored energy is recovered elastically while another part is dissipated.
The ratio between the dissipated energy density 𝑊dis and the recovered
(or stored) energy density 𝑊st is referred to as relative dissipation,
𝑊dis∕𝑊st , and can be calculated based on the stress–stretch curves.
Thereby, 𝑊dis is the area between the loading and unloading curve and
𝑊st is the area under the unloading curve (see Fig. 4). Due to energy
considerations, however, engineering stress 𝜎Ewith the stretch ratio
𝜆as its energy conjugate (Shergold et al.,2006) were used for these
calculations (Fig. 4).
Journal of the Mechanical Behavior of Biomedical Materials 112 (2020) 104038
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S.-J. Estermann et al.
Fig. 5. Generalized Maxwell model, consisting of the longterm elastic modulus 𝐸∞
and the 𝑁parallel Maxwell branches with springs 𝐸1,𝐸2, . . . 𝐸𝑁and dashpots 𝜇1,𝜇2,
. . . 𝜇𝑁.
Stress relaxation. Assuming a step displacement, where the holding
phase is reached instantaneously, stress 𝜎and the constant strain 𝜀0are
connected via the time-dependent relaxation function 𝐸(𝑡)according to
𝜎(𝑡) = 𝐸(𝑡)𝜀0.(8)
The relaxation function can be approximated in terms of Prony series,
𝐸(𝑡) = 𝐸∞+
𝑁
𝑖=1
𝐸𝑖𝑒−𝑡
𝜏𝑖,(9)
which correspond to the generalized Maxwell model depicted in Fig. 5,
with the longterm elastic modulus 𝐸∞, the elastic moduli of the springs
𝐸𝑖, and the characteristic relaxation times 𝜏𝑖(which are related to the
dashpot viscosities 𝜇𝑖via 𝜏𝑖=𝜇𝑖∕𝐸𝑖) (Findley et al.,1989). Replacing
𝐸∞by
𝐸∞=𝐸0−
𝑁
𝑖=1
𝐸𝑖,(10)
in Eq. (9), expresses the relaxation function in terms of the initial
elasticity 𝐸0(the ratio between stress and strain at the beginning of
the holding phase) according to
𝐸(𝑡) = 𝐸0−
𝑁
𝑖=1
𝐸𝑖(1 − 𝑒−𝑡
𝜏𝑖).(11)
After inserting Eq. (11), Eq. (8) was utilized to approximate the
experimental stress relaxation, applying a non-linear least squares
method, implemented with a trust region reflective algorithm, which
limited fitting parameters to positive values. Thereby, the curve fit was
performed over the complete holding time of 300 s, as discussed below
and in Appendix B. Furthermore, based on considerations presented
in Appendix B,𝑁= 3 was chosen for further calculations, as was also
done by Ocal et al. (2010) for compression and Estermann et al. (2020)
for indentation of hepatic tissue in the context of relaxation within the
same magnitude of holding times.
Obtained, model parameters 𝐸𝑖and 𝜏𝑖were then utilized to calcu-
lated the storage and loss moduli 𝐸′and 𝐸′′ according to Gutierrez-
Lemini (2014) with
𝐸′=𝐸∞+
𝑁
𝑖=1
𝐸𝑖𝜔2𝜏2
𝑖
1 + 𝜔2𝜏2
𝑖
(12)
and
𝐸′′ =
𝑁
𝑖=1
𝐸𝑖𝜔𝜏𝑖
1 + 𝜔2𝜏2
𝑖
,(13)
where 𝜔is the angular frequency. The ratio of loss modulus to storage
modulus
𝐸′′
𝐸′= t an 𝛿=
𝑁
𝑖=1
𝜔𝜇𝑖𝐸2
𝑖
𝐸∞(𝐸2
𝑖+𝜔2𝜇2
𝑖) + 𝜔2𝐸𝑖𝜇2
𝑖
(14)
is called loss tangent t an 𝛿, which is a characteristic viscoelastic property
that describes the frequency-dependent material damping behaviour.
Concerning the Prony series curve fit the question arises whether
holding duration influences the resulting viscoelastic properties. The
reason for this being the following: Data was sampled equidistantly
throughout the holding phase of 300 s. Thus, the portion of rapid
stress decline at the beginning of the stress-time curve is emphasized
less in the curve fit, in contrast to the portion of nearly constant
stress in the middle and end part of the curve, where many more data
points are available. Further examination of the influence of holding
time is described in Appendix B, showing that tan 𝛿is fairly consistent
throughout the tested time span. Thus, if not further specified, tan 𝛿is
reported for a holding time of 300 s in the following sections.
Additionally to the frequency-dependent damping behaviour ex-
pressed by tan 𝛿and the dynamic moduli 𝐸′and 𝐸′′ , relative dissipation
𝑊dis
𝑊st is calculated for the performed experiment. The work density,
necessary for initially reaching 𝜀0during the fast ramp loading, can
be expressed as
𝑊0=1
2𝐸0𝜀2
0,(15)
with 𝐸0=𝐸∞+𝐸1+𝐸2+𝐸3(Eq. (10)) referring to the parallel springs
in Fig. 5. After waiting for the holding period to pass and the viscosities
to dissipate, the only spring contributing to the stress response is the
long-term stiffness 𝐸∞, meaning that the work, which can be elastically
recovered if the material were unloaded at this point, is
𝑊st =1
2𝐸∞𝜀2
0.(16)
Thus, the dissipated energy density, being the difference between 𝑊0
and 𝑊∞, can be expressed as
𝑊dis =1
2(𝐸1+𝐸2+𝐸3)𝜀2
0.(17)
Finally, the relative dissipation 𝑊dis∕𝑊st results in
𝑊dis
𝑊st
=𝐸1+𝐸2+𝐸3
𝐸∞
.(18)
Dynamic cyclic test. The analysis of dynamic cyclic tests consists of
finding tan 𝛿,𝐸′, and 𝐸′′ from sinusoidal data. For a linear viscoelastic
material exposed to a sinusoidal displacement
𝑢(𝑡) = 𝑢Asin(𝜔𝑡)(19)
with the displacement amplitude 𝑢A, angular frequency 𝜔, and time 𝑡,
the responding force is also sinusoidal with the same frequency and
amplitude 𝐹Abut shifted by a phase 𝛿(Fig. 6a)
𝐹(𝑡) = 𝐹Asin(𝜔𝑡 +𝛿).(20)
Due to the fact that phase shifts are expected to be in the range of a
few milliseconds, special attention needs to be taken concerning the
synchronicity of the force and displacement measurement. Thus, the
loss tangent tan 𝛿was based on LVDT displacements. Non-linear least
squares method was utilized to fit Eqs. (19) and (20) to 𝐹and 𝑢LVDT,
yielding the fitting parameters 𝐹A,𝑢A,𝜔, and 𝛿. This method was
applied to the last 30 cycles of each frequency test, leaving the first
70 cycles for preconditioning the material to a steady state oscillation.
Furthermore, after utilizing 𝑢LVDT for calculating tan 𝛿, storage and
loss modulus 𝐸′and 𝐸′′ were calculated using strains 𝜀based on the
DIC measurement with
𝐸′=𝜎A
𝜀A
cos 𝛿(21)
and
𝐸′′ =𝜎A
𝜀A
sin 𝛿 , (22)
with 𝜀Abeing the amplitude of engineering strain and 𝜎Abeing the
amplitude of true stress. Description of the measured quantities can be
found in Fig. 6b.
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Fig. 6. Schematic overview of the analysis of the dynamic tests: (a) the LVDT
displacement and the load cell force are utilized to identify the phase shift 𝛿due to
their perfect measuring synchronicity; (b) accurate signal amplitudes, 𝜀Aand 𝜎A, are
extracted from the strain, based on the DIC measurement, and the true stress, based
on the load cell force.
In order to describe the loss factor in terms of energy dissipation,
tan 𝛿can be written as the ratio between dissipated energy density and
maximum stored energy density (Roylance,2001):
2𝜋tan 𝛿=𝑊dis
𝑊st
.(23)
Thus, the relative dissipation 𝑊dis∕𝑊st per cycle is calculated, base on
the obtained tan 𝛿. In case of linear viscoelasticity, it is assumed that
the energy dissipation stems from viscous effects only and not plastic
behaviour.
2.4. Statistical analysis
Mean values and standard deviations were calculated over 12
bovine and 12 porcine samples for every result:
•from ramp tests the pseudoelastic parameters (𝑐load,𝛽load ,𝑐unload,
𝛽unload), elastic moduli (𝐸I,𝐸II ,𝐸III), and 𝑊dis ∕𝑊st
•from relaxation the Prony parameters (𝐸0,𝐸𝑖,𝜏𝑖for 𝑖= 1,2,3),
viscoelastic properties (𝐸′,𝐸′′,t an 𝛿), and 𝑊dis∕𝑊st
•and from DMA at 4 frequencies the viscoelastic properties (𝐸′,
𝐸′′,t an 𝛿), and 𝑊dis∕𝑊st.
For comparing the two methods of viscoelastic characterization, the
resulting viscoelastic properties tan 𝛿,𝐸′, and 𝐸′′ of relaxation and
DMA were checked for normal distribution (Shapiro–Wilk test). Then,
given normality, Welch’s 𝑡-tests were performed to identify significant
differences (𝛼= 0.05) between the measured properties for tan 𝛿,𝐸′,
and 𝐸′′ at 𝑓= 0.5,𝑓= 1.0,𝑓= 1.5, and 𝑓= 2.0Hz, depending on
the experimental modality without the requirement of homogeneous
variances.
Fig. 7. (a) Mean true stress 𝜎Tplotted over the mean stretch 𝜆for the ramp tests with
the shaded areas being the standard deviations of stress and stretch (𝑛= 12 for bovine
and 𝑛= 12 for porcine) and (b) the mean true stress 𝜎Tplotted over time 𝑡for the
relaxation tests, with the shaded area being the standard deviation of stress (𝑛= 12 for
bovine and 𝑛= 12 for porcine).
3. Results
Ramp test. The average values and standard deviations of true stress
and stretch resulting from the loading–unloading sequence were plotted
for all samples of the two tissue types in Fig. 7a. The characteristic
mean stress–stretch curves of porcine and bovine tissue are visibly
different from each other with the average maximum stress for porcine
tissue being 19.15 kPa and for bovine tissue 10.69 kPa. The maximum
average strain applied to the samples was ∼11 %.
The experimental curves – exhibiting non-linearity and hysteresis –
were interpreted with a pseudoelastic Veronda–Westmann model with
the resulting parameters for the loading and unloading part given in
Table 1. The coefficient of determination being around 0.998 for bovine
as well as porcine tissue, signifies an excellent agreement between
experimental data and model. Parameters 𝑐load and 𝑐unload can be inter-
preted as shear-like moduli with 𝛽load and 𝛽unload being dimensionless
exponential parameters (Limbert,2019). The initial zero-strain shear-
like modulus 𝑐load was higher for bovine tissue (0.542 ± 0.190 kPa) than
for porcine tissue (0.353 ± 0.237 kPa).
Based on the pseudoelastic model, elastic moduli at different strain
values were calculated (Table 1). While, the initial stiffness 𝐸Iwas
nearly the same for porcine and bovine tissue, the average elastic
moduli for larger strains 𝐸II and unloading 𝐸III were more than twice
as high for porcine liver. For the ramp loading and unloading sequence,
the ratio between dissipated and stored energy density was calculated
and results are given in Table 1. The relative energy dissipation being
Journal of the Mechanical Behavior of Biomedical Materials 112 (2020) 104038
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Table 1
Results of the ramp tests for bovine and porcine tissue, given as mean values and standard deviations: parameters of the pseudoelastic Veronda-Westmann model 𝑐load,𝛽load ,𝑐unload,
and 𝛽unload, the coefficient of determination 𝑟2, elastic moduli 𝐸I,𝐸II , and 𝐸III, and relative dissipation 𝑊dis ∕𝑊st.
Tissue 𝑐load [kPa] 𝛽load [–] 𝑐unload [kPa] 𝛽unload [–] 𝑟2[–] 𝐸I[kPa] 𝐸II [kPa] 𝐸III [kPa] 𝑊dis∕𝑊st [–]
Bovine 0.542 ± 0.190 21.76 ± 8.953 0.013 ± 0.007 85.13 ± 23.97 0.998 32.00 ± 11.18 283.9 ± 92.94 689.6 ± 221.5 1.129 ± 0.177
Porcine 0.353 ± 0.237 35.11 ± 15.39 0.009 ± 0.008 101.50 ± 26.54 0.998 29.77 ± 14.79 669.9 ± 299.7 1428.4 ± 512.9 1.052 ± 0.285
Table 2
Results of the stress relaxation tests for bovine and porcine tissue, given as mean values and standard deviations: parameters of the Prony series fit (𝑁= 3)𝐸0,𝐸1,𝜏1,𝐸2,𝜏2,𝐸3,
and 𝜏3, the resulting root-mean-square error RMSE, and the relative dissipation 𝑊dis∕𝑊st .
Tissue 𝐸0[kPa] 𝐸1[kPa] 𝜏1[s] 𝐸2[kPa] 𝜏2[s] 𝐸3[kPa] 𝜏3[s] RMSE [kPa] 𝑊dis∕𝑊st [–]
Bovine 117.96 ± 30.24 33.92 ± 8.85 0.58 ± 0.15 20.16 ± 5.67 11.07 ± 0.83 26.61 ± 6.81 140.48 ± 6.84 0.03 ± 0.01 2.17 ± 0.14
Porcine 129.89 ± 64.91 44.19 ± 20.34 0.34 ± 0.19 22.08 ± 12.04 7.33 ± 2.04 24.98 ± 13.83 115.80 ± 22.29 0.04 ± 0.01 2.64 ± 0.71
Table 3
Relative dissipation 𝑊dis∕𝑊st , loss tangent tan 𝛿, storage modulus 𝐸′, and loss modulus
𝐸′′ for the tested frequencies of bovine and porcine tissue.
𝑓= 0.5Hz 𝑓= 1.0Hz 𝑓= 1.5Hz 𝑓= 2.0 Hz
Bovine
𝑊dis∕𝑊st [–] 0.603 ± 0.183 0.683 ± 0.335 n.a. n.a.
tan 𝛿[–] 0.096 ± 0.029 0.121 ± 0.067 n.a. n.a.
𝐸′[kPa] 172.9 ± 88.74 158.4 ± 83.34 n.a. n.a.
𝐸′′ [kPa] 17.08 ± 9.975 17.55 ± 9.565 n.a. n.a.
Porcine
𝑊dis∕𝑊st [–] 0.611 ± 0.086 0.662 ± 0.283 0.696 ± 0.241 0.675 ± 0.254
tan 𝛿[–] 0.097 ± 0.012 0.114 ± 0.061 0.117 ± 0.017 0.102 ± 0.046
𝐸′[kPa] 527.2 ± 172.2 488.3 ± 163.9 556.3 ± 205.6 509.8 ± 163.5
𝐸′′ [kPa] 50.76 ± 16.75 52.23 ± 28.91 64.59 ± 25.49 48.00 ± 20.31
around 𝑊dis∕𝑊st ∼ 1.1for both tissue types, signifies that a similar
amount of energy was dissipated due to viscosity and plasticity as
stored elastically.
Stress relaxation. Characteristic stress relaxation curves were plotted in
Fig. 7b for bovine and porcine tissue by averaging the true stress of all
tested samples for each time instance. Prony series were fit to these
experimental curves, yielding the model parameters given in Table 2.
The resulting relaxation times represent 3 orders of magnitude with
average 𝜏1= 0.58 s, 𝜏2= 11.07 s, and 𝜏3= 140.48 s for bovine tissue and
average 𝜏1= 0.34 s, 𝜏2= 7.33 s, and 𝜏3= 115.80 s for porcine tissue.
Relaxation times were shorter for porcine tissue compared to bovine
tissue for each time scale.
Relative dissipation for bovine and porcine tissue was 𝑊dis∕𝑊st >2
(Table 2), meaning that more than twice as much energy was dissipated
during relaxation than stored elastically in the material. Porcine hepatic
tissue exhibited 𝑊dis∕𝑊st around 20% higher than bovine tissue which
is also mirrored in the trend that porcine tan 𝛿is higher than bovine
tan 𝛿(Fig. 8). Thus, the relaxation results indicate that porcine hepatic
tissue exhibits a higher viscous contribution than bovine tissue.
Dynamic cyclic test. For identifying the phase shift 𝛿between force and
displacement, sine curves were fit to the experimental force data 𝐹and
the displacement of the position sensor 𝑢LVDT. Due to the noisiness of
the force readings for the two higher frequencies and due to the fact
that the force level of bovine samples was even lower than porcine
samples, the analysis for bovine tissue at 𝑓= 1.5 Hz and 𝑓= 2.0 Hz
was not possible in the current setup. Dynamic viscoelastic properties
are thus given for 𝑓= 0.5,𝑓= 1.0,𝑓= 1.5, and 𝑓= 2.0 Hz for porcine
tissue and for 𝑓= 0.5and 𝑓= 1.0 Hz for bovine tissue in the following
sections. Due to the pre-stress and the small oscillatory amplitudes,
samples were under tension throughout the whole experiment.
Table 3 lists the relative dissipation 𝑊dis∕𝑊st depending on the
tested frequency for both tissue types, showing that on average
𝑊dis∕𝑊st = 0.64 for bovine and 𝑊dis ∕𝑊st = 0.66 for porcine liver.
𝑊dis∕𝑊st <1means that more energy was recovered elastically than
dissipated due to viscosity for each cycle.
Furthermore, given in Table 3, are the mean values and standard
deviations of the viscoelastic properties tan 𝛿,𝐸′, and 𝐸′′ found for
bovine and porcine hepatic tissue in the dynamic tests.
Comparison of the viscoelastic parameters. Loss tangent, storage modulus
and loss modulus were extracted from relaxation, as well as DMA tests.
Figs. 8–10 depict the frequency dependence of tan 𝛿,𝐸′, and 𝐸′′ as
was calculated based on the generalized Maxwell model for relaxation
and measured in DMA. Porcine and bovine loss tangents correspond
very well for the different testing methods throughout the examined
frequency range (Figs. 8a and b). When regarding the porcine storage
modulus (Fig. 9a), however, a discrepancy between relaxation (𝐸′∼
100 kPa) and dynamic cyclic testing (𝐸′∼ 500 k Pa) becomes apparent
for all tested frequencies. Bovine tissue, on the other hand, yielded
storage moduli that matched well for the testing methods (Fig. 9b). A
similar trend can be observed, concerning 𝐸′′: While for porcine tissue,
the dynamic loss modulus was more than two times higher than that
found in relaxation (Fig. 10a), bovine tissue exhibited 𝐸′′ relatively
independent of the testing method (Fig. 10b).
Regarding a given frequency, for instance 1 Hz, the loss tangent
tan 𝛿was not significantly different depending on the type of test for
porcine (𝑝= 0.3) or bovine tissue (𝑝= 0.07). The viscoelastic moduli 𝐸′
and 𝐸′′ found in dynamic tests, however, were significantly higher than
the relaxation 𝐸′and 𝐸′′ for porcine tissue at 1 Hz (𝑝= 0.000006 and
𝑝= 0.002). For bovine tissue at 1 Hz, only 𝐸′′ was significantly different
depending the testing method (𝑝= 0.01), while 𝐸′was statistically the
same for relaxation and cyclic testing (𝑝= 0.1).
4. Discussion
For evaluating liver mechanical properties, different tests were per-
formed in tension on bovine and porcine hepatic parenchyma sam-
ples. In order to describe non-linear as well as viscoelastic behaviour,
mechanical testing consisted of ramp loading and unloading, stress
relaxation, and DMA. In the following section, the resulting properties
from the different tests are discussed and finally the two viscoelastic
testing methods (stress relaxation and DMA) are compared with each
other.
Ramp test. The ramp stress–stretch curves were interpreted based on a
hyperelastic modelling approach, utilizing a Veronda-Westmann strain
energy function.
Chui et al. (2004) examined different strain energy functions for
hyperelastic modelling of combined compression and elongation of
porcine liver parenchyma at a much faster strain rate of ∼0.03 s−1 than
the current study (∼0.001 s−1), however without modelling unloading.
The Veronda-Westmann model thereby yielded average parameters of
𝑐= 0.07 kPa and 𝛽= 4.5, which are lower than the current results
(𝑐load ∼ 0.35 kPa and 𝛽load ∼ 35.1for porcine tissue). The differing
model parameters could be attributed to the higher strain rate, used
by Chui et al. (2004). Interestingly, it is counterintuitive that the
stiffness would be lower for higher strain rates. The results furthermore
differ when comparing the magnitude of stress found for similar strains,
with Chui et al. (2004) obtaining much smaller stresses than stresses
measured in the current study. Other publications (Lu et al.,2014;
Dunford et al.,2018) reported similar stress magnitudes as found in the
Journal of the Mechanical Behavior of Biomedical Materials 112 (2020) 104038
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Fig. 8. Loss tangent tan 𝛿and standard deviation measured in dynamic cyclic tests for different frequencies plotted alongside t an 𝛿, based on relaxation data, calculated for different
frequencies with the generalized Maxwell model for (a) porcine hepatic tissue (𝑛= 12 for relaxation and 𝑛= 12 for dynamic) and (b) bovine tissue (𝑛= 12 for relaxation and 𝑛= 12
for dynamic), asterisks marking frequencies at which the properties found in the two testing methods were significantly different (𝛼= 0.05).
Fig. 9. Storage modulus 𝐸′and standard deviation measured in dynamic cyclic tests for different frequencies plotted alongside 𝐸′, based on relaxation data, calculated for different
frequencies with the generalized Maxwell model for (a) porcine hepatic tissue (𝑛= 12 for relaxation and 𝑛= 12 for dynamic) and (b) bovine tissue (𝑛= 12 for relaxation and 𝑛= 12
for dynamic), asterisks marking frequencies at which the properties found in the two testing methods were significantly different (𝛼= 0.05).
Fig. 10. Loss modulus 𝐸′′ and standard deviation measured in dynamic cyclic tests for different frequencies plotted alongside 𝐸′′, based on relaxation data, calculated for different
frequencies with the generalized Maxwell model for (a) porcine hepatic tissue (𝑛= 12 for relaxation and 𝑛= 12 for dynamic) and (b) bovine tissue (𝑛= 12 for relaxation and 𝑛= 12
for dynamic), with asterisks marking frequencies at which the properties found in the two testing methods were significantly different (𝛼= 0.05).
current study, even though strain rates were also higher with 0.01 s−1.
This discrepancy between stress response found in Chui et al. (2004)
and other publications could be explained by differences in sample
fixation strategies during testing: For example, Chui et al. (2004) glued
the samples to movable plates while Lu et al. (2014), Dunford et al.
(2018), and the here-presented study utilized tissue clamps. Another
difference is that Chui et al. (2004) reported stretch based on the ma-
chine displacement, while the other mentioned studies utilized optical
measurement.
The current loading curves of the ramp tests were nearly linear for
small strains up to ∼3% (see Fig. 7a), which is similar to behaviour
reported by Hollenstein et al. (2006) and Snedeker et al. (2005), who
calculated initial stiffness up to 2% and 5% for hepatic and renal
capsule tissue, which means that a single modulus 𝐸Ican describe
tensile behaviour of liver up to a strain of 3% with sufficient accuracy.
Furthermore, it can be observed that the parenchyma is much softer
than the capsule when comparing 𝐸I∼ 30 kPa for bovine and porcine
tissue, found in the current study, with the values reported for capsule
of ∼1100 kPa (Hollenstein et al.,2006). This huge difference shows that
for complete liver characterization, capsule as well as parenchyma need
be examined.
While 𝐸Iwas the same for porcine and bovine tissue, stiffness differ-
ences between the two tissues only became apparent for higher strains
in 𝐸II and 𝐸III. This could be due to histological differences of the
tissues: Porcine hepatic tissue exhibits a higher content of collagen than
bovine tissue (Neuman and Logan,1950), which is the main structural
protein in healthy liver due to elastin content being very low (Kanta,
2016). While elastin mainly contributes to the initial stiffness in the
initial linear region of the stress–stretch curve, collagen characterizes
the tissue stiffness for higher strains (Duong et al.,2015). Thus, the
Journal of the Mechanical Behavior of Biomedical Materials 112 (2020) 104038
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higher stiffness in 𝐸II and 𝐸III of porcine liver could be explained by its
higher collagen content compared to bovine liver.
Regarding the comparison of animal and human hepatic tissue,
Kemper et al. (2010) tested human liver parenchyma in uniaxial tension
at different strain rates, showing that failure stress and stretch were
very similar to bovine but significantly different to porcine hepatic
tissue by comparing their results with Santago et al. (2009) for bovine
and with Uehara (1995) for porcine tissue. Thus, it would be interesting
to further investigate the similarity between the tissue types, not only
for failure properties but also for stiffness in sub-failure strain ranges.
Concerning its collagen distribution, human liver corresponds better
to bovine than porcine liver (Zhang,1999;Eurell and Frappier,2006;
Lowe and Anderson,2015), which might lead to similar stiffness of
human and bovine tissue, especially in terms of 𝐸II and 𝐸III. This,
however, still remains to be shown in future research.
The resulting relative dissipation of around 1 for porcine and bovine
tissue indicated that liver tissue is highly viscoelastic, which motivated
the detailed analysis via relaxation and dynamic cyclic testing.
Stress relaxation. In order to quantify the viscous properties of liver
tissue that already became visible as hysteresis and, with that, en-
ergy dissipation in the ramp tests, stress relaxation experiments were
performed.
The stress decline over time was modelled with a 3-element Prony
series and viscoelastic properties tan 𝛿,𝐸′, and 𝐸′′ were calculated in
the frequency domain. Resulting mean tan 𝛿= 0.07 − 0.22 for porcine
and mean tan 𝛿= 0.05 − 0.17 for bovine tissue corresponded very
well to previously published values. For instance, the loss tangent has
previously been reported to be in a range of 0.20 − 0.25 for dynamic
compression of canine liver (Kiss et al.,2004), dynamic shear of porcine
and murine liver (Wex et al.,2014;Zhang et al.,2017), and indentation
relaxation of porcine and bovine liver (Estermann et al.,2020).
Average storage and loss moduli for porcine as well as bovine
hepatic tissue of around 𝐸′∼ 100 kPa and 𝐸′′ = 5 − 20 k Pa found in
the current study match the values reported by Ocal et al. (2010) for
bovine liver in compression (𝐸′∼ 75 kPa and 𝐸′′ ∼ 10 k Pa after 48 h
preservation for comparable frequencies) well.
Furthermore, the absolute value of the complex modulus 𝐸∗=
𝐸′2 +𝐸′′2 = 10 − 40 kPa was reported by Zhang et al. (2007) for
compression tests on bovine liver, which is lower than the 𝐸∗∼
100 kPa of the current study. However, Zhang et al. (2007) refrigerated
the cut samples over night in saline solution before testing, which could
have lead to a decrease in stiffness (Dunford et al.,2018).
Kiss et al. (2004) reported 𝐸′= 50 kPa and 𝐸′′ = 10 kPa for
𝑓= 1.0 Hz in dynamic compression of canine liver, with 𝐸′being lower
than the current values and 𝐸′′ matching very well. The lower stiffness
concerning the storage modulus could be associated with differences
between the hepatic tissue types or differences between compression
and tension.
For all conducted experiments, the relative dissipation was calcu-
lated, yielding the highest values of 𝑊dis∕𝑊st for relaxation, compared
to ramp testing and DMA. 𝑊dis∕𝑊st describes tissue behaviour in a
specific experimental framework and is not understood as material
property. Differences in relative dissipation are caused by inherent
differences in the loading methods and cannot be compared directly
between the methods. For instance, relative dissipation of stress re-
laxation is not frequency-dependent and describes viscoelasticity in a
temporal sense. Relative dissipation of DMA, on the other hand, was
calculated depending on the frequency. However, 𝑊dis∕𝑊st can be
used as a straightforward parameter for comparing materials that were
tested with the same method.
Dynamic cyclic testing. Additionally to stress relaxation, DMA was con-
ducted as the gold standard in terms of measuring viscoelastic proper-
ties. The large strain ramp tests yielded notable hyperelastic behaviour
in their stress–stretch plots. However, due to the fact that the strain am-
plitudes used for DMA were very small (0.6%), linear elastic behaviour
was assumed for the given prestrain level (8%) for the DMA tests in the
range of the amplitude.
The sinusoidal stress and strain curves directly yielded the viscoelas-
tic properties tan 𝛿,𝐸′, and 𝐸′′ for different frequencies (𝑓= 0.5,
𝑓= 1.0,𝑓= 1.5, and 𝑓= 2.0 Hz for porcine tissue and 𝑓= 0.5and
𝑓= 1.0 Hz for bovine tissue). No clear trend, concerning the influence
of frequency on the measured properties, could be observed (Figs. 8–
10). A much larger range of frequencies is necessary to accurately
evaluate the frequency-dependent behaviour, for example using a shear
rheometer (Zhu et al.,2013;Wex et al.,2014;Zhang et al.,2017).
Nevertheless, the here-presented method proved feasible for probing
a few distinct frequencies of interest.
The tensile dynamic loss tangent of the current study matches
tan 𝛿for compression, shear, and indentation found in literature (Kiss
et al.,2004;Wex et al.,2014;Zhang et al.,2017;Estermann et al.,
2020). However, no other studies conducted in tension were found, for
comparing the current results of tensile storage 𝐸′and loss moduli 𝐸′′
directly.
As discussed above, storage and loss moduli from compressive ex-
periments corresponded to the current results for 𝐸′and 𝐸′′ agreeably
well for stress relaxation of bovine tissue. Furthermore, bovine tissue
yielded similar results in DMA as in stress relaxation. However, results
based on DMA for porcine tissue were higher than previously published
results for 𝐸′and 𝐸′′ in compression (Kiss et al.,2004;Zhang et al.,
2007;Ocal et al.,2010). This discrepancy could mean that there is
a larger difference between tensile and compressive properties for
porcine tissue than bovine liver.
Experiments, concerning viscoelasticity of human liver, by Lim
et al. (2009) yielded tan 𝛿∼ 0.6for 1 Hz. However, these results
were obtained via dynamic indentation on the whole intact organ,
including contributions from the capsule and under different boundary
conditions. Thus, to compare tan 𝛿,𝐸′, and 𝐸′′ of human hepatic tissue
to the current results, human liver needs to be tested with the same or
similar method.
Concerning relative dissipation, 𝑊dis∕𝑊st was calculated for each
tested frequency. When comparing 𝑊dis∕𝑊st between porcine and
bovine tissue, no significant difference can be observed between the
tissue types. Thus, based on the here-presented dynamic results alone,
it cannot be concluded which tissue type is more viscous.
Comparison of the viscoelastic parameters. Hysteresis found in the ramp
stress–stretch curves, motivated further examination of liver viscoelas-
tic behaviour. Thus, stress relaxation and DMA were conducted for
extracting storage and loss moduli, as well as loss tangent. While DMA
is the gold standard, concerning the evaluation of these viscoelastic
properties, it is connected to experimental difficulties. The following
section should answer the question whether relaxation experiments can
yield comparable results to DMA.
Stress relaxation experiments, consisting of a single holding phase,
were easier to conduct compared to the dynamic tests: For example, in
DMA the temporal stress and strain resolution was of essence as phase
shifts of ∼15 ms had to be measured, thus demanding an additional
LVDT position sensor. The DIC system which yields the accurate mag-
nitude of strains does not necessarily coincide perfectly in a temporal
sense with the displacement of the machine.
Usually, DMA is conducted on rheometers designed specifically for
this type of experiment. However, the current setup consisted of a
universal testing machine. Even though only limited testing frequencies
and no temperature variations were possible, we were able to conduct
the experiments without using a rheometer.
Ocal et al. (2010) also calculated storage and loss moduli based on
relaxation data as well as on dynamic cyclic testing for bovine hepatic
tissue, finding a good agreement between the two testing methods.
However their experiments were conducted in compression.
Concerning both porcine and bovine tissue, tan 𝛿was statistically
indiscernible depending on the testing method in the current study
Journal of the Mechanical Behavior of Biomedical Materials 112 (2020) 104038
11
S.-J. Estermann et al.
(Fig. 8). Big differences in testing method, however, became visible
for porcine liver when regarding 𝐸′and 𝐸′′ (Figs. 9a and 10a), while
agreement for 𝐸′and 𝐸′′ of bovine tissue was good (Figs. 9b and
10b). Stress relaxation was conducted at 6% tissue strain while dynamic
cyclic testing was done at a strain level of 8%. This difference in strain
seems to be irrelevant for bovine tissue, which can be explained by the
fairly linear behaviour in terms of elasticity in Fig. 7a; not however
for porcine tissue which exhibits a more pronounced non-linear stress
relation for 6–8% strain. This means that the difference between testing
methods in porcine storage and loss moduli could be explained by the
non-linear elastic behaviour found in the tested strain range, while loss
tangent tan 𝛿was unaffected by the difference in strain level.
In conclusion, the comparison of stress relaxation and DMA results
is feasible as long as elasticity non-linearities are kept in mind. The
results showed that tan 𝛿is quite robust in terms of slight variations of
strain level between relaxation and DMA, while 𝐸′and 𝐸′′ are more
severely affected. The here presented viscoelastic investigation is novel
in its application to hepatic parenchyma and provides a guideline for
evaluation of soft tissues mechanical properties in tension.
5. Limitations
Following limitations of this study should be pointed out:
•Concerning the stress relaxation experiments, a perfect step dis-
placement was assumed where the tissue is stretched instanta-
neously. A more sophisticated modelling approach, as suggested
for example by Oyen (2005), would include the influence of the
finite ramp time.
•The methods for evaluating tan 𝛿,𝐸′, and 𝐸′′ are valid for linear
viscoelastic materials, meaning that the viscoelastic properties are
independent of strain up to the limit of linear viscoelasticity. The
limit of linear viscoelasticity reported for liver in literature is
based on oscillatory shear experiments at different compressive
preloads (Liu and Bilston,2000;Tan et al.,2013;Wex et al.,2013;
Ayyildiz et al.,2014). For example, Ayyildiz et al. (2014) found a
linear shear strain limit of 1% at a prestrain of 5% for bovine liver.
In the current study, DMA was performed at 8% prestrain, which
was as low as possibly achievable with the given setup, while
still avoiding compression of the sample throughout the whole
test. Even though, the prestrain was higher than the limit for
linear elasticity of 3% found in the ramp tests, the amplitude of
oscillation (0.6%) was small enough to assume linear behaviour at
the given prestrain level. However, we did not perform amplitude
sweep experiments in tension to verify the linear viscoelastic
limit.
•Given the long duration of the experiments (4 min for ramp tests,
5 min for relaxation, and 23 min for DMA), tissue dehydration
cannot be ruled out completely. Wetting the samples during
testing was not possible, as water drops on the point markers
would have impeded the DIC measurement. Nicolle and Palierne
(2010) found an increase in stiffness and damping behaviour due
to dehydration in kidney tissue after a few minutes of dynamic
shear testing.
•Furthermore, it should be noted that the tissue was not perfused,
thus poroelastic behaviour was not modelled in the current study.
According to (Kerdok et al.,2006) excised unperfused samples are
stiffer and more viscous, compared to in vivo conditions.
6. Conclusion
Hepatic parenchyma tissue is non-linear and viscoelastic. Thus, for
comprehensively describing liver mechanical properties, both aspects
must be considered. First, the non-linear behaviour of porcine and
bovine hepatic tissue was analysed via ramp tests and interpreted in the
framework of a pseudoelastic Veronda-Westmann model for extracting
Fig. A.11. True stress–stretch curve of one bovine loading sequence fitted with
different hyperelastic models.
strain-specific elastic moduli. Next, a method was presented to measure
viscoelastic properties (tan 𝛿,𝐸′′ , and 𝐸′′) in tension utilizing stress
relaxation, as well as DMA. For each testing method, considerations
about dissipated and stored energy were presented.
DMA is the gold standard for examining linear viscoelastic prop-
erties, but is associated with many experimental difficulties (e.g. the
necessity of an extremely high temporal resolution of stress and strain
measurement). We were able to answer the question whether loss
tangent, storage modulus, and loss modulus, based on the relaxation
experiments, could approximate the dynamic viscoelastic properties,
affirmatively. However, special attention must be paid to the strain
level at which the experiments are conducted when comparing the two
methods, in order to avoid differing properties due to non-linearities.
The interesting discrepancies that were found regarding mechanical
properties of bovine and porcine tissue call for further investigation on
human hepatic tissue.
CRediT authorship contribution statement
Sarah-Jane Estermann: Methodology, Software, Formal analysis,
Investigation, Visualization, Writing - original draft. Dieter H. Pahr:
Resources, Funding acquisition, Project administration, Writing - re-
viewing & editing, Supervision. Andreas Reisinger: Conceptualiza-
tion, Methodology, Validation, Data curation, Project administration,
Writing - reviewing & editing, Supervision.
Declaration of competing interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Acknowledgements
The research was funded by the Niederösterreich Forschung &
Bildung, Austria Science Call Dissertations 2017 (SC17-016), and the
Austrian Center for Medical Innovation and Technology (funded in
the framework of COMET by BMVIT, BMDW, the Federal State of
Lower Austria and Standortagentur Tyrol). The authors acknowledge
TU Wien University Library for financial support through its Open
Access Funding Programme.
Journal of the Mechanical Behavior of Biomedical Materials 112 (2020) 104038
12
S.-J. Estermann et al.
Table A.4
Overview of material models with corresponding true stress expressions and model parameters, and the corresponding coefficient of
determination 𝑟2for a typical bovine hepatic sample.
Material model Stress expression Model parameters 𝑟2
Neo-Hookean 𝜎NH = 2𝜆2−1
𝜆𝑐1𝑐10.9054
Mooney–Rivlin 𝜎MR = 2𝜆2−1
𝜆𝑐1+𝑐2
1
𝜆𝑐1,𝑐20.9954
Veronda-Westmann 𝜎VW = 2𝜆2−1
𝜆𝑐𝛽e𝛽(𝐼1−3) −1
2𝜆𝑐,𝛽0.9980
Yeoh 𝜎Y= 2𝜆2−1
𝜆(𝑐1+ 2𝑐2(𝐼1− 3) + 3𝑐3(𝐼1− 3)2)𝑐1,𝑐2,𝑐30.9998
Fig. B.12. (a) The root-mean-square (RSME) and standard deviation of curve fitting
Prony series with different numbers of elements to the 300 s of stress relaxation of
bovine and porcine tissue (𝑓= 1 Hz); (b) the loss tangent t an 𝛿plotted for varying
relaxation durations (𝑁= 3,𝑓= 1 Hz).
Appendix A. Ramp test — Consideration of appropriate hyperelas-
tic models
For selecting an appropriate hyperelastic model to represent the
experimental ramp stress–stretch data, four different models varying
in complexity were tested (Neo-Hookean, Mooney–Rivlin, Veronda-
Westmann, and Yeoh). All four models have previously been used to
describe hepatic tissue (Chui et al.,2004;Hollenstein et al.,2006;
Umale et al.,2013). Table A.4 summarizes the models, applied in a
hyperelastic framework for an isotropic, incompressible material under
uniaxial tension with 𝐼1being the first strain invariant. The resulting
stress–stretch curves for the different models and the experimental data
are plotted in Fig. A.11 for a bovine hepatic sample during extension.
The coefficients of determination ranged from 𝑟2= 0.9054 for the Neo-
Hookean model to 𝑟2= 0.9998 for the Yeoh model (see Table A.4). The
standard deviation of 𝑟2of each model was about 0.0005, when calcu-
lated for all samples in loading and unloading. Thus the curves depicted
in Fig. A.11 are considered representative of the other samples.
The Yeoh model exhibited the highest coefficient of determina-
tion, however also has the largest number of fitting parameters. The
Veronda-Westmann model also provided excellent correlation with the
experimental data (𝑟2= 0.9980) and has only two fitting parameters.
Furthermore, the Veronda–Westmann model was originally developed
specifically for soft tissues (Veronda and Westmann,1970). Thus, the
Veronda–Westmann model was chosen for further calculations.
Detailed description of the different models and derivations of the
equations can be found elsewhere; for example in Holzapfel (2000)
or Martins et al. (2006).
Appendix B. Stress relaxation — Influence of Prony series ele-
ments and holding time
Prony series elements. Concerning the choice of how many terms are
necessary in Eq. (11) to accurately represent the experimental stress
relaxation, the root-mean-square error (RMSE) – which describes the
goodness of fit – was plotted for 𝑁= {1,2,3,4,5} in Fig. B.12a. After
reaching an average level of ∼0.03 kPa for 𝑁= 3, the RMSE did
not decrease significantly when the number of terms was increased to
𝑁= {4,5}. Compared to 𝑁= 3, the RSME was about ten times higher
for 𝑁= 1 and two times higher for 𝑁= 2. Thus, the three element
Prony series approach was selected.
When fitting the Prony series to the experimental data, the duration
of the holding time could influences the resulting viscoelastic proper-
ties. Thus, the loss tangent was examined by calculating tan 𝛿for cut-off
times ranging from 10 to 300 s. The behaviour of tan 𝛿depending on
changes in test duration, can be seen in Fig. B.12b. Even though the
loss factors, calculated for different holding times, were statistically not
distinguishable from each other, the influence of the finite ramp time
may not be negligible for very short holding times (Liu and Bilston,
2002). For further investigation, tan 𝛿is reported for 300 s.
Appendix C. Supplementary data
Supplementary material related to this article can be found online
at https://doi.org/10.1016/j.jmbbm.2020.104038.
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