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Poisson's ratio of bovine meniscus determined combining unconfined and confined compression
Journal of Biomechanics xxx (2018) xxx–xxx
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Short communication
Poisson’s ratio of bovine meniscus determined combining unconfined and confined compression E.K. Danso a,b,⇑, P. Julkunen b,c, R.K. Korhonen b a
Department of Mechanical Engineering, Colorado State University, Fort Collins, CO, USA Department of Applied Physics, University of Eastern Finland, POB 1627, FI-70211 Kuopio, Finland c Department of Clinical Neurophysiology, Kuopio University Hospital, POB 100, FI-70029, KYS, Kuopio, Finland b
a r t i c l e
i n f o
Article history: Accepted 4 July 2018 Available online xxxx Keywords: Poisson’s ratio Meniscus Stress relaxation Confined Unconfined Aggregate modulus Young’s modulus
a b s t r a c t Poisson’s ratio has not been experimentally measured earlier for meniscus in compression. It is however an important intrinsic material property needed in biomechanical analysis and computational models. In this study, equilibrium Poisson’s ratio of bovine meniscus (n = 6) was determined experimentally by combining stress-relaxation measurements in unconfined and confined compression geometries. The average Young’s modulus, aggregate modulus and Poisson’s ratio were 0.182 ± 0.086 MPa, 0.252 ± 0.089 MPa and 0.316 ± 0.040, respectively. These moduli are consistent with previously determined values, but the Poisson’s ratio is higher than determined earlier for meniscus in compression through biomechanical modelling analysis. This new experimentally determined Poisson’s ratio value could be used in the analysis of biomechanical data as well as in computational finite element analysis when the Poisson’s ratio is needed as an input for the analysis. Ó 2018 Published by Elsevier Ltd.
1. Introduction Meniscus plays an important role in the knee joint to distribute and absorb forces. This role is strongly affected by the biomechanical properties of the tissue. Certain elastic, viscoelastic and poroelastic biomechanical properties of meniscus have been characterized (Danso et al., 2015; Pereira et al., 2014; Sweigart et al., 2004; Tissakht and Ahmed, 1995). However, experimentally measured Poisson’s ratio has never been presented for meniscus in compression. In experimental biomechanical testing, the Poisson’s ratio is required for instance when calculating the Young’s modulus from indentation testing (Fig. 1). Also in finite element modelling simulating biomechanical behaviour of soft tissues, the Poisson’s ratio affects the results (Fig. 2). Some attempts have been made to determine the compressive Poisson’s ratio of meniscus using finite element optimization from creep response (Sweigart et al., 2004), determining the mean Poisson’s ratio for bovine meniscus between 0 and 0.01 depending on the location of meniscus. Similar values were also presented for different species including baboon, canine, human, lapine and porcine, with the highest reported mean value of 0.08 at the anterior
⇑ Corresponding author at: Department of Mechanical Engineering, Colorado State University, Fort Collins, CO, USA. E-mail address: [email protected] (E.K. Danso).
segment from the tibial aspect of lapine meniscus (Sweigart et al., 2004; Sweigart and Athanasiou, 2005). However, none of these values were directly measured from experiments. Goertzen et al. determined experimentally the Poisson’s ratio of meniscus being close to 1 (Goertzen et al., 1997), but the measurements were done in the tension. Values of the Poisson’s ratio in tension are typically much higher than those in compression. The aim of this study was to determine the bulk equilibrium Poisson’s ratio of bovine meniscus by the combination of stressrelaxation tests in unconfined and confined compression measurement geometries. This approach has been shown earlier to produce the same Poisson’s ratio with direct optical measurements of articular cartilage (Korhonen et al., 2002).
2. Methods 2.1. Sample preparation Mature bovine knee joints were obtained from a local slaughterhouse and stored at 25 °C until the experiment. Prior to the biomechanical tests, menisci were thawed in water bath at room temperature (21 °C). Cylindrical samples (n = 6) were prepared with a biopsy punch and a razor blade from random locations of both lateral and medial menisci. First, using a biopsy punch (diameter = 4 mm), a cylindrical plug was cut perpendicularly from the
https://doi.org/10.1016/j.jbiomech.2018.07.001 0021-9290/Ó 2018 Published by Elsevier Ltd.
Please cite this article in press as: Danso, E.K., et al. Poisson’s ratio of bovine meniscus determined combining unconfined and confined compression. J. Biomech. (2018), https://doi.org/10.1016/j.jbiomech.2018.07.001
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E.K. Danso et al. / Journal of Biomechanics xxx (2018) xxx–xxx
thickness of the samples was obtained by cutting the samples from the slanted bottom. The thickness (1.54 ± 0.31 mm) was measured with a digital caliper and the diameter (3.82 ± 0.02 mm) with a stereomicroscope. 2.2. Biomechanical testing
Fig. 1. Examples of the effect of Poisson’s ratio in the analysis of Young’s modulus from indentation testing. Indentation moduli have been analyzed with Hayes equation (for plane-ended indenter (Hayes et al., 1972)) and Hertzian contact equation (for spherical indenter) (Fischenich et al., 2018)), and plotted against ð1v 21 ÞpaE1 Poisson’s ratio. Hayes equation (Es ¼ ) with a = 0.5 mm, h = 2.5 mm, E1 = 2jh 0.83 MPa, j is Poisson’s ratio specific (Hayes et al., 1972). Hertzian contact equation ð1v 21 Þ (Es ¼ 1v 2 ) with v 2 = 0.3, a = 0.80 mm, d = 0.25 mm, E2 = 210 GPa, F = 4a1=2 d3=2 3F
E2
Biomechanical testing was conducted by using a custom-made material testing system equipped with a high precision load cell (model 31/AL311AR, Honeywell, Columbus, OH, USA; resolution: 0.005 N) and an actuator (PM1A1798, Newport Corporation, Irvine, CA, USA; resolution: 0.1 µm) (Korhonen et al., 2002). For the unconfined compression measurements, the samples were placed between an upper steel rod and the bottom of the measuring chamber filled with phosphate buffered saline (Fig. 4). For the confined compression measurement, the samples were placed into a cylindrical metallic chamber sealed at the bottom, and the upper meniscal surface was in contact with a porous platen. The only means of fluid flow was through the porous platen at the top. Unconfined compression measurements were first conducted for
2
0.048 N. ES = Young’s modulus of the tissue, E1 = stress-strain ratio from experiments, E2 = Young’s modulus of indenter, v 1 = Poisson’s ratio of tissue, v 2 = Poisson’s ratio of indenter, a = radius of indenter, j = theoretical scaling factor (Hayes et al., 1972), h = tissue thickness, d = indentation depth, F = applied force.
Fig. 3. Schematic of meniscus showing a cylindrical plug and the portion used for biomechanical testing.
surface to the bottom of the tissue (Fig. 3). This results in a cylindrical disc having a horizontally flat surface and a slanted bottom because of the wedge shape of meniscus (Fig. 3). The superficial surfaces of the samples were then kept intact, and the uniform
Fig. 4. Schematic of experimental set-up with parameters obtained. Confined compression measurement geometry (A) and unconfined compression measurement geometry (B) were combined to determine the Poisson’s ratio.
Fig. 2. Examples of the effect of Poisson’s ratio in finite element analysis. Cylindrical-shaped Neo-Hookean hyperelastic sample of 1 mm in radius and 1 mm in height, and meshed with axisymmetric 4-node elements (CAX4), was compressed by 30% of its original thickness in unconfined (A) and confined (B) compression geometries. The bottom of the sample and the axis of symmetry were fixed in axial direction and lateral direction, respectively. In addition to these, in confined compression, the movement of the sample edge was restricted in the lateral direction. The Young’s modulus (E) was 0.5 MPa and the Poisson’s ratios (m) were between 0 and 0.4. By calculating the shear (µ) and 6ð12mÞ E 2 bulk (j) moduli from E and m, the implemented material constants C10 and D1 of the strain energy density function can be calculated as follows: C10 = l2= 4ð1þ . mÞ; D1 = j = E Static analysis was conducted with nonlinear geometry (nlgeom) in Abaqus 6.13 (Dassault Systèmes, Providence, RI, USA).
Please cite this article in press as: Danso, E.K., et al. Poisson’s ratio of bovine meniscus determined combining unconfined and confined compression. J. Biomech. (2018), https://doi.org/10.1016/j.jbiomech.2018.07.001
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each sample. After these tests and following the release of the load (at least 1 h), the relaxed state of the tissue was assumed and confined compression measurements were conducted. After reaching contact with the meniscus, a steady-state prestress of 6 kPa was applied. Fifteen minutes after this, dynamic pre-conditioning (sinusoidal loading, 2% strain, 4 cycles) was performed and followed by 15 min of relaxation. When the force was the same as obtained after pre-stress, we assumed a good contact was reached. Using a stress-relaxation testing protocol, four steps (5% of the remaining meniscus thickness at each step) with a 100%/sec (unconfined) or 0.09 ± 0.02%/sec strain rate (confined) and <10 Pa/min relaxation criterion (to initiate the next step) were applied (Fig. 5A). In order to avoid exceeding the limit of the load cell, the ramp rate in confined compression was set to be slower. However, this choice does not affect the results as only the equilibrium response was analyzed. The Young’s and aggregate moduli were calculated from the slopes of the equilibrium stress-strain curves (Fig. 5B) obtained from unconfined and confined compression measurements, respectively. Subsequently, the Poisson’s ratio was determined indirectly for the samples by the known relationship between the Young’s and aggregate modulus (Colombo et al., 2013; Korhonen et al., 2002; Mahmoodian et al., 2009),
v¼
Es HA
1þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Es 1 8 HEsA 1 HA 4
ð1Þ
where Es is the Young’s modulus, determined in unconfined compression, and HA is the aggregate modulus, determined in confined compression. For comparing paired-samples, we applied non-parametric Wilcoxon signed rank test and bootstrapping was applied to determine 95% confidence interval for the mean mechanical parameters (SPSS software v. 19.0.0.2, SPSS Inc., IBM Company, Armonk, NY, USA). 3. Results Fig. 5A shows typical examples of stress-relaxation curves for unconfined and confined compression measurements. At equilibrium, linear stress-strain relationships were exhibited in both unconfined and confined compression geometries (Fig. 5B). The average Young’s modulus (ES ) and aggregate modulus (HA ) were 0.182 ± 0.086 MPa and 0.252 ± 0.089 MPa, respectively, and they were significantly different (Table 1, p < 0.05). The average Poisson’s ratio (v ), determined indirectly from Eq. (1), was 0.316 ± 0.040 (Table 1). Based on 1000 bootstrapped samples, the means ofES , HA and v were between 0.121 and 0.248 MPa, 0.188 and 0.320 MPa, as well as between 0.287 and 0.344, respectively, with 95% confidence intervals. 4. Discussion Fluid-independent elastic properties (Young’s and aggregate moduli) of bovine meniscus were determined by stressrelaxation tests in unconfined and confined compression measurement geometries of the same samples. Results from these data were used to indirectly determine the equilibrium Poisson’s ratio of meniscus. Although finite element analysis has been used earlier to determine the Poisson’s ratio of meniscus in compression, to the best of our knowledge, this is the first experimental study to determine the Poisson’s ratio of bovine meniscus in compression. The equilibrium Poisson’s ratio of articular cartilage in compression is well known and it varies greatly. Reported values range between 0.09 and 0.40 for different species (bovine, dog, monkey, rabbit, human) at different locations (femoral condyles, tibial plateau, ankle talus, humeral head, patella) (Athanasiou et al., 1991; Colombo et al., 2013; Jurvelin et al., 1997; Kiviranta et al., 2006; Korhonen et al., 2002; Mahmoodian et al., 2009; Wong et al., 2000). Our new values for the meniscus Poisson’s ratio are comparable with those of cartilage. For meniscus, there exist only limited information about the Poisson’s ratio in compression. Using finite element analyses, an earlier study (Sweigart et al., 2004) determined indirectly the drained Poisson’s ratio of meniscus for
Table 1 Material parameters of bovine meniscus, obtained through unconfined compression, confined compression, and mathematical approaches.
Fig. 5. Examples of stress relaxation curves in unconfined and confined compression geometries (A). Equilibrium stress-strain responses (mean ± SD) in unconfined and confined compression geometries (B). The slopes of the equilibrium, fluidindependent stress-strain responses in unconfined and confined compression geometries (B) results in the Young’s and aggregate moduli, respectively.
Sample number
Young’s modulus, ES (MPa)
Aggregate modulus, HA (MPa)
Poisson’s ratio,v (–)
Sample Sample Sample Sample Sample Sample
0.091 0.135 0.123 0.288 0.165 0.291
0.139 0.201 0.212 0.361 0.241 0.359
0.338 0.331 0.365 0.271 0.326 0.264
0.182 ± 0.086*
0.252 ± 0.089
0.316 ± 0.040
1 2 3 4 5 6
Mean ± SD
ES determined in unconfined compression, HA determined in confined compression, determined indirectly from ES and HA (Eq. (1)). * p < 0.05, compared to HA.
v
Please cite this article in press as: Danso, E.K., et al. Poisson’s ratio of bovine meniscus determined combining unconfined and confined compression. J. Biomech. (2018), https://doi.org/10.1016/j.jbiomech.2018.07.001
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different species (baboon, bovine, canine, human, lapine, porcine) to be between 0 and 0.08. With a similar approach, another study (Abdelgaied et al., 2015) also predicted the Poisson’s ratio for decellularised and natural porcine meniscus to be zero. Compared to the present study, our Poisson’s ratio is higher (Table 1), and there are a number of possible reasons for this difference. The first reason could be attributed to the different techniques used in determining the Poisson’s ratio. The previous studies determined the Poisson’s ratio using a computational finite element analysis and optimization method from creep indentation (Abdelgaied et al., 2015; Sweigart et al., 2004; Sweigart and Athanasiou, 2005), while we combined experimental stress-relaxation tests from unconfined and confined compression geometries. Due to the presence of proteoglycans in the tissue, our values may not be exactly the same as the drained Poisson’s ratio because of residual swelling pressure resulting in fluid pressure even at steady state. Also, Sweigart et al. (2004) modelled the meniscus tissue as an axisymmetric isotropic biphasic cylinder, without considering the meniscal architecture. It is well known that the collagen network architecture contributes significantly to the Poisson’s ratio of biological materials (Chegini and Ferguson, 2010; Ficklin et al., 2007; Kiviranta et al., 2006). Another possible reason for the difference is that their indentation geometry tends to test more the superficial layer properties than the bulk tissue properties. Therefore, we believe that our experimentally defined values represent more realistically the actual behavior of the entire meniscus. This is supported by an earlier study showing the method, used in the present study, to produce the same Poisson’s ratios of cartilage with directly measured values from optimal measurements (Korhonen et al., 2002). This presented method is also simple and easily available. It is however important to note that the results obtained in this study are valid only under the assumptions of linear elasticity and isotropy in both the radial and circumferential planes. Radial and circumferential Poisson’s ratios could be different, as the lateral expansion of the tissue largely depends on the collagen orientation (Changoor et al., 2011). Similarly as shown earlier for soft biological tissues (Fortin et al., 2000; Korhonen et al., 2002; Mäkelä et al., 2012; Martin Seitz et al., 2013), the equilibrium stress-strain relationship was linear in both unconfined and confined compression geometries (constant Young’s or aggregate modulus with strain), as well as in indentation for human meniscus (Danso et al., 2015). Possible nonlinear and strain-dependent behavior could be observed in the meniscus instantaneous modulus (from the peak forces) due to the nonlinear collagen fibril network. However, since we were interested in the equilibrium response of meniscus and the ramp rate was different in confined and unconfined compression measurements, we did not pursue in comparing these nonlinearities between the measurement geometries. For the same reason, the Poisson’s ratio was not analyzed in a time-dependent manner, even though viscoelastic materials like the meniscus have a time- and frequency-dependent Poisson’s ratio (Chegini and Ferguson, 2010; Lakes, 1992; Lakes and Wineman, 2006). A previous study (Langelier and Buschmann, 2003) showed the history-dependence on the biomechanical properties of cartilage in unconfined compression. In that study, quite small strains and increasing strain rates affected subsequent responses because of tissue softening. However, these effects were mainly seen in the peak stiffness, while the equilibrium stiffness remained more stable. Here, we studied only the equilibrium responses with similar strains used before (Korhonen et al., 2002) and the relaxation time between different tests was more than 1 h. Nonetheless, we cannot fully rule out the possibility of tissue alterations between the unconfined and confined compression tests. However, if there were tissue damages, it should affect negatively to the tissue, that is, the later measured aggregate modulus in confined compression
would become smaller and the Poisson’s ratio would approach zero. This is not what we observed. The samples were cut from the slanted bottom to make them horizontally flat and to enable tissue testing (Fig. 3). However, we do not think that cutting had a significant impact on the results, as most of the tissue remained intact from all the three main layers of meniscus (Petersen and Tillmann, 1998). The superior surface was preserved, instead of the inferior, as physiological joint loads from the femoral condyle are experienced through the top of the tissue. We though acknowledge that the results might not be exactly the same if cutting would have been done to the superior surface of the samples. As expected and due to the Poisson’s ratio being over zero, the isotropic elastic Young’s moduli from the unconfined compression measurements were lower than the aggregate moduli obtained from the confined compression measurements (Table 1). This result was not a surprise but to some extent confirms that there was a good fit of the sample in the confining chamber. These modulus values were also consistent with previous studies (Abdelgaied et al., 2015; Danso et al., 2015; Martin Seitz et al., 2013). Obviously the confined chamber could cause some friction between the sample and the walls of the chamber. However, friction should not affect the measured equilibrium parameters. On the other hand, a small gap between the specimen and the metallic chamber might affect the measured properties (Buschmann et al., 1997). In our study, although we cannot know for sure whether there were gaps or not, it took a slight effort to push the samples into the metallic chamber. If the fit was not good, the aggregate and Young’s moduli would be close to equal and the Poisson’s ratio would approach zero. In this study, we analyzed a ‘‘general” Poisson’s ratio for bovine meniscus and did not characterize site-specific differences. Therefore, all the samples both from the lateral and medial menisci were combined. For the same reason, the number of samples was not an issue. Furthermore, as can be seen from Table 1, the Poisson’s ratios of all the samples were really close to each other. Also, the range of values within 95% confidence interval was found reasonable. These support the analysis of one ‘‘generalized” value for the Poisson’s ratio of bovine meniscus in compression. Although Sweigart et al. obtained smaller Poisson’s ratio than we did, they also showed only small location-specific differences (Sweigart et al., 2004). In conclusion, we propose that in the analysis of biomechanical data, such as indentation moduli, and in finite element models where the equilibrium Poisson’s ratio of meniscus is needed as an input (Figs. 1 and 2), our experimentally analyzed compressive Poisson’s ratio could be used, under the assumption of linear elasticity and isotropy. For anisotropic analyses, this value would represent the out-of-plane, compressive Poisson’s ratio if the tissue is compressed perpendicularly to the plane of isotropy. Conflict of interest statement None. Acknowledgment The research leading to these results has received funding from the Finnish Cultural Foundation (Central Fund), Academy of Finland (grant 286526), State Research Funding, Kuopio, Finland (project 5041763). References Abdelgaied, A., Stanley, M., Galfe, M., Berry, H., Ingham, E., Fisher, J., 2015. Comparison of the biomechanical tensile and compressive properties of
Please cite this article in press as: Danso, E.K., et al. Poisson’s ratio of bovine meniscus determined combining unconfined and confined compression. J. Biomech. (2018), https://doi.org/10.1016/j.jbiomech.2018.07.001
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Please cite this article in press as: Danso, E.K., et al. Poisson’s ratio of bovine meniscus determined combining unconfined and confined compression. J. Biomech. (2018), https://doi.org/10.1016/j.jbiomech.2018.07.001
Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Short communication
Poisson’s ratio of bovine meniscus determined combining unconfined and confined compression E.K. Danso a,b,⇑, P. Julkunen b,c, R.K. Korhonen b a
Department of Mechanical Engineering, Colorado State University, Fort Collins, CO, USA Department of Applied Physics, University of Eastern Finland, POB 1627, FI-70211 Kuopio, Finland c Department of Clinical Neurophysiology, Kuopio University Hospital, POB 100, FI-70029, KYS, Kuopio, Finland b
a r t i c l e
i n f o
Article history: Accepted 4 July 2018 Available online xxxx Keywords: Poisson’s ratio Meniscus Stress relaxation Confined Unconfined Aggregate modulus Young’s modulus
a b s t r a c t Poisson’s ratio has not been experimentally measured earlier for meniscus in compression. It is however an important intrinsic material property needed in biomechanical analysis and computational models. In this study, equilibrium Poisson’s ratio of bovine meniscus (n = 6) was determined experimentally by combining stress-relaxation measurements in unconfined and confined compression geometries. The average Young’s modulus, aggregate modulus and Poisson’s ratio were 0.182 ± 0.086 MPa, 0.252 ± 0.089 MPa and 0.316 ± 0.040, respectively. These moduli are consistent with previously determined values, but the Poisson’s ratio is higher than determined earlier for meniscus in compression through biomechanical modelling analysis. This new experimentally determined Poisson’s ratio value could be used in the analysis of biomechanical data as well as in computational finite element analysis when the Poisson’s ratio is needed as an input for the analysis. Ó 2018 Published by Elsevier Ltd.
1. Introduction Meniscus plays an important role in the knee joint to distribute and absorb forces. This role is strongly affected by the biomechanical properties of the tissue. Certain elastic, viscoelastic and poroelastic biomechanical properties of meniscus have been characterized (Danso et al., 2015; Pereira et al., 2014; Sweigart et al., 2004; Tissakht and Ahmed, 1995). However, experimentally measured Poisson’s ratio has never been presented for meniscus in compression. In experimental biomechanical testing, the Poisson’s ratio is required for instance when calculating the Young’s modulus from indentation testing (Fig. 1). Also in finite element modelling simulating biomechanical behaviour of soft tissues, the Poisson’s ratio affects the results (Fig. 2). Some attempts have been made to determine the compressive Poisson’s ratio of meniscus using finite element optimization from creep response (Sweigart et al., 2004), determining the mean Poisson’s ratio for bovine meniscus between 0 and 0.01 depending on the location of meniscus. Similar values were also presented for different species including baboon, canine, human, lapine and porcine, with the highest reported mean value of 0.08 at the anterior
⇑ Corresponding author at: Department of Mechanical Engineering, Colorado State University, Fort Collins, CO, USA. E-mail address: [email protected] (E.K. Danso).
segment from the tibial aspect of lapine meniscus (Sweigart et al., 2004; Sweigart and Athanasiou, 2005). However, none of these values were directly measured from experiments. Goertzen et al. determined experimentally the Poisson’s ratio of meniscus being close to 1 (Goertzen et al., 1997), but the measurements were done in the tension. Values of the Poisson’s ratio in tension are typically much higher than those in compression. The aim of this study was to determine the bulk equilibrium Poisson’s ratio of bovine meniscus by the combination of stressrelaxation tests in unconfined and confined compression measurement geometries. This approach has been shown earlier to produce the same Poisson’s ratio with direct optical measurements of articular cartilage (Korhonen et al., 2002).
2. Methods 2.1. Sample preparation Mature bovine knee joints were obtained from a local slaughterhouse and stored at 25 °C until the experiment. Prior to the biomechanical tests, menisci were thawed in water bath at room temperature (21 °C). Cylindrical samples (n = 6) were prepared with a biopsy punch and a razor blade from random locations of both lateral and medial menisci. First, using a biopsy punch (diameter = 4 mm), a cylindrical plug was cut perpendicularly from the
https://doi.org/10.1016/j.jbiomech.2018.07.001 0021-9290/Ó 2018 Published by Elsevier Ltd.
Please cite this article in press as: Danso, E.K., et al. Poisson’s ratio of bovine meniscus determined combining unconfined and confined compression. J. Biomech. (2018), https://doi.org/10.1016/j.jbiomech.2018.07.001
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thickness of the samples was obtained by cutting the samples from the slanted bottom. The thickness (1.54 ± 0.31 mm) was measured with a digital caliper and the diameter (3.82 ± 0.02 mm) with a stereomicroscope. 2.2. Biomechanical testing
Fig. 1. Examples of the effect of Poisson’s ratio in the analysis of Young’s modulus from indentation testing. Indentation moduli have been analyzed with Hayes equation (for plane-ended indenter (Hayes et al., 1972)) and Hertzian contact equation (for spherical indenter) (Fischenich et al., 2018)), and plotted against ð1v 21 ÞpaE1 Poisson’s ratio. Hayes equation (Es ¼ ) with a = 0.5 mm, h = 2.5 mm, E1 = 2jh 0.83 MPa, j is Poisson’s ratio specific (Hayes et al., 1972). Hertzian contact equation ð1v 21 Þ (Es ¼ 1v 2 ) with v 2 = 0.3, a = 0.80 mm, d = 0.25 mm, E2 = 210 GPa, F = 4a1=2 d3=2 3F
E2
Biomechanical testing was conducted by using a custom-made material testing system equipped with a high precision load cell (model 31/AL311AR, Honeywell, Columbus, OH, USA; resolution: 0.005 N) and an actuator (PM1A1798, Newport Corporation, Irvine, CA, USA; resolution: 0.1 µm) (Korhonen et al., 2002). For the unconfined compression measurements, the samples were placed between an upper steel rod and the bottom of the measuring chamber filled with phosphate buffered saline (Fig. 4). For the confined compression measurement, the samples were placed into a cylindrical metallic chamber sealed at the bottom, and the upper meniscal surface was in contact with a porous platen. The only means of fluid flow was through the porous platen at the top. Unconfined compression measurements were first conducted for
2
0.048 N. ES = Young’s modulus of the tissue, E1 = stress-strain ratio from experiments, E2 = Young’s modulus of indenter, v 1 = Poisson’s ratio of tissue, v 2 = Poisson’s ratio of indenter, a = radius of indenter, j = theoretical scaling factor (Hayes et al., 1972), h = tissue thickness, d = indentation depth, F = applied force.
Fig. 3. Schematic of meniscus showing a cylindrical plug and the portion used for biomechanical testing.
surface to the bottom of the tissue (Fig. 3). This results in a cylindrical disc having a horizontally flat surface and a slanted bottom because of the wedge shape of meniscus (Fig. 3). The superficial surfaces of the samples were then kept intact, and the uniform
Fig. 4. Schematic of experimental set-up with parameters obtained. Confined compression measurement geometry (A) and unconfined compression measurement geometry (B) were combined to determine the Poisson’s ratio.
Fig. 2. Examples of the effect of Poisson’s ratio in finite element analysis. Cylindrical-shaped Neo-Hookean hyperelastic sample of 1 mm in radius and 1 mm in height, and meshed with axisymmetric 4-node elements (CAX4), was compressed by 30% of its original thickness in unconfined (A) and confined (B) compression geometries. The bottom of the sample and the axis of symmetry were fixed in axial direction and lateral direction, respectively. In addition to these, in confined compression, the movement of the sample edge was restricted in the lateral direction. The Young’s modulus (E) was 0.5 MPa and the Poisson’s ratios (m) were between 0 and 0.4. By calculating the shear (µ) and 6ð12mÞ E 2 bulk (j) moduli from E and m, the implemented material constants C10 and D1 of the strain energy density function can be calculated as follows: C10 = l2= 4ð1þ . mÞ; D1 = j = E Static analysis was conducted with nonlinear geometry (nlgeom) in Abaqus 6.13 (Dassault Systèmes, Providence, RI, USA).
Please cite this article in press as: Danso, E.K., et al. Poisson’s ratio of bovine meniscus determined combining unconfined and confined compression. J. Biomech. (2018), https://doi.org/10.1016/j.jbiomech.2018.07.001
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each sample. After these tests and following the release of the load (at least 1 h), the relaxed state of the tissue was assumed and confined compression measurements were conducted. After reaching contact with the meniscus, a steady-state prestress of 6 kPa was applied. Fifteen minutes after this, dynamic pre-conditioning (sinusoidal loading, 2% strain, 4 cycles) was performed and followed by 15 min of relaxation. When the force was the same as obtained after pre-stress, we assumed a good contact was reached. Using a stress-relaxation testing protocol, four steps (5% of the remaining meniscus thickness at each step) with a 100%/sec (unconfined) or 0.09 ± 0.02%/sec strain rate (confined) and <10 Pa/min relaxation criterion (to initiate the next step) were applied (Fig. 5A). In order to avoid exceeding the limit of the load cell, the ramp rate in confined compression was set to be slower. However, this choice does not affect the results as only the equilibrium response was analyzed. The Young’s and aggregate moduli were calculated from the slopes of the equilibrium stress-strain curves (Fig. 5B) obtained from unconfined and confined compression measurements, respectively. Subsequently, the Poisson’s ratio was determined indirectly for the samples by the known relationship between the Young’s and aggregate modulus (Colombo et al., 2013; Korhonen et al., 2002; Mahmoodian et al., 2009),
v¼
Es HA
1þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Es 1 8 HEsA 1 HA 4
ð1Þ
where Es is the Young’s modulus, determined in unconfined compression, and HA is the aggregate modulus, determined in confined compression. For comparing paired-samples, we applied non-parametric Wilcoxon signed rank test and bootstrapping was applied to determine 95% confidence interval for the mean mechanical parameters (SPSS software v. 19.0.0.2, SPSS Inc., IBM Company, Armonk, NY, USA). 3. Results Fig. 5A shows typical examples of stress-relaxation curves for unconfined and confined compression measurements. At equilibrium, linear stress-strain relationships were exhibited in both unconfined and confined compression geometries (Fig. 5B). The average Young’s modulus (ES ) and aggregate modulus (HA ) were 0.182 ± 0.086 MPa and 0.252 ± 0.089 MPa, respectively, and they were significantly different (Table 1, p < 0.05). The average Poisson’s ratio (v ), determined indirectly from Eq. (1), was 0.316 ± 0.040 (Table 1). Based on 1000 bootstrapped samples, the means ofES , HA and v were between 0.121 and 0.248 MPa, 0.188 and 0.320 MPa, as well as between 0.287 and 0.344, respectively, with 95% confidence intervals. 4. Discussion Fluid-independent elastic properties (Young’s and aggregate moduli) of bovine meniscus were determined by stressrelaxation tests in unconfined and confined compression measurement geometries of the same samples. Results from these data were used to indirectly determine the equilibrium Poisson’s ratio of meniscus. Although finite element analysis has been used earlier to determine the Poisson’s ratio of meniscus in compression, to the best of our knowledge, this is the first experimental study to determine the Poisson’s ratio of bovine meniscus in compression. The equilibrium Poisson’s ratio of articular cartilage in compression is well known and it varies greatly. Reported values range between 0.09 and 0.40 for different species (bovine, dog, monkey, rabbit, human) at different locations (femoral condyles, tibial plateau, ankle talus, humeral head, patella) (Athanasiou et al., 1991; Colombo et al., 2013; Jurvelin et al., 1997; Kiviranta et al., 2006; Korhonen et al., 2002; Mahmoodian et al., 2009; Wong et al., 2000). Our new values for the meniscus Poisson’s ratio are comparable with those of cartilage. For meniscus, there exist only limited information about the Poisson’s ratio in compression. Using finite element analyses, an earlier study (Sweigart et al., 2004) determined indirectly the drained Poisson’s ratio of meniscus for
Table 1 Material parameters of bovine meniscus, obtained through unconfined compression, confined compression, and mathematical approaches.
Fig. 5. Examples of stress relaxation curves in unconfined and confined compression geometries (A). Equilibrium stress-strain responses (mean ± SD) in unconfined and confined compression geometries (B). The slopes of the equilibrium, fluidindependent stress-strain responses in unconfined and confined compression geometries (B) results in the Young’s and aggregate moduli, respectively.
Sample number
Young’s modulus, ES (MPa)
Aggregate modulus, HA (MPa)
Poisson’s ratio,v (–)
Sample Sample Sample Sample Sample Sample
0.091 0.135 0.123 0.288 0.165 0.291
0.139 0.201 0.212 0.361 0.241 0.359
0.338 0.331 0.365 0.271 0.326 0.264
0.182 ± 0.086*
0.252 ± 0.089
0.316 ± 0.040
1 2 3 4 5 6
Mean ± SD
ES determined in unconfined compression, HA determined in confined compression, determined indirectly from ES and HA (Eq. (1)). * p < 0.05, compared to HA.
v
Please cite this article in press as: Danso, E.K., et al. Poisson’s ratio of bovine meniscus determined combining unconfined and confined compression. J. Biomech. (2018), https://doi.org/10.1016/j.jbiomech.2018.07.001
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different species (baboon, bovine, canine, human, lapine, porcine) to be between 0 and 0.08. With a similar approach, another study (Abdelgaied et al., 2015) also predicted the Poisson’s ratio for decellularised and natural porcine meniscus to be zero. Compared to the present study, our Poisson’s ratio is higher (Table 1), and there are a number of possible reasons for this difference. The first reason could be attributed to the different techniques used in determining the Poisson’s ratio. The previous studies determined the Poisson’s ratio using a computational finite element analysis and optimization method from creep indentation (Abdelgaied et al., 2015; Sweigart et al., 2004; Sweigart and Athanasiou, 2005), while we combined experimental stress-relaxation tests from unconfined and confined compression geometries. Due to the presence of proteoglycans in the tissue, our values may not be exactly the same as the drained Poisson’s ratio because of residual swelling pressure resulting in fluid pressure even at steady state. Also, Sweigart et al. (2004) modelled the meniscus tissue as an axisymmetric isotropic biphasic cylinder, without considering the meniscal architecture. It is well known that the collagen network architecture contributes significantly to the Poisson’s ratio of biological materials (Chegini and Ferguson, 2010; Ficklin et al., 2007; Kiviranta et al., 2006). Another possible reason for the difference is that their indentation geometry tends to test more the superficial layer properties than the bulk tissue properties. Therefore, we believe that our experimentally defined values represent more realistically the actual behavior of the entire meniscus. This is supported by an earlier study showing the method, used in the present study, to produce the same Poisson’s ratios of cartilage with directly measured values from optimal measurements (Korhonen et al., 2002). This presented method is also simple and easily available. It is however important to note that the results obtained in this study are valid only under the assumptions of linear elasticity and isotropy in both the radial and circumferential planes. Radial and circumferential Poisson’s ratios could be different, as the lateral expansion of the tissue largely depends on the collagen orientation (Changoor et al., 2011). Similarly as shown earlier for soft biological tissues (Fortin et al., 2000; Korhonen et al., 2002; Mäkelä et al., 2012; Martin Seitz et al., 2013), the equilibrium stress-strain relationship was linear in both unconfined and confined compression geometries (constant Young’s or aggregate modulus with strain), as well as in indentation for human meniscus (Danso et al., 2015). Possible nonlinear and strain-dependent behavior could be observed in the meniscus instantaneous modulus (from the peak forces) due to the nonlinear collagen fibril network. However, since we were interested in the equilibrium response of meniscus and the ramp rate was different in confined and unconfined compression measurements, we did not pursue in comparing these nonlinearities between the measurement geometries. For the same reason, the Poisson’s ratio was not analyzed in a time-dependent manner, even though viscoelastic materials like the meniscus have a time- and frequency-dependent Poisson’s ratio (Chegini and Ferguson, 2010; Lakes, 1992; Lakes and Wineman, 2006). A previous study (Langelier and Buschmann, 2003) showed the history-dependence on the biomechanical properties of cartilage in unconfined compression. In that study, quite small strains and increasing strain rates affected subsequent responses because of tissue softening. However, these effects were mainly seen in the peak stiffness, while the equilibrium stiffness remained more stable. Here, we studied only the equilibrium responses with similar strains used before (Korhonen et al., 2002) and the relaxation time between different tests was more than 1 h. Nonetheless, we cannot fully rule out the possibility of tissue alterations between the unconfined and confined compression tests. However, if there were tissue damages, it should affect negatively to the tissue, that is, the later measured aggregate modulus in confined compression
would become smaller and the Poisson’s ratio would approach zero. This is not what we observed. The samples were cut from the slanted bottom to make them horizontally flat and to enable tissue testing (Fig. 3). However, we do not think that cutting had a significant impact on the results, as most of the tissue remained intact from all the three main layers of meniscus (Petersen and Tillmann, 1998). The superior surface was preserved, instead of the inferior, as physiological joint loads from the femoral condyle are experienced through the top of the tissue. We though acknowledge that the results might not be exactly the same if cutting would have been done to the superior surface of the samples. As expected and due to the Poisson’s ratio being over zero, the isotropic elastic Young’s moduli from the unconfined compression measurements were lower than the aggregate moduli obtained from the confined compression measurements (Table 1). This result was not a surprise but to some extent confirms that there was a good fit of the sample in the confining chamber. These modulus values were also consistent with previous studies (Abdelgaied et al., 2015; Danso et al., 2015; Martin Seitz et al., 2013). Obviously the confined chamber could cause some friction between the sample and the walls of the chamber. However, friction should not affect the measured equilibrium parameters. On the other hand, a small gap between the specimen and the metallic chamber might affect the measured properties (Buschmann et al., 1997). In our study, although we cannot know for sure whether there were gaps or not, it took a slight effort to push the samples into the metallic chamber. If the fit was not good, the aggregate and Young’s moduli would be close to equal and the Poisson’s ratio would approach zero. In this study, we analyzed a ‘‘general” Poisson’s ratio for bovine meniscus and did not characterize site-specific differences. Therefore, all the samples both from the lateral and medial menisci were combined. For the same reason, the number of samples was not an issue. Furthermore, as can be seen from Table 1, the Poisson’s ratios of all the samples were really close to each other. Also, the range of values within 95% confidence interval was found reasonable. These support the analysis of one ‘‘generalized” value for the Poisson’s ratio of bovine meniscus in compression. Although Sweigart et al. obtained smaller Poisson’s ratio than we did, they also showed only small location-specific differences (Sweigart et al., 2004). In conclusion, we propose that in the analysis of biomechanical data, such as indentation moduli, and in finite element models where the equilibrium Poisson’s ratio of meniscus is needed as an input (Figs. 1 and 2), our experimentally analyzed compressive Poisson’s ratio could be used, under the assumption of linear elasticity and isotropy. For anisotropic analyses, this value would represent the out-of-plane, compressive Poisson’s ratio if the tissue is compressed perpendicularly to the plane of isotropy. Conflict of interest statement None. Acknowledgment The research leading to these results has received funding from the Finnish Cultural Foundation (Central Fund), Academy of Finland (grant 286526), State Research Funding, Kuopio, Finland (project 5041763). References Abdelgaied, A., Stanley, M., Galfe, M., Berry, H., Ingham, E., Fisher, J., 2015. Comparison of the biomechanical tensile and compressive properties of
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Please cite this article in press as: Danso, E.K., et al. Poisson’s ratio of bovine meniscus determined combining unconfined and confined compression. J. Biomech. (2018), https://doi.org/10.1016/j.jbiomech.2018.07.001