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Mechanical properties of the porcine growth plate and its three zones from unconfined compression tests
ARTICLE IN PRESS Journal of Biomechanics 42 (2009) 510–516
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Mechanical properties of the porcine growth plate and its three zones from unconfined compression tests Kim Sergerie a,b, Marc-Olivier Lacoursie`re a,b, Martin Le´vesque a, Isabelle Villemure a,b, a b
´ cole Polytechnique of Montreal, PO Box 6079, Station ‘‘Centre-Ville’’ Montre ´al, Que´bec, Canada H3C 3A7 Department of Mechanical Engineering, E ˆte-Ste-Catherine Road Montre ´al, Que´bec, Canada H3T 1C5 Sainte-Justine University Hospital Center 3175 Co
a r t i c l e in f o
a b s t r a c t
Article history: Accepted 10 November 2008
The aim of the study was to determine intrinsic mechanical properties of the complete growth plate and its reserve, proliferative and hypertrophic zones. Growth plate disk samples from newborn swine’s ulnae were tested using stress relaxation tests under unconfined compression. The Transversely Isotropic Biphasic Model (TIBPE) derived by [Cohen, B., Lai, W. M., Mow, V. C., 1998. A transversely isotropic biphasic model for unconfined compression of growth plate and chondroepiphysis. Journal of Biomechanical Engineering, 120, pp. 491–496] was used to extract intrinsic mechanical properties using a four-parameter optimization procedure. Significant differences were found for the transverse permeability k1, the Poisson’s ratio in the transverse plane n21, the out-of-plane Poisson’s ratio n31 and the out-of-plane Young’s modulus E3 between the reserve zone and the proliferative zone as well as between the reserve zone and the hypertrophic zone. The same trends were obtained for the Young’s modulus in the transverse plane E1, but significant differences were also found between the reserve zone and the complete growth plate. The proliferative and hypertrophic zones are half as stiff as the reserve zone along the compression axis and about three times less stiff than the reserve zone in the transverse plane. These two zones are also three times as permeable as the reserve zone in the radial direction. The mechanical behavior of the newborn porcine distal ulna growth plate is non-uniform along its thickness. The reserve zone, with its greater zonal component at that development stage, has noteworthy effects on the complete growth plate intrinsic mechanical properties. This study provides, for the very first time, an investigation of the intrinsic mechanical properties of the reserve, proliferative and hypertrophic zones of the growth plate. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Growth plate Mechanical loading Unconfined compression Biphasic model Mechanobiology
1. Introduction The longitudinal growth of long bones and vertebrae occurs in the growth plates. Within the growth plate, chondrocytes evolve through three morphologically distinct zones. The reserve, proliferative and hypertrophic zones play specific roles in the growth process and present differences in cell arrangement, cell size, matrix composition as well as cell/matrix volume ratio (Hunziker and Schenk, 1989; Farnum and Wilsman, 1998) (Fig. 1). Growth plates are also sensitive to their surrounding mechanical environment. Experimental studies have previously demonstrated that mechanical loading can modulate longitudinal bone growth (Farnum et al., 2000; Stokes, 2002; Wang and Mao, 2002; Stokes
Corresponding author at: E´cole Polytechnique of Montreal, Department of Mechanical Engineering, PO Box 6079, Station ‘‘Centre-Ville’’ Montreal, Quebec Canada H3C 3A7. Tel.: +1 514 340 4711x4900; fax: +1 514 340 4176. E-mail addresses: [email protected] (K. Sergerie), [email protected] (M.-O. Lacoursie`re), [email protected] (M. Le´vesque), [email protected] (I. Villemure).
0021-9290/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2008.11.026
et al., 2006). This phenomenon has key implications in the pathogenesis and treatment of infant and juvenile musculoskeletal deformities, such as adolescent idiopathic scoliosis, hyperkyphosis and genus varus/valgus (LeVeau and Bernhardt, 1984; Frost, 1990; Mao and Nah, 2004). Previous in vitro studies have determined growth plate intrinsic mechanical properties (elastic moduli, Poisson’s ratio and permeability coefficients) under uniaxial compression (Cohen et al., 1994). Other properties of growth plates (ultimate stress and strain, tangent modulus) were also investigated in vitro using bone-growth plate-bone specimens under uniaxial traction (Cohen et al., 1992; Williams et al., 2001). In a study by Cohen et al. (1998), the growth plate was partitioned in two adjacent regions, the chondroepiphysis/reserve zone region and the proliferative/hypertrophic zones region, and further tested under unconfined compression. Using the transversely isotropic biphasic model (TIBPE), this study showed that the chondroepiphysis/ reserve zone was about twice as stiff as the proliferative/ hypertrophic zone along the longitudinal axis as well as in the transverse plane. These mechanical properties represent the bulk
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Nomenclature Parameters for biphasic model E1 E3 k1 k3
n21 n31
transverse Young’s modulus out-of-plane Young’s modulus transverse permeability axial permeability transverse Poisson’s ratio out-of-plane Poisson’s ratio
511
Parameters for FE and three-spring models Ti E3i E¯ 3
thickness of zone i out-of-plane modulus of zone i overall out-of-plane Young’s modulus predicted by the models (FE and three-spring models)
Fig. 1. Histological section showing the growth plate structure and the corresponding mean zonal proportions of the reserve, proliferative and hypertrophic zones.
behavior of the entire growth plate or growth plate regions but the intrinsic mechanical properties of each of the three zones have not been fully characterized. With an experimental setup combining confocal microscopy and mechanical loading, Villemure et al. (2007) characterized continuous strain fields along the growth plate thickness and observed non-uniform strain distributions among the histomorphological zones of the growth plate. However, the overall mechanical properties of the three histomorphological zones of the growth plate, considered separately, have not been characterized yet. The aim of this study is to determine intrinsic mechanical properties of the complete growth plate and its reserve, proliferative and hypertrophic zones. Stress relaxation tests under unconfined compression are performed on each of these four materials taken separately and the obtained mechanical properties are those best fitting a TIBPE model through the experimental data. Results provide an understanding of the growth plate intrinsic mechanical properties variation along its thickness for a specific animal model.
Growth plate Radius
∅ 4 mm
Ulna
[mm] 2. Methods
Growth plate cylindrical sample
Fig. 2. Growth plate cylindrical sample extraction using 4 mm diameter biopsy punches. The growth plate is located between the metaphysis and epiphysis.
2.1. Animal model and histological study Distal ulnae from newborn swines were obtained within 3 h of slaughter and dissected to extract growth plate samples using 4 mm diameter biopsy punches (Fig. 2). A histological study (n ¼ 8) was first performed in order to determine the proportions of the three zones of the newborn porcine growth plate. Following dissection, growth plate samples were fixed in 4% paraformaldehyde and
embedded in paraffin. Ten histological sections (5 mm) taken at different depth within the sample and stained using toluidine blue were viewed (25 ) using a microscope (Leica DMR with a Retiga Qimaging Camera). Measurements (10 per section) of growth plate and zones thicknesses were made with an image analysis software (Digimizer v3.0).
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Growth plate samples used for biomechanical testing were divided in two groups. In one sample group, upper and lower surfaces of the growth plate disks were trimmed using a Vibratome (Vibratome 1500 Sectioning System) to obtain two parallel surfaces and to provide thickness of the complete growth plate sample (repeatable height adjustment of 5 mm). In the other sample group, the three zones were sequentially trimmed from the complete growth plate based on mean zone proportions previously obtained from the histological study. Unconfined compression tests were performed for their good repeatability and because they allowed extraction of Poisson’s ratios. Disk samples were placed between two impermeable smooth platens allowing for their lateral expansion and subjected to stress relaxation tests using a micromechanical testing system (MACH-1, Biosyntech Inc.) (Fig. 3). Loads were recorded with a load cell, whose range and resolution were, respectively, 17 and 0.026 N. Axial displacements were controlled via an encoder of 100 mm range and 0.5 mm resolution. Specimens were initially preloaded at 5% strain, followed by 15% strain at a strain rate of 1.5 103 s1. A relaxation criterion of 0.05 g/min was used to allow samples to reach equilibrium. All samples were bathed in Hank’s balanced salt solution (HBSS) throughout the test.
experimental data from unconfined compression stress relaxation tests were then fitted with analytical curves for the complete growth plate (N ¼ 23), the reserve zone (N ¼ 26), the proliferative zone (N ¼ 13) and the hypertrophic zone (N ¼ 16). Specific ranges for initial values were determined for each of the four parameters based on previous experimental studies on the growth plate (Cohen et al., 1998). For each experimental data set, 81 different combinations of initial values (three values for each parameter) were used to fit the data. The optimal solution was the result showing the minimal error and simultaneously meeting the restrictions listed in Eqs. (1) and (2) imposed by thermodynamics on the engineering constants for a transversely isotropic material (Lemprie`re, 1968) E1 1 n231 (1) ; ð1 n221 Þ40 E3 1 n221 2n231
E1 E1 2n21 n231 40 E3 E3
(2)
2.2. Determination of intrinsic mechanical properties
2.3. Statistical analysis
The transversely isotropic biphasic model derived by Cohen et al. (1998) was used to extract intrinsic mechanical properties of growth plates and the three zones. This model is a modified version of the the linear biphasic poroelastic model (BPE), initially developed by Mow et al. (1980), and adapted for transverse isotropy. In the TIBPE model, the properties are the same in all directions within the transverse plane of isotropy (1–2). This model involves five parameters: E1 is the Young’s modulus in the transverse plane, E3 is the out-of-plane Young’s modulus, n21 is the Poisson’s ratio in the transverse plane, n31 is the out-of-plane Poisson’s ratio, and k1 is the transverse permeability coefficient. The out-of-plane permeability coefficient k3 could be eventually obtained from confined compression stress relaxation tests. A four-parameter optimization procedure was used to extract intrinsic mechanical properties (k1, E1, n21 and n31), while moduli E3 were evaluated based on the applied strains and relaxed stress values obtained from the experimental curves. Using MATLAB software and the root mean square error (RMSE) method,
Basic statistical analysis (mean, standard deviation) was performed for each parameter (thickness, E3, E1, k1, n21, n31) and on each set of data (complete growth plate, reserve, proliferative and hypertrophic zones). A one-way ANOVA for repeated measures was carried out to determine whether any difference existed between the mean parameter values obtained for each of the three zones as well as for the complete growth plate. The post-hoc comparisons between groups were done using the Tukey’s method. The results are presented as mean values7standard deviations. For all tests, the level of significance was fixed at po0.05. The software used for the statistical analysis was STATISTICA 7.0 from StatSoft Inc.
axis 3 to load cell applied impermeable loading platen
displacement
HBSS sample
axes 1, 2 Glass Petri dish Fig. 3. Schematic of the unconfined compression test. Samples (4 mm in diameter) were initially preloaded at 5% strain, followed by 15% strain at a strain rate of 1.5 103 s1. A relaxation criterion of 0.05 g/min was used to allow samples to reach equilibrium. All samples were bathed in Hank’s balanced salt solution (HBSS) throughout the test.
3. Results Histological analyses are summarized in Fig. 1. The resulting proportions of the reserve, proliferative and hypertrophic zones are 7073%, 1772% and 1371%, respectively. These proportions were used to sequentially trim the three zones from the complete growth plate. Complete growth plate thickness varied between 2500 and 4200 mm with a mean value of 3510 mm, while reserve, proliferative and hypertrophic zones indicated, respectively, 2450, 600 and 460 mm for average thicknesses (Table 1). Typical experimental stress relaxation curves are presented in Fig. 4(a) for a complete growth plate as well as samples of the three zones. Fig. 4(b) shows theoretical stress relaxation time histories based on mean mechanical properties obtained from the optimization procedure. The growth plate curve showed great similarity to the reserve zone curve for comparable peak and relaxation stresses. The proliferative and hypertrophic zones indicated different curve profiles with considerably lower peak stresses and slightly smaller relaxation stresses and times. Representative curve-fits between experimental data and analytical model are illustrated in Fig. 5. Very good agreement was observed between experimental and theoretical values at the equilibrium state, as moduli E3 were predetermined from the experimental curves and fixed in the model. Experimental peak stress was generally higher than in the analytical curve. Intrinsic mechanical properties of the complete growth plate and its three zones optimized with the TIBPE model as well as calculated equilibrium Young’s moduli are summarized in Table 1 and
Table 1 Transversely isotropic biphasic properties of porcine distal ulna growth plate and its three zones from unconfined compression tests (mean values7standard deviations). Experiment
Complete growth plate (N ¼ 23) Reserve zone (N ¼ 26) Proliferative zone (N ¼ 13) Hypertrophic zone (N ¼ 16)
Optimization
Thickness (mm)
E3 (MPa)
E1 (MPa)
n21 (–)
n31 (–)
k1 ( 1015 m4/Ns)
35107370 24507210 600740 460740
0.5170.12 0.4870.11 0.2570.12 0.2770.10
8.6571.72 11.0872.86 3.4273.49 3.2673.28
0.2470.07 0.2570.07 0.3270.05 0.3470.05
0.0870.03 0.0770.03 0.1370.04 0.1370.05
1.8270.67 1.4170.94 5.6074.74 5.9875.67
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0.40
0.30
Reserve zone
0.25
Proliferative zone
0.20
Hypertrophic zone
0.15
0.30
Hypertrophic zone
0.15
0.05
0.05 60 120 180 240 300 360 420 480 540 600 660 720 780
Proliferative zone
0.20
0.10
0
Reserve zone
0.25
0.10
0.00
Complete growth plate
0.35
Stress (MPa)
Stress (MPa)
0.40
Complete growth plate
0.35
513
0.00
0
60 120 180 240 300 360 420 480 540 600 660 720 780
Time (s)
Time (s)
Fig. 4. (a) Experimental stress relaxation time histories from unconfined compression tests on the complete growth plate and the reserve, proliferative and hypertrophic zones for typical samples; (b) theoretical stress relaxation time histories based on average mechanical properties obtained from the optimization procedure.
0.40
0.40
0.35
Complete Growth Plate
0.25 0.20 0.15 0.10
Proliferative Zone
0.35 Stress [mPa]
Stress [MPa]
0.30
0.05
0.30
Experimental Analytical
0.20 0.15 0.10 0.05
0
0
60 120 180 240 300 360 420 480 540 600 660 720 780
60 120 180 240 300 360 420 480 540 600 660 720 780
Time [s]
Time [s] 0.40
0.40 0.35 0.25 0.20 0.15 0.10
Hypertrophic Zone
0.35
Reserve Zone
0.30
Stress [MPa]
Stress [MPa]
X
0.25
0.30 0.25 0.20 0.15 0.10 0.05
0.05 0
60 120 180 240 300 360 420 480 540 600 660 720 780 Time [s]
0
60 120 180 240 300 360 420 480 540 600 660 720 780 Time [s]
Fig. 5. Typical stress relaxation histories in response to a 15% ramped displacement with curve-fits of the transversely isotropic biphasic model.
graphically shown in Fig. 6. According to the ANOVA analysis described at the previous section, statistically significant differences were found in transverse permeability coefficients (k1), transverse Poisson’s ratios (n21), out-of-plane Poisson’s ratios (n31) and out-of-plane Young’s modulus (E3) between the reserve zone and the proliferative zone as well as between the reserve zone and the hypertrophic zone. Parameters k1, n21, n31, E1 and E3 also significantly differed between the complete growth plate zone and the proliferative and hypertrophic zones. Significant difference was also found between the reserve zone and the complete growth plate for the transverse Young’s moduli (E1). The proliferative and hypertrophic zones are half as stiff as the reserve zone along the compression axis and about three times less stiff than the reserve zone in the transverse plane. These two zones are three times as permeable as the reserve zone in the radial direction. The out-of-plane Poisson’s ratio of the reserve zone is half the ones of the two other zones, and its transverse Poisson’s ratio is smaller than in the proliferative and hypertrophic zones.
4. Discussion The aim of this study was to determine intrinsic mechanical properties of the complete growth plate and its reserve,
proliferative and hypertrophic zones using stress relaxation tests under unconfined compression combined with an optimization procedure of the TIBPE model. The reserve zone, with its greater zonal component at the newborn development stage (70% of the total thickness), has noteworthy effects on the growth plate intrinsic mechanical properties. Newborn porcine growth plate indicates non-homogeneous mechanical properties along its thickness. The proliferative and hypertrophic zones show very similar mechanical behavior and differ from the reserve zone, which indicates comparable mechanical properties with the complete growth plate. During early development, the ossification process is highly active and the reserve zone and the chondroepiphysis are merged in the upper growth plate zone. Therefore, the reserve zone would be a good indicator of the complete growth plate intrinsic mechanical properties in the newborn porcine ulnae. The obtained mechanical properties are, however, dependent on the optimized model selected to extract the intrinsic tissue parameters. The TIBPE model was chosen for its very good agreement with experimental data for growth plate tissues (Cohen et al., 1998). Results show that E3 for the complete growth plate is greater than the moduli of the three zones. In order to discuss this result, an unconfined compression test was simulated by finite element (FE) simulations on a three-section transversely isotropic cylinder
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Transverse Young's Modulus E1
18 16
0.7
14
0.6
12
0.5 [MPa]
10
[MPa]
Out-of-Plane Young's Modulus E3
0.8
8
0.4 0.3
6
0.2
4
0.1
2 Complete
Reserve
Prolif.
Hyper.
Complete
0
Reserve
Prolif.
Hyper.
0.0
Transverse Poisson's Ratio 0.50
Out-of-Plane Poisson's Ratio
21
0.50
0.45
0.45
0.40
0.40
0.35
0.35
0.30
0.30
0.25
0.25
0.20
0.20
0.15
0.15
0.10
0.10
31
0.05
0.05 Complete
Reserve
Prolif.
Complete
Hyper.
Reserve
Prolif.
Hyper.
0.00
0.00
Permeability k1
16
LEGEND 14
Significant difference (p < 0.05) between the complete growth plate and one of the three zones
x 1E-15 [m4/Ns]
12
Significant difference (p < 0.05) between two of the three zones
10 8 6 4 2 0
Complete
Reserve
Prolif.
Hyper.
Fig. 6. Intrinsic mechanical properties of the complete growth plate and its three zones (mean values7standard deviations). Significant differences are marked with connecting lines (p-valueo0.05).
representing the complete growth plate. Each section was attributed the average mechanical properties (Table 1). The overall E3 of this assemblage, denoted by E¯ 3, was computed as the ratio between the applied pressure and the resulting effective strain (overall displacement divided by sample height). E¯ 3 ¼ 0.39 MPa was obtained. Then, another simplified model that consisted of three cylinders piled on the top of each other with frictionless interfaces (i.e. three springs in series) was developed. For this unidimensional model, 3 X 1 1 Ti ¼ P3 E E¯ 3 3i i¼1 T i i¼1
(3)
where Ti and E3i are, respectively, the thickness and E3 of zone i. E¯ 3 ¼ 0.38 MPa was obtained when inserting the corresponding average values (Table 1). It can be seen that both E¯ 3 obtained from these two models were very close. This suggests that this simplified model can predict relatively well E¯ 3 for the range of mechanical properties simulated with a minimal computational effort. This model was therefore used for estimating the statistical variability in E¯ 3 . In order to estimate the mean of E¯ 3 obtained with
the simplified model, all the raw data that led to Table 1 was used. Each T i E3i pair was considered as a building block for the threesprings model and 5408 ¼ (N ¼ 26) (N ¼ 13) (N ¼ 16) threesprings models were created. E¯ 3 ¼ 0.35370.02 MPa was obtained, while experimental E3 ¼ 0.5170.05 MPa was obtained for the complete growth plate (95% confidence intervals on the mean). This suggests that there is a statistically significant difference between E¯ 3 and E3 for the complete growth plate. Possible causes of such discrepancy could be that the micro-structural changes caused by the sectioning of complete growth plate in its three zones (involving interruption of the collagen and proteoglycan networks) are not taken into account. Hence, the complete growth plate, with its continuous fibrils networks along the three zones, could provide a greater resistance than three zones taken separately. Sectioning the growth plate could induce damage in the three zones thus leading to decreased mechanical properties. Such results are a motivation for future studies and remind the difficulty of obtaining meaningful in-situ mechanical properties. Obtained mechanical properties are comparable to reported studies on growth plate mechanics. Transverse and out-of-plane Young’s moduli obtained for the complete growth plate by (Cohen
ARTICLE IN PRESS K. Sergerie et al. / Journal of Biomechanics 42 (2009) 510–516
et al., 1994) indicate similar magnitudes than the moduli of the present study. Young’s moduli of 4-month-old calves for composite samples combining the proliferative and hypertrophic zones, as well as for samples referred to as the chondroepiphysis, result in similar magnitudes and trends, with higher moduli in chondroepiphysis samples as compared to proliferative/hypertrophic samples (Cohen et al., 1998). However, our experimental trends do not corroborate with the results obtained from strain patterns of loaded rat tibial growth plates, where the proliferative zone was identified as more rigid when compared to the reserve and hypertrophic zones (Villemure et al., 2007). Differences might be related to the different developmental stage (adolescent) of these growth plates, as growth plates might present different relative mechanical behavior at different development stage. Differences in the tissue histomorphometry within the proliferative/hypertrophic zones could explain their different mechanical behavior as compared to the reserve zone. Major differences in matrix/cell volume ratio, cell arrangement as well as cell size exist between the three zones. Indeed, the matrix/cell volume ratio significantly decreases when approaching the chondro-osseous junction. The ratio reaches values greater than 9 in the reserve zone, remains bounded between 4 and 8 in the proliferative zone and decreases substantially to values smaller than 1 in the hypertrophic zone (Farnum et al., 2002). Considering that the extracellular matrix has the most important structural role in the mechanical behavior of the soft tissue, it is expected that the zone with higher matrix/cell volume ratio would offer stiffer property. In addition, chondrocytes are arranged in columns in the proliferative and hypertrophic zones while they are randomly dispersed in the reserve zone (Hunziker and Schenk, 1989). The chondrocytes volume also increases by an approximate factor of 9 from the proliferative to the hypertrophic phases (Hunziker and Schenk, 1989). These volume and spatial characteristics, which involve a sequence of very soft cells (as compared to the surrounding extracellular matrix) in the proliferative and hypertrophic zones, might contribute to a weaker mechanical support in these lower zones. The non-uniform matrix composition within the different zones cannot fully explain their different mechanical behavior. In a study carried out by Alvarez et al. (2000), the distribution of collagens was characterized in rat tibial growth plates divided into five horizontal strata of equal heights. The type II collagen expression showed peak values in the proliferative zone, while the type X collagen expression attained maximum values approaching the upper hypertrophic zone and decreased thereafter to the lower hypertrophic zone (Alvarez et al., 2000). A qualitative evaluation realized by Sandell et al. (1994) showed comparable trends for both type II and type X collagen. As for aggrecan, its localization within the zones presented a more or less uniform distribution (Sandell et al., 1994). Since structural elements (type II collagen, type X collagen, aggrecan) of the extracellular matrix are predominantly contained in the proliferative zone, this zone could be expected to offer higher mechanical properties. However, our results show that the proliferative zone is less stiff than the reserve zone in both, tension and compression. Therefore, depth-dependant relationships between structural elements of the extracellular matrix and their localization in the growth plate do not entirely explain the non-homogeneous mechanical properties among the three zones of the growth plate. It should be noted however that the distribution of matrix components detailed above refer to different animal models and to varying developmental stages. This study provides, for the very first time, an investigation of the intrinsic mechanical properties of the reserve, proliferative and hypertrophic zones of the growth plate. The mechanical behavior of the newborn porcine ulnar growth plate is non-
515
uniform along its thickness; the proliferative and hypertrophic zones are half as stiff as the reserve zone along the compression axis and about three times less stiff than the reserve zone in the transverse plane. This knowledge on the zone mechanical behavior is very important for growth plate mechanobiology, where the effects of mechanical loading on the biological process of endochondral growth, with each zone playing its specific roles in the process, are investigated. Based on the results of this study, the more compliant proliferative and hypertrophic zones might be more prone to trigger abnormal endochondral growth upon compressive mechanical loading. More studies are required to confirm if these two lower zones are major actors in the growth plate mechanobiology in this early developmental stage. Similar studies should aim at investigating the mechanical behavior of the growth plate at different developmental stages. This scientific knowledge will also further be very valuable in growth plate modeling as well as in tissue engineering, where quantitative data is required both at the structural and compositional levels for engineering of this living tissue.
Conflict of interest None.
Acknowledgements The authors acknowledge the participation of Marjorie Beausoleil in the experimental design and Jose´e De´poˆt for her technical assistance in histology. This study was funded by the Fonds Que´be´cois de la Recherche sur la Nature et les Technologies (FQRNT, I.V.), the Natural Sciences and Engineering Research Council of Canada (NSERC, I.V.) and the Canada Research Chair in Mechanobiology of the Pediatric Musculoskeletal System (I.V.).
Appendix A. Supplementary materials The online version of this article contains additional supplementary data. Please visit doi:10.1016/j.jbiomech.2008.11.026.
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Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Mechanical properties of the porcine growth plate and its three zones from unconfined compression tests Kim Sergerie a,b, Marc-Olivier Lacoursie`re a,b, Martin Le´vesque a, Isabelle Villemure a,b, a b
´ cole Polytechnique of Montreal, PO Box 6079, Station ‘‘Centre-Ville’’ Montre ´al, Que´bec, Canada H3C 3A7 Department of Mechanical Engineering, E ˆte-Ste-Catherine Road Montre ´al, Que´bec, Canada H3T 1C5 Sainte-Justine University Hospital Center 3175 Co
a r t i c l e in f o
a b s t r a c t
Article history: Accepted 10 November 2008
The aim of the study was to determine intrinsic mechanical properties of the complete growth plate and its reserve, proliferative and hypertrophic zones. Growth plate disk samples from newborn swine’s ulnae were tested using stress relaxation tests under unconfined compression. The Transversely Isotropic Biphasic Model (TIBPE) derived by [Cohen, B., Lai, W. M., Mow, V. C., 1998. A transversely isotropic biphasic model for unconfined compression of growth plate and chondroepiphysis. Journal of Biomechanical Engineering, 120, pp. 491–496] was used to extract intrinsic mechanical properties using a four-parameter optimization procedure. Significant differences were found for the transverse permeability k1, the Poisson’s ratio in the transverse plane n21, the out-of-plane Poisson’s ratio n31 and the out-of-plane Young’s modulus E3 between the reserve zone and the proliferative zone as well as between the reserve zone and the hypertrophic zone. The same trends were obtained for the Young’s modulus in the transverse plane E1, but significant differences were also found between the reserve zone and the complete growth plate. The proliferative and hypertrophic zones are half as stiff as the reserve zone along the compression axis and about three times less stiff than the reserve zone in the transverse plane. These two zones are also three times as permeable as the reserve zone in the radial direction. The mechanical behavior of the newborn porcine distal ulna growth plate is non-uniform along its thickness. The reserve zone, with its greater zonal component at that development stage, has noteworthy effects on the complete growth plate intrinsic mechanical properties. This study provides, for the very first time, an investigation of the intrinsic mechanical properties of the reserve, proliferative and hypertrophic zones of the growth plate. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Growth plate Mechanical loading Unconfined compression Biphasic model Mechanobiology
1. Introduction The longitudinal growth of long bones and vertebrae occurs in the growth plates. Within the growth plate, chondrocytes evolve through three morphologically distinct zones. The reserve, proliferative and hypertrophic zones play specific roles in the growth process and present differences in cell arrangement, cell size, matrix composition as well as cell/matrix volume ratio (Hunziker and Schenk, 1989; Farnum and Wilsman, 1998) (Fig. 1). Growth plates are also sensitive to their surrounding mechanical environment. Experimental studies have previously demonstrated that mechanical loading can modulate longitudinal bone growth (Farnum et al., 2000; Stokes, 2002; Wang and Mao, 2002; Stokes
Corresponding author at: E´cole Polytechnique of Montreal, Department of Mechanical Engineering, PO Box 6079, Station ‘‘Centre-Ville’’ Montreal, Quebec Canada H3C 3A7. Tel.: +1 514 340 4711x4900; fax: +1 514 340 4176. E-mail addresses: [email protected] (K. Sergerie), [email protected] (M.-O. Lacoursie`re), [email protected] (M. Le´vesque), [email protected] (I. Villemure).
0021-9290/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2008.11.026
et al., 2006). This phenomenon has key implications in the pathogenesis and treatment of infant and juvenile musculoskeletal deformities, such as adolescent idiopathic scoliosis, hyperkyphosis and genus varus/valgus (LeVeau and Bernhardt, 1984; Frost, 1990; Mao and Nah, 2004). Previous in vitro studies have determined growth plate intrinsic mechanical properties (elastic moduli, Poisson’s ratio and permeability coefficients) under uniaxial compression (Cohen et al., 1994). Other properties of growth plates (ultimate stress and strain, tangent modulus) were also investigated in vitro using bone-growth plate-bone specimens under uniaxial traction (Cohen et al., 1992; Williams et al., 2001). In a study by Cohen et al. (1998), the growth plate was partitioned in two adjacent regions, the chondroepiphysis/reserve zone region and the proliferative/hypertrophic zones region, and further tested under unconfined compression. Using the transversely isotropic biphasic model (TIBPE), this study showed that the chondroepiphysis/ reserve zone was about twice as stiff as the proliferative/ hypertrophic zone along the longitudinal axis as well as in the transverse plane. These mechanical properties represent the bulk
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Nomenclature Parameters for biphasic model E1 E3 k1 k3
n21 n31
transverse Young’s modulus out-of-plane Young’s modulus transverse permeability axial permeability transverse Poisson’s ratio out-of-plane Poisson’s ratio
511
Parameters for FE and three-spring models Ti E3i E¯ 3
thickness of zone i out-of-plane modulus of zone i overall out-of-plane Young’s modulus predicted by the models (FE and three-spring models)
Fig. 1. Histological section showing the growth plate structure and the corresponding mean zonal proportions of the reserve, proliferative and hypertrophic zones.
behavior of the entire growth plate or growth plate regions but the intrinsic mechanical properties of each of the three zones have not been fully characterized. With an experimental setup combining confocal microscopy and mechanical loading, Villemure et al. (2007) characterized continuous strain fields along the growth plate thickness and observed non-uniform strain distributions among the histomorphological zones of the growth plate. However, the overall mechanical properties of the three histomorphological zones of the growth plate, considered separately, have not been characterized yet. The aim of this study is to determine intrinsic mechanical properties of the complete growth plate and its reserve, proliferative and hypertrophic zones. Stress relaxation tests under unconfined compression are performed on each of these four materials taken separately and the obtained mechanical properties are those best fitting a TIBPE model through the experimental data. Results provide an understanding of the growth plate intrinsic mechanical properties variation along its thickness for a specific animal model.
Growth plate Radius
∅ 4 mm
Ulna
[mm] 2. Methods
Growth plate cylindrical sample
Fig. 2. Growth plate cylindrical sample extraction using 4 mm diameter biopsy punches. The growth plate is located between the metaphysis and epiphysis.
2.1. Animal model and histological study Distal ulnae from newborn swines were obtained within 3 h of slaughter and dissected to extract growth plate samples using 4 mm diameter biopsy punches (Fig. 2). A histological study (n ¼ 8) was first performed in order to determine the proportions of the three zones of the newborn porcine growth plate. Following dissection, growth plate samples were fixed in 4% paraformaldehyde and
embedded in paraffin. Ten histological sections (5 mm) taken at different depth within the sample and stained using toluidine blue were viewed (25 ) using a microscope (Leica DMR with a Retiga Qimaging Camera). Measurements (10 per section) of growth plate and zones thicknesses were made with an image analysis software (Digimizer v3.0).
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Growth plate samples used for biomechanical testing were divided in two groups. In one sample group, upper and lower surfaces of the growth plate disks were trimmed using a Vibratome (Vibratome 1500 Sectioning System) to obtain two parallel surfaces and to provide thickness of the complete growth plate sample (repeatable height adjustment of 5 mm). In the other sample group, the three zones were sequentially trimmed from the complete growth plate based on mean zone proportions previously obtained from the histological study. Unconfined compression tests were performed for their good repeatability and because they allowed extraction of Poisson’s ratios. Disk samples were placed between two impermeable smooth platens allowing for their lateral expansion and subjected to stress relaxation tests using a micromechanical testing system (MACH-1, Biosyntech Inc.) (Fig. 3). Loads were recorded with a load cell, whose range and resolution were, respectively, 17 and 0.026 N. Axial displacements were controlled via an encoder of 100 mm range and 0.5 mm resolution. Specimens were initially preloaded at 5% strain, followed by 15% strain at a strain rate of 1.5 103 s1. A relaxation criterion of 0.05 g/min was used to allow samples to reach equilibrium. All samples were bathed in Hank’s balanced salt solution (HBSS) throughout the test.
experimental data from unconfined compression stress relaxation tests were then fitted with analytical curves for the complete growth plate (N ¼ 23), the reserve zone (N ¼ 26), the proliferative zone (N ¼ 13) and the hypertrophic zone (N ¼ 16). Specific ranges for initial values were determined for each of the four parameters based on previous experimental studies on the growth plate (Cohen et al., 1998). For each experimental data set, 81 different combinations of initial values (three values for each parameter) were used to fit the data. The optimal solution was the result showing the minimal error and simultaneously meeting the restrictions listed in Eqs. (1) and (2) imposed by thermodynamics on the engineering constants for a transversely isotropic material (Lemprie`re, 1968) E1 1 n231 (1) ; ð1 n221 Þ40 E3 1 n221 2n231
E1 E1 2n21 n231 40 E3 E3
(2)
2.2. Determination of intrinsic mechanical properties
2.3. Statistical analysis
The transversely isotropic biphasic model derived by Cohen et al. (1998) was used to extract intrinsic mechanical properties of growth plates and the three zones. This model is a modified version of the the linear biphasic poroelastic model (BPE), initially developed by Mow et al. (1980), and adapted for transverse isotropy. In the TIBPE model, the properties are the same in all directions within the transverse plane of isotropy (1–2). This model involves five parameters: E1 is the Young’s modulus in the transverse plane, E3 is the out-of-plane Young’s modulus, n21 is the Poisson’s ratio in the transverse plane, n31 is the out-of-plane Poisson’s ratio, and k1 is the transverse permeability coefficient. The out-of-plane permeability coefficient k3 could be eventually obtained from confined compression stress relaxation tests. A four-parameter optimization procedure was used to extract intrinsic mechanical properties (k1, E1, n21 and n31), while moduli E3 were evaluated based on the applied strains and relaxed stress values obtained from the experimental curves. Using MATLAB software and the root mean square error (RMSE) method,
Basic statistical analysis (mean, standard deviation) was performed for each parameter (thickness, E3, E1, k1, n21, n31) and on each set of data (complete growth plate, reserve, proliferative and hypertrophic zones). A one-way ANOVA for repeated measures was carried out to determine whether any difference existed between the mean parameter values obtained for each of the three zones as well as for the complete growth plate. The post-hoc comparisons between groups were done using the Tukey’s method. The results are presented as mean values7standard deviations. For all tests, the level of significance was fixed at po0.05. The software used for the statistical analysis was STATISTICA 7.0 from StatSoft Inc.
axis 3 to load cell applied impermeable loading platen
displacement
HBSS sample
axes 1, 2 Glass Petri dish Fig. 3. Schematic of the unconfined compression test. Samples (4 mm in diameter) were initially preloaded at 5% strain, followed by 15% strain at a strain rate of 1.5 103 s1. A relaxation criterion of 0.05 g/min was used to allow samples to reach equilibrium. All samples were bathed in Hank’s balanced salt solution (HBSS) throughout the test.
3. Results Histological analyses are summarized in Fig. 1. The resulting proportions of the reserve, proliferative and hypertrophic zones are 7073%, 1772% and 1371%, respectively. These proportions were used to sequentially trim the three zones from the complete growth plate. Complete growth plate thickness varied between 2500 and 4200 mm with a mean value of 3510 mm, while reserve, proliferative and hypertrophic zones indicated, respectively, 2450, 600 and 460 mm for average thicknesses (Table 1). Typical experimental stress relaxation curves are presented in Fig. 4(a) for a complete growth plate as well as samples of the three zones. Fig. 4(b) shows theoretical stress relaxation time histories based on mean mechanical properties obtained from the optimization procedure. The growth plate curve showed great similarity to the reserve zone curve for comparable peak and relaxation stresses. The proliferative and hypertrophic zones indicated different curve profiles with considerably lower peak stresses and slightly smaller relaxation stresses and times. Representative curve-fits between experimental data and analytical model are illustrated in Fig. 5. Very good agreement was observed between experimental and theoretical values at the equilibrium state, as moduli E3 were predetermined from the experimental curves and fixed in the model. Experimental peak stress was generally higher than in the analytical curve. Intrinsic mechanical properties of the complete growth plate and its three zones optimized with the TIBPE model as well as calculated equilibrium Young’s moduli are summarized in Table 1 and
Table 1 Transversely isotropic biphasic properties of porcine distal ulna growth plate and its three zones from unconfined compression tests (mean values7standard deviations). Experiment
Complete growth plate (N ¼ 23) Reserve zone (N ¼ 26) Proliferative zone (N ¼ 13) Hypertrophic zone (N ¼ 16)
Optimization
Thickness (mm)
E3 (MPa)
E1 (MPa)
n21 (–)
n31 (–)
k1 ( 1015 m4/Ns)
35107370 24507210 600740 460740
0.5170.12 0.4870.11 0.2570.12 0.2770.10
8.6571.72 11.0872.86 3.4273.49 3.2673.28
0.2470.07 0.2570.07 0.3270.05 0.3470.05
0.0870.03 0.0770.03 0.1370.04 0.1370.05
1.8270.67 1.4170.94 5.6074.74 5.9875.67
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0.40
0.30
Reserve zone
0.25
Proliferative zone
0.20
Hypertrophic zone
0.15
0.30
Hypertrophic zone
0.15
0.05
0.05 60 120 180 240 300 360 420 480 540 600 660 720 780
Proliferative zone
0.20
0.10
0
Reserve zone
0.25
0.10
0.00
Complete growth plate
0.35
Stress (MPa)
Stress (MPa)
0.40
Complete growth plate
0.35
513
0.00
0
60 120 180 240 300 360 420 480 540 600 660 720 780
Time (s)
Time (s)
Fig. 4. (a) Experimental stress relaxation time histories from unconfined compression tests on the complete growth plate and the reserve, proliferative and hypertrophic zones for typical samples; (b) theoretical stress relaxation time histories based on average mechanical properties obtained from the optimization procedure.
0.40
0.40
0.35
Complete Growth Plate
0.25 0.20 0.15 0.10
Proliferative Zone
0.35 Stress [mPa]
Stress [MPa]
0.30
0.05
0.30
Experimental Analytical
0.20 0.15 0.10 0.05
0
0
60 120 180 240 300 360 420 480 540 600 660 720 780
60 120 180 240 300 360 420 480 540 600 660 720 780
Time [s]
Time [s] 0.40
0.40 0.35 0.25 0.20 0.15 0.10
Hypertrophic Zone
0.35
Reserve Zone
0.30
Stress [MPa]
Stress [MPa]
X
0.25
0.30 0.25 0.20 0.15 0.10 0.05
0.05 0
60 120 180 240 300 360 420 480 540 600 660 720 780 Time [s]
0
60 120 180 240 300 360 420 480 540 600 660 720 780 Time [s]
Fig. 5. Typical stress relaxation histories in response to a 15% ramped displacement with curve-fits of the transversely isotropic biphasic model.
graphically shown in Fig. 6. According to the ANOVA analysis described at the previous section, statistically significant differences were found in transverse permeability coefficients (k1), transverse Poisson’s ratios (n21), out-of-plane Poisson’s ratios (n31) and out-of-plane Young’s modulus (E3) between the reserve zone and the proliferative zone as well as between the reserve zone and the hypertrophic zone. Parameters k1, n21, n31, E1 and E3 also significantly differed between the complete growth plate zone and the proliferative and hypertrophic zones. Significant difference was also found between the reserve zone and the complete growth plate for the transverse Young’s moduli (E1). The proliferative and hypertrophic zones are half as stiff as the reserve zone along the compression axis and about three times less stiff than the reserve zone in the transverse plane. These two zones are three times as permeable as the reserve zone in the radial direction. The out-of-plane Poisson’s ratio of the reserve zone is half the ones of the two other zones, and its transverse Poisson’s ratio is smaller than in the proliferative and hypertrophic zones.
4. Discussion The aim of this study was to determine intrinsic mechanical properties of the complete growth plate and its reserve,
proliferative and hypertrophic zones using stress relaxation tests under unconfined compression combined with an optimization procedure of the TIBPE model. The reserve zone, with its greater zonal component at the newborn development stage (70% of the total thickness), has noteworthy effects on the growth plate intrinsic mechanical properties. Newborn porcine growth plate indicates non-homogeneous mechanical properties along its thickness. The proliferative and hypertrophic zones show very similar mechanical behavior and differ from the reserve zone, which indicates comparable mechanical properties with the complete growth plate. During early development, the ossification process is highly active and the reserve zone and the chondroepiphysis are merged in the upper growth plate zone. Therefore, the reserve zone would be a good indicator of the complete growth plate intrinsic mechanical properties in the newborn porcine ulnae. The obtained mechanical properties are, however, dependent on the optimized model selected to extract the intrinsic tissue parameters. The TIBPE model was chosen for its very good agreement with experimental data for growth plate tissues (Cohen et al., 1998). Results show that E3 for the complete growth plate is greater than the moduli of the three zones. In order to discuss this result, an unconfined compression test was simulated by finite element (FE) simulations on a three-section transversely isotropic cylinder
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Transverse Young's Modulus E1
18 16
0.7
14
0.6
12
0.5 [MPa]
10
[MPa]
Out-of-Plane Young's Modulus E3
0.8
8
0.4 0.3
6
0.2
4
0.1
2 Complete
Reserve
Prolif.
Hyper.
Complete
0
Reserve
Prolif.
Hyper.
0.0
Transverse Poisson's Ratio 0.50
Out-of-Plane Poisson's Ratio
21
0.50
0.45
0.45
0.40
0.40
0.35
0.35
0.30
0.30
0.25
0.25
0.20
0.20
0.15
0.15
0.10
0.10
31
0.05
0.05 Complete
Reserve
Prolif.
Complete
Hyper.
Reserve
Prolif.
Hyper.
0.00
0.00
Permeability k1
16
LEGEND 14
Significant difference (p < 0.05) between the complete growth plate and one of the three zones
x 1E-15 [m4/Ns]
12
Significant difference (p < 0.05) between two of the three zones
10 8 6 4 2 0
Complete
Reserve
Prolif.
Hyper.
Fig. 6. Intrinsic mechanical properties of the complete growth plate and its three zones (mean values7standard deviations). Significant differences are marked with connecting lines (p-valueo0.05).
representing the complete growth plate. Each section was attributed the average mechanical properties (Table 1). The overall E3 of this assemblage, denoted by E¯ 3, was computed as the ratio between the applied pressure and the resulting effective strain (overall displacement divided by sample height). E¯ 3 ¼ 0.39 MPa was obtained. Then, another simplified model that consisted of three cylinders piled on the top of each other with frictionless interfaces (i.e. three springs in series) was developed. For this unidimensional model, 3 X 1 1 Ti ¼ P3 E E¯ 3 3i i¼1 T i i¼1
(3)
where Ti and E3i are, respectively, the thickness and E3 of zone i. E¯ 3 ¼ 0.38 MPa was obtained when inserting the corresponding average values (Table 1). It can be seen that both E¯ 3 obtained from these two models were very close. This suggests that this simplified model can predict relatively well E¯ 3 for the range of mechanical properties simulated with a minimal computational effort. This model was therefore used for estimating the statistical variability in E¯ 3 . In order to estimate the mean of E¯ 3 obtained with
the simplified model, all the raw data that led to Table 1 was used. Each T i E3i pair was considered as a building block for the threesprings model and 5408 ¼ (N ¼ 26) (N ¼ 13) (N ¼ 16) threesprings models were created. E¯ 3 ¼ 0.35370.02 MPa was obtained, while experimental E3 ¼ 0.5170.05 MPa was obtained for the complete growth plate (95% confidence intervals on the mean). This suggests that there is a statistically significant difference between E¯ 3 and E3 for the complete growth plate. Possible causes of such discrepancy could be that the micro-structural changes caused by the sectioning of complete growth plate in its three zones (involving interruption of the collagen and proteoglycan networks) are not taken into account. Hence, the complete growth plate, with its continuous fibrils networks along the three zones, could provide a greater resistance than three zones taken separately. Sectioning the growth plate could induce damage in the three zones thus leading to decreased mechanical properties. Such results are a motivation for future studies and remind the difficulty of obtaining meaningful in-situ mechanical properties. Obtained mechanical properties are comparable to reported studies on growth plate mechanics. Transverse and out-of-plane Young’s moduli obtained for the complete growth plate by (Cohen
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et al., 1994) indicate similar magnitudes than the moduli of the present study. Young’s moduli of 4-month-old calves for composite samples combining the proliferative and hypertrophic zones, as well as for samples referred to as the chondroepiphysis, result in similar magnitudes and trends, with higher moduli in chondroepiphysis samples as compared to proliferative/hypertrophic samples (Cohen et al., 1998). However, our experimental trends do not corroborate with the results obtained from strain patterns of loaded rat tibial growth plates, where the proliferative zone was identified as more rigid when compared to the reserve and hypertrophic zones (Villemure et al., 2007). Differences might be related to the different developmental stage (adolescent) of these growth plates, as growth plates might present different relative mechanical behavior at different development stage. Differences in the tissue histomorphometry within the proliferative/hypertrophic zones could explain their different mechanical behavior as compared to the reserve zone. Major differences in matrix/cell volume ratio, cell arrangement as well as cell size exist between the three zones. Indeed, the matrix/cell volume ratio significantly decreases when approaching the chondro-osseous junction. The ratio reaches values greater than 9 in the reserve zone, remains bounded between 4 and 8 in the proliferative zone and decreases substantially to values smaller than 1 in the hypertrophic zone (Farnum et al., 2002). Considering that the extracellular matrix has the most important structural role in the mechanical behavior of the soft tissue, it is expected that the zone with higher matrix/cell volume ratio would offer stiffer property. In addition, chondrocytes are arranged in columns in the proliferative and hypertrophic zones while they are randomly dispersed in the reserve zone (Hunziker and Schenk, 1989). The chondrocytes volume also increases by an approximate factor of 9 from the proliferative to the hypertrophic phases (Hunziker and Schenk, 1989). These volume and spatial characteristics, which involve a sequence of very soft cells (as compared to the surrounding extracellular matrix) in the proliferative and hypertrophic zones, might contribute to a weaker mechanical support in these lower zones. The non-uniform matrix composition within the different zones cannot fully explain their different mechanical behavior. In a study carried out by Alvarez et al. (2000), the distribution of collagens was characterized in rat tibial growth plates divided into five horizontal strata of equal heights. The type II collagen expression showed peak values in the proliferative zone, while the type X collagen expression attained maximum values approaching the upper hypertrophic zone and decreased thereafter to the lower hypertrophic zone (Alvarez et al., 2000). A qualitative evaluation realized by Sandell et al. (1994) showed comparable trends for both type II and type X collagen. As for aggrecan, its localization within the zones presented a more or less uniform distribution (Sandell et al., 1994). Since structural elements (type II collagen, type X collagen, aggrecan) of the extracellular matrix are predominantly contained in the proliferative zone, this zone could be expected to offer higher mechanical properties. However, our results show that the proliferative zone is less stiff than the reserve zone in both, tension and compression. Therefore, depth-dependant relationships between structural elements of the extracellular matrix and their localization in the growth plate do not entirely explain the non-homogeneous mechanical properties among the three zones of the growth plate. It should be noted however that the distribution of matrix components detailed above refer to different animal models and to varying developmental stages. This study provides, for the very first time, an investigation of the intrinsic mechanical properties of the reserve, proliferative and hypertrophic zones of the growth plate. The mechanical behavior of the newborn porcine ulnar growth plate is non-
515
uniform along its thickness; the proliferative and hypertrophic zones are half as stiff as the reserve zone along the compression axis and about three times less stiff than the reserve zone in the transverse plane. This knowledge on the zone mechanical behavior is very important for growth plate mechanobiology, where the effects of mechanical loading on the biological process of endochondral growth, with each zone playing its specific roles in the process, are investigated. Based on the results of this study, the more compliant proliferative and hypertrophic zones might be more prone to trigger abnormal endochondral growth upon compressive mechanical loading. More studies are required to confirm if these two lower zones are major actors in the growth plate mechanobiology in this early developmental stage. Similar studies should aim at investigating the mechanical behavior of the growth plate at different developmental stages. This scientific knowledge will also further be very valuable in growth plate modeling as well as in tissue engineering, where quantitative data is required both at the structural and compositional levels for engineering of this living tissue.
Conflict of interest None.
Acknowledgements The authors acknowledge the participation of Marjorie Beausoleil in the experimental design and Jose´e De´poˆt for her technical assistance in histology. This study was funded by the Fonds Que´be´cois de la Recherche sur la Nature et les Technologies (FQRNT, I.V.), the Natural Sciences and Engineering Research Council of Canada (NSERC, I.V.) and the Canada Research Chair in Mechanobiology of the Pediatric Musculoskeletal System (I.V.).
Appendix A. Supplementary materials The online version of this article contains additional supplementary data. Please visit doi:10.1016/j.jbiomech.2008.11.026.
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