Elastic and physicochemical relationships within cortical bone Sean S. Kohles,1,2,3 Daniel A. Martinez4 1 Departments of Biomedical Engineering, Mechanical Engineering, and Biology & Biotechnology, Worcester Polytechnic Institute, 100 Institute Road, Worcester, Massachusetts 01609-2280 USA 2 Department of Clinical Science, Tufts University School of Veterinary Medicine, 200 Westboro Road, North Grafton, Massachusetts 01536 USA 3 Departments of Physiology and Orthopedics, University of Massachusetts Medical School, 55 Lake Avenue North, Worcester, Massachusetts 01655 USA 4 Department of Biology and Biochemistry, University of Houston, 4800 Calhoun Road, Houston, Texas 77204 USA Received 24 July 1998; revised 10 March 1999; accepted 20 July 1999 Abstract: The purpose of this study was to examine the relationships that exist between the elastic properties and the physicochemical properties of cortical bone in two groups of experimental animals. The animal model was the immature mutant dwarf rat, and the groups consisted of rats treated and not treated with recombinant human growth hormone (rhGH). The objective was to establish and broaden the quantifiable link between the three-dimensional form and function of bone beyond the typical unidirectional measures. This study was based on previously reported work that refined the ultrasonic elasticity technique for use with small specimens ( 0.80, respectively. All analyses were performed with commercially available software packages (SAS Institute Inc., Cary, North Carolina, and Mathematica, Wolfram Research Inc., Champaign, Illinois). Means, sample standard deviations (unbiased), and standard errors of the mean (SE) were calculated and used for descriptive comparisons.

RESULTS Ultrasonic elasticity

Figure 1. Locations from which test samples were acquired from the rat femora for elastic and physicochemical property analysis. This posterior view shows the diaphyseal locations of cortical samples used for (A) ultrasonic elasticity, (A) microscopy, (A) porosity, (B) density, (B) crystal size, (C) morphology, (D) collagen content, (D) crosslink content, and (D) mineral content.

As noted previously,45 the cortical elastic material properties generally decreased in magnitude in the dwarf rat bone after a 14-day treatment with rhGH (Table II). Overall, the increased accretion rate caused by rhGH altered the mechanical properties. This trend was statistically consistent (p < 0.05) in all (−26.74% average) of the Young’s moduli (Eii). Shear moduli (Gij ) also demonstrated a general decrease (−16.67% average) in value after rhGH treatment, with statistical significance (p < 0.05) demonstrated in G31 and G12. The ratio of transverse strain to axial strain (Poisson’s ratio, ␯ij) had less consistent trends (−30% to +104%) in response to the rhGH treatment of dwarfism. Generally there was a decrease in value after treatment, with

CONSTITUTIVE RELATIONSHIPS IN CORTICAL BONE

TABLE II Summary of the Mean (±SE) Orthotropic Elastic Material Properties (Units) As Measured Using the Ultrasonic Elasticity Technique45 Property

Dwarf

Dwarf w/GH

E11 (GPa)* E22 (GPa)* E33 (GPa)* G23 (GPa) G31 (GPa)* G12 (GPa)* ␯31 (ratio)* ␯21 (ratio)* ␯32 (ratio) ␯13 (ratio) ␯12 (ratio)* ␯23 (ratio)

11.68 (0.299) 14.91 (0.630) 17.98 (0.794) 3.65 (0.110) 4.27 (0.090) 4.37 (0.159) 0.373 (0.027) 0.314 (0.046) 0.269 (0.018) 0.247 (0.021) 0.257 (0.041) 0.228 (0.022)

9.48 (0.428) 9.17 (0.614) 13.84 (0.757) 3.37 (0.161) 3.50 (0.141) 3.31 (0.120) 0.260 (0.030) 0.484 (0.017) 0.308 (0.031) 0.184 (0.025) 0.523 (0.047) 0.200 (0.017)

*Statistical differences at p < 0.05.

a significant decrease seen in ␯31. However, a significant increase was measured concurrently in ratios ␯21 and ␯12. Overall, rhGH had a dramatic effect on the elastic properties of growth-delayed cortical bone in rats after 14 days of treatment.

Physicochemical properties The use of rhGH as a treatment for dwarfism caused some significant changes in both the morphologic and biochemical organization of rat cortical bone (Table III), as presented earlier.44,45 This evidence of cortical

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remodeling within the existing organic systems in the brief 14-day period is an indicator that rhGH supplementation given to GH-deficient rats accelerates growth in protein systems and also can impact inorganic components. Generally, a significant decrease in density (−2.42%), in mineral crystal length (−8.95%), and in mineral width (−11.47%) was detected while a significant increase in surface and volume porosity (+19% and +125%, respectively) was determined (p < 0.05). Apparent densities, calculated using surface (−5.48%) and volume (−3.45%) porosities, reiterated a similar decrease in density (p < 0.05). Increases in cortical area (+19.71%) and the moments of inertia (+30 to +35%) were measured (p < 0.05) while medullary area (+3.90%) remained statistically unchanged, suggesting cortical growth took place periosteally and in a direction away from the cross-sectional centroid. Concomitantly, no statistical changes in mineral concentration via calcium (−2.86%) and phosphorous (−1.63%) measures and nonmineral content via collagen concentration (−5.47%) or crosslinking (−1.07%) were determined. Crystallinity was preserved, as seen in the maintenance of the crystal aspect ratio (length:width) at 2:1. However, the crystal size was statistically smaller in the rhGH group (p < 0.05). In general, the administration of rhGH as a treatment to accelerate growth, altered the organization of bone at both the macro (morphologic) and micro (biochemical) levels. Microradiographs demonstrate throughout the thickness alterations in tissue organization via changes in porosity among treatment groups (Fig. 2).

TABLE III Summary of the Measured Cortical Bone Physicochemical Properties from Dwarf Rats and Dwarf Rats Treated with rhGH44,45 Property (Units) 3

†

Density (g/cm )* Apparent surface density (g/cm3)*‡ Apparent volumetric density (g/cm3)*‡ Surface porosity (ratio)*‡ Volume porosity (ratio)‡ Crystal length (Å)*† Crystal width (Å)*† Crystal area (Å2)* Crystal aspect (ratio) Calcium (mg/mg dry mass)† Phosphorous (mg/mg dry mass)† Total mineral (mg/mg dry mass)† Collagen (mg/mg dry mass)† HP crosslink (mol/mol collagen)† LP crosslink (mol/mol collagen)† Total crosslinks (mol/mol collagen)† Cortical area (mm2)*† Medullary area (mm2)† Moment of inertia, I1 (mm4)* Moment of inertia, I2 (mm4)* Polar moment of inertia, J (mm4)*

Dwarf

Dwarf w/GH

1.777 (0.011) 1.734 (0.010) 1.682 (0.012) 0.0241 (0.012) 0.0531 (0.003) 125.94 (3.85) 65.11 (2.46) 8213.3 (421.1) 1.955 (0.090) 0.245 (0.004) 0.184 (0.006) 0.429 (0.009) 0.020 (0.001) 0.163 (0.005) 0.210 (0.010) 0.374 (0.013) 2.663 (0.120) 3.358 (0.090) 2.233 (0.155) 1.708 (0.093) 3.941 (0.247)

1.734 (0.016) 1.639 (0.021) 1.624 (0.012) 0.0542 (0.006) 0.0630 (0.004) 114.67 (2.55) 57.64 (2.16) 6606.0 (262.6) 2.011 (0.088) 0.238 (0.003) 0.181 (0.004) 0.419 (0.006) 0.019 (0.001) 0.165 (0.007) 0.205 (0.011) 0.369 (0.016) 3.188 (0.093) 3.489 (0.088) 3.013 (0.158) 2.133 (0.094) 5.146 (0.250)

*Statistical differences at p < 0.05; †as reported in Martinez et al., 199644; ‡as reported in Kohles et al., 199745; values are means ± SEM.

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properties were analyzed and added to the equation with each iteration. Additive constants and variable coefficients were determined. The resulting linear regression models (Table IV) attempted to maximize the fit between the generated equations and the acquired data. Relationships were considered statistically strong when R2 > 0.80 and demonstrated by some combinations (e.g., R2 = 0.961 for ␯12). However, most equations formed were limited in their power to predict a relationship (e.g., R2 = 0.351 for E33). Continued analysis of the elastic and physicochemical relationships was undertaken via nonlinear regressions. Initial template equations incorporated the results of the linear regression analysis. The final nonlinear regression models (Table V) determined the specific additive constants, coefficients, and exponents based on the initial form. Again, optimization was attempted via numerous iterations and resulted in a range of predictive fits (e.g., R2 = 0.758 for G12 to R2 = 0.020 for ␯32). Generally, apparent density proved to be the best predictor of the elastic and shear moduli while crystal size offered the best prediction of Poisson’s ratios.

DISCUSSION AND CONCLUSIONS

Figure 2. Representative microradiographs of (a) untreated and (b) growth hormone-treated dwarf rat femoral cortical bone. Specimens were cut to this shape for ultrasonic evaluation. Specimens are approximately 2.0 × 2.0 × 0.4 mm (height × width × thickness). Note the difference in tissue organization between the two samples. Magnification ×20.

Constitutive modeling Analysis of the relationships between the dependence of the elastic properties on the physicochemical characteristics initially was combined via multiple linear regressions. Linear equations were constructed by equationally comparing each elastic property to all of the physicochemical properties individually until the best fit was determined. Additional physicochemical

In this study, the relationships between the elastic and physicochemical properties of cortical bone were examined. The presented findings expand upon previously reported work that determined that the administration of rhGH can counter the degeneration of the elastic and physicochemical characteristics of cortical bone resulting from hormone-suppressed downregulation.44,45 The value of this analysis is that it shows mathematical models that relate changes in the cortical microstructure to changes in the orthotropic elastic properties of the tissue. The results clearly indicate that rhGH, when used as a treatment for dwarfism, has a limited yet significant effect on the structure and composition of cortical bone. This influence does not extend into the mineralization of bone but is expressed in the architectural arrangement of the mineralized material via the production of new porous bone and the turnover of the existing mature bone. Although extensive in the number of physicochemical properties gathered, further analysis, including determination of the mechanisms associated with an rhGH response, the addition of histological markers distinguishing old and new bone, comparisons with age-matched control bone, and more detailed X-ray and microscopic evaluation, would further the understanding of growth and its effect on bone elasticity. A temporal examination of these properties via an increase in the number of timeinterval groups would aid in describing the time-

CONSTITUTIVE RELATIONSHIPS IN CORTICAL BONE

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TABLE IV Equational Models Demonstrating the Dependence of Ultrasonic Elastic Properties Upon the Measured Physicochemical Characteristics of Rat Cortical Bone E11 E22 E33 G23 G31 G12 ␯31

␯21

␯32 ␯13

␯12

␯23

= −2.147 + (4.468) × (Apparent density, surface)2 = −323.464 × (9.269) × (Apparent density, surface)2 + (36.401) × Log(Crystal area) + (−0.0000003) × (Crystal area)2 = −7.317 + (8.153) × (Apparent density, surface)2 = 6.535 + (3.275) × Log(Total mineral) + (−5.445) × (Surface porosity) = 4.753 + (−179.245) × (Volume porosity)2 + (−6.153) × (Surface porosity) = −10.752 + (1.912) × (Apparent density, surface)2 + (2.224) × Log(Crystal width) = −1.657 + (−48.047) × (Volume porosity)2 + (0.482) × Log(Crystal width) + (−75.968) × (Collagen) × (Total cross-links) + (0.518) × (Apparent density, surface) + (−4.859) × (Phosphorous)2 = 3.546 + (−0.778) × Log(Crystal width) + (−0.342) × (Apparent density, surface)2 + (86.776) × (Collagen) × Total cross-links) + (37.043) × (Volume porosity)2 + (32.100) × (Collagen) × (Total mineral) No comparison where p < 0.1 = −2.362 + (0.784) × Log(Crystal width) + (−43.339) × (Volume porosity)2 + (−47.898) × (Collagen) × (Total cross-links) + (−0.000000005) × (Crystal area)2 = 6.206 + (−1.597) × Log(Crystal width) + (77.069) × (Volume porosity)2 + (0.000000005) × (Crystal area)2 + (−0.416) × (Apparent density, surface)2 + (92.218) × (Collagen) × (Total cross-links) + (3.193) × (Total mineral)2 + (7.414) × (HP cross-links)2 = 0.113 + (0.00003) × (Crystal width)2

(p (p (p (p (p (p (p (p (p (p (p (p (p (p (p (p (p (p (p (p (p

= = = = = = = = = = = = = = = = = = = = =

0.0020, R2 0.0008) 0.0231) 0.0492, R2 0.0059, R2 0.0644) 0.0904, R2 0.0235) 0.0831, R2 0.0001) 0.0061, R2 0.0003) 0.0007) 0.0073) 0.0373) 0.0504, R2 0.0001) 0.0006) 0.0053) 0.0058) 0.0489, R2

(p (p (p (p (p (p (p (p (p (p (p (p

= = = = = = = = = = = =

0.0002) 0.0001) 0.0130) 0.0159, R2 = 0.762) 0.0001) 0.0001) 0.0065) 0.0001) 0.0012) 0.0007) 0.0144, R2 = 0.961) 0.0653, R2 = 0.176)

= 0.421)

= 0.678) = 0.351) = 0.448) = 0.440) = 0.755)

= 0.793)

= 0.856)

Relationships were created using multiple linear regression analysis of up to 200 iterations. The p values of