YOUNG S MODULUS OF TRABECULAR AND CORTICAL BONE MATERIAL: ULTRASONIC AND MICROTENSILE MEASUREMENTS*

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1 J. Biomechanics Vol. 26, No. 2, pp. 11 I / Printed in Great Britain #C:, 1993 Perpamon Press Ltd YOUNG S MODULUS OF TRABECULAR AND CORTICAL BONE MATERIAL: ULTRASONIC AND MICROTENSILE MEASUREMENTS* JAE YOUNG RHO,~$ RICHARD B.ASHMAN and CHARLES H. TURNER~~ IDepartment of Orthopaedic Surgery, State University of New York at Buffalo, Buffalo, New York, U.S.A.; Research Department, Texas Scottish Rile Hospital, Dallas, TX 75219, U.S.A. and //Department of Orthopaedic Surgery and the Biomechanics and Biomaterials Research Center, Indiana University, Indianapolis, IN46202, U.S.A. Abstract-An ultrasonic technique and microtensile testing were used to determine the Young s modulus of individual trabeculae and micro-specimens of cortical bone cut to similar size as individual trabeculae. The average trabecular Young s modulus measured ultrasonically and mechanically was 14.8 GPa (S.D. 1.4) and 10.4 (S.D. 3.5) and the average Young s modulus of microspecimens of cortical bone measured ultrasonically and mechanically was 20.7 GPa (S.D. 1.9) and 18.6 GPa (S.D. 3.5). With either testing technique the mean trabecular Young s modulus was found to be significantly less than that of cortical bone (p <0.0001). However, the specimens were dried before microtensile testing so Young s modulus values may have been greater than those of trabeculae in viuo. Using Young s modulus measurements obtained from 450 cubes of cancellous bone and 256 cubes of cortical bone, Wolffs hypothesis that cortical bone is simply dense cancellous bone was tested. A multiple regression analysis that controlled for group membership showed that Young s modulus of cortical bone cannot be extrapolated from the Young s modulus vs density relationship for cancellous bone, yet the Young s modulus of trabeculae can be predicted by extrapolation from the relationship between Young s modulus vs density of the cancellous bone. These results suggest that when considered mechanically, cortical and trabecular bone are not the same material. INTRODUCTION In 1892, Wolff suggested that cortical bone was simply more dense cancellous bone (Wolff, 1892). Some investigators (Carter and Hayes, 1976,1977; Keller et al., 1990) have produced data that indirectly support Wolff s argument. Carter and Hayes (1977) suggested that the mechanical properties of cortical bone could be extrapolated from those of cancellous bone; however, this result has been disputed (Rice et al., 1988). Keller et al. (1990) note that the distinction between very porous cortical bone and very dense cancellous bone is somewhat arbitrary; therefore, the transition between cancellous bone and cortical bone is continuous. Yet, direct measurements of the elastic properties of individual trabeculae show them to be considerably less than those of cortical bone (Ashman and Rho, 1988; Choi et al., 1989, 1991; Ku et al., 1987; Kuhn et al., 1989; Mente and Lewis, 1989; Townsend and Rose, 1975). Furthermore, a statistical analysis employed by Rice et al. (1988) using pooled data obtained from several different studies contradicted WolfFs hypothesis by showing that the Young s modulus of cortical bone could not be extrapolated from a modulus-density relationship for cancellous bone. Received in jinal form 5 July t Author to whom correspondence should be. addressed at: Department of OrthopaedG Surgery, SUNY at Buffalo, 162 Farber Hall. Buffalo. NY U.S.A. *Presenteh in pa;t at the 3ith Annual Meeting of the Orthopaedic Research Society, Anaheim, California, March In the literature survey outlined in Table 1, the measured and estimated values of the Young s modulus of trabecular bone material were found to range from 0.76 to 20 GPa. The variance in the reported values indicates that there is still a lingering controversy about Young s modulus of trabecular material. Using ultrasonic techniques, we previously reported the Young s modulus of trabecular bone material to be 12.7 GPa (Ashman and Rho, 1988). More recent studies using mechanical testing of trabeculae have reported considerably lower values for Young s modulus (Choi et al., 1989,199l; Kuhn et al., 1989; Ryan and Williams, 1989). Addressed in this study are three questions: (1) Is the Young s modulus of individual trabeculae the same as that of microspecimens of cortical bone? (2) Can the Young s modulus of trabecular bone material be extrapolated from the Young s modulus vs density relationship for cancellous bone? (3) Can the Young s modulus of cortical bone be extrapolated from the Young s modulus vs density relationship for cancellous bone? MATERIALS AND TEST PROTOCOLS The studies done here have been divided into two parts for clarity. Part 1 is directed toward answering whether trabeculae and cortical bone have the same Young s modulus by comparing Young s modulus of individual trabeculae and microspecimens of cortical bone using both microtensile testing and ultrasonic techniques. Part 2 is directed toward answering whether Young s modulus of trabecular bone material BM 26:2-E 111

2 112 J. Y. RHO et al. Table 1. Estimates and determinations of the elastic modulus of trabecular bone material. Table presented by Rice et al. (1988) was extended Source Type of bone Test method Wolff (1892) Pugh et al. (1973) Runkle and Pugh (1975) Townsend et al. (1975) Human Human, distal femur Human, distal femur Buckling Human, proximal tibia Inelastic buckling Williams and Lewis (1982) Human, proximal tibia Ryan and Williams (1986) Ku et al. (1987) Mente and Lewis (1987) Ashman and Rho (1988) Choi et al. (1989) Kuhn et al. (1989) Mente and Lewis (1989) Ryan and Williams (1989) Williams and Johnson (1989) Jensen et al. (1990) Choi et al. (1991) Present study Fresh bovine femur Fresh frozen human tibia Dried human femur, fresh human tibia Bovine femur, human femur Human tibia Human iliac crests Dried human femur, fresh human tibia Bovine femur Bovine tibia formed from PMMA bone cement Human vertebra (L3) Human tibia Human tibia Hypothesis Finite element method Experiment with two-dimensional finite element method Tensile testing Three-point bending Cantilever bending with finite element analysis Ultrasonic test method Three-point bending Thfee-point bending Cantilever bending with finite element analysis Tensile testing Predicted by extrapolation from modulus of composite of trabecular bone and PMMA using ultrasonic test method Structural analysis by threedimensional mode1 Four-point bending Tensile testing Ultrasonic test method Estimate of elastic modulus of trabecular bone material GPa (wet) Concluded the modulus of the trabeculae as less than the modulus of cortical bone 8.69 (3.17) GPa (dry) GPa (wet) GPa (dry) 1.30 GPa 0.76 (0.39) GPa 3.17 (1.5) GPa 5.3 (2.6) GPa 10.9 (1.6) GPa 12.7 (2.0) GPa 4.59 GPa 3.81 GPa 7.8 (5.4) GPa 1.0 GPa 8.9 GPa 3.8 GPa 5.35 (1.36) GPa (wet) 10.4 (3.5) GPa (dry) 14.8 (1.4) GPa (wet) and cortical bone can be extrapolated from those of cancellous bone using a regression analysis similar to that employed by Rice et al. (1988). Throughout this paper, the term cancellous bone will refer to the porous cancellous structure (cubic specimens). Trabecular bone will be used when speaking of the trabecular material (bone tissue within cancellous bone). Individual trabeculae will refer to a single strut taken from cancellous bone. Part 1. Young s modulus measurements in individual trabeculae and microspecimens of cortical bone Specimens. Twenty individual trabeculae (approximate diameter 0.18 mm, length 2.3 mm) were isolated using a stereomicroscope, scalpel, and forceps from the proximal region of a single human tibia (age, sex, and cause of death unknown). Specimens consisted of a long, uniform, rod-like central region and nonuniform branching ends (Fig. 1). The nonuniform portions were inserted into grips so as to provide a free length of the specimen with as little variation in the cross section as possible. The dimensions of the specimens were measured using a 100 x calibrated eyepiece. Cross-sectional area was determined by averaging several sections along the free length. The diameter of the typical specimens in the present study varied by less than 10% along the free length. The specimens were assumed to be circular in cross section. Also, 20 microspecimens of cortical bone-axially oriented parallelepiped specimens (approximate dimensions 0.3 x 0.3 x 2.2 mm-were obtained from the diaphyseal region of the same tibia. Microspecimens of cortical bone were cut by using the following method (Choi et al., 1989): a series of bone sections were cut on a low-speed diamond blade saw (Model 1 l Isomet, Buehler), each cortical bone section was then placed between two plates of plexiglass and cut sequentially, thus all four surfaces of rectangular parallelepiped specimens were produced by the saw. Other than bone specimens, ten standard samples of stainless steel wire (diameter 0.3 mm) and aluminium rectangular parallelepipeds (approximate dimensions

3 Fig. 1. Typical individual trabecular specimens viewed by ordinary light microscope (67 x ), 113

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5 Young s modulus of trabecular and cortical bone material x 0.3 mm) were cut by a similar method. These specimens were tested to verify the microtensile procedure. Aspect ratios (length/width or diameter) of these specimens were kept constant (L/d = 10). Ultrasonic tests. A pulse transmission ultrasonic technique, similar to that previously described (Ashman et al., 1984; Ashman and Rho, 1988) was used to measure the bar velocity, v, of individual trabeculae and microspecimens of cortical bone, aluminum, and stainless steel. Ultrasonic waves were propagated through the bone at a frequency of 2.25 MHz. Young s modulus was determined using E = pv, where p is the specimen s density. Young s modulus of individual trabeculae was calculated with an assumed constant density of 1764 kgme3 for all specimens from the average density measured in previous work (Ashman and Rho, 1988). Young s modulus of microspecimens of cortical bone was calculated with an assumed constant density of 1817 kg m-3 for all specimens from the average density measured from 256 cortical cubes. All bone specimens were fully saturated with physiological saline during the tests. Microtensile tests. Tensile tests were performed using a specially designed microtensile apparatus (Fig. 2). The ends of each specimen were attached to 6.4 mm diameter brass rods by gluing the specimen ends into 1 mm diameter holes under a stereomicroscope and aligned to minimize the bending moments. The load cell was fabricated of 1.62 mm thick aluminium channel with two biaxial strain gages connected in a full bridge configuration. Calibration of the load cell was performed using dead weights. The maximum applied tensile load was about 1 N. A 12 mm gauge length clip-on extensometer was attached to the brass rods to measure strain. The stiffness of the testing apparatus was much greater than the stiffness of any of the specimens so the errors in strain measurement associated with compliance of the grips was negligible. This extensometer was also calibrated using another 25.4 mm extensometer. Tensile deflection was applied to the specimens with a lead screw at approximately 5.5 microstrain s - r. The signal from the load cell and extensometer were amplified through a strain gage conditioner (2100 system, Measurement Group, Inc., Raleigh, NC). Individual trabeculae, microspecimens of cortical bone, aluminum, and stainless steel specimens were tested in tension after the ultrasonic tests were completed. Each specimen was glued into the testing apparatus using a cyanoacrylate glue. The bone speci- Detail Breather hole Note: The breather holes 0110~ the cac.pc of entrapped air during glue injection. Grip assembly Load cell Guide block / r Strain gage (4) J Detail L Specimen I \ cl 1 Fig. 2. Schematic diagram of the microtensile testing apparatus (Copyright J. Y. Rho, all rights reserved).

6 116 J. Y. RHO et al. mens had to be dried so they could be bonded to the testing apparatus. The bone specimens were not rewetted before testing. Also, the effects of gluing the specimens to the tensile grips were investigated. Ultrasonic measurements were made on a separate group of ten trabeculae before and after soaking with glue and allowing them to dry. Velocities were found to average 2898ms- (SD. 84.5) and 2950ms- (SD. 68.2), respectively for the unglued and glued specimens. Again assuming a density of 1764 kg m- 3, the change in modulus would be from 14.8 to 15.4 GPa, i.e. the glue increased Young s modulus by approximately 4%. This difference was not statistically significant (P=O.15). Part 2. Young s modulus measurements in cortical bone, cancellous bone, and trabecular bone Specimens. Ten millimeter cubic cancellous specimens from eight unembalmed human cadavers (45-68 years, mean 60 years) were obtained. A total of 450 specimens of cancellous bone were obtained from the proximal tibiae (144), proximal femora (128), distal femora (146), and proximal humeri (32). To address whether cortical bone properties can be extrapolated from cancellous bone properties, 5 mm cubic cortical bone specimens were obtained from the same cadavers. A total of 256 cortical bone specimens were obtained from tibiae (96), femora (96), and humeri (64). To address whether trabecular material properties can be extrapolated from cancellous bone properties, trabecular material properties obtained from 53 cylindrical cancellous specimens (5 mm diameter by 15 mm length) analyzed previously (Ashman and Rho, 1988) were used. Young s modulus of the trabecular material was measured in these specimens using high frequency (2.25 MHz) ultrasound that propagates though individual trabeculae and gives an average Young s modulus for the trabecular bone within the cancellous structure. We chose to use these data because trabecular density measurements were made, whereas we were unable to measure trabecular density in our current studies of individuals trabeculae. It should be noted that the trabecular density measures for these specimens represent an average trabecular density for the entire specimen. Ultrasonic tests. Ultrasonic tests of cortical and cancellous bone cubes were performed using previously reported techniques (Ashman et al., 1984, 1987). Both cortical and cancellous bone cubes were tested in the longitudinal direction. The density of cortical bone cubes was determined by Archimedes principle (Ashman et al., 1984). The apparent density of cancellous bone cubes was determined using the volume of the overall physical dimensions of the specimen which takes into account the porosity of trabecular bone (Ashman et al., 1987). ultrasonically. Differences between the two measurement techniques were evaluated using a paired t-test. Regression analysis also was used to examine the relationship between microtensile measurements and ultrasonic measurements. A t-test was employed to test the hypothesis that the slope of the regression equals one and the intercept equals zero. Comparison of the Young s moduli of trabeculae with those of cortical bone specimens was done using a t-test. To test whether the Young s modulus of trabeculae or cortical bone can be extrapolated from the density vs cancellous bone relationship, we performed a multiple regression analysis similar to that employed by Rice et al. (1988). A strength of this analysis is that the same measurement technique was used for all specimens. We employed a multiple regression analysis which is described in Draper and Smith (1981). This regression included a group variable and density as independent variables and Young s modulus as the dependent variable. the basic model we used was E=a,+aJ+a,p+a,pl, (1) where E is Young s modulus, p is density, and I is the group variable. For our analyses I = 0 for cancellous bone specimens and I = 1 for either cortical bone cubes or trabeculae. Therefore, equation (1) becomes for cancellous bone specimens and E=a, +a,p (2) E=(a,+a,)+(a,+a& (3) for either cortical bone or trabeculae. If the Young s modulus vs density relationship for cancellous bone can be extrapolated to predict Young s moduli of individual trabeculae, then equations (2) and (3) must be equivalent, and a, and a4 must necessarily be equal to zero. Besides equation (1) we also fit the data to the following equations: E = a, + a,1 + a,p* + a,p*l, E=a,+a,Z+a,p+a,pl+a,p2+a6p21, E=a,+a,I+a,p+a,p~+a,p*+a,p*1 + a,p31 + a,p31. RESULTS Is Young s modulus of individual trabeculae the same as that of microspecimens of cortical bone? Significant differences between the Young s modulus of trabeculae and Young s modulus of microspecimens of cortical bone were found with both ultrasound and miciotensile testing (p < for each technique). The average Young s moduli measured using the Statistical methods. Young s moduli for individual trabeculae, cortical bone, aluminium, and stainless microtensile test and ultrasound for individual trabesteel specimens were measured in both tension and culae, microspecimens of cortical bone, aluminum and (4) (3 (6)

7 Young s modulus of trabecular and cortical bone material 117 stainless steel are shown in Table 2. Significant differences were found between ultrasonic and microtensile tests of trabeculae (p <O.OOOl) and microspecimens of cortical bone (p < 0.006), but not for stainless steel (p = 0.16) or aluminum (p = 0.70). In each case the average value of Young s modulus measured ultrasonically was greater than Young s modulus measured by microtensile testing. Separate regression analyses were employed for the data from individual trabeculae and microspecimens of cortical bone. For trabeculae, E met,, = 1.64-L, GPa, r2=0.46, p<o.ool, the slope was not significantly different from one (p= 0.14), but the intercept was significantly different from zero (p < 0.001). For microspecimens of cortical bone, E mech = 157E,,,, GPa, r2=0..55, p<o.ool, the slope was not significantly different from l), but again the intercept was significantly different from zero (~~0.01). Therefore, the main difference between the two measurement techniques for testing bone specimens was a shift in the intercept rather than a difference in slope. Can the Young s modulus of trabecular bone material be extrapolated from the Young s modulus us density relationship for cancellous bone? Results form the multiple regression analysis showed that the Young s modulus of trabecular bone material could be extrapolated from that of cancellous bone using a quadratic relationship [equation (5)]. The results of the multiple regression to equation (5) are summarized in Table 3. Note that the constants a,, a4, and a6 are not significant, meaning their values are not different than zero. Regressions using a linear relationship [equation (l)], a squared relationship Table 2. The average Young s moduli measured mechanically and ultrasonically from metal, individual trabeculae, and microspecimens of cortical bone (units = GPa, standard deviations in parentheses) Aluminum Stainless steel Individual trabeculae Microspecimens of cortical bone Mechanical Ultrasonic N (tensile) (2.25 MHz) (12.7) 65.8 (4.0) (18.6) (6.6) (3.5) 14.8 (1.4) (3.5) 20.7 (1.9) The ultrasonic Young s modulus of individual trabeculae was calculated by assumed constant density (1764 kgme3, S.D. 157) based on a previous study (Ashman and Rho, 1988) whereas that of microspecimens of cortical bone (1817 kgme3, SD. 138) was calculated by assumed constant density based on measurements from 5 mm cortical cubes. The measured velocities of individual trabeculae was 2892 m s-l (SD. 139), whereas that of microspecimens of cortical bone was 3369 ms- (SD. 152). (7) (8) Table 3. The statistical significance of the indicator variables for model predicting Young s modulus from apparent density for pooled cancellous bone specimens combined with Young s modulus measurements of trabecular material (Ashman and Rho, 1988) Constants a1 a2 a3 a4 a5 a6 Coefficient l.l08e E-6 t-value Probability, p [equation (411 or a cubic relationship [equation (6)] did not allow the extrapolation (i.e. at least one coefficient associated with a group variable was significant for each of these regressions). For the data measured in this study such an extrapolation is valid provided a quadratic relationship between Young s modulus and density is used. The following equation is valid for both cancellous bone cubes and trabecular bone material: E= p+1.8x10-6p2, rz=0.97 (9) where E is in GPa and p is in kg m- 3. This equation is represented in an expanded form including group variables in Table 3*. Figure 3 shows the results of a regression using a quadratic equation for cancellous bone specimens only. The grey lines represent 95% confidence intervals of the regression coefficients. The majority of the values for Young s modulus of trabeculae fall between the confidence intervals. Can the Young s modulus of cortical bone be extrapolated from the Young s modulus us density relationship for cancellous bone alone? Young s modulus values for cortical bone fall outside the 95% confidence intervals. The multiple regression results showed that the Young s modulus of cortical bone could not be extrapolated from cancellous bone data (i.e. a regression between Young s modulus of cortical bone and density was no equivalent to a regression between Young s modulus of cancellous bone and density). DISCUSSION The Young s modulus of individual trabeculae was significantly less than that of cortical bone (p < ) regardless of testing technique (microtensile or ultrasound). This is probably due to microstructure, mineralization, and collagen fiber orientation differences of the two tissues (Martin, 1991). The measured tensile Young s modulus of individual trabeculae and cortical bone was 10.4 and *The coefficients in equation (9) have slightly different values than those in Table 3 because of the addition of group variables in the regression reported in Table 3. -

8 118 J. Y. RHO et al Mx) 800 LOW 12W 14CO wO CO 2600 Apparent Density (kg/m3) Fig. 3. Data from ultrasonic measurements of Young s modulus in cancellous bone, cortical bone and trabecular bone material. The black line represents a regression using a quadratic equation for cancellous bone specimens only. The grey lines represent 95% confidence intervals of the regression coefficients. The majority of the values of Young s modulus of trabecular bone material fall between the confidence intervals, while Young s modulus values for cortical bone fall outside the confidence intervals. In other words, the Young s modulus of trabecular bone material could be extrapolated from that of cancellous bone using a quadratic relationship, while Young s modulus of cortical bone could not be extrapolated from cancellous bone data (Copyright J. Y. Rho, all rights reserved) GPa, respectively. However, drying specimens before testing may have caused an overestimation of Young s modulus by about 24% (Townsend and Rose, 1975). Also, glue was applied to the ends of the specimens which could cause as much as a 4% increase in the measured Young s modulus. If these data are corrected for the effects of drying and glue, the resulting value for Young s modulus is about 7.6 GPa. Therefore, the Young s modulus values of microtensile specimens, as tested, were probably greater than those of trabeculae in uiuo. The range of Young s modulus values of trabecular bone material measured by bending tests is GPa, while Young s modulus measured by buckling and ultrasound is GPa (Table 1). Previous tensile measurements of Young s modulus of trabeculae produced values close to 1 GPa (Ryan and Williams, 1989) and the present microtensile tests produced a value of 10.4 GPa for dry trabeculae. With the exception of Ryan and Williams data, Young s modulus values measured using bending tests tend to be lower than those measured using other techniques (see Table 1). Bending tests may underestimate the actual Young s modulus because local deformation in the specimen at the loading supports causes an overestimation of bending strain. Assuming that bending studies of trabecular properties underestimate Young s modulus and the tensile measurements in the present study overestimated Young s modulus (due to drying and glue effects), it is reasonable to assume that the actual Young s modulus of trabecular bone material falls nearer to the upper end of values obtained by bending tests ( GPa) and toward the lower end of the range found using microtensile and other tests ( GPa). There is probably no single value for the Young s modulus of trabeculae, as the properties may-vary depending upon the location, age, species, and state of health. Choi et al. (1990) found that the bending Young s modulus of cortical bone was dependent upon the specimen size with the Young s modulus decreasing in value as the dimensions of the specimen were decreased. For instance, they found the Young s modulus of a specimen with a 0.5 mm cross section to be 14.5 GPa and that of a specimen with a 0.3 mm cross section to be 11.6 GPa. Using microtensile testing, Young s modulus did not appear to be dependent upon specimen size. This is demonstrated by the fact that the Young s modulus of microtensile specimens (0.3 mm cross section) is the same as the Young s modulus of larger tensile specimens tested by others (Reilly et al., 1974). The Young s modulus of cortical bone could not be extrapolated from relationships between density and Young s modulus of cancellous bone. This results is in agreement with the results of Rice et al. (1988). However, the analysis of Rice et al. was performed with pooled data from different sources obtained by different testing methods. In the present study, a single technique, ultrasonic measurement, was used to address this question. Ultrasonic techniques gave higher values for Young s modulus for both microspecimens of cortical bone and individual trabeculae. It is possible that this discrepancy is due to the dependence of Young s modulus in bone upon strain rate (Carter and Hayes, 1977) and may indicate that the very slow strain rate used in the microtensile tests (5.5 x 10m6 s-i) was not equivalent to the strain rate associated with elastic wave propagation. Conversely, in another study using a similar strain rate, one to one agreement was found between mechanical and ultrasonic testing (Ashman et al., 1987). It should be noted that, in metal specimens where there is no strain rate dependence, there was no significant difference between Young s modulus measured by microtensile and ultrasonic tests.

9 Young s modulus of trabecular and cortical bone material 119 Moreover, the measured values for Young s modulus of stainless steel and aluminum were within 7% of textbook values (190 GPa for stainless steel and 69 GPa for aluminum, MacGregor et al., 1978) using both microtensile and ultrasonic tests. Limitations in the techniques used here include the assumption of a single density value of individual trabecular and microspecimens of cortical bone. We were unable to accurately measure fhe volume of trabecular and microspecimens of cortical bone due to the very miniature dimensions of specimens. The values for density used represented an average of specimens from any different skeletal sites and were compared well to the density of trabecular and cortical bone found by Gong et al. (1964). Nevertheless, significant differences were found between Young s moduli measured in tension and those measured ultrasonically for both cortical bone specimens and individual trabeculae. The ultrasonic and mechanical moduli differed by 11% for the microspecimens of cortical bone in comparison to 42% for the individual trabeculae. This may indicate that the density values of individual trabeculae had more variability than those of microspecimens of cortical bone. Large regional changes in the mineral content and microstructure of trabeculae have been measured, even in one location of the same bone (Obrant and Odselius, 1984; Choi et al., 1990). Since the trabecular bone undergoes a considerably higher rate of turnover than cortical bone (Parfitt, 1983), one would expect to find a great deal of variation in its properties. Acknowledgements-This work was supported by the National Institute of Arthritis. Diabetes. Digestive and Kidney Diseases AM36257 and the Research Fund of Texas Scottish Rite Hospital for Children, Dallas, TX. REFERENCES Ashman, R. B., Corin, J. D. and Turner, C. H. (1987) Elastic properties of cancellous bone: measurement by an ultrasonic technique. 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