X-ray quantitative computed tomography: The relations to physical properties of proximal tibial trabecular bone specimens

@X-9290/89

J. Biomechanics Vol. 22, No. 8/9, pp. 837-844. 1989. Prmted in Great Britain

$3.00+ .I0

Pergamon Press

plc

X-RAY QUANTITATIVE COMPUTED TOMOGRAPHY: THE RELATIONS TO PHYSICAL PROPERTIES OF PROXIMAL TIBIAL TRABECULAR BONE SPECIMENS IVAN HVID, WREN M. BENTZEN?, FRANK LINDE*, LIS MOSEKILDE~ and BUNTOING PONGSOIPETCH * *Biomechanics Laboratory, The Orthopaedic Hospital, Randersvej 1, DK-8200 Aarhus N, Denmark; TRadiophysics Laboratory and Department of Diagnostic Radiology, Aarhus Municipal Hospital, Aarhus, Denmark; iDepartment of Anatomy C, University of Aarhus, Aarhus, Denmark Abstract-Cylindrical bone specimens from the proximal epiphysis of ten normal human proximal tibiae were randomly assigned to a destructive axial compression test-series (N=94) or to a protocol of standardized mechanical conditioning followed by non-destructive repeated testing to 0.6% strain and a final destructive test (N= 121). Specimen X-ray quantitative computed tomography (QCT) obtained at different scanning energies (100, 120 and 140 kVp) yielded closely related results (r = 1.00). Accordingly, predictions of physically measured densities or mechanical properties were not improved by using more than one scanning energy. QCT and physically measured densities were intimately related (QCT at 140 kVp to apparent density using linear regression: r=0.94, and to apparent ash density: r=0.95) and did not differ significantly in their ability to predict the mechanical properties, thus favouring the more easily implemented QCT for routine work. Evaluation of the relation of apparent density to Young’s modulus and ultimate strength suggested that a power law regression model is preferable to a linear model, although linear model prediction of mechanical properties does not have significantly worse accuracy within the narrow density range investigated. The effect of conditioning on the behaviour of bone specimens subjected to destructive compression tests was to increase the stiffness and strength by approximately 50 and 20% respectively.

INTRODUCTION X-ray quantitative been extensively

computed employed

tomography (QCT) has to assay bone mineral

(Bradley et al., 1978; Cann et al., 1980; Cann et al., 1985; Genant et al., 1985; Liliequist et al., 1979; Posner and Griffiths, 1977; Pullan and Roberts, 1978) and experimentally to predict vertebral body fracture load (Brassow et al., 1982; McBroom et al., 1985). Trabecular bone strength assay using CT-scanning of intact tibiae (in situ measurement) subsequently machined to small axial cylinders, rescanned and mechanically tested, was reported by Bentzen et al. (1987). Fair correlations were obtained with mechanical parameters, better when the cylindrical specimens were scanned, which was attributed primarily to limitations in establishing the same levels of cross-sectioning during in situ scanning and specimen preparation. The proximal tibia appears to be particularly difficult in this respect since there is a steep density gradient within a few mm deep to the subchondral bone plate. In spite of this difficulty, reproducibility of in situ scans is good, approximately + 4% (95 % confidence limits) close to the subchondral bone plate and+ 1% 8-24 mm deep to the subchondral bone plate. McBroom et al. (1985) scanned intact vertebral bodies and examined the relation to the apparent density of samples taken from them. In this less heterogeneous region, they reported a coefficient of determination of

Received

in ,jinal form

11 November 1988.

0.89, but results of mechanical tests on machined specimens as related to QCT were not reported. In biomechanical investigations of trabecular bone, it is often practical to include measurements of apparent density or ash apparent density as an explanatory variable. Such data are tedious to obtain. QCT, on the other hand, is easily implemented, and a large number of specimens may be scanned simultaneously. This study explores the relation of QCT to apparent density and ash apparent density of trabecular bone specimens, and relates these density parameters to the results of destructive and non-destructive axial compression tests. In addition, the possibility of improving the empirical relation of QCT to densities and mechanical characteristics by using more than one scanning energy is investigated. MATERIAL

AND METHODS

The material consisted of ten knee specimens obtained at routine autopsy. There were five male and five female donors ranging in age from 60 to 83 yr. Only macroscopically normal joints were accepted for the study. The joints were excised within 6-24 h after death and immediately deep frozen to below -21°C. Specimen preparation

After thawing, the proximal tibiae were cleared of soft tissues. The plane of the condylar subchondral bone plates was identified according to the method of Linde et al. (1988), and three transverse bone cuts were made using a diamond band saw with continuous 837

838

I. HVID

water cooling (EXAKTa cutting-grinding system), the proximal one being 2 mm deep to the articular surface of the subchondral bone plate, to produce two 7.5 mm slices of epiphyseal trabecular bone. The slices were refrozen, and then, in the frozen state, a 7.5 mm core drill was used to produce cylindrical specimens, approximately 25 from each tibia, all taken from the condylar areas thus avoiding the weak central tibia1 bone (Goldstein et al., 1983; Hvid and Hansen, 1985). The exact dimensions of the cylinders were measured by a micrometer screw. Between subsequent steps, specimens were refrozen and thawed in physiological saline at room temperature before measurement. Phantom QCT

X-ray computed tomography was performed on an EM1 7070 commercial scanner using an acrylic phantom containing 49 specimens embedded in physiological saline as previously described (Bentzen et al., 1987). Scans were performed at 140 kVp nominal energy and 40mA tube current (QCT,,,), 120 kVp and 50mA (QCT,,,), and 100 kVp and 60mA

(QCTm). The CT-number (H) in Hounsfield units or the relative attenuation coefficient (i-c= 1 +O.OOl H) was used as the density parameter (Bentzen et al., 1987). The scanner was routinely calibrated against water and a CaCl, solution contained in Plexiglas phantoms. Once calibrated, the drift of CT values was minimal. Destructive mechanical

tests

One hundred specimens, five from each of the two levels from each of the ten tibiae, were randomly allocated to destructive testing. The tests were run in an INSTRON 1195 universal test machine using a calibrated 1 kN load cell and an extensometer as described earlier (Bentzen et al., 1987). Axial compression testing was performed at a cross-head speed of 5 mm min- ‘, corresponding to a strain rate of approximately 0.01 s-l. The tests were run at room temperature and humidity (time from removal of specimens from saline to completion of test was 15-20s). Forcedeformation tracings were obtained on an X-Y recorder and converted to stress-strain diagrams from which ultimate stress (u,, MPa), Young’s modulus [maximal curve slope (E, MPa)], ultimate strain [relative deformation at the ultimate failure point, which was defined as the first curve point showing a load maximum (s”)] and energy absorption related to original cylinder volume at the ultimate failure point (_&I,, kJ m- 3, were read or calculated. Six tests failed (see below) leaving 94 specimens for evaluation. Non-destructive

mechanical

tests

The remaining 149 trabecular bone specimens were allocated to non-destructive testing (Linde and Hvid,

et al.

1987). These tests were performed using an INSTRON 4302 universal test machine, otherwise the test set-up was identical to that used in the destructive tests. The specimens were conditioned by repeated (0.2 Hz) compressions at 0.01 strain rate to 0.6% strain, the strain channel being reset between compression cycles. Zero strain was defined as the strain at 5 N compressive load (corresponding to 0.10 MPa). Conditioning was complete when the specimens had reached steady state regarding viscoelastic properties, i.e. zero strain was reproducible. This required approximately ten loading cycles (7-15). After conditioning, five compressive loading cycles (0.2 Hz) between 5 N and 0.6% strain were recorded at 0.01 strain rate. The properties obtained were the stress (cr”, MPa) and stiffness (E,, MPa) normalized with respect to cylinder cross-sectional area and initial length (comparable to Young’s modulus) at 0.6% strain, the hysteresis energy [EA,, viscoelastic energy calculated from the area enclosed by the hysteresis loop (kJ mm3)] and the elastic energy [EA,, calculated from the area below the hysteresis loop (kJ me3)]. Furthermore, the damping modulus [D=(EA,/(EA, + EA,)] (Frost, 1973) was calculated. These parameters were averaged from the five loading cycles. Immediately after the non-destructive load cycles, a further test was run to compressive destruction. Some specimens showed fatigue behaviour during conditioning or the subsequent five loading cycles (decreasing stiffness), or they exhibited a biphasic behaviour during the final destructive test (temporary decrease of stiffness before the failure point, or a sharp increase in stiffness), and a few specimens were lost due to errors of operation of the test machine. A total of 28 tests were rejected for these reasons, leaving 121 specimens for evaluation. Measurement

of apparent density and ash

The specimens were cleansed under air jet, defatted in 70% alcohol for 24 h, then recleansed under air jet. The specimens appeared macroscopically clean after this procedure. After 24 h evaporation at room temperature, the specimens were weighed (tissue weight), and apparent density (p, g cmm3) was calculated. Finally, the specimens were ashed in a muffle furnace at 105°C for 2 h, then at 580°C for 24 h, weighed, and apparent ash density (p,, g cm- 3, calculated. The ashweight per cent was calculated taking the ashweight as a percentage of the tissue weight. Statistical

analyses

Regression analyses were by the least squares method using raw data or In-ln transformed data with one or more independent variables. Comparisons of correlation coefficients were by the method described by Hald (1952). Student’s t-test for unpaired data was used to compare group means.

X-ray quantitative computed tomography

839

RESULTS Relation of QCT to physical density measurements

The relation of QCTIbo to apparent density is shown in Fig. l(a). Except for a systematic shift in CT values, similar relations were obtained using QCT,,, and QCT,,, [Fig. l(b)] with correlation coefficients of 0.935,0.938 and 0.941 for the three energies. There was a very close intercorrelation between QCT measurements at different energies (r= 1.00). Figure 2(a) shows the relation of QCT,,, to apparent ash density, and the systematic shift in CT values with 120 kVp and 100 kVp scans is shown in Fig. 2(b) (QCT,,,: r=0.952; QCT12,,: r=0.955; r=0.959. For QCT,,, the correlation to QCT,,,: apparent ash density is better than to apparent density, p < 0.05.) Alternative regression models including

,“I F

0.6++

\ s

+

0.4-

E c

0.3-

(a)

,...,...r...(.60 160 240

320

’

IO

OCT(140)(HU)

'/'

/.' a/;

I

++

&

,... 0

1; ++++I,+ +

0.5-

-60

+ 0.0625 N = 215 r P 0.952 SEE - 0.019p/ccm

+

;5 % 8

0.2-

a 0.1.

-60

A

y 5 0.00130 x + 0.103 215

N=

0

60

160

320

400

QCT(140)(HU)

(a)

0.7

II

1

(b)

240

QCT(HU)

Fig. 1. (a) Scatter diagram showing the relation of QCT,,, to apparent density. (b) Diagram showing the dependency on scanning energy of the relation between QCT Cl00 kVp: dotted line (y=0.00106x+0.0949); 120 kVp: broken line (y =0.00120x+0.101); 140 kVp: unbroken line (y=O.O013Ox + 0.103)] and apparent density.

(‘4

o.“T-----r--QCT(HU)

-1

Fig. 2. (a) Scatter diagram showing the relation of QCT,,, to apparent ash density. (b) Diagram showing the dependency on scanning energy of the relation between QCT Cl00 kVp: dotted line (y =0.000610x+0.0578); 120 kVp: broken line (y=O.OOO688x+O.O613); 140 kVp: unbroken line (y =0.000746x+0.0625)] and apparent ash density.

power law modelling and multiple regression did not result in better prediction of apparent density or apparent ash density from QCT. Apparent density and apparent ash density were intimately related (Fig. 3). There was a weak negative linear correlation between apparent density and ashweight per cent (ashweight per cent = -9.32 x appar ent density + 61.5; r = - 0.283; p < O.OOl), signifying that less porous trabecular bone seems to have less mineral per unit mass of bone material, while apparent ash density did not correlate significantly to the ashweight per cent (r = - 0.111, p > 0.05). Destructive mechanical

tests

The empirical relations of apparent density and apparent ash density to mechanical properties are outlined in Table 1. There is a general tendency for

840

I.

HVID

et al. Non-destructive

0.0 0.0

0.1

0.2

0.3

0.4

APPARENT DENSllY

0.5

0.6

0.7

(g/cm-)

Fig. 3. Scatter diagram showing the relation of apparent density to apparent ash density.

power law correlations to be better than linear correlations. However, none of the differences are statistically significant (~~0.05 in each case). Likewise, the differences between correlation coefficients obtained using apparent density and ash apparent density are not statistically significant. The ashweight per cent did not correlate significantly to ultimate stress, ultimate strain or elastic modulus, but did show a weak negative correlation to energy absorption at ultimate failure (I= -0.215, pCO.05). Table 2 shows the empirical relations between QCT and the mechanical data. Although there is a general tendency for power law regression to produce higher correlation coefficients than linear regression, the differences are not statistically significant. Multiple regression models using QCT at different energies as non-dependent variables did not result in improved correlations. There were no statistically significant differences between the correlation coefficients obtained using direct or indirect density measurement.

mechanical

tests

Predictions of non-destructive mechanical properties from apparent density and apparent ash density are outlined in Table 3. All correlation coefficients shown are statistically significant (p
Table 1. Destructive compression tests: estimation of mechanical parameters (y) from physically measured densities (x) Mechanical oarameter E*

Et 0” * 0.t E”* s,t EA,”

EAut

a 1173 1371 18.99 25.30 0.0114 0.0256 296 432

Apparent density r b -44.38 1.33 - 1.141 1.494 0.0175 0.177 -25.34 1.69

0.744 0.794 0.863 0.892 0.271 0.329 0.859 0.896

S.E.E. 130 1.55 1.37 1.39 0.005 1.25 21.7 1.43

a 2278 3473 36.19 70.38 0.0200 0.0282 555 1344

Apparent ash density r b

S.E.E.

-79.74 1.43 - 1.600 1.596

0.776 0.815 0.884 0.908

122 1.52 1.27 1.35

0.0175 0.176 - 30.95 1.80

0.255 0.313 0.866 0.906

*Linear correlation: y = ax + b. t Power law correlation: y = CIX(exp b). S.E.E. is the standard error of estimate. With power law modelling, the figure given is a factor (estimate +/-S.E.E. linear model corresponds to estimate x /+ S.E.E. in the power law model). r is the correlation coefficient.

0.005 1.25 21.2 1.42

in the

X-ray quantitative

computed

tomography

841

In both series, there was a close relationship between stiffness and strength (destructive series: a,=0.0129 E+0.573, r=0.921, S.E.E.=1.05 MPa; non-destructive series: c, = 0.0106 E + 0.543, r = 0.924, S.E.E. = 1.12 MPa).

DISCUSSION

Apparent density and ash apparent density are intimately related (Fig. 3) with S.E.E.s of approximately 6% when related to the respective mean values. Therefore, as baseline parameters in biomechanical studies, either may be used, which is further supported by the finding of practically equal accuracies of prediction of mechanical properties. Although there was a uniform tendency for power law modelling to yield higher correlation coefficients than linear modelling in the regression analyses involving apparent density and mechanical properties, the differences were not statistically significant. The power exponents differ from the widely cited values (2 for ultimate strength, 3 for Young’s modulus) reported by Carter and Hayes (1977). However, their study utilized specimens with a significantly different geometry and included human tibia1 and bovine femoral specimens together with literature data on compact bone. Since trabecular architecture has a significant influence on mechanical behaviour, more or less independent of apparent density (Gibson, 1985; Harrigan and Mann, 1984; Harrigan et al., 1981; Mosekilde et nl., 1987), it is problematic to pool data from different anatomical sources. The power exponents that we report differ statistically significantly from linearity (the standard errors of the coefficients are all smaller than 0.09) which is likely to indicate that the structural organization of trabeculae at the tibia1 condyles deviates somewhat from the ‘ideal’ columnar architecture (Gibson, 1985). Further, our linear regression models in most instances predict mechanical property values below zero for zero apparent density, which obviously does not make sense. Power law modelling may be preferred for theoretical reasons (Currey, 1986), and the statistical analyses of our data support this concept. As to the finding of higher power law exponents in predicting ultimate strength than in predicting Young’s modulus from apparent density, this was also noted in a study of human and goat trabecular bone from the femoral head (Favenesi et al., 1984). The degree of mineralization of bone matrix, reflected in this study by the ashweight per cent, has very little influence on the mechanical properties of trabecular bone. This is in agreement with previous findings (Ducheyne et al., 1977; Galante et al., 1970) and contrasting to the finding for compact bone (Currey, 1969). Galante et al. (1970) reported an inverse relation of real density (i.e. the density of bone matrix) to the compressive strength of vertebral trabecular bone, although only in one of several experimental series. Ashweight per cent did not signifi-

I. HVID et al.

842

Table 3. Non-destructive compression tests: estimation of mechanical parameters (y) from physically measured densities (x) Mechanical parameter Ellf Emt

EA,* EA,t EA,*

EAet D*

Dt

Apparent density a

b

1689 2132 5.18 6.16 1.94 1.90 8.61 8.98 -0.137 0.177

-99.4 1.46 - 0.060 1.13 0.24 0.73 0.43 0.95 0.262 -0.156

r

0.752 0.782 0.737 0.765 0.572 0.599 0.676 0.695 -0.214 -0.258

Apparent ash density S.E.E.

a

149 1.54 0.53 1.42 0.28 1.43 0.94 1.43 0.048 1.25

2963 4996 10.20 12.03 3.46 2.97 15.31 16.11 -0.244 0.160

b

r

_ 124.6 0.776 1.52 0.801 -0.153 0.765 1.18 0.787 0.20 0.600 0.77 0.623 0.27 0.708 1.00 0.724 0.264 -0.288 -0.166 -0.271

S.E.E.

143 1.51 0.51 1.40 0.27 1.43 0.91 1.42 0.048 1.25

See footnotes for Table 1.

cantly influence strength in this study, but a weak negative correlation to apparent density was noted. These observations may be a reflection of a mechanism tending to compensate a relative loss of the amount of bone (e.g. with aging) by increasing the mineral content. QCT was a good predictor of apparent density and apparent ash density. The accuracy of prediction is adequate to replace these measurements with the much more easily implemented QCT measurement when all that is wanted is a baseline parameter, e.g. to secure comparability between groups of specimens subjected to mechanical tests. The relation of QCT to mechanical properties derived from destructive testing confirmed previous findings (Bentzen et al., 1987). Transformation of QCT to the relative attenuation coefficient did result in power coefficients somewhat different from those previously reported. However, the relative attenuation coefficient varies close to unity. Therefore, small systematic changes in the density of the bone under investigation may influence the power coefficients significantly. In this study, only condylar bone specimens were used whereas in our previous study, intercondylar specimens were included. Thus, specimens in the lower density range are likely to differ markedly in their internal structure between the two studies. In accordance with this, predicted values are practically equal in the high density range, while in the present study higher values are predicted in the low density range. Density measurements and QCT related to the mechanical parameters derived from non-destructive tests with almost equal correlation coefficients. Predictions of stiffness were slightly less accurate than predictions from destructive tests. The correlations to elastic and hysteresis energies were too weak to permit meaningful predictions. In general, the amount of variation of mechanical properties not explained by variation of density-related data (1 - Rz) is considerable. For instance, the correlations of apparent density to ultimate strength and Young’s modulus, using power law regression, reveal that unexplained vari-

ation amounts to 20 and 37%. Conceivably, this residual variation can be attributed largely to variation of trabecular architecture (Harrigan et al., 1981) which is not reflected by the apparent density, although the inherent errors in compression testing of trabecular bone specimens may be quite significant (Odgaard et al., 1988). With the spatial resolution currently available on commercial CT scanners, and certainly with the scanner used in this study, it would not seem possible to extract information on the structural arrangement of trabeculae in sufficient detail to improve the prediction of mechanical properties. However, it is possible to improve spatial resolution to the extent that three-dimensional reconstruction of trabecular structures is possible (Goldstein et al., 1987), an option that may eventually become available on commercial scanners. Despite its theoretical superiority, dual energy CT has not yet proven to be advantageous in practice. This is also the conclusion in the present study, where the inclusion of CT data recorded at more than one nominal X-ray energy did not significantly improve the accuracy of prediction of apparent densities or mechanical properties. The most likely explanation in our view is that differences in the chemical composition of the specimen [e.g. variations in fat content in the marrow space (Faul et al., 1982; Genant et al., 1985)] is not the accuracy-limiting factor in QCT prediction of mechanical properties of tibia1 trabecular bone. The variation in apparent density or apparent ash density unexplained by the regression on CT value is only about 10%. This variation is probably due to the imperfect matching between CT measuring volume and the bone specimen and to the inherent experimental uncertainties in both the CT and the density measurements. Problems related to different amounts of red marrow (Mazess, 1983) are not a concern in studies on the peripheral skeleton of mature individuals. The sensitivity of dual energy techniquts increases with the range of available X-ray energies. However, the application of very low energies is counter-indi-

843

X-ray quantitative computed tomography

la’

Scld

4 I

. +

I

I

1

,

,

I,,,!

7*10-W APPARENT DENSITY (g/ccm)

lo”

Fig. 4. Scatter diagram showing the relation of apparent density to Young’s modulus in the final destructive test of the nonrdestructive series (filled circles, broken line, y = 2560x (exp 1.47), r =0.84) and in the destructive series (crosses, unbroken line, y= 1371x (exp 1.33), r=0.79).

.

,” ,/.+

. .+.

/‘,’

l’, ,J

++

*

,‘/

5LlO&

+ r

,

7*10-a10-’ APPARENT

I

I

I,,

4

DENSITY (g/ccm)

Fig. 5. Scatter diagram showing the relation of apparent density to ultimate strength in the final destructive tests of the non-destructive series (filled circles, broken line, y = 34.2x (exp 1.56), r =0.89) and in the destructive series (crosses, unbroken line, y = 25.3x (exp 1.49), T= 0.89).

cated by increasing image noise and/or (in clinical applications) increasing absorbed dose to the patient. The effect of conditioning the specimens before testing to failure was to increase the Young’s moduli by more than SO%, confirming our previous findings (Linde et al., 1985; Linde and Hvid, 1987) and to increase the ultimate strength by approximately 20%.

I.

844

HVID et al.

The computation of Young’s modulus and the energy absorptions were based on the original length of the test cylinders, measured prior to the non-destructive tests. From previous data, where the absolute length of the test cylinders was recorded throughout the test sequence, we found that the creep (change in cylinder length) before execution of the final destructive test amounted to 0.3%, which would not significantly influence the calculation of Young’s modulus or the energy absorptions. In conclusion, QCT was found adequate to substitute for measurements of apparent density or apparent ash density of trabecular bone specimens in applications not directly concerned with these parameters. Use of more than one scanning energy within the range from 100 to 140 kVp did not enhance predictions of density or mechanical properties.

Acknowledgements-This work was supported by a grant from Gigtforeningen (the Danish Rheumatism Association), grant No. 233-273.

REFERENCES

Bentzen, S. M., Hvid, I. and Jorgensen, J. (1987) Mechanical strength of tibia1 trabecular bone evaluated by X-ray computed tomography. J. Biomechonics 20,743-752. Bradley, J. G., Huang, H. K. and Ledley, R. S. (1978) Evaluation of calcium concentration in bones from CT scans. Radiology 128, 103-107. Brassow, F., Crone-Miinzebrock, W., Weh, L., Kranz, R. and Eggers-Stroeder, G. (1982) Correlations between breaking load and CT absorption values of vertebral bodies. Eur. J. Radio/. 2, 99-101. Cann C. E., Genant, H. K., Kolb, F. 0. and Ettinger, B. (1985) Quantitative computed tomography for prediction of vertebral fracture risk. Bone 6, l-7. Cann, C. E., Genant, H. K. and Young, D. R. (1980) Comparison of vertebral and peripheral mineral losses in disuse osteoporosis in monkeys. Radiology 134, 525-529. Carter, D. R. and Hayes, W. C. (1977) The compressive behavior of bone as a two-phase porous structure. J. Bone Jt Surg. 59-A, 954962. Currey, J. D. (1969) The mechanical consequences of variation in the mineral content of bone. J. Biomechanics 2, l-11. Currey, J. D. (1986) Power law models for the mechanical properties of cancellous bone. Engng Med. 15, 153-154. Ducheyne, P., Heymanns, L., Martens, M., Aernoudt, E., De Meester, P. and Mulier, J.C. (1977) The mechanical behaviour of intracondylar cancellous bone of the femur at different loading rates. J. Biomechanics 10, 747-762. Faul. D. D.. Couch. J. L.. Cann. C. E.. Bovd. D. B. and G&ant, fi. K. (1982) domposition-sdlecti&‘reconstruction for mineral content in the axial and appendicular skeleton. J. Comput. Assist. Tomogr. 6, 202-204. Favenesi, J. A., Gardeniers, J. W. M., Huiskes, R. and Slooff, T. J. J. H. (1984) Mechanical properties of normal and avascular cancellous bone. Trans. O.R.S. 8, 198.

Frost, H. M. (1973) Orthopaedic Biomechanics, pp. 57-58. Charles C. Thomas, Springfield, IL. Galante, J., Rostoker, W. and Ray, R. D. (1970) Physical properties of trabecular bone. Calcif: Tissue Res. 5, 236-246. Genant, H. K., Ettinger, B., Cann, C. E., Reiser, U., Gordan, G. S. and Kolb, F. 0. (1985) Osteoporosis: assessment by quantitative computed tomography. Orthop. Clin. North Am. 16, 557-568.

Gibson, L. J. (1985) The mechanical behavior of cancellous bone. J. Biomechanics 18, 317-328. Goldstein, S. A., Ku, J. L., Hollister, S., doulet, R., Champlain, F. W. and Matthews, L. S. (1987) Experimentally controlled trabecular bone remodelling: effects of applied stress. Trans. 0. R. S. 12, 461. Goldstein, S. A., Wilson, D. L., Sonstegard, D. A. and Matthews, L. S. (1983) The mechanical properties of human tibia1 trabecular bone as a function m&aphyseal location. J. Biomechanics 16. 965-969. Hald, A. (1952) Statistical Theory with Engineering Applications, pp. 394-401, 522-584 and 608-613. John Wiley, New York. Harrigan, T. P., Carter, D. R., Mann, R. W. and Harris, W. H. (1981) The influence of apparent density and trabecular orientation on the elastic modulus of cancellous bone.

of

Trans. O.R.S. 6, 277.

Harrigan, T. P. and Mann, R. W. (1984) Characterization of microstructural anisotropy in orthotropic materials using a second rank tensor. J. Mater Sci. 19, 761-767. Hvid, I. and Hansen, S. L. (1985) Trabecular bone strength patterns at the proximal tibia1 epiphysis. J. orthop. Res. 3, 464-472.

Liliequist, B., Larsson, S.-E., Sjiigren, I., Wickman, G. and Wing, K. (1979) Bone mineral content in the proximal tibia measured by computer tomography. Acta Radial. Diagn. 20,957-965. Linde, F., GBthgen, C. B., Hvid, I., Pongsoipetch, B. and Bentzen, S. M. (1988) Mechanical properties of trabecular bone by a non-destructive compression testing- approach. __ Engng Med. 17, 23-29.

Linde. F. and Hvid. I. (1987) Stiffness behaviour of trabecular bone specimens.‘J. ‘Biom>chanics 20, 83-89. Linde, F., Hvid, I. and Jensen, N. C. (1985) Material properties of cancellous bone in repetitive axial loading. Engng Med. 14, 173-177.

Mazess, R. B. (1983) Errors in measuring trabecular bone by computed tomography due to marrow and bone composition. Calcif: Tissue fnt. 35, 148-152. McBroom, R. J., Hayes, W. C., Edwards, W. T., Goldberg, R. P. and White, A. A., III (1985) Prediction of vertebral body compressive fracture using quantitative computed tomography. J. Bone Jt Surg. 67-A, 1206-1214. Mosekilde, L., Mosekilde, L. and Danielsen, C. C. (1987) Biomechanical competence of vertebral trabecular bone in relation to ash density and age in normal individuals. Bone 8, 79-85.

Odgaard, A., Hvid, I. and Linde, F. (1989) Strain distributions in cancellous bone specimens. J. Biomechanics (in press). Posner, I. and Griffiths, H. J. (1977) Comparison of ct scanning with photon absorptiometric measurement of bone mineral content in the appendicular skeleton. Invest. Radial. 12, 524-544.

Pullan, B. R. and Roberts, T. E. (1978) Bone mineral measurement using an EMI scanner and standard methods: a comparative study. Br. J. Radial. 51, 24-28.