1. Bunnrchunv~ Vol. 23. No. 8. pp. 837-W. Pnntal III Great Bnta~n
oo?I 929090 noo+.al Pergamon PWM pk
1990.
PHYSICAL CHARACTERISTICS AFFECTING THE TENSILE FAILURE PROPERTIES OF COMPACT BONE J. D. CURREY Department of Biology, University of York, York YOI SDD, U.K. Abstract-Compact bone specimens from a Fide variety of reptiles, birds, and mammals were tested in tension, and their failure properties related to mineral volume fraction. porosity and histological orientation. The principal findings were that the ultimate strain and the work under the stress-strain curve declined sharply with mineralisation. as did the stress and strain appearing after the specimen had yielded. Ultimate tensile strength was not simply related to any combination of the possible explanatory variables, but some relatively poorly mineral&d bones. notably antlers. had high stresses at failure. These high strengths were allowed by a great increase in stress after the bones had yielded at quite low stresses.
INTRODUCTION
Recently (Currey, 1988a). I showed the value of using a large data set from different animals to discover what determines the elastic properties of compact bone. I discussed results from 103 specimens from 24 bones from I8 speciesof tetrapod vertebrates. About 80% of the variance in Young’s modulus could be explained by variation in two variables: calcium content and porosity. The value of Young’s modulus increased monotonically with calcium content and inversely with porosity. In the present paper. I examine the failure propertics of bone loaded in tension. Unlike the case for Young’s modulus, some failure properties do not change monotonically with any combination of explanatory variables. Nevertheless, a number of clear dcpendcncies between failure properties and cxplanatory variablesemerge, giving insight into tcnsilc failure in bone.
undum paper, and tested wet in an Instron I I22 table testing machine. An extensometer measured strain over a gauge length of 11 mm. The cross-head produced a strain rate over the gauge length of about 0.2 s- ‘. The few specimens that failed outside the gauge length were ignored. Young’s modulus of elasticity, yield stress, yield strain, ultimate stress, ultimate strain, and work under the curve, were obtained from the output of a storage oscilloscope. Most of these properties arc obvious; two need some explanation. Yield stress und sfruirl
Many specimens showed a fairly sharp yield point, but some had a rather gently curving load-dcformation curve between straight prc- and post-yield regions. The yield point was taken as the point where the curve had deviated by a strain of 0.002 from the straight line describing the initial part of the curve (Fig. I).
MATERIALS AND METHODS Explunutory
Table I lists the bones and their mechanical properties; 93 specimens from 28 bones from I7 species were tested. Most specimens were collected fairly fresh, but a few were dry museum specimens. All specimens, when tested, had been thoroughly soaked with water. Although, of course, drying must have had some effect on the bones, particularly the collagenous component, Currey (1988b) shows that the effect of drying and re-wetting on the mechanical properties of bone is small. Examination of residuals (see below) showed no evidence that museum specimens behaved differently. as a group, from the fresher specimens. The specimens were cut from the bone with a bandsaw, and shaped on a milling machine guided by a template. The specimens had expanded ends leading by rounded shoulders to a central uniform section of length 16 mm, and a I.8 mm square cross-section, They were smoothed with No. 400 grade carborReceived in final form 3 January 1990.
uariuhles
In tests of the elastic properties of bone (Currey.
1988a) I examined the elfects of two explanatory variables: calcium content and porosity (Currey, 1987). Briefly, calcium, expressed in mg calcium per g of dry, de-fatted bone was determined colorimetrically. This was turned into a measure of mineral volume fraction in the dry bone using [0.78Ca/ (0.91-1.50 Ca)] where’ca’ is the calcium content in kg per kg. The justification for this formula is given in the Appendix. Porosity was determined by pointcounting on undecalcified cross-sections taken just behind the fracture surface. Blood channels and Haversian spaces were considered as pores; osteocytc lacunae and canaliculi were not. In what follows, I use ‘apparent density’ (I -porosity). These two explanatory variables were determined for each of the specimens reported here. In some types of bone. the direction of loading, in relation to the grain of the bone, can have an important effect on the failure properties (Reilly and Burstein,
837
838
J. D.
CURREY
Table 1. Median values of various properties for each bone of each species tested. ‘MVF’: mineral volume fraction; ‘p’: apparent density: ‘E’: Young’s modulus of elasticity (GPa); ‘u,,,‘: . ulttmate stress (MPa); cv,,‘: ultimate strain; ‘IV’: work under stress-strain curve (MJ m-‘k ‘N’: sample size
Galapagos tortoise Geochelone midas Femur 0.340 0.928 13.8 Fibula 0.322 0.960 13.2 Tibia 0.337 0.817 Il.5 Alligator Alligoror missipiensis Femur 0.365 0.940 I28 King penguin Aptenodytes patugonicu Humerus 0.453 0.901 22.8 Radius 0.394 0.91 I 22.1 Ulna 0.42 I 0.923 22.9 Sarus crane Grus anligone Tarsometatarsus 0.341 0.923 23.1 Tibiotarsus 0.382 0.94 I 23.5 Flamingo Phcwnicoprerus phornicopterus Tibiotarsus 0.382 0.942 28.2 Red-necked wallaby Protemnodon rufoeiseu Femur 0.437 0.968 21.8 Tibia 0.402 0.965 25.4 Dugnng Dugong dugon Radius 0.371 0.910 7.5 Scapula 0.388 0.908 6.1 Ulna 0.344 0.847 4.9 Whitcsidcd dolphin Lugentwhyncus acutus Rib. outer cortex 0.3n9 0.696 Rib, inner cortex 0.372 0.799 !::: Whim bcakcd dolphin Laqcnorhyncus olhimsrris Ulna 0.294 0.660 2.7 Fin w halt UY/UCVII~~~C~~U phyxuluz Tympanic bulla 0.560 0.964 34. I Donkey Equtcscuhcllus Radius 0.381 0.944 IS.3 Horse Equus cuhullus Femur 0.395 0.954 24.5 Fallow dczr Da~nu rlumu Tibia 0.430 0.970 26.8 Radius 0.360 0.96 I 25.5 Red dcTr Cervus r/up/~us Antler. mature 0.2&t? Antler, immature 0.28 1 0.932 0.910 1:: Roe deer Cupret~luseuprcolus Femur 0.383 0.943 18.4 Reindeer Rung&v rurundus Antler 0.300 0.829 8.1 Cow Bos taurus One year old Tibia 0.364 0.944 19.7 Cow Bm tuurus Nine years old Femur 0.410 0.948 26. I
1975). Here specimenswere taken as nearly as possible with their grain coincident with the direction of loading. Some bones, however, have a chaotic arrangcment of their lamellae, possibly affecting failure properties. To try to allow for this, I made an estimate of alignment, using this as an explanatory variable, to see whether it was important. Undecalcified longitudinal sections, oriented in the radial and tangential directions in relation to the bone from which they came, were examined using ordinary and polarised light. An estimate of the degree of orientation was made on a
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scale ofO(for bone showing no preferred orientation in any direction) to 6, for bone etfectively perfectly aligned with the long axis of the specimen. No specimen had an orientation at a large angle to the long axis of the specimen. Subjective estimates are, of course, not as satisfactory as objective measurements. However, (as shown below) although orientation does have some statistically significant explanatory effect on some mechanical variables, it is much smaller than those produced by the other explanatory variables. It is
839
Tensile failure propertiesof compactbone
unlikely that refining the estimate of orientation would have much effect on the amount of variance explained. The histology of 67 of the 92 specimens was examined, and also of 77 bending specimens from the same bones. For the specimens for which histological sections were not made, the degree of orientation was taken as the mean value for all specimens from that bone, both tensile and bending (not otherwise used here), that were examined. Specimens from the same bone tend to be similar in their degree of orientation. The explanatory variables listed above are just some of a large number of possibilities. They were chosen because they were reasonably easy to measure, and because they accorded with what common sense dictated might affect the failure properties of bone. This matter is raised again at the end of the discussion.
RESULTS
Table I gives the characteristics of the specimens from different bones, and Table 2 gives the statistical relationships expressed as regressions. The explanatory variables and the dependent variables are expressed in logarithms, except for orientation. (which has arbitrary differences between the values). The reasons for this procedure are explained in Currey (1988a): the multiplicative model, which is implied by the use of logarithms, is as reasonable a model as the additive one, in particular the multiplicative model does not imply large negative values for most of the mechanical variables when mineral volume fraction is zero, a characteristic of the additive model. Also, the statistical explanatory power of the multiplicative model (R*) is greater than that of the additive model. Yield and ultimate
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Figure 2 shows the relationship between mineral volume fraction and both yield strain and ultimate strain. There is a marked dimerence in the way these two properties vary with mineral volume fraction, yield strain showing a slight negative relationship, and ultimate strain showing a very strong negative relationship. A regression analysis (I) using log mineral volume fraction, log apparent density, and orientation as explanatory variables, showed that only mineral volume fraction has a significant effect on ultimate strain. HZ shows that 74% of the variance is explained by the rcgrcssion. (In Table I. ths value of R2 is given both adjusted and unadjusted for the number of degrecx of freedom; in the text the adjusted value is used.) Pos-
Table 2. Regression equations showing the relationships between mechanical propdes and various explanarory variables. ‘MVT: mineral volume fraction; ‘p’: apparent density; &,‘: ultimate strain; ‘8,‘: yield strain; ‘0”‘: yield slrcss;‘c,,‘: post-yield slroin;‘a P,‘: post-yield stress;‘Ori’: orientation. Two values for K’ are given, the lirsl is undjusd. the second (in brackets) is adjusted for the number of degrees of freedom MVI II lug c.,, =-3.96-5.16log I= 16.21 2) log &.I, = - 3.92 - 5.011log MVI I= 14.39 3) log E, = -2.81 - I.12 log MVf+0.0354 Ori 1= 8.4 I 4.36 4) log EY = -2.57 -0.85 log MVT+0.360 log p +0.0107 Ori I= 6.09 1.40 1.22 5) log uY =2.16 3.53 log p +0.0925 Ori +0.X19 log MVI I= 6.14 4.73 2.62 6) log zp, = -4.94 -6.86 log MVf f= 13.61 7) log cp, = - 4.86 - 6.7 I log M VT I= I I.RS 8) log a,, = -1.44 -6.3Olog MVf+264logp I= 9.54 2.07 logMVf+2.94logp +O.O47Ori 9) log up, =-0.92-5.01 I= 8.35 2.R3 I.31 IO) log work = - 1.97 -6.66 log MVT+4.12 logp +0.0989 Ori I= 12.68 3.90 3.17 I I) log work = - 1.63 -6.24 log MV1+4.39 logp +0.0643 Ori I= 10.92 4.19 I .RO
(With bulla) R1 = 74.5% (74.2%) (No bulla) R* =70.2% (69.8%) (With bulla) R’=49.7% (48.5%) (No bullr) R’ = 30.2% (27.7%) (No bulla) R’ = 60.0% (58.6%) (With bulla) R2 -ii 67.0% (66.9)% (No bulla or brittles) R’ = 63.4% (63.0%) (With bulla) R* = 51.0% (49.9%) (No bulla or brittles) R’ = 47.6% (45.6%) (With bulla) R’ = 67.0% (65.9%) (No bulla) R2 = 58.6 (57.1%)
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sibly the two extreme values (of the bulla), on the bottom right of Fig. 2, might be having an undue influence on the form of the regression. However, if they are omitted from the regression the results are little changed (2). The coefficient for mineral volume fraction is barely altered, from -5.16 to -5.08. However, the two bulla values certainly have an important, and possibly distorting, effect on the relationship for yield strain. If the bullae are included (3). RZ = 48%. and both mineral volume fraction and orientation are significant explanatory variables. The coefftcient for mineral volume fraction is - 1.12. If the bullae are excluded (4). R2 drops to 28%. apparent density is also just worth keeping as an explanatory variable, and the coefficient for mineral volume fraction is -0.85. It is obvious that whereas ultimate strain is extremely strongly dependent upon the mineral content, the strain at yield is much less so. Ultimate strain is far more variable than yield strain. Ignoring the bullae, probably a special case, yield strain varies from 0.0045 to 0.0108. a factor of x 2.4, while ultimate strain varies from 0.0045 to 0.1340. a factor of x 28.5. The very weak bullae have been removed from the following analysis of strength, although pairs of equations with and without them are sometimes shown, but their effect on whatever conclusions that can be drawn will be included in the discussion.
Plots of the log of yield stress against possible explanatory variables are shown in Fig. 3a. b, and c. There is no obvious relationship with mineral volume fraction, but apparent density and orientation appear to have a positive relationship, although that for density depends heavily upon three specimens with values of zero for orientation. The correlation matrix (Table 3) shows that the correlation between log apparent density and orientation is only 0.33, which makes it reasonable to consider these explanatory variables to be more or less independent. A multiple
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Fig. 2. Relationship between mineral volume fraction and both yield strain and ultimate strain. Solid circles: ultimate strain; open circles: yield strain; diagonal crosses:specimens in which there was essentially no post-yield strain. In this and the succeedingdiagrams. both ordinate and abscissaare on a logarithmic scale. except the scale for orientation.
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linear regression [equation (S)]. using the three explanatory variables, shows mineral volume fraction to have some explanatory ability, and apparent density and orientation to be more important explanatory
Tensile failure properties of compact bone
variables. The three variables together explain 59% of the variation in log stress at yield.
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slightly greater than yield stress, the relationship between the explanatory variables and both ultimate and yield stressshould be roughly the same. However, many less mineral&d specimens, particularly of antler, showed a considerable post-yield increase in stress.As a result (Fig. 4) there is no clear relationship between mineral volume fraction and ultimate tensile strength, nor does bringing in other possible explanatory variables help statistically.
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Equations (6)-(9) both include and omit the specimens that behaved in an apparently brittle manner. Excluding the brittle specimens has little effect on coefficients.Orientation and apparent density have no significant effect on post-yield strain, but apparent density does have some effect on post-yield stress. Figure 5c shows the relationship between the two variables themselves. Although post-yield stress and post-yield strain co-vary, it is not reasonable to choose
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Stress and strain in the yield region The post-yield strain is usually considerably greater than the elastic strain. However, the post-yield stressis usually less, often very much less, than the elastic stress. The mechanical properties discussed above were transformed to logarithms. Several specimens apparently behaved in a brittle way, showing no postyield increase in stress or strain (within the limits of measurement precision). Log 0 = - a, so for purposes of converting to logarithms, I have performed the standard procedure (Atkinson, 1985, p. 184) of adding a small value (post-yield stress: I MPa; post-yield strain: 0.001) to all specimens. These values are at about the limits of measurement error. Figure 5a andb show the log of post-yield stressand the log of post-yield strain plotted against the log of mineral volume fraction. It is clear that both of these variables undergo a marked reduction with an increase in mineral volume fraction.
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one as being a function of the other. We are attempting to determine afinctional relationship. In such a case, where the co-variates are expressed in logarithms, it is sensible to use the reduced major axis (the exponent of the slope divided by the correlation coefficient) as an estimate of the exponent (Paget and Harvey, 1988). if the brittle specimens are included: post-yield strain a (post-yield stress)t.o’; if they are excluded: post-yield strain a (post-yield stress)“.92.The implication of this is that whatever affects post-yield stress and strain affects them about equally.
842
J. D. CURREY
properties and the explanatory variables. This is certainly sometimes the case. For instance, the antler ... bones seem to have the ability to bear considerably increased stress after they have yielded, compared ma .. . . . with other bones of the same amount of mineralisation .r=-. 0. . . . . 1. and porosity. Similarly, the bulla seemsto behave in a . way almost qualitatively different from the other bones. It is important to be reasonably sure that other . such unusual bone types are not lost in the mass of data. It is in fact difficult to test this. Each type of bone has rather few specimens, and a very limited range of Fig. 6. Relationship berween work under the curve and values for mineral volume fraction and other exmineral volume hction. planatory variables. As a result, regressions for individual types of bone have very large standard errors for the coefficients, so it is not possible to make useful Work comparisons between dimerent bones. However I The work done in breaking a specimen in tension is examined the residuals for those overall regressions some measure of toughness. It is not a rigorous that had reasonably high explanatory power fracture mechanics parameter. However, because it (R’ > 50%) to see if any types of bone stood out. This includes all the work done on the specimen after it has examination showed no remarkable types of bone. yielded, it does give some idea of how resistant the (Antler is remarkable, but its anomolous post-yield material is to impact loading. Furthermore, all specistress ruins the explanatory power of the regression, mens tested here had the same size and shape, so the and no residual analysis is required to identify it.) cflicts of geometry are not confusing the issue. The Another test is to compare the coefficients of the shape of the stress-strain curve implies that the work equations before and after points having a high value will be very nearly proportional to the product of for Cook’s D are removed. Cook’s D mcasurcs instress at failure and strain at failure. llucntial outlicrs in a distribution (Cook, 1977). I Figure 6 shows a strong negative relationship berecalculated all the equations with reasonably high tween mineral volume fraction and the log of work. explanatory power, after excluding the live or six Multiple regressions [equations (IO) and (I l)] show points with the highest values of Cook’s D. In no case that mineral volume fraction seems to be the most did this exclusion have much elTccton the coeflicients important determinant of work, having a negative of the equation. These analyses did not, therefore. effect on it, but that apparent density and. to a lesser show up types of bone that departed markedly from extent, orientation, have a positive elTcct. the rest in their response to variation in the explanatory variables, nor did they suggest that the DISCUSSION coefficients of the equations of the regressions were sensitive to the behaviour of a few anomalous bones. The aim of this paper is to produce a model relating The compact bone of reptiles, birds, and mammals failure properties or bone to other, non-mechanical shows considerable variation in properties when properties, the explanatory variables. There are, of loaded in tension. Mineral volume fraction, apparent course. an infinity of variables that are potentially of density and orientation often ‘explain’. in statistical importance, of which the great majority are un- terms, a considerable proportion of this variation. The measurable. However, the model can be considered most striking of the findings are the large amounts of reasonably successful if the variables that are used post-yield stressshown by the less mineralized bones, accord with common sense. For instance. it has been particularly antler, and the very great variation in the shown that calcium content (converted to mineral ultimate strain of different bones. and the extent to volume fraction) is often important. In reality, of which this is related to mineralisation. Increased course, at the very least it must be calcium phosphate, mineralisation strongly reduces the extra stress and rather than calcium alone, that is important. Fur- strain reached after yield. This shows itself also in a thermore, it is almost certainly the interaction of marked etTecton ultimate strain, but not on ultimate mineral with the other major components, collagen stress, because the latter value is dominated by the and water, that alTectsthe mechanical properties, and yield stress, which is much less affected by mineralisso if coilagen content had been measured. it might ation and, insofar as it is affcctcd, is altered in the have proved to be just as good an explanatory variable opposite, positive, direction. as calcium or mineral volume fraction. This paper is based on data from a great variety of speciesand bones. A possible matter of concern is that Yield stress and strain there may be a number of difTerent types of bone that Compared with its effects on post-yield stress and have dilTcrcnt relationships between the mechanical strain, the effects of variation in mineral volume
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Tensile failure properties of compact
fraction on yield stress and strain are small though real. Changing the mineral volume fraction from the least to the greatest value found in the present data set, while keeping the other explanatory variables constant (at their mean value), halves yield strain, decrcasing it from 0.0094 to 0.0045. Conversely, yield stressis doubled, increasing from 77 to 154 M Pa. The implication of this, that the specimens yield when the strain energy density reaches some particular value, is false, as shown in Fig. 7, which shows (yield stress x yield strain) plotted against mineral volume fraction. Although, in agreement with the implication above, there is no relation with mineral volume fraction, there is nevertheless a great deal of variation in strain energy; it is clearly not a constant. Mineralisation
and ultimate strain
The increased strain at fracture associated with reduced mineral volume fraction is produced by general&d strain occurring throughout the specimen, and not just by processes occurring near to the fracture line. No necking occurs in bone specimens, the cross-sectional dimensions of which are virtually unaltered after fracture. even in those specimens that had shown 10% or so of ultimate strain. Nor is the strain associated with the fracture process itself. The lnstron testing machine applies deformation at a
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constant rate. This loading system allows the possibility of a specimen fracturing in a piecemeal way at one level, without the fracture travelling at once across the whole cross-section of the specimen. This in turn would allow the possibility of an increase in gauge length. and therefore apparent strain, as the remaining piecesof unfractured bone at the cross-section became strung out between the two intact parts of the specimen, resulting in a decrease in load as the specimen broke up. This does not happen; in fact, the load always increases monotonically as deformation increases,and there are no irregularities in the trace as the fracture point is reached. The specimen always breaks instantaneously. The fracture surface. though not smooth, particularly in the specimensthat showed large strains at failure, shows no very long pullouts or tendrils such as would appear if the large strains were associated with crack travel during fracture. The large ultimate strains associated with lower mineral volume fraction must, then, occur diffusely throughout the specimen. The largest strains are also associated with large increases with post-yield stress. Figure 2 and equation (I) show that the ultimate strain falls very sharply with increasing mineral volume fraction. Mincralisation
and ulbmatc stress
The post-yield behaviour of the lcast and the most highly mincraliscd specimens differ greatly. In highly miner&cd specimens the stress--strain curve is ;Ilmost flat in the post-yield region. In the lessminoraliscd bones. the stress continues IO incrcasc quite markedly with the strain. This is carried to an rxtreme in the antlers. which as a result have a very high ultimates~rcss,although their yield stressis lower than that of highly mineralised bone (Fig. 8). The tympanic bullae of the whale have fracture properties that take them out of the mainstream of the relationship that are found in the other bones. Not only are they brittle, as might bc expected of such highly mineraliscd bone, but their tensile strength is very low. The fracture surfaces of the specimens discussed in this paper (which will be discussed elsewhere) show considerable variation in roughness.
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J. D. CURREY
844
However, remarkable mechanisms
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ultimate properties of compact bone tissue. J. Eiomrchanics 8. 393-405.
for arresting cracks when a crack starts
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APPENDIX
specimen. Justification for the formula: Acknowledgements-1 thank Dn Julian Bryant, David White and anonymous referees for their helpful criticism. Dr Caroline Pond kindly provided many of the specimens. This work was supported by a grant (GR/DO4076) from the Science and Engineering Research Council.
REFERENCES
Atkinson, A. C. (1985) Plots, Transjbrmations, wtd Regression. Oxford University Pmss. U.K. Cook, R. D. (1977) Detection of influential observations in linear regression. Technomefrics 19, 15-18. Currey. J. D. (1987) The evolution of the mechanical properties of amniote bone. J. Biomechanics 20. 1035-1044. Currey. J. D. (1988a) The effect of porosity and mineral content on the Young’s modulus of elasticity of compact bone. 1. Biomechanics 21. 131-139. Currey. J. D. (1988b) The e&t of drying and rc-wetting on some mechanical properties of cortical bone. J. Eiomechanics 21.439-441. Pagct. M. D. and Harvey, P. H. (1988) The taxon level problem in the evolution or mammalian brain size: facts and artifacts. Am. Nar. 132. 344-359. Reilly, D. T. and Burstcin. A. H. (1975) The elastic and
Mineral volume fraction =(0.78 Ca/(0.91- 1.50) Ca). where ‘Ca’ is the calcium content in kg per kg of dry bone. Assume that the mineral in bone is apatite Ca,0(PW,(OH)2. Molecular weight of apatite- 1004. Therefore calcium mass = 39.8% of mineral mass. 1 kg of calcium is equivalent to 110.398= 2.51 kg of mineral. Density of apatite=3200 kgm-‘. Assume density of dry collagen = I I.000kg m-j. Therefore if mass of calcium =X kn kx- ’ bone: mass of apatite=2.51 Xkg kg-’ bone. and mass of collagen = ( I - 2.5 IX kg) kg- ’ bone. Volume of apatite=2.51 X/3200 m’ -0.000784X m’. Volume of collagen =( I -2.51 X)/l IO0m’ =0.000909-0.00228Xm’. Volume fraction of mineral =0.000784X/(0.000784X +0.000909 -0.00228 X) =0.78X/(0.91 - 1.50X). This formula makes a number of not fully justitiable assump tions. such as the fact that the bone mineral is assumed to be all apatite. whereas in fact it is a mixture of minerals of slightly dilTerent densities. However. the formula produns a non-linear relationship between calcium content and mineral volume Iraction that is roughly of the right form. The values for R’ using mineral volume fraction are always slightly greater than those obtrincd using calcium content.
oo?I 929090 noo+.al Pergamon PWM pk
1990.
PHYSICAL CHARACTERISTICS AFFECTING THE TENSILE FAILURE PROPERTIES OF COMPACT BONE J. D. CURREY Department of Biology, University of York, York YOI SDD, U.K. Abstract-Compact bone specimens from a Fide variety of reptiles, birds, and mammals were tested in tension, and their failure properties related to mineral volume fraction. porosity and histological orientation. The principal findings were that the ultimate strain and the work under the stress-strain curve declined sharply with mineralisation. as did the stress and strain appearing after the specimen had yielded. Ultimate tensile strength was not simply related to any combination of the possible explanatory variables, but some relatively poorly mineral&d bones. notably antlers. had high stresses at failure. These high strengths were allowed by a great increase in stress after the bones had yielded at quite low stresses.
INTRODUCTION
Recently (Currey, 1988a). I showed the value of using a large data set from different animals to discover what determines the elastic properties of compact bone. I discussed results from 103 specimens from 24 bones from I8 speciesof tetrapod vertebrates. About 80% of the variance in Young’s modulus could be explained by variation in two variables: calcium content and porosity. The value of Young’s modulus increased monotonically with calcium content and inversely with porosity. In the present paper. I examine the failure propertics of bone loaded in tension. Unlike the case for Young’s modulus, some failure properties do not change monotonically with any combination of explanatory variables. Nevertheless, a number of clear dcpendcncies between failure properties and cxplanatory variablesemerge, giving insight into tcnsilc failure in bone.
undum paper, and tested wet in an Instron I I22 table testing machine. An extensometer measured strain over a gauge length of 11 mm. The cross-head produced a strain rate over the gauge length of about 0.2 s- ‘. The few specimens that failed outside the gauge length were ignored. Young’s modulus of elasticity, yield stress, yield strain, ultimate stress, ultimate strain, and work under the curve, were obtained from the output of a storage oscilloscope. Most of these properties arc obvious; two need some explanation. Yield stress und sfruirl
Many specimens showed a fairly sharp yield point, but some had a rather gently curving load-dcformation curve between straight prc- and post-yield regions. The yield point was taken as the point where the curve had deviated by a strain of 0.002 from the straight line describing the initial part of the curve (Fig. I).
MATERIALS AND METHODS Explunutory
Table I lists the bones and their mechanical properties; 93 specimens from 28 bones from I7 species were tested. Most specimens were collected fairly fresh, but a few were dry museum specimens. All specimens, when tested, had been thoroughly soaked with water. Although, of course, drying must have had some effect on the bones, particularly the collagenous component, Currey (1988b) shows that the effect of drying and re-wetting on the mechanical properties of bone is small. Examination of residuals (see below) showed no evidence that museum specimens behaved differently. as a group, from the fresher specimens. The specimens were cut from the bone with a bandsaw, and shaped on a milling machine guided by a template. The specimens had expanded ends leading by rounded shoulders to a central uniform section of length 16 mm, and a I.8 mm square cross-section, They were smoothed with No. 400 grade carborReceived in final form 3 January 1990.
uariuhles
In tests of the elastic properties of bone (Currey.
1988a) I examined the elfects of two explanatory variables: calcium content and porosity (Currey, 1987). Briefly, calcium, expressed in mg calcium per g of dry, de-fatted bone was determined colorimetrically. This was turned into a measure of mineral volume fraction in the dry bone using [0.78Ca/ (0.91-1.50 Ca)] where’ca’ is the calcium content in kg per kg. The justification for this formula is given in the Appendix. Porosity was determined by pointcounting on undecalcified cross-sections taken just behind the fracture surface. Blood channels and Haversian spaces were considered as pores; osteocytc lacunae and canaliculi were not. In what follows, I use ‘apparent density’ (I -porosity). These two explanatory variables were determined for each of the specimens reported here. In some types of bone. the direction of loading, in relation to the grain of the bone, can have an important effect on the failure properties (Reilly and Burstein,
837
838
J. D.
CURREY
Table 1. Median values of various properties for each bone of each species tested. ‘MVF’: mineral volume fraction; ‘p’: apparent density: ‘E’: Young’s modulus of elasticity (GPa); ‘u,,,‘: . ulttmate stress (MPa); cv,,‘: ultimate strain; ‘IV’: work under stress-strain curve (MJ m-‘k ‘N’: sample size
Galapagos tortoise Geochelone midas Femur 0.340 0.928 13.8 Fibula 0.322 0.960 13.2 Tibia 0.337 0.817 Il.5 Alligator Alligoror missipiensis Femur 0.365 0.940 I28 King penguin Aptenodytes patugonicu Humerus 0.453 0.901 22.8 Radius 0.394 0.91 I 22.1 Ulna 0.42 I 0.923 22.9 Sarus crane Grus anligone Tarsometatarsus 0.341 0.923 23.1 Tibiotarsus 0.382 0.94 I 23.5 Flamingo Phcwnicoprerus phornicopterus Tibiotarsus 0.382 0.942 28.2 Red-necked wallaby Protemnodon rufoeiseu Femur 0.437 0.968 21.8 Tibia 0.402 0.965 25.4 Dugnng Dugong dugon Radius 0.371 0.910 7.5 Scapula 0.388 0.908 6.1 Ulna 0.344 0.847 4.9 Whitcsidcd dolphin Lugentwhyncus acutus Rib. outer cortex 0.3n9 0.696 Rib, inner cortex 0.372 0.799 !::: Whim bcakcd dolphin Laqcnorhyncus olhimsrris Ulna 0.294 0.660 2.7 Fin w halt UY/UCVII~~~C~~U phyxuluz Tympanic bulla 0.560 0.964 34. I Donkey Equtcscuhcllus Radius 0.381 0.944 IS.3 Horse Equus cuhullus Femur 0.395 0.954 24.5 Fallow dczr Da~nu rlumu Tibia 0.430 0.970 26.8 Radius 0.360 0.96 I 25.5 Red dcTr Cervus r/up/~us Antler. mature 0.2&t? Antler, immature 0.28 1 0.932 0.910 1:: Roe deer Cupret~luseuprcolus Femur 0.383 0.943 18.4 Reindeer Rung&v rurundus Antler 0.300 0.829 8.1 Cow Bos taurus One year old Tibia 0.364 0.944 19.7 Cow Bm tuurus Nine years old Femur 0.410 0.948 26. I
1975). Here specimenswere taken as nearly as possible with their grain coincident with the direction of loading. Some bones, however, have a chaotic arrangcment of their lamellae, possibly affecting failure properties. To try to allow for this, I made an estimate of alignment, using this as an explanatory variable, to see whether it was important. Undecalcified longitudinal sections, oriented in the radial and tangential directions in relation to the bone from which they came, were examined using ordinary and polarised light. An estimate of the degree of orientation was made on a
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scale ofO(for bone showing no preferred orientation in any direction) to 6, for bone etfectively perfectly aligned with the long axis of the specimen. No specimen had an orientation at a large angle to the long axis of the specimen. Subjective estimates are, of course, not as satisfactory as objective measurements. However, (as shown below) although orientation does have some statistically significant explanatory effect on some mechanical variables, it is much smaller than those produced by the other explanatory variables. It is
839
Tensile failure propertiesof compactbone
unlikely that refining the estimate of orientation would have much effect on the amount of variance explained. The histology of 67 of the 92 specimens was examined, and also of 77 bending specimens from the same bones. For the specimens for which histological sections were not made, the degree of orientation was taken as the mean value for all specimens from that bone, both tensile and bending (not otherwise used here), that were examined. Specimens from the same bone tend to be similar in their degree of orientation. The explanatory variables listed above are just some of a large number of possibilities. They were chosen because they were reasonably easy to measure, and because they accorded with what common sense dictated might affect the failure properties of bone. This matter is raised again at the end of the discussion.
RESULTS
Table I gives the characteristics of the specimens from different bones, and Table 2 gives the statistical relationships expressed as regressions. The explanatory variables and the dependent variables are expressed in logarithms, except for orientation. (which has arbitrary differences between the values). The reasons for this procedure are explained in Currey (1988a): the multiplicative model, which is implied by the use of logarithms, is as reasonable a model as the additive one, in particular the multiplicative model does not imply large negative values for most of the mechanical variables when mineral volume fraction is zero, a characteristic of the additive model. Also, the statistical explanatory power of the multiplicative model (R*) is greater than that of the additive model. Yield and ultimate
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Figure 2 shows the relationship between mineral volume fraction and both yield strain and ultimate strain. There is a marked dimerence in the way these two properties vary with mineral volume fraction, yield strain showing a slight negative relationship, and ultimate strain showing a very strong negative relationship. A regression analysis (I) using log mineral volume fraction, log apparent density, and orientation as explanatory variables, showed that only mineral volume fraction has a significant effect on ultimate strain. HZ shows that 74% of the variance is explained by the rcgrcssion. (In Table I. ths value of R2 is given both adjusted and unadjusted for the number of degrecx of freedom; in the text the adjusted value is used.) Pos-
Table 2. Regression equations showing the relationships between mechanical propdes and various explanarory variables. ‘MVT: mineral volume fraction; ‘p’: apparent density; &,‘: ultimate strain; ‘8,‘: yield strain; ‘0”‘: yield slrcss;‘c,,‘: post-yield slroin;‘a P,‘: post-yield stress;‘Ori’: orientation. Two values for K’ are given, the lirsl is undjusd. the second (in brackets) is adjusted for the number of degrees of freedom MVI II lug c.,, =-3.96-5.16log I= 16.21 2) log &.I, = - 3.92 - 5.011log MVI I= 14.39 3) log E, = -2.81 - I.12 log MVf+0.0354 Ori 1= 8.4 I 4.36 4) log EY = -2.57 -0.85 log MVT+0.360 log p +0.0107 Ori I= 6.09 1.40 1.22 5) log uY =2.16 3.53 log p +0.0925 Ori +0.X19 log MVI I= 6.14 4.73 2.62 6) log zp, = -4.94 -6.86 log MVf f= 13.61 7) log cp, = - 4.86 - 6.7 I log M VT I= I I.RS 8) log a,, = -1.44 -6.3Olog MVf+264logp I= 9.54 2.07 logMVf+2.94logp +O.O47Ori 9) log up, =-0.92-5.01 I= 8.35 2.R3 I.31 IO) log work = - 1.97 -6.66 log MVT+4.12 logp +0.0989 Ori I= 12.68 3.90 3.17 I I) log work = - 1.63 -6.24 log MV1+4.39 logp +0.0643 Ori I= 10.92 4.19 I .RO
(With bulla) R1 = 74.5% (74.2%) (No bulla) R* =70.2% (69.8%) (With bulla) R’=49.7% (48.5%) (No bullr) R’ = 30.2% (27.7%) (No bulla) R’ = 60.0% (58.6%) (With bulla) R2 -ii 67.0% (66.9)% (No bulla or brittles) R’ = 63.4% (63.0%) (With bulla) R* = 51.0% (49.9%) (No bulla or brittles) R’ = 47.6% (45.6%) (With bulla) R’ = 67.0% (65.9%) (No bulla) R2 = 58.6 (57.1%)
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sibly the two extreme values (of the bulla), on the bottom right of Fig. 2, might be having an undue influence on the form of the regression. However, if they are omitted from the regression the results are little changed (2). The coefficient for mineral volume fraction is barely altered, from -5.16 to -5.08. However, the two bulla values certainly have an important, and possibly distorting, effect on the relationship for yield strain. If the bullae are included (3). RZ = 48%. and both mineral volume fraction and orientation are significant explanatory variables. The coefftcient for mineral volume fraction is - 1.12. If the bullae are excluded (4). R2 drops to 28%. apparent density is also just worth keeping as an explanatory variable, and the coefficient for mineral volume fraction is -0.85. It is obvious that whereas ultimate strain is extremely strongly dependent upon the mineral content, the strain at yield is much less so. Ultimate strain is far more variable than yield strain. Ignoring the bullae, probably a special case, yield strain varies from 0.0045 to 0.0108. a factor of x 2.4, while ultimate strain varies from 0.0045 to 0.1340. a factor of x 28.5. The very weak bullae have been removed from the following analysis of strength, although pairs of equations with and without them are sometimes shown, but their effect on whatever conclusions that can be drawn will be included in the discussion.
Plots of the log of yield stress against possible explanatory variables are shown in Fig. 3a. b, and c. There is no obvious relationship with mineral volume fraction, but apparent density and orientation appear to have a positive relationship, although that for density depends heavily upon three specimens with values of zero for orientation. The correlation matrix (Table 3) shows that the correlation between log apparent density and orientation is only 0.33, which makes it reasonable to consider these explanatory variables to be more or less independent. A multiple
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Tensile failure properties of compact bone
variables. The three variables together explain 59% of the variation in log stress at yield.
Ultimate stress
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slightly greater than yield stress, the relationship between the explanatory variables and both ultimate and yield stressshould be roughly the same. However, many less mineral&d specimens, particularly of antler, showed a considerable post-yield increase in stress.As a result (Fig. 4) there is no clear relationship between mineral volume fraction and ultimate tensile strength, nor does bringing in other possible explanatory variables help statistically.
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Equations (6)-(9) both include and omit the specimens that behaved in an apparently brittle manner. Excluding the brittle specimens has little effect on coefficients.Orientation and apparent density have no significant effect on post-yield strain, but apparent density does have some effect on post-yield stress. Figure 5c shows the relationship between the two variables themselves. Although post-yield stress and post-yield strain co-vary, it is not reasonable to choose
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Stress and strain in the yield region The post-yield strain is usually considerably greater than the elastic strain. However, the post-yield stressis usually less, often very much less, than the elastic stress. The mechanical properties discussed above were transformed to logarithms. Several specimens apparently behaved in a brittle way, showing no postyield increase in stress or strain (within the limits of measurement precision). Log 0 = - a, so for purposes of converting to logarithms, I have performed the standard procedure (Atkinson, 1985, p. 184) of adding a small value (post-yield stress: I MPa; post-yield strain: 0.001) to all specimens. These values are at about the limits of measurement error. Figure 5a andb show the log of post-yield stressand the log of post-yield strain plotted against the log of mineral volume fraction. It is clear that both of these variables undergo a marked reduction with an increase in mineral volume fraction.
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one as being a function of the other. We are attempting to determine afinctional relationship. In such a case, where the co-variates are expressed in logarithms, it is sensible to use the reduced major axis (the exponent of the slope divided by the correlation coefficient) as an estimate of the exponent (Paget and Harvey, 1988). if the brittle specimens are included: post-yield strain a (post-yield stress)t.o’; if they are excluded: post-yield strain a (post-yield stress)“.92.The implication of this is that whatever affects post-yield stress and strain affects them about equally.
842
J. D. CURREY
properties and the explanatory variables. This is certainly sometimes the case. For instance, the antler ... bones seem to have the ability to bear considerably increased stress after they have yielded, compared ma .. . . . with other bones of the same amount of mineralisation .r=-. 0. . . . . 1. and porosity. Similarly, the bulla seemsto behave in a . way almost qualitatively different from the other bones. It is important to be reasonably sure that other . such unusual bone types are not lost in the mass of data. It is in fact difficult to test this. Each type of bone has rather few specimens, and a very limited range of Fig. 6. Relationship berween work under the curve and values for mineral volume fraction and other exmineral volume hction. planatory variables. As a result, regressions for individual types of bone have very large standard errors for the coefficients, so it is not possible to make useful Work comparisons between dimerent bones. However I The work done in breaking a specimen in tension is examined the residuals for those overall regressions some measure of toughness. It is not a rigorous that had reasonably high explanatory power fracture mechanics parameter. However, because it (R’ > 50%) to see if any types of bone stood out. This includes all the work done on the specimen after it has examination showed no remarkable types of bone. yielded, it does give some idea of how resistant the (Antler is remarkable, but its anomolous post-yield material is to impact loading. Furthermore, all specistress ruins the explanatory power of the regression, mens tested here had the same size and shape, so the and no residual analysis is required to identify it.) cflicts of geometry are not confusing the issue. The Another test is to compare the coefficients of the shape of the stress-strain curve implies that the work equations before and after points having a high value will be very nearly proportional to the product of for Cook’s D are removed. Cook’s D mcasurcs instress at failure and strain at failure. llucntial outlicrs in a distribution (Cook, 1977). I Figure 6 shows a strong negative relationship berecalculated all the equations with reasonably high tween mineral volume fraction and the log of work. explanatory power, after excluding the live or six Multiple regressions [equations (IO) and (I l)] show points with the highest values of Cook’s D. In no case that mineral volume fraction seems to be the most did this exclusion have much elTccton the coeflicients important determinant of work, having a negative of the equation. These analyses did not, therefore. effect on it, but that apparent density and. to a lesser show up types of bone that departed markedly from extent, orientation, have a positive elTcct. the rest in their response to variation in the explanatory variables, nor did they suggest that the DISCUSSION coefficients of the equations of the regressions were sensitive to the behaviour of a few anomalous bones. The aim of this paper is to produce a model relating The compact bone of reptiles, birds, and mammals failure properties or bone to other, non-mechanical shows considerable variation in properties when properties, the explanatory variables. There are, of loaded in tension. Mineral volume fraction, apparent course. an infinity of variables that are potentially of density and orientation often ‘explain’. in statistical importance, of which the great majority are un- terms, a considerable proportion of this variation. The measurable. However, the model can be considered most striking of the findings are the large amounts of reasonably successful if the variables that are used post-yield stressshown by the less mineralized bones, accord with common sense. For instance. it has been particularly antler, and the very great variation in the shown that calcium content (converted to mineral ultimate strain of different bones. and the extent to volume fraction) is often important. In reality, of which this is related to mineralisation. Increased course, at the very least it must be calcium phosphate, mineralisation strongly reduces the extra stress and rather than calcium alone, that is important. Fur- strain reached after yield. This shows itself also in a thermore, it is almost certainly the interaction of marked etTecton ultimate strain, but not on ultimate mineral with the other major components, collagen stress, because the latter value is dominated by the and water, that alTectsthe mechanical properties, and yield stress, which is much less affected by mineralisso if coilagen content had been measured. it might ation and, insofar as it is affcctcd, is altered in the have proved to be just as good an explanatory variable opposite, positive, direction. as calcium or mineral volume fraction. This paper is based on data from a great variety of speciesand bones. A possible matter of concern is that Yield stress and strain there may be a number of difTerent types of bone that Compared with its effects on post-yield stress and have dilTcrcnt relationships between the mechanical strain, the effects of variation in mineral volume
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fraction on yield stress and strain are small though real. Changing the mineral volume fraction from the least to the greatest value found in the present data set, while keeping the other explanatory variables constant (at their mean value), halves yield strain, decrcasing it from 0.0094 to 0.0045. Conversely, yield stressis doubled, increasing from 77 to 154 M Pa. The implication of this, that the specimens yield when the strain energy density reaches some particular value, is false, as shown in Fig. 7, which shows (yield stress x yield strain) plotted against mineral volume fraction. Although, in agreement with the implication above, there is no relation with mineral volume fraction, there is nevertheless a great deal of variation in strain energy; it is clearly not a constant. Mineralisation
and ultimate strain
The increased strain at fracture associated with reduced mineral volume fraction is produced by general&d strain occurring throughout the specimen, and not just by processes occurring near to the fracture line. No necking occurs in bone specimens, the cross-sectional dimensions of which are virtually unaltered after fracture. even in those specimens that had shown 10% or so of ultimate strain. Nor is the strain associated with the fracture process itself. The lnstron testing machine applies deformation at a
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constant rate. This loading system allows the possibility of a specimen fracturing in a piecemeal way at one level, without the fracture travelling at once across the whole cross-section of the specimen. This in turn would allow the possibility of an increase in gauge length. and therefore apparent strain, as the remaining piecesof unfractured bone at the cross-section became strung out between the two intact parts of the specimen, resulting in a decrease in load as the specimen broke up. This does not happen; in fact, the load always increases monotonically as deformation increases,and there are no irregularities in the trace as the fracture point is reached. The specimen always breaks instantaneously. The fracture surface. though not smooth, particularly in the specimensthat showed large strains at failure, shows no very long pullouts or tendrils such as would appear if the large strains were associated with crack travel during fracture. The large ultimate strains associated with lower mineral volume fraction must, then, occur diffusely throughout the specimen. The largest strains are also associated with large increases with post-yield stress. Figure 2 and equation (I) show that the ultimate strain falls very sharply with increasing mineral volume fraction. Mincralisation
and ulbmatc stress
The post-yield behaviour of the lcast and the most highly mincraliscd specimens differ greatly. In highly miner&cd specimens the stress--strain curve is ;Ilmost flat in the post-yield region. In the lessminoraliscd bones. the stress continues IO incrcasc quite markedly with the strain. This is carried to an rxtreme in the antlers. which as a result have a very high ultimates~rcss,although their yield stressis lower than that of highly mineralised bone (Fig. 8). The tympanic bullae of the whale have fracture properties that take them out of the mainstream of the relationship that are found in the other bones. Not only are they brittle, as might bc expected of such highly mineraliscd bone, but their tensile strength is very low. The fracture surfaces of the specimens discussed in this paper (which will be discussed elsewhere) show considerable variation in roughness.
200 F s1ramr
(MP4)
-
Cow’s 1*mur
Fig. 8. Characteristic
load-derormation
843
l1l*r
curves of cow bone and anller.
J. D. CURREY
844
However, remarkable mechanisms
the surfaces of the bulla
specimens
are
for their smoothness. There seem to be no
ultimate properties of compact bone tissue. J. Eiomrchanics 8. 393-405.
for arresting cracks when a crack starts
to travel in a bulla specimen it travels easily across the
APPENDIX
specimen. Justification for the formula: Acknowledgements-1 thank Dn Julian Bryant, David White and anonymous referees for their helpful criticism. Dr Caroline Pond kindly provided many of the specimens. This work was supported by a grant (GR/DO4076) from the Science and Engineering Research Council.
REFERENCES
Atkinson, A. C. (1985) Plots, Transjbrmations, wtd Regression. Oxford University Pmss. U.K. Cook, R. D. (1977) Detection of influential observations in linear regression. Technomefrics 19, 15-18. Currey. J. D. (1987) The evolution of the mechanical properties of amniote bone. J. Biomechanics 20. 1035-1044. Currey. J. D. (1988a) The effect of porosity and mineral content on the Young’s modulus of elasticity of compact bone. 1. Biomechanics 21. 131-139. Currey. J. D. (1988b) The e&t of drying and rc-wetting on some mechanical properties of cortical bone. J. Eiomechanics 21.439-441. Pagct. M. D. and Harvey, P. H. (1988) The taxon level problem in the evolution or mammalian brain size: facts and artifacts. Am. Nar. 132. 344-359. Reilly, D. T. and Burstcin. A. H. (1975) The elastic and
Mineral volume fraction =(0.78 Ca/(0.91- 1.50) Ca). where ‘Ca’ is the calcium content in kg per kg of dry bone. Assume that the mineral in bone is apatite Ca,0(PW,(OH)2. Molecular weight of apatite- 1004. Therefore calcium mass = 39.8% of mineral mass. 1 kg of calcium is equivalent to 110.398= 2.51 kg of mineral. Density of apatite=3200 kgm-‘. Assume density of dry collagen = I I.000kg m-j. Therefore if mass of calcium =X kn kx- ’ bone: mass of apatite=2.51 Xkg kg-’ bone. and mass of collagen = ( I - 2.5 IX kg) kg- ’ bone. Volume of apatite=2.51 X/3200 m’ -0.000784X m’. Volume of collagen =( I -2.51 X)/l IO0m’ =0.000909-0.00228Xm’. Volume fraction of mineral =0.000784X/(0.000784X +0.000909 -0.00228 X) =0.78X/(0.91 - 1.50X). This formula makes a number of not fully justitiable assump tions. such as the fact that the bone mineral is assumed to be all apatite. whereas in fact it is a mixture of minerals of slightly dilTerent densities. However. the formula produns a non-linear relationship between calcium content and mineral volume Iraction that is roughly of the right form. The values for R’ using mineral volume fraction are always slightly greater than those obtrincd using calcium content.
