J. Bmmchanics
0021-9290/91
Vol. 24. No. I. pp. 57 61, 1991.
$3.00+ .W
Pergamonpressplc
Pnntedin Great Britain
ORIENTATION OF MINERAL IN BOVINE BONE AND THE ANISOTROPIC MECHANICAL PROPERTIES OF PLEXIFORM BONE NAOKI
SASAKI, TETSU IKAWA and AKEHARU FUKUDA
Department of Applied Materials Science, Muroran Institute of Technology, 27-l Mizumoto, Muroran, Japan Abstract-Angular dependent Young’s modulus E, presented by Bonfield and Grynpas [Nature 270, 453454 (1977)] was simulated by using the distribution function of the orientation of mineral in plexiform bone introduced on the basis of an X-ray pole figure analysis (XPFA) and a small angle X-ray scattering (SAXS) results. Calculations were performed with the aid of a simple model which expresses well the geometrical characteristic of plexiform bone. Estimated angular dependent Young’s modulus in terms of the distribution of mineral orientation reproduced the experimental results. The suitable aspect ratio of bone mineral for the reproduction of the empirical data was a reasonable value compared with the morphological study of bone mineral. It is concluded that the angular dependence of mechanical properties of plexiform bone is explained by the distribution of bone mineral orientation and its morphology.
platelet shape. Modulus of the mineral is as large as 100 times of that of collagen. According to these facts, Bone is both anisotropic and heterogeneous in its it is expected that the mechanical properties of bone mechanical properties. One of the anisotropic properwould be affected by the size of mineral crystallite and ties of bone still to be explained is the angular its orientation. In the theoretical consideration on the dependence of the modulus, the angle between bone mechanical properties of bone, Bundy (1985) claimed (longitudinal) and stress axes. Young’s moduli of bone the necessity of the information of the orientational in the direction along and perpendicular to the bone distribution of bone mineral as well as the effectiveness axis are 20 GPa and 10 GPa, respectively. Currey in mineral-organic bonding in bone. Empirically we (1969) had tried to interpret this anisotropy by using also demonstrated the important role of collagenthe mode1 of a simple two phase composite (unimineral interaction in the mineral content-dependent directionally hydroxyapatite-reinforced collagen Young’s modulus of cortical bone (Sasaki et al., 1986, plate) on the basis of a morphology of hydroxyapatite 1989a). (HAP) crystal in bone. However, it was revealed by Our basic aim is to introduce an empirical strucBonfield and Grynpas (1977) that the model did not tural information at collagen-mineral level into provide a proper prediction; at angles between the understanding of anisotropic mechanical properties of bone and stress axes greater than approximately 30”, bone. Among structural information, we think that the measured Young’s moduli exceeded the predicted orientation and its distribution of bone mineral is values considerably. They concluded by stating the particularly important because it has been regarded as need for an alternative model. a direct consequence of WolR’s law in the growth and The current trend in the field of mechanical properremodeling of bone (Bacon et al., 1977, 1979, 1980). In ties of bone is to establish more suitable model of bone this context, we have emphasized the importance of than the conventional two phase composite model. Katz the orientational distribution of bone mineral in con(1980,198l) and Lipson and Katz (1984) have pointed sidering mechanical anisotropic properties of bone, by out the necessity of the microstructural information in reproducing the transverse anisotropic mechanical establishing the alternative mechanical model of bone properties of plexiform bone from bovine femur only on the basis of the ultrasonic wave propagation on the basis of the results of X-ray pole figure analysis measurements (Van Buskirk et al., 1981; Katz et al., (XPFA) of the orientation of a long axis of bone 1983). They have proposed that organization on the mineral (Sasaki et al., 1989b). microscopic level can also be modeled as a fiber Here we intended to provide another example of the reinforced composite. In their fiber reinforced com- introduction of structural information into the mechposite model at a microscopic level, it is found that anical calculation. In this paper, we simulated the information about the orientational distribution of angular dependent anisotropic mechanical properties collagen fiber and then bone mineral are needed (Katz, in bone axial-tangential plane of plexiform bone 1980). Mineral in bone is in a rod-like or elongated (Bonfield and Grynpas, 1977) on the basis of the orientational distribution function (ODF) of bone mineral and intended to demonstrate the importance Received in final form 12 July 1990. INTRODUCTION
57
58
N.
SASAKI
of introducing the structural information such as ODF into mechanical calculation. We have used ODF which had been obtained from results of X-ray pole figure analysis (XPFA) (Sasaki et al., 1989b) and small angle X-ray scattering (SAXS) of bone (Matsushima et al., 1982). In actual application of data on the size and orientation of bone mineral, it is by means of the simple uniform strain composite model that we can most easily introduce the structural information into the mechanical calculations. So long as the orientational distribution function of bone mineral can be regarded as a direct consequence of Wolffs law, such a composite model should be primarily suitable for the situation. The geometry of a plexiform bone in bone axial-tangential plane was considered to be well characterized by the uniform strain model. For different types of bone, a more suitable model is needed. In the case of Haversian bone, the model which has taken account of the geometric characteristics of a Haversian system was proposed to succeed in explaining the anisotropy of bone (Katz, 1980, 1981). The model might not be perfect, but we believe that our approach will be a stimulus for designing new biomimetic composite materials, in particular an artificial bone-like material, because of the simplicity of the approach.
TWO-PHASE
COMPOSITE
et af.
the sample 4, a direction pointed by the longer axis of bone mineral, is defined by angle 0 on the basis of sample axis of a specimen [Fig. l(b)]. A plexiform bone consists of a number of laminar layers. Each laminar plane is almost parallel to the bone axis. Mechanical property is homogeneous in a plane but changes when passing through from one lamina to the next. In a lamina (by light microscopy) orientation of collagen fiber, and also of bone mineral, are generally uniform (Cowin, 1981; Currey, 1984); by electron microscopy, a lamina is regarded as a mosaic work of unidirectionally HAP-reinforced collagen plate (Boyde and Hobdell, 1969; Frasca et al., 1977). BA t
MODEL
In order to estimate the angular dependence of the moduli of bone, which was empirically presented by Bonfield and Grynpas (1977), by using only the structural data, we have made use of our previous results of XPFA in combination with SAXS data (Sasaki et al., 1989b). Mineral is assumed to be of the rod-like shape. As a bone, we assumed a bovine femoral bone, because both XPFA (Sasaki et al., 1989b) amd the mechanical measurements by Bonfield and Grynpas (1977) were made for bovine femoral bone. The bovine femoral bone is generally a plexiform bone and in our XPFA experiments, optical microscopic assessment at least proved the sample was generally plexiform (Sasaki et al., 1989b). As Bonfield and Grynpas have made measurements in the bone axial-tangential (BA-TA) plane, in order to simulate the angular dependent anisotropic Young’s modulus in the BA-TA plane, we considered a set of plates cut from femoral cortical bone as in Fig. l(a). Such a plane has been assumed perpendicular to the bone sample plate and contains the sample axis, the bone axis and the tangential axis. Through all of the samples for mechanical measurements, it is only the mineral orientation parallel to this plane that changes successively with the cutting angle 4, which would contribute to the angular dependence of Young’s modulus of bone. Therefore, it may also be sufficient to take into account the orientation of minerals within this plane (BA-TA plane). In the sample of the cutting angle 4,
Fig. 1. (a) Rectangular specimen assumed cut from bovine femur at any angle, 4, for example, against bone axis (BA). TA and RA represent the tangential and radial directions, respectively.
t
BA
Fig. I. (b) Relationship between the cutting angle I#Jand a direction pointed by mineral, characterized by angle 0. In the diagram, BA, SA, and M represent bone axis, sample axis, and longitudinal axis of the mineral, respectively.
Mineral
orientation
59
in bovine bone
a=2(L/d) =0.5 = 1
for for
for
E,
(5)
E,
(61
G,,
(7)
where E,, and E, are Young’s moduli of HAP and collagen, respectivebr. G, is shear modulus of collagen and the volume fractions of HAP, $, and collagen, 1 -tj, were assumed to be 0.5 each, L and d are respectively a length and a diameter of rod-like shape mineral, and v is the Poisson’s ratio assuming to be 0.33 (Currey, 1969). In this estimation, values of E, = 114 GPa (Gilmore and Katz, 1982), E, = 1.47 GPa (Currey, 1969) and G,= 1 GPa (Tanioka et al., 1974) are used. p is the parameter relating to the packing density of HAP rod and defined as
SA b
03)
P=expC(l -hJ~hGl
where $, is the maximum packing density of fiber. We take the value of II/, =0.82, assuming the parallel random packing (Nielsen, 1975), then p = 1.14. Fig. 1. (c) Mechanical unit representing the mechanical properties of the plate in Fig. l(b). The mechanical unit is a
cube whose edges are parallel to those of sample plate and is assumed to be composed of elementary HAP reinforced collagen plates where HAP orientation is unidirectional. The arrow in each elementary plate points in the direction of HAP orientation. BA and SA are the directions of the long axis of bone (bone axis) and the sample axis, respectively.
Figure l(c) shows a mechanical unit which represents the mechanical properties of the plexiform bone sample plate as in Fig. l(b). The mechanical unit is a cube whose edges are parallel to those of the sample plate and it is considered to be made up of elementary HAP-reinforced collagen plates in each of which orientation of HAP is uniaxial. We assumed the Voigt type combination of the elementary plates in order to express the mechanical characteristics of a layer structure of a plexiform bone in the BA-TA plane as explained above. Geometric characteristics of plexiform bone is expressed by this model inclusive of Voigt type combination.
MISALIGNMENT
FUNCTION
Young’s modulus E, in each elementary plate at each angle 0 was estimated by the misalignment function (Currey, 1969; Nielsen, 1975);
ORIENTATIONAL
DISTRIBUTION
FUNCTION
As has been revealed, the orientation of HAP crystal in bone mineral or bone mineral itself is not unidirectional but has an appreciable distribution of direction around the bone axis. So, for sample 4, in this work, according to the model introduced above, the Young’s modulus, E,, was estimated as: 90
E,=
i
E,CA(& @)+A($,
--@IId@
(9)
-90
where A(4,O) is the orientational distribution function (ODF) of bone mineral in the BA-TA plane, obtained by the XPFA or the SAXS intensity I(0) as;
where as r(e), for 0,O” < 0 < 50”, the scattering intensity from (002) plane of HAP in bone mineral, which generally coincides with a longitudinal axial orientation of mineral, was used and for SO”< 6 < 90”, the orientation distribution of a longitudinal axis of rod-like bone mineral by SAXS was used. Figure 2 shows I(0) for 4 =o”, actually used in this calculation, obtained by XPFA and SAXS (Sasaki et al., 1989b Matsushima et al., 1982).
E,/E,=cos46+(E,/E,)sin48 RESULTS AND DISCUSSION
+[(E,/G,,)-2v]sinZBcos20 and the modified 1975), E Lor r/4=(1
Halpin-Tsai +nBW(1
equation
(1) (Nielsen,
--BPd4
GLTIG~=(~+~BI~~)/(~--P~)
E={(E,IE,)-
1 }I{(E,IE,)+aJ
(2)
(3) (4)
In Fig. 3, angular dependent Young’s moduli estimated here are shown with empirical data points presented by Bonfield and Grynpas (1977). Solid lines are the calculated angular dependent Young’s moduli for mineral with aspect ratios of L/d= 10, 5 and 2.5, respectively. In the figure, the dashed line represents the result of an estimation on the basis of Currey’s
60
N. SASAKI et al.
modulus value in Currey’s model (dashed line), assuming a unidirectional orientation of bone mineral to the direction of bone axis, reduces rapidly in the range of
I$ from 30 to 90”, while that around O”,i.e. the direction of bone axis, and in this case general orientational direction of bone mineral, accords with the measured Young’s moduli of bone in this direction (Currey, 0.5 1969; Bonfield and Grynpas, 1977). By using an actual distribution function of bone mineral orientation, in our estimation, calculated E, at the &range from 30 to 90” as well, generally traces the empirical data points. It is notable that the shape of mineral in bone which can explain the mechanical properties of bone in our 0 estimation is rather stumpy. Regarding the shape and size of bone mineral, a number of works have been BA performed. Bone mineral has been considered to have 3 (degrees) a shape of rod (Arsenault, 1988) or elongated platelet Fig. 2. Normalized (002) scattering intensity plotted against (Traub et al., 1989). There are some inconsistencies in 19for the sample of $ =O”, actually used in the simulation of its size depending on the origin of the sample and the E,. 0=0” is the direction of 4 and in this case, 4 =o”, this measuring method. Using an X-ray diffraction direction is identical with that of bone longer axis. Open method, Wheeler and Lewis (1977) have reported that circles are data points taken from X-ray pole figure diagram (Sasaki et al., 1989b) and solid line from SAXS data (Matsuin bovine femur, mineral was of the rod-like shape and shima et al., 1982). the average size was 220 A in length and the diameter 70 A. This means that the aspect ratio is about 3. Cuisinier et al. (1987) studied the morphology of human alveolar bone mineral by means of transmission electron microscopy. It was revealed that there were two types of minerals; minerals of the platelet shape and the rod-like shape. In the platelet shaped mineral, the average ratio of width ( W) against thickness (T), W/T, was found to be 6.9. In the rod-like shaped mineral, the average aspect ratio (= length/ diameter) was about 7.7. The smaller value of our aspect ratio when compared with this latter example is due to the difference in the source of bone samples, though the interpretation of a structural model on the basis of the result should be made with many reservations. Compared with these works, our resultant value of the aspect ratio of bone mineral can be stated to be not unreasonable. As mentioned in the Model section, our estimation was limited to the calculation for the angular depend75 90 15 30 45 SO 0 ent anisotropy in the BA-TA plane. The plexiform bone has been recognized as orthotropic in nature. In # (degrees) our previous publication, we estimated the anisotropy Fig. 3. 4 dependence of moduli; Young’s moduli for dry (0) in the tangential-radial (TA-RA) plane (transverse and wet (0) bone according to Bonfield and Grynpas (1977). anisotropy), where the Young’s modulus of TA direcSolid lines are estimated angular dependent Young’s mod& for mineral with aspect ratio L/d= 10,5 and 2.5. Dashed line tion is 1.1 times larger than that of RA direction. It is likely that the mechanical properties in bone reproduces the result on the basis of the conventional twophase composite model. axial-radial (BA-RA) plane is different from that in BA-TA plane estimated here. A Voigt model is not the best approximation of the bone microstructure for BA-RA plane. For the estimation in BA-RA plane calculation procedures on the basis of a more plausible conventional two phase composite model, employing model with the ODF should be required. There have a uniaxial orientation of bone mineral. been, however, no empirical data of angular dependIt is found that the estimated angular dependent ent anisotropic mechanical properties in BA-RA nature of Young’s modulus using the ODF obtained from XPFA and SAXS generally reproduces the em- plane. Detailed estimation and its empirical confirmation are now in progress. pirical moduli as a function of cutting angles. Young’s
Mineral orientation in bovine bone At any rate, these results indicate that in considering an anisotropic mechanical properties of bone, a nonlongitudinal (off-bone axial) orientation of bone mineral is one of many important factors. This means that bone is multi-directionally reinforced material as well as in the direction of bone axis. We believe that these results are sufficiently encouraging to place reliance upon the approach of estimating mechanical properties of bone on the basis of empirical structural information, in particular orientational distribution function of bone mineral.
REFERENCES
Arsenault, A. L. (1988) Crystal-collagen in calcified turkey leg tendons visualized by selected-area dark field electron microscopy. CalcijI Tissue Int. 43, 202-212. Bacon, G. E., Bacon, P. J. and Griffiths, R. K. (1977) The study of bones by neutron diffraction. J. apgl. Cryst. 10, 124- 126. Bacon, G. E., Bacon, P. J. and Griffiths, R. K. (1979) Stress distribution in the scapula studied by neutron diffraction. Proc. R. Sot. Lond. B. 204, 355-362. Bacon, G. E., Bacon, P. J. and Griffiths, R. K. (1980) Orientation of apatite crystals in relation to muscle attachment in the mandible. J. Biomechanics 13, 725-729. Bonfield, W. and Grynpas, M. D. (1977) Anisotropy of the Young’s modulus of bone. Nature 270, 453454. Boyde, A. and Hobdell, M. H. (1969) Scanning electron microscopy of lamellar bone. 2. Zellforsch 93, 2 13-23 1. Bundy, K. J. (1985) Determination of mineral-organic bonding effectiveness in bone; theoretical consideration. Ann. biomed. Engng 13, fi9-136. Cowin, S. C. (1981) Introduction to the symposium on the mechanical properties of bone. In Mechanical Properties of Bone (Edited by Cowin, S. C.), Vol. 45, pp. 1-12. American Society of Mechanical Engineers, New York. Cuisinier, F., Bres, E. F., Hemmerle, J., Voegel, J. C. and Frank, R. M. (1987) Transmission electron microscopy of lattice planes in human alveolar bone apatite crystals. Cal& Tissue Int. 40, 332-338. Currey, J. D. (1969) The relationship between the stiffness and the mineral content of bone. J. Biomechanics 2, 477480.
Currey,
61
J. D. (1984) The Mechanical Adaptations of Bones.
Priketon University Press, Princeton, NJ. Frasca, P., Harper, R. A. and Katz, J. L. (1977) Collagen fiber orientations in human secondary osteons. Acta anat. 98, I-17 _ ._. Gilmore, R. S. and Katz, J. L. (1982) Elastic properties of apatites. J. Mater. Sci. 17, 1131-1141. Katz, J. L. (1980) Anisotropy of Young’s modulus of bone. Nature 283, 106-107. Katz, J. L. (1981) Composite material models for cortical bone. In Mechanical Properties ofBone (Edited by Cowin, S. C.), Vol. 45, pp. 171-184. American Society of Mechanical Engineers, New York. Katz, J. L., Yoon, H. S.. Lipson, S., Maharidge, R., Meunier, A. and Christel. P. (1983) The effect of remodeling on the elastic properties of bone. Calcif: Tissue Int. 36, S31-S36. Lipson, S. F. and Katz, J. L. (1984) The relationship between elastic properties and microstructure of bovine cortical bone. J. Biomechanics 17, 231-240. Matsushima, N., Akiyama, M. and Terayama. Y. (1982) Quantitative analysis of orientation of mineral in bone from small angle X-ray scattering patterns. Japan J. appl. Phys. 21, 186-189. Nielsen, L. E. (1975) Mechanical Properties qf Polymers and Composites. Marcel Dekker, New York. Sasaki, N.. Matsushima, N., Ikawa, T., Yamanura, H. and Fukuda, A. (1989b) Orientation of bone mineral and its role in the anisotropic mechanical properties of bonetransverse anisotropy. J. Biomechanics 22, 157-164. Sasaki, N., Umeda, H., Okada, S., Kojima, R. and Fukuda, A. (1989a) Mechanical properties of hydroxyapatite-reinforced gelatin as a model system of bone. Biomaterials 10, 129-132. Sasaki, N., Yamamura, H. and Matsushima, N. (1986) Is there a relation between bone strength and percolation? J. theoret. Biol. 122, 25-31. Tanioka, A., Tazawa, T., Miyasaka, K. and Ishikawa, K. (1974) Effects of water on the mechanical properties of gelatin films. Biopolymers 13, 735-746. Traub, W., Arad, T. and Weiner, S. (1989) Three-dimensional ordered distribution of crystals in turkey tendon collagen fibers. Proc. Nat. Acad. Sci., U.S.A. 86, 9822-9826. Van Buskirk, W. C., Cowin, S. C. and Ward, R. N. (1981) Ultrasonic measurements of orthotropic elastic constants of bovine femoral bone. J. biomech. Engng 103, 67-72. Wheeler, E. J. and Lewis, D. (1977) An X-ray study of the paracrystalline nature of bone apatite. Calcif: Tissue Rex 24, 243-248.
0021-9290/91
Vol. 24. No. I. pp. 57 61, 1991.
$3.00+ .W
Pergamonpressplc
Pnntedin Great Britain
ORIENTATION OF MINERAL IN BOVINE BONE AND THE ANISOTROPIC MECHANICAL PROPERTIES OF PLEXIFORM BONE NAOKI
SASAKI, TETSU IKAWA and AKEHARU FUKUDA
Department of Applied Materials Science, Muroran Institute of Technology, 27-l Mizumoto, Muroran, Japan Abstract-Angular dependent Young’s modulus E, presented by Bonfield and Grynpas [Nature 270, 453454 (1977)] was simulated by using the distribution function of the orientation of mineral in plexiform bone introduced on the basis of an X-ray pole figure analysis (XPFA) and a small angle X-ray scattering (SAXS) results. Calculations were performed with the aid of a simple model which expresses well the geometrical characteristic of plexiform bone. Estimated angular dependent Young’s modulus in terms of the distribution of mineral orientation reproduced the experimental results. The suitable aspect ratio of bone mineral for the reproduction of the empirical data was a reasonable value compared with the morphological study of bone mineral. It is concluded that the angular dependence of mechanical properties of plexiform bone is explained by the distribution of bone mineral orientation and its morphology.
platelet shape. Modulus of the mineral is as large as 100 times of that of collagen. According to these facts, Bone is both anisotropic and heterogeneous in its it is expected that the mechanical properties of bone mechanical properties. One of the anisotropic properwould be affected by the size of mineral crystallite and ties of bone still to be explained is the angular its orientation. In the theoretical consideration on the dependence of the modulus, the angle between bone mechanical properties of bone, Bundy (1985) claimed (longitudinal) and stress axes. Young’s moduli of bone the necessity of the information of the orientational in the direction along and perpendicular to the bone distribution of bone mineral as well as the effectiveness axis are 20 GPa and 10 GPa, respectively. Currey in mineral-organic bonding in bone. Empirically we (1969) had tried to interpret this anisotropy by using also demonstrated the important role of collagenthe mode1 of a simple two phase composite (unimineral interaction in the mineral content-dependent directionally hydroxyapatite-reinforced collagen Young’s modulus of cortical bone (Sasaki et al., 1986, plate) on the basis of a morphology of hydroxyapatite 1989a). (HAP) crystal in bone. However, it was revealed by Our basic aim is to introduce an empirical strucBonfield and Grynpas (1977) that the model did not tural information at collagen-mineral level into provide a proper prediction; at angles between the understanding of anisotropic mechanical properties of bone and stress axes greater than approximately 30”, bone. Among structural information, we think that the measured Young’s moduli exceeded the predicted orientation and its distribution of bone mineral is values considerably. They concluded by stating the particularly important because it has been regarded as need for an alternative model. a direct consequence of WolR’s law in the growth and The current trend in the field of mechanical properremodeling of bone (Bacon et al., 1977, 1979, 1980). In ties of bone is to establish more suitable model of bone this context, we have emphasized the importance of than the conventional two phase composite model. Katz the orientational distribution of bone mineral in con(1980,198l) and Lipson and Katz (1984) have pointed sidering mechanical anisotropic properties of bone, by out the necessity of the microstructural information in reproducing the transverse anisotropic mechanical establishing the alternative mechanical model of bone properties of plexiform bone from bovine femur only on the basis of the ultrasonic wave propagation on the basis of the results of X-ray pole figure analysis measurements (Van Buskirk et al., 1981; Katz et al., (XPFA) of the orientation of a long axis of bone 1983). They have proposed that organization on the mineral (Sasaki et al., 1989b). microscopic level can also be modeled as a fiber Here we intended to provide another example of the reinforced composite. In their fiber reinforced com- introduction of structural information into the mechposite model at a microscopic level, it is found that anical calculation. In this paper, we simulated the information about the orientational distribution of angular dependent anisotropic mechanical properties collagen fiber and then bone mineral are needed (Katz, in bone axial-tangential plane of plexiform bone 1980). Mineral in bone is in a rod-like or elongated (Bonfield and Grynpas, 1977) on the basis of the orientational distribution function (ODF) of bone mineral and intended to demonstrate the importance Received in final form 12 July 1990. INTRODUCTION
57
58
N.
SASAKI
of introducing the structural information such as ODF into mechanical calculation. We have used ODF which had been obtained from results of X-ray pole figure analysis (XPFA) (Sasaki et al., 1989b) and small angle X-ray scattering (SAXS) of bone (Matsushima et al., 1982). In actual application of data on the size and orientation of bone mineral, it is by means of the simple uniform strain composite model that we can most easily introduce the structural information into the mechanical calculations. So long as the orientational distribution function of bone mineral can be regarded as a direct consequence of Wolffs law, such a composite model should be primarily suitable for the situation. The geometry of a plexiform bone in bone axial-tangential plane was considered to be well characterized by the uniform strain model. For different types of bone, a more suitable model is needed. In the case of Haversian bone, the model which has taken account of the geometric characteristics of a Haversian system was proposed to succeed in explaining the anisotropy of bone (Katz, 1980, 1981). The model might not be perfect, but we believe that our approach will be a stimulus for designing new biomimetic composite materials, in particular an artificial bone-like material, because of the simplicity of the approach.
TWO-PHASE
COMPOSITE
et af.
the sample 4, a direction pointed by the longer axis of bone mineral, is defined by angle 0 on the basis of sample axis of a specimen [Fig. l(b)]. A plexiform bone consists of a number of laminar layers. Each laminar plane is almost parallel to the bone axis. Mechanical property is homogeneous in a plane but changes when passing through from one lamina to the next. In a lamina (by light microscopy) orientation of collagen fiber, and also of bone mineral, are generally uniform (Cowin, 1981; Currey, 1984); by electron microscopy, a lamina is regarded as a mosaic work of unidirectionally HAP-reinforced collagen plate (Boyde and Hobdell, 1969; Frasca et al., 1977). BA t
MODEL
In order to estimate the angular dependence of the moduli of bone, which was empirically presented by Bonfield and Grynpas (1977), by using only the structural data, we have made use of our previous results of XPFA in combination with SAXS data (Sasaki et al., 1989b). Mineral is assumed to be of the rod-like shape. As a bone, we assumed a bovine femoral bone, because both XPFA (Sasaki et al., 1989b) amd the mechanical measurements by Bonfield and Grynpas (1977) were made for bovine femoral bone. The bovine femoral bone is generally a plexiform bone and in our XPFA experiments, optical microscopic assessment at least proved the sample was generally plexiform (Sasaki et al., 1989b). As Bonfield and Grynpas have made measurements in the bone axial-tangential (BA-TA) plane, in order to simulate the angular dependent anisotropic Young’s modulus in the BA-TA plane, we considered a set of plates cut from femoral cortical bone as in Fig. l(a). Such a plane has been assumed perpendicular to the bone sample plate and contains the sample axis, the bone axis and the tangential axis. Through all of the samples for mechanical measurements, it is only the mineral orientation parallel to this plane that changes successively with the cutting angle 4, which would contribute to the angular dependence of Young’s modulus of bone. Therefore, it may also be sufficient to take into account the orientation of minerals within this plane (BA-TA plane). In the sample of the cutting angle 4,
Fig. 1. (a) Rectangular specimen assumed cut from bovine femur at any angle, 4, for example, against bone axis (BA). TA and RA represent the tangential and radial directions, respectively.
t
BA
Fig. I. (b) Relationship between the cutting angle I#Jand a direction pointed by mineral, characterized by angle 0. In the diagram, BA, SA, and M represent bone axis, sample axis, and longitudinal axis of the mineral, respectively.
Mineral
orientation
59
in bovine bone
a=2(L/d) =0.5 = 1
for for
for
E,
(5)
E,
(61
G,,
(7)
where E,, and E, are Young’s moduli of HAP and collagen, respectivebr. G, is shear modulus of collagen and the volume fractions of HAP, $, and collagen, 1 -tj, were assumed to be 0.5 each, L and d are respectively a length and a diameter of rod-like shape mineral, and v is the Poisson’s ratio assuming to be 0.33 (Currey, 1969). In this estimation, values of E, = 114 GPa (Gilmore and Katz, 1982), E, = 1.47 GPa (Currey, 1969) and G,= 1 GPa (Tanioka et al., 1974) are used. p is the parameter relating to the packing density of HAP rod and defined as
SA b
03)
P=expC(l -hJ~hGl
where $, is the maximum packing density of fiber. We take the value of II/, =0.82, assuming the parallel random packing (Nielsen, 1975), then p = 1.14. Fig. 1. (c) Mechanical unit representing the mechanical properties of the plate in Fig. l(b). The mechanical unit is a
cube whose edges are parallel to those of sample plate and is assumed to be composed of elementary HAP reinforced collagen plates where HAP orientation is unidirectional. The arrow in each elementary plate points in the direction of HAP orientation. BA and SA are the directions of the long axis of bone (bone axis) and the sample axis, respectively.
Figure l(c) shows a mechanical unit which represents the mechanical properties of the plexiform bone sample plate as in Fig. l(b). The mechanical unit is a cube whose edges are parallel to those of the sample plate and it is considered to be made up of elementary HAP-reinforced collagen plates in each of which orientation of HAP is uniaxial. We assumed the Voigt type combination of the elementary plates in order to express the mechanical characteristics of a layer structure of a plexiform bone in the BA-TA plane as explained above. Geometric characteristics of plexiform bone is expressed by this model inclusive of Voigt type combination.
MISALIGNMENT
FUNCTION
Young’s modulus E, in each elementary plate at each angle 0 was estimated by the misalignment function (Currey, 1969; Nielsen, 1975);
ORIENTATIONAL
DISTRIBUTION
FUNCTION
As has been revealed, the orientation of HAP crystal in bone mineral or bone mineral itself is not unidirectional but has an appreciable distribution of direction around the bone axis. So, for sample 4, in this work, according to the model introduced above, the Young’s modulus, E,, was estimated as: 90
E,=
i
E,CA(& @)+A($,
--@IId@
(9)
-90
where A(4,O) is the orientational distribution function (ODF) of bone mineral in the BA-TA plane, obtained by the XPFA or the SAXS intensity I(0) as;
where as r(e), for 0,O” < 0 < 50”, the scattering intensity from (002) plane of HAP in bone mineral, which generally coincides with a longitudinal axial orientation of mineral, was used and for SO”< 6 < 90”, the orientation distribution of a longitudinal axis of rod-like bone mineral by SAXS was used. Figure 2 shows I(0) for 4 =o”, actually used in this calculation, obtained by XPFA and SAXS (Sasaki et al., 1989b Matsushima et al., 1982).
E,/E,=cos46+(E,/E,)sin48 RESULTS AND DISCUSSION
+[(E,/G,,)-2v]sinZBcos20 and the modified 1975), E Lor r/4=(1
Halpin-Tsai +nBW(1
equation
(1) (Nielsen,
--BPd4
GLTIG~=(~+~BI~~)/(~--P~)
E={(E,IE,)-
1 }I{(E,IE,)+aJ
(2)
(3) (4)
In Fig. 3, angular dependent Young’s moduli estimated here are shown with empirical data points presented by Bonfield and Grynpas (1977). Solid lines are the calculated angular dependent Young’s moduli for mineral with aspect ratios of L/d= 10, 5 and 2.5, respectively. In the figure, the dashed line represents the result of an estimation on the basis of Currey’s
60
N. SASAKI et al.
modulus value in Currey’s model (dashed line), assuming a unidirectional orientation of bone mineral to the direction of bone axis, reduces rapidly in the range of
I$ from 30 to 90”, while that around O”,i.e. the direction of bone axis, and in this case general orientational direction of bone mineral, accords with the measured Young’s moduli of bone in this direction (Currey, 0.5 1969; Bonfield and Grynpas, 1977). By using an actual distribution function of bone mineral orientation, in our estimation, calculated E, at the &range from 30 to 90” as well, generally traces the empirical data points. It is notable that the shape of mineral in bone which can explain the mechanical properties of bone in our 0 estimation is rather stumpy. Regarding the shape and size of bone mineral, a number of works have been BA performed. Bone mineral has been considered to have 3 (degrees) a shape of rod (Arsenault, 1988) or elongated platelet Fig. 2. Normalized (002) scattering intensity plotted against (Traub et al., 1989). There are some inconsistencies in 19for the sample of $ =O”, actually used in the simulation of its size depending on the origin of the sample and the E,. 0=0” is the direction of 4 and in this case, 4 =o”, this measuring method. Using an X-ray diffraction direction is identical with that of bone longer axis. Open method, Wheeler and Lewis (1977) have reported that circles are data points taken from X-ray pole figure diagram (Sasaki et al., 1989b) and solid line from SAXS data (Matsuin bovine femur, mineral was of the rod-like shape and shima et al., 1982). the average size was 220 A in length and the diameter 70 A. This means that the aspect ratio is about 3. Cuisinier et al. (1987) studied the morphology of human alveolar bone mineral by means of transmission electron microscopy. It was revealed that there were two types of minerals; minerals of the platelet shape and the rod-like shape. In the platelet shaped mineral, the average ratio of width ( W) against thickness (T), W/T, was found to be 6.9. In the rod-like shaped mineral, the average aspect ratio (= length/ diameter) was about 7.7. The smaller value of our aspect ratio when compared with this latter example is due to the difference in the source of bone samples, though the interpretation of a structural model on the basis of the result should be made with many reservations. Compared with these works, our resultant value of the aspect ratio of bone mineral can be stated to be not unreasonable. As mentioned in the Model section, our estimation was limited to the calculation for the angular depend75 90 15 30 45 SO 0 ent anisotropy in the BA-TA plane. The plexiform bone has been recognized as orthotropic in nature. In # (degrees) our previous publication, we estimated the anisotropy Fig. 3. 4 dependence of moduli; Young’s moduli for dry (0) in the tangential-radial (TA-RA) plane (transverse and wet (0) bone according to Bonfield and Grynpas (1977). anisotropy), where the Young’s modulus of TA direcSolid lines are estimated angular dependent Young’s mod& for mineral with aspect ratio L/d= 10,5 and 2.5. Dashed line tion is 1.1 times larger than that of RA direction. It is likely that the mechanical properties in bone reproduces the result on the basis of the conventional twophase composite model. axial-radial (BA-RA) plane is different from that in BA-TA plane estimated here. A Voigt model is not the best approximation of the bone microstructure for BA-RA plane. For the estimation in BA-RA plane calculation procedures on the basis of a more plausible conventional two phase composite model, employing model with the ODF should be required. There have a uniaxial orientation of bone mineral. been, however, no empirical data of angular dependIt is found that the estimated angular dependent ent anisotropic mechanical properties in BA-RA nature of Young’s modulus using the ODF obtained from XPFA and SAXS generally reproduces the em- plane. Detailed estimation and its empirical confirmation are now in progress. pirical moduli as a function of cutting angles. Young’s
Mineral orientation in bovine bone At any rate, these results indicate that in considering an anisotropic mechanical properties of bone, a nonlongitudinal (off-bone axial) orientation of bone mineral is one of many important factors. This means that bone is multi-directionally reinforced material as well as in the direction of bone axis. We believe that these results are sufficiently encouraging to place reliance upon the approach of estimating mechanical properties of bone on the basis of empirical structural information, in particular orientational distribution function of bone mineral.
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