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Mechanical properties of hydroxyapatite-reinforced gelatin as a model system of bone
Mecha~c~ propertiesof hy~o~apatite-reinforcedgelatinas a modelsystem of bone Departmenr of Applied Mater&is Science, Muroran lnsfifufe of Tecbnoiogy, Mororan 050, Japan /Received 7 October 1987; accepfed 6 Februaty 1988)
The elastic Young’s modulus of hydroxyapatite-reinforced gelatin as a mechanical model system of bone was measured as a function of the volume fraction of hydroxyapatite. &. Initially, the Voung’s modulus gradually increased with an increase in q& and then increased rapidly in the vicinity of #,, - 0.2. The en dependence of the Young’s modulus was analysed by means of the theory of composite materials. It was found that with the increase in Cpithe initial uniform stress defo~etion mode of the sample changed to the uniform strain deformation mode. The non-linear character in $J~dependence of the Young’s modulus of this system was considered to reproduce well the novel behaviour of the mechanical properties of bone as a function of the mineral fraction. The situation wes considered to be similar to a percolation problem. A preliminary analysis revealed that the critical exponent about the viscosity of the system accorded with the theoretically expected value. The result may present the evidence that the discontinuous point in mechanical properties of bone would be originated from an inte~ction such as a percolation of mineral particles on a matrix protein. Keywords: Bone, hydroxyapa
fife, composiie
materials, mechanical properties
The major constituents of bone are a rigid apatite-like mineral and pliant collagen. Bone has been regarded for some time as a composite two-phase structure consisting of components with different mechanical properties. In order to know the composite structure of the two components, mechanical properties of bone have been investigated as a function of mineral content, &,,. It has been found that the mechanical properties change with 41, and there is a discontinuous point in the $,-dependent variation of the mechanical properties at certain (p,.r,‘-3.Hypotheses about the origin of the discontinuous point have been proposed. Lees2, by measuring the ultrasonic velocity in hard tissues of several mineral contents, proposed that in the vicinity of the discontinuous point, the number of collagenmineral network per unit volume may increase remarkably. In our previous work, we proposed the percolating structure of bone mineral on to collagen matrix (mineral-mineral interaction)3. It is not known if the discontinuous point of the mechanical properties, as a function of #,,, is explained only by the process of the mineral-mineral or mineral-collagen interactions in bone, or if there are any other mechanisms involving higher level structures of bone, such as osteons. If, according to these hypotheses, the discontinuous point is caused only by the mineral-mineral and mineralcollagen interactions, in particular the interfacial interaction Correspondence to Dr N. Sasaki. 0 1989
between apatite particle and polypeptide chain of collagen, the same discontinuous point should appear when the mineral-collagen system is reconstructed. We examined the mechanical properties of hydroxyapatite (HAP) dispersed gelatin as a reconstructed bone and measured the Young’s modulus as the mechanical properties of the HAP-gelatin system. Observed dependence of the mechanical properties of the system on the volume fraction of HAP, @,, was discussed by means of the mechanical composite structure of mineral and apatite. The system examined here is important for biomedical applications since the material examined will have an excellent bi~ompatibili~ m~hanically as well as bi~hemicalfy4.
MATERIALS
AND
METHODS
Hydroxyapatite (HAP) powder was prepared by the reaction of Ca(OH), and H3P04 in aqueous solution at pH 8.0, After a few days maturation, the resultant substance was diffractometri~lly assigned to be HAP. The apatite particle size was also assessed by an SK Laser Micron Sizer. Figure 1 shows the distribution of apatite particle size. The distribution curve generally confirms a Gaussian function and gives a value of 3.5pm as a median. This value of median was slightly controlled bythe ageing period of the aqueous suspension. Gelatin powder (distributed by Merck Co. Ltd). was
Butterworth b Co (Publishers) Ltd. 0142-9612/89/020129-04$03.00 Biomafer&s
1989, Vat 10 March
129
Bone model: N. Sasaki et al.
50 u 40 ijy $ 30 e c rc -.
employed for the introduction of networks has been known to be produced only among collagen chains. It is not certain at this point if the networks so introduced are related to those proposed by Lees’ et a/.3 model. In Figure 2, dashed lines 1 and 2 represent the most elementary equations determining the Young’s modulus for two-phase composite system; the linear combination of Young’s moduli of both components (Voigt model):
0
20 = h
E = Q&H + (1 - @‘h)EG
,OZ
0
5
10
Particle
15
Size
0
20
(pm)
Figure 1 Particle size distribution measured by Laser Micron Sizer, where HAP particles are dispersed in water.
dissolved in the aqueous suspension of apatite powder. Aqueous solution of gelatin with HAP dispersed was cast on a cooled Teflon@ plate for fast gelatin and preventing the sedimentation of HAP particles. The gelatin gel was dried in the air at r.t. and then submitted to vacuum drying at r.t. In order to make a cross-link in gelatin film, obtained film was incubated in air at 80°C for 24 h. An isotropic dispersion of HAP in the gelatin film in the direction of thickness was assessed qualitatively by optical microscope observation. Young’s moduli of the gelatin-apatite sample were measured in terms of the vibrating reed method made in an atmosphere of r.h. of 84% at 20°C.
(l/E) = k#~,dEd + [(I - @‘h)hl
130
Biomaterials
1989. Vol 10 March
(4
respectively. In these equations, EH and EG are, respectively, the Young’s moduli of HAP and gelatin. The Voigt and Reuss models represent arrangements of components in a composite in such a manner that deformation occurs predominantly under uniform strain and uniform stress, respectively. The Voigt model of uniform strain and Reuss model of uniform stress are generally known to establish, respectively, the upper and the lower boundaries for the elastic constants of a composite. It is clear that the oh dependence of E obtained here for cross-linked samples is not expressed by these extreme equations. In the system, the mechanical composite structure of HAP and gelatin will
1
I
I
I
15
I
cl
I
I
RESULTS AND DISCUSSION Because of the macroscopic inhomogeneous mixing of HAP powder in gelatin matrix, good samples of &, > 0.25 were not obtained in this work, though a new technique of homogeneous mixing is now in development in our laboratory. In Figure 2, Young’s modulus, E, is plotted as a function of the volume fraction of HAP&, for uncross-linked and cross-linked samples measured at r.t. and r.h. of 84%. It is clear that E for cross-linked samples are larger than those for uncross-linked samples. The observed maximum value of Efor this system, E - 6 GPa, is compared with that of human bone in the transverse direction5, though oh of the sample is far smaller than that of bone. In this range of &, in crosslinked samples, it is found that as the solid line Efor small @,, increases gradually and then rapidly in the vicinity of oh of 0.2. Such a non-linear dependence of E on @, has been also observed for fluorapatite (FAP) -epoxy resine composite system by Lees eta/.’ as well as for bone. On the other hand, our results for cross-linked samples and those of Lees, contrast with the results of corresponding measurements for HAP-reinforced polyethylene4, bone-particle impregnated polymethyl methacrylate (PMMA)‘, crystabolite (CB)-epoxy and CB-PMMA systems7, where E of the system almost linearly related to HAP, bone particle and CB fraction, respectively. It is also found that uncross-linked samples belong to the latter group of linear relation in E and &,. The introduction of networks in to the gelatin matrix seems to be an important factor for the reconstituted system to manifest the particular mechanical properties. The procedure
(1)
and inverse linear combination (Reuss model):
1
2 0
10
I
’
I
w
I I I
5
1 0
d
2 /’
-4 ----
-+--
I
I
I
0.1
0.2
0.3
I
0.4
I
0.5
8, Figure 2 Young’s modulus of HAP-gelatin composite for cross-linked system (0)and uncross-linked system (@)plotted against &,. (0) represents the Young’s modulus of bovine femur. Dashed lines 1 and 2 represent the Voigt and Reuss models, respectively.
Bone model: N. Sasaki et al.
1.0
be more complicated than these equations indicate. In order to express the case, a more sophisticated model has to be employed. One such model had been presented by Hirsch’ for expressing the elastic properties of concrete as a function of a constitution of cement paste and aggregates. It is the weighted combination of the Voigt and Reuss model: (l/E)
= x/f&$, + If1 -
+ (1 -#,)&I #hf/EGl)
0.8
+ (1 - X)&W~H)
0.6
(3)
wherex is the mixing weight of thevoigt model and takes the value between 1 (Voigt model) and 0 (Reuss model). This model had been applied for bone samples of several mineral fractions by Piekarskig. He repot-ted that the calculated @,, dependence of the moduli fitted much more closely to the observed values with x = 0.925, which suggested that the collagen and mineral are working at almost equal strain, in other words, arranging parallel. This result accorded with the result obtained by neutron diffraction investigation of bone”. In the gelatin-apatite system investigated here, the morphol~y of the two components is quite similar to those in the system on which the model Equation (3) had been established*; the structure of both systems is regarded as a pliant continuum impregnated with hard particles. It may therefore be reasonable to use this model for mechanical analysis of our system. In our result, however, rapid increase in E in the vicinity of &, - 0.2 suggests that x would vary with #,,. The value of x was estimated for each cross-linked sample. In Figure 3, estimated x values for cross-linked samples are plotted against (p,. It is clear that the initially small x value increases linearly with &, and will be saturated to be unity at a certain &, between 0.3 and 0.4, as long as the linear relation was supposed to hold at these &, values. The small values of x for samples of small &, indicates that a uniform stress deformation predominantly occurs, where the applied tensile force would be supported generally by the gelatin phase. The increase inx with &, also implies that the domain of a parallel arrangement of the mineral phase and the gelatin phase to a tensile force, which guarantees the uniform strain deformation, increases with &,. In that case, for a test piece of samples at any cutting direction, parallel portions in mineral and gelatin arrangement to a force of any direction would increase with (Ph.The saturation of x = 1 at &, between 0.3 and 0.4 expected here means that in those $,, regions the sample would be deformed in uniform strain condition where the applied tensile force would be supported mainly by the mineral phase (Figure 4). Mechanical properties of this system just below the saturation point will be qualitatively different from those just above the saturation point. The value of the saturation point accords with the discontinuous point of bone, by which we considered that our system may reproduce well the mineral fraction dependence of the mechanical properties of bone though studied for HAP volume fractions below that of bone. It is noted that such a non-linear behaviour in &, dependence of E has been observed for the FAP-epoxy system, and not for the HAP-PE system, bone panicle-impregnated PMMA system, CB-epoxy system, CB-PMMA system and so on. Some thing like an affinity between mineral and matrix may cause such an special aggregation of mineral particles in the matrix as indicated by the &, dependence of x. Changes in an arrangement of mineral particles indicated by the linear dependence of x on (p, suggest that the discontinuous point expected in &, dependence of E for HAP-gelatin system would not be attributed only to the
x 0.4
0.2
0.0 0.t3 L
I
I
I
I
1
0.4
0.2
0.6
P(h Figure 3 q%,,dependence of Voigt mode fraction, x in Hirsch’s e~ua?ion applied to cross-linked HAP-gelatin system. fCl] represents the x value for bone obtained by Piekarski.
I
’
’
1
l
100
10
1
0.0
0.5 Fbh
Figure 4 Log E for cross-linked sample is p&ted against @,. A come co~oect~og data points represents the linear increase of x with & in Hirsh’s equation. f0) represents the Young’s modulus of bovine femur.
Biomaterials
1989, Vol 10 March
13t
Bone model: N. Sasaki et al.
increasing cross-link density in the gelatin matrix, though the cross-links introduced into gelatin may be one of the most important factors responsible for the appearance of the discontinuous point. The situation for cross-linked samples discussed above has a characteristic reminiscent of the percolation structural model for the composite structure of mineral and protein in bone3,“. Indicated composite structure by the saturation inx in thevicinityof @,,- 0.4 may have a topological analogy with the percolating system near the percolation threshold”. Standard scaling arguments allow us to show asymptotically’*, 13: ‘I cc (4; - &l)-S f =
(@h -
@i)’
ech@h -
@h x
@“h e
1 1
ACKNOWLEDGEMENTS The authors would like to thank Dr M. Tokita of Hokkaido University for his helpful discussions and suggestions and M r T. Hayasaka of Nittetsu-Cement Co. Ltd for his cooperation.
REFERENCES
(4) (5)
where 17represents the viscosity of the system, s and t are critical exponents and @F;is a percolation threshold. In three dimensions, numerical and renormalization group calculations give s = 0.7-0.9 and t = 1.7-l .914,15, which agree quite well with experimental data16. For an ideal continuous random mixture,& = 0.15 in threedimensions”. However, & can be affected by extrinsic factors such as the state of dispersion, wetting characteristics and particle size, shape, distribution and orientation’* . Therefore 4: is best determined empirically. In our case, q may be related to the plastic deformation or energy dissipation in the system, which are specified by an imaginary part of the Young’s modulus, E”. There is a well known relation between q and F: ‘J’(w) = F(o)/w
be attributed to such an interaction at the level between two components as the percolation transition discussed above.
4
5
6
7 8
(6) 9
wherew is the measuring frequency. Then,s is approximately determined as the slope of the log (F/w) versus log (4: - oh) plot. Our preliminary analysis using an apparent threshold &, - 0.4 presents the value of s - 0.6, which accords well with the s value having been calculated in the literature. These results underscore the percolation transition nature and signifies the switching of the roles (fillers versus host) for the two components, although we do not have enough compositions at @h> 0.3 to pinpoint an exact value of the threshold and determine exactlythe critical exponents. Further work is contemplated. We may conclude that, as a model system of bone, the cross-linked gelatin-HAP system examined here qualitatively reproduces the novel behaviour in the mechanical properties of bone although it has only been investigated for low volume fractions of HAP. The conclusion may indicate that the discontinuous point which has been observed in mineral fraction dependence of mechanical properties of bone would
132
Biometerials
1989, Vol 10 March
10
11 12 13 14 15 16
17 18
Wainwrite, S.A., Biggs, W.D., Currey, J.D. and Gosline, J.M., Mechanical Design in Organisms Edward Arnold, 1976, Chap. 5 Lees, S. and Davidson, C.L., The role of collagen in the elastic properties of calcified tissues, J. Biomech. 1977, 10, 473-486 Sasaki, N., Yamamura. H. and Matsushima, N.. Is there a relation between bone strength and percolation? J. Thor. Biol. 1986. 122. 25-31 Bonfield. W., Grynpas, M.D., Tully, A.E., Bowman, J. and Abram, J., Hydroxyapatite reinforced polyethylene - a mechanically compatible implant material for bone replacement, Biomaterials 1981, 2, 185-l 86 Knets, I.V., Krauya, U.E. and Vilks, Yu.K., Acoustic emission in human bone tissue subjected to longitudinal extension, Mechanika Polimerov 1975.11.685-690 Park, H.C., Liu, Y.K. and Lakes, R.S., The mechanical properties of bone-particle impregnated PMMA, J. Biomed. Engng. 1986, 108, 141-148 Lees, S. and Davidson, C.L., Ultrasonic measurement of some mineral filled plastics, IEEE Trans. Sonics Ultrasonics 1977, SU-24.222-225 Hirsch, J., Modulus of elasticity of concrete affected by elastic moduli of cement paste matrix and aggregate, Proc. Am. Concr. inst. 1962, 59,427-45 1 Piekarski, K., Analysis of bone as a composite material, lnt. J. Engng. Sci. 1973, 11, 557-565 Bacon, G.E., Bacon, P.J. and Griffith, R.K., Stress distribution in the scapula studied by neutron diffraction, Proc. Roy. Sot. Lond. B 1979, 204. 355-362 Stauffer, D., Scaling theory of percolation clusters, Phys. Rep. 1979, 54, l-74 Stanley, H.E., Introduction to Phase Transitions and Critical Phenomena Clarendon Press, Oxford (197 1). Chap. 3 de Gennes, P.G., Scaling Concepts in Polymer Physics Cornell University Press, Ithaca, New York (1979). Chap. 5 Straley, J.P., Critical exponents for the conductivity of random resistor lattices, Phys. Rev., B 1977. 15, 5733-5737 Kirkpatric, S., Percolation thresholds In lsing magnets and conducting mixtures, Phys. Rev., B 1977, 15, 1533-l 538 Hsu, W.Y., Barkley, JR. and Meakin, P., Ion percolation and insulatorto-conductor transition in nafion perfluorosulfonic acid, Macromolecules 1980. 13. 198-200 Powell, M.J.. Site percolation in randomly packed spheres, Phys. Rev., B 1979,20,4194-4198 Hsu, W.Y., Giri, M.R. and Ikeda, R.M., Percolation transition and elastic properties of block copolymers, Macromolecules 1982, 15, 1210-1212
The elastic Young’s modulus of hydroxyapatite-reinforced gelatin as a mechanical model system of bone was measured as a function of the volume fraction of hydroxyapatite. &. Initially, the Voung’s modulus gradually increased with an increase in q& and then increased rapidly in the vicinity of #,, - 0.2. The en dependence of the Young’s modulus was analysed by means of the theory of composite materials. It was found that with the increase in Cpithe initial uniform stress defo~etion mode of the sample changed to the uniform strain deformation mode. The non-linear character in $J~dependence of the Young’s modulus of this system was considered to reproduce well the novel behaviour of the mechanical properties of bone as a function of the mineral fraction. The situation wes considered to be similar to a percolation problem. A preliminary analysis revealed that the critical exponent about the viscosity of the system accorded with the theoretically expected value. The result may present the evidence that the discontinuous point in mechanical properties of bone would be originated from an inte~ction such as a percolation of mineral particles on a matrix protein. Keywords: Bone, hydroxyapa
fife, composiie
materials, mechanical properties
The major constituents of bone are a rigid apatite-like mineral and pliant collagen. Bone has been regarded for some time as a composite two-phase structure consisting of components with different mechanical properties. In order to know the composite structure of the two components, mechanical properties of bone have been investigated as a function of mineral content, &,,. It has been found that the mechanical properties change with 41, and there is a discontinuous point in the $,-dependent variation of the mechanical properties at certain (p,.r,‘-3.Hypotheses about the origin of the discontinuous point have been proposed. Lees2, by measuring the ultrasonic velocity in hard tissues of several mineral contents, proposed that in the vicinity of the discontinuous point, the number of collagenmineral network per unit volume may increase remarkably. In our previous work, we proposed the percolating structure of bone mineral on to collagen matrix (mineral-mineral interaction)3. It is not known if the discontinuous point of the mechanical properties, as a function of #,,, is explained only by the process of the mineral-mineral or mineral-collagen interactions in bone, or if there are any other mechanisms involving higher level structures of bone, such as osteons. If, according to these hypotheses, the discontinuous point is caused only by the mineral-mineral and mineralcollagen interactions, in particular the interfacial interaction Correspondence to Dr N. Sasaki. 0 1989
between apatite particle and polypeptide chain of collagen, the same discontinuous point should appear when the mineral-collagen system is reconstructed. We examined the mechanical properties of hydroxyapatite (HAP) dispersed gelatin as a reconstructed bone and measured the Young’s modulus as the mechanical properties of the HAP-gelatin system. Observed dependence of the mechanical properties of the system on the volume fraction of HAP, @,, was discussed by means of the mechanical composite structure of mineral and apatite. The system examined here is important for biomedical applications since the material examined will have an excellent bi~ompatibili~ m~hanically as well as bi~hemicalfy4.
MATERIALS
AND
METHODS
Hydroxyapatite (HAP) powder was prepared by the reaction of Ca(OH), and H3P04 in aqueous solution at pH 8.0, After a few days maturation, the resultant substance was diffractometri~lly assigned to be HAP. The apatite particle size was also assessed by an SK Laser Micron Sizer. Figure 1 shows the distribution of apatite particle size. The distribution curve generally confirms a Gaussian function and gives a value of 3.5pm as a median. This value of median was slightly controlled bythe ageing period of the aqueous suspension. Gelatin powder (distributed by Merck Co. Ltd). was
Butterworth b Co (Publishers) Ltd. 0142-9612/89/020129-04$03.00 Biomafer&s
1989, Vat 10 March
129
Bone model: N. Sasaki et al.
50 u 40 ijy $ 30 e c rc -.
employed for the introduction of networks has been known to be produced only among collagen chains. It is not certain at this point if the networks so introduced are related to those proposed by Lees’ et a/.3 model. In Figure 2, dashed lines 1 and 2 represent the most elementary equations determining the Young’s modulus for two-phase composite system; the linear combination of Young’s moduli of both components (Voigt model):
0
20 = h
E = Q&H + (1 - @‘h)EG
,OZ
0
5
10
Particle
15
Size
0
20
(pm)
Figure 1 Particle size distribution measured by Laser Micron Sizer, where HAP particles are dispersed in water.
dissolved in the aqueous suspension of apatite powder. Aqueous solution of gelatin with HAP dispersed was cast on a cooled Teflon@ plate for fast gelatin and preventing the sedimentation of HAP particles. The gelatin gel was dried in the air at r.t. and then submitted to vacuum drying at r.t. In order to make a cross-link in gelatin film, obtained film was incubated in air at 80°C for 24 h. An isotropic dispersion of HAP in the gelatin film in the direction of thickness was assessed qualitatively by optical microscope observation. Young’s moduli of the gelatin-apatite sample were measured in terms of the vibrating reed method made in an atmosphere of r.h. of 84% at 20°C.
(l/E) = k#~,dEd + [(I - @‘h)hl
130
Biomaterials
1989. Vol 10 March
(4
respectively. In these equations, EH and EG are, respectively, the Young’s moduli of HAP and gelatin. The Voigt and Reuss models represent arrangements of components in a composite in such a manner that deformation occurs predominantly under uniform strain and uniform stress, respectively. The Voigt model of uniform strain and Reuss model of uniform stress are generally known to establish, respectively, the upper and the lower boundaries for the elastic constants of a composite. It is clear that the oh dependence of E obtained here for cross-linked samples is not expressed by these extreme equations. In the system, the mechanical composite structure of HAP and gelatin will
1
I
I
I
15
I
cl
I
I
RESULTS AND DISCUSSION Because of the macroscopic inhomogeneous mixing of HAP powder in gelatin matrix, good samples of &, > 0.25 were not obtained in this work, though a new technique of homogeneous mixing is now in development in our laboratory. In Figure 2, Young’s modulus, E, is plotted as a function of the volume fraction of HAP&, for uncross-linked and cross-linked samples measured at r.t. and r.h. of 84%. It is clear that E for cross-linked samples are larger than those for uncross-linked samples. The observed maximum value of Efor this system, E - 6 GPa, is compared with that of human bone in the transverse direction5, though oh of the sample is far smaller than that of bone. In this range of &, in crosslinked samples, it is found that as the solid line Efor small @,, increases gradually and then rapidly in the vicinity of oh of 0.2. Such a non-linear dependence of E on @, has been also observed for fluorapatite (FAP) -epoxy resine composite system by Lees eta/.’ as well as for bone. On the other hand, our results for cross-linked samples and those of Lees, contrast with the results of corresponding measurements for HAP-reinforced polyethylene4, bone-particle impregnated polymethyl methacrylate (PMMA)‘, crystabolite (CB)-epoxy and CB-PMMA systems7, where E of the system almost linearly related to HAP, bone particle and CB fraction, respectively. It is also found that uncross-linked samples belong to the latter group of linear relation in E and &,. The introduction of networks in to the gelatin matrix seems to be an important factor for the reconstituted system to manifest the particular mechanical properties. The procedure
(1)
and inverse linear combination (Reuss model):
1
2 0
10
I
’
I
w
I I I
5
1 0
d
2 /’
-4 ----
-+--
I
I
I
0.1
0.2
0.3
I
0.4
I
0.5
8, Figure 2 Young’s modulus of HAP-gelatin composite for cross-linked system (0)and uncross-linked system (@)plotted against &,. (0) represents the Young’s modulus of bovine femur. Dashed lines 1 and 2 represent the Voigt and Reuss models, respectively.
Bone model: N. Sasaki et al.
1.0
be more complicated than these equations indicate. In order to express the case, a more sophisticated model has to be employed. One such model had been presented by Hirsch’ for expressing the elastic properties of concrete as a function of a constitution of cement paste and aggregates. It is the weighted combination of the Voigt and Reuss model: (l/E)
= x/f&$, + If1 -
+ (1 -#,)&I #hf/EGl)
0.8
+ (1 - X)&W~H)
0.6
(3)
wherex is the mixing weight of thevoigt model and takes the value between 1 (Voigt model) and 0 (Reuss model). This model had been applied for bone samples of several mineral fractions by Piekarskig. He repot-ted that the calculated @,, dependence of the moduli fitted much more closely to the observed values with x = 0.925, which suggested that the collagen and mineral are working at almost equal strain, in other words, arranging parallel. This result accorded with the result obtained by neutron diffraction investigation of bone”. In the gelatin-apatite system investigated here, the morphol~y of the two components is quite similar to those in the system on which the model Equation (3) had been established*; the structure of both systems is regarded as a pliant continuum impregnated with hard particles. It may therefore be reasonable to use this model for mechanical analysis of our system. In our result, however, rapid increase in E in the vicinity of &, - 0.2 suggests that x would vary with #,,. The value of x was estimated for each cross-linked sample. In Figure 3, estimated x values for cross-linked samples are plotted against (p,. It is clear that the initially small x value increases linearly with &, and will be saturated to be unity at a certain &, between 0.3 and 0.4, as long as the linear relation was supposed to hold at these &, values. The small values of x for samples of small &, indicates that a uniform stress deformation predominantly occurs, where the applied tensile force would be supported generally by the gelatin phase. The increase inx with &, also implies that the domain of a parallel arrangement of the mineral phase and the gelatin phase to a tensile force, which guarantees the uniform strain deformation, increases with &,. In that case, for a test piece of samples at any cutting direction, parallel portions in mineral and gelatin arrangement to a force of any direction would increase with (Ph.The saturation of x = 1 at &, between 0.3 and 0.4 expected here means that in those $,, regions the sample would be deformed in uniform strain condition where the applied tensile force would be supported mainly by the mineral phase (Figure 4). Mechanical properties of this system just below the saturation point will be qualitatively different from those just above the saturation point. The value of the saturation point accords with the discontinuous point of bone, by which we considered that our system may reproduce well the mineral fraction dependence of the mechanical properties of bone though studied for HAP volume fractions below that of bone. It is noted that such a non-linear behaviour in &, dependence of E has been observed for the FAP-epoxy system, and not for the HAP-PE system, bone panicle-impregnated PMMA system, CB-epoxy system, CB-PMMA system and so on. Some thing like an affinity between mineral and matrix may cause such an special aggregation of mineral particles in the matrix as indicated by the &, dependence of x. Changes in an arrangement of mineral particles indicated by the linear dependence of x on (p, suggest that the discontinuous point expected in &, dependence of E for HAP-gelatin system would not be attributed only to the
x 0.4
0.2
0.0 0.t3 L
I
I
I
I
1
0.4
0.2
0.6
P(h Figure 3 q%,,dependence of Voigt mode fraction, x in Hirsch’s e~ua?ion applied to cross-linked HAP-gelatin system. fCl] represents the x value for bone obtained by Piekarski.
I
’
’
1
l
100
10
1
0.0
0.5 Fbh
Figure 4 Log E for cross-linked sample is p&ted against @,. A come co~oect~og data points represents the linear increase of x with & in Hirsh’s equation. f0) represents the Young’s modulus of bovine femur.
Biomaterials
1989, Vol 10 March
13t
Bone model: N. Sasaki et al.
increasing cross-link density in the gelatin matrix, though the cross-links introduced into gelatin may be one of the most important factors responsible for the appearance of the discontinuous point. The situation for cross-linked samples discussed above has a characteristic reminiscent of the percolation structural model for the composite structure of mineral and protein in bone3,“. Indicated composite structure by the saturation inx in thevicinityof @,,- 0.4 may have a topological analogy with the percolating system near the percolation threshold”. Standard scaling arguments allow us to show asymptotically’*, 13: ‘I cc (4; - &l)-S f =
(@h -
@i)’
ech@h -
@h x
@“h e
1 1
ACKNOWLEDGEMENTS The authors would like to thank Dr M. Tokita of Hokkaido University for his helpful discussions and suggestions and M r T. Hayasaka of Nittetsu-Cement Co. Ltd for his cooperation.
REFERENCES
(4) (5)
where 17represents the viscosity of the system, s and t are critical exponents and @F;is a percolation threshold. In three dimensions, numerical and renormalization group calculations give s = 0.7-0.9 and t = 1.7-l .914,15, which agree quite well with experimental data16. For an ideal continuous random mixture,& = 0.15 in threedimensions”. However, & can be affected by extrinsic factors such as the state of dispersion, wetting characteristics and particle size, shape, distribution and orientation’* . Therefore 4: is best determined empirically. In our case, q may be related to the plastic deformation or energy dissipation in the system, which are specified by an imaginary part of the Young’s modulus, E”. There is a well known relation between q and F: ‘J’(w) = F(o)/w
be attributed to such an interaction at the level between two components as the percolation transition discussed above.
4
5
6
7 8
(6) 9
wherew is the measuring frequency. Then,s is approximately determined as the slope of the log (F/w) versus log (4: - oh) plot. Our preliminary analysis using an apparent threshold &, - 0.4 presents the value of s - 0.6, which accords well with the s value having been calculated in the literature. These results underscore the percolation transition nature and signifies the switching of the roles (fillers versus host) for the two components, although we do not have enough compositions at @h> 0.3 to pinpoint an exact value of the threshold and determine exactlythe critical exponents. Further work is contemplated. We may conclude that, as a model system of bone, the cross-linked gelatin-HAP system examined here qualitatively reproduces the novel behaviour in the mechanical properties of bone although it has only been investigated for low volume fractions of HAP. The conclusion may indicate that the discontinuous point which has been observed in mineral fraction dependence of mechanical properties of bone would
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