**5.2. Change in the relaxation modulus of cortical bone by the change in the mineral fraction**

118 Viscoelasticity – From Theory to Biological Applications

sample code

sample size

relaxation modulus curve.

E0

(GPa) A1 <sup>τ</sup>1 (sec) <sup>β</sup> <sup>τ</sup><sup>2</sup>

(Iyo et al., J. Biomechanics 37, 1433-1437 (2004). With permission.)

P 14.2 1.19 13.0

N 11.6 1.28 10.3

(Iyo et al., J. Biomechanics 37, 1433-1437 (2004). With permission.)

processes and anisotropic parameters.

mechanically isotropic.

bone.

In the figure, lines represent the relaxation modulus of the KWW1 process, *E*1(*t*), and that of the KWW2 process, *E*2(*t*), for P and N specimens decomposed from the data according to eq. 18 using parameters of the best fit results listed in Table 2. Despite a difference in structural anisotropy of the specimens, the KWW1 relaxation process of a P specimen is indistinguishable from that of an N specimen at this magnification. In order to quantify the anisotropic mechanical properties of cortical bone, anisotropy ratio (AR) has been defined as the ratio of Young's modulus of bone in the direction parallel to the BA, *E*P, against that normal to the BA, *E*N, AR=*E*P/*E*N (Hasegawa *et al*., 1994). AR values estimated from our results using average values are listed in Table 3, where the AR value for *E*0 was listed as

AR0, and AR values for *E*1, AR1= *E*P1/*E*N1, and *E*2, AR2=*E*P2/*E*N2, were also estimated.

P 7 14.2 0.08 49 0.28 9.3 0.35 0.67 0.66 0.66 0.65 0.99989 0.0086 N 5 11.6 0.11 50 0.26 6.4 0.37 0.64 0.63 0.62 0.63 0.99986 0.0195

**Table 2.** Relaxation parameters according to the empirical equation (18) determined for the average

sample code E0 (GPa) E1(GPa) E2(GPa) AR0 AR1 AR2

**Table 3.** Decomposition of initial Young's modulus value into those of the KWW1 and KWW2

The *E*1 value for a P specimen was almost equal to that for an N specimen (p>0.6, ANOVA), and AR1 (=0.93) was close to 1, indicating that relaxation Young's modulus in the KWW1 process was insensitive to anisotropic morphology of bone. An elementary process of KWW1 relaxation processes was thought to be attributed to a component of bone that was

AR2 (=1.26) for the KWW2 process was similar to that of AR0 (=1.22) for the whole bone, indicating that an elementary process of the KWW2 relaxation process originates from a component causing the anisotropy of the whole bone. The difference between the whole relaxation Young's modulus value of a P sample from that of an N sample is represented by the difference in the respective KWW2 relaxation modulus values. The relaxation time for the KWW2 process, *τ*2, for P-specimen was larger than that for N-specimen. Values of for P and N specimens were similar but larger than *β* values. This indicates that the KWW2 process is attributable to a mode that is governed by the structural anisotropy in

Standard Error (GPa)

1.22 0.93 1.26

(X106 sec) γ 10 sec 102 sec 103 sec 104 sec R2 s

It has been regarded that the stiffness of bone is originated from minerals because modulus value of HAp minerals is almost 100 times larger than that of collagen. In the application as artificial bone materials, materials are required to have bio-compatibility, resistance to corrosion, adequate fracture toughness and fatigue strength. As for the bio-mechanicalcompatibility, in order to obtain the matching in modulus, it is possible to fabricate a composite material of stiff materials with pliant matrix. Changing the stiff component, we will be able to have materials with similar modulus as bone. At the same time, with the change in stiff component, reinforcement state of the matrix can be changed. This means the viscoelastic properties of the matrix changes with the stiff component. We aimed to investigate the viscoelastic properties of bone with changing mineral content (Sasaki & Yoshikawa, 1993). Demineralization of bone specimens was performed in 0.5 M EDTA, pH8.0 at 4oC. Mineral fraction was determined by weighing EDTA treated bone specimens. Fig. 19 shows the relaxation moduli for bovine femoral cortical bone specimens of five different mineral contents. The set of relaxation modulus curves appear to be different parts

**Figure 19.** Relaxation modulus curves for bone specimens with various mineral contents; B *φ*M =0.41 (volume fraction), B1 *φ*M =0.35, B2 *φ*M =0.33, B3 *φ*M =0.24, and BC =0 (bone collagen). (Sasaki et al., J. Biomechanics 26, 77-83 (1993). With permission.)

of a single large master curve, suggesting the reduced variable method, that is, the timemineral content superposition principle could be applicable. Fig. 20 shows the synthetic master curve constructed for bone specimens with different mineral contents. The synthetic curve looks smooth and the scatter of the data points is small. Fig. 21 shows the vertical shift factor, *b*m, plotted against the mineral content. The filled circles are taken from the mineral content dependence of the elastic modulus of bone after Katz (1971). Mineral content dependence of *b*m accords well with that of elastic modulus itself. This fact indicates that the superposition procedure was carried out correctly. Then, the result indicates the timemineral content superposition principle. A polymer-filler system has been considered to have the same reinforcing mechanism as the model discussed. But in the usual polymerfiller system, the time-filler-fraction superposition principle does not hold. The size of the commercially available filler is at least of the order of a few μm. By the analysis of the horizontal shift factor, the reinforcing effect depends on the filler-matrix surface area, not on the filler size. The mineral particle in bone, where the time-mineral content reduction was concluded to be applicable, has been recognized to be of the size of a few hundred Å at most. The reason why the time-filler fraction superposition principle does not hold in the polymer-filler system is deduced to be related to the very large filler size compared with the mineral particles in bone, as well as an adhesive weakness between filler and matrix. This fact leads to the suggestion that, in order to improve the relaxation properties of mineralresin composite as artificial bone, the mineral size should be reduced to, say, submicron level.

Viscoelastic Properties of Biological Materials 121

**Figure 21.** Vertical shift factor as a function of mineral content *φ*M (volume fraction) (○). Mineral content dependence of elastic modulus is also plotted (●). (Sasaki et al., J. Biomechanics 26, 77-83 (1993).

Buschmann, M. D., Gluzband, Y. A., Grodzinsky, A. J., Hunziker, E. B. (1995). Mechanical compression modulates matrix biosynthesis in chondrocyte/agarose culture. *Journal of* 

Currey, JD. (1965) Anelasticity in bone and echinoderm skeletons. *Journal of Experimental* 

Currey, J. D. (1964) Three analogies to explain the mechanical properties of bone. *Biorheology*

Fanter, G., Hassenkam, T., Kindt, J., Weaver, J. C., Birkedal, H., Pechenik, L., Cutroni, J. A., Cidade, G. A. G., Stucky, G. D., Morse, D. E., Hansma, P. K. (2005) Sacrificial bonds and hidden length dissipate energy as mineralized fibrils separate during bone fracture.

Foord, S. A., Atkins, E. D. T. (1989). New X-ray diffraction results from agarose: extended single helix structure and implications for gelation mechanism. *Biopolymers* 28(8), 1345-1365. Hanazaki, Y., Ito, D., Furusawa, K., Fukui, A., Sasaki, N. (2012) Change in the viscoelastic properties of agarose gel by Hap precipitation by osteoblasts cultured in agarose gel

Ferry, J. D. (1980) *Viscoelastic Properties of Polymers.* Wiley & Sons, New York.

matrix. *Journal of Biorhelogy*, in press, DOI 10.1007/s12573-011-0043-2.

*Department of Interdisciplinary Sciences, Hokkaido University, Japan* 

With permission.)

**Author details** 

**6. References** 

2, 1-10.

*Faculty of Advanced Life Science,* 

*Cell Science* 108, 1497-1508.

*Nature Materials* 4, 612-615.

*Biology* 43, 279-292.

Naoki Sasaki

**Figure 20.** A master curve constructed by superimposing the relaxation modulus curves in Fig. 19. For the successful superposition, both the vertical and horizontal shifts were needed. (Sasaki et al., J. Biomechanics 26, 77-83 (1993). With permission.)

**Figure 21.** Vertical shift factor as a function of mineral content *φ*M (volume fraction) (○). Mineral content dependence of elastic modulus is also plotted (●). (Sasaki et al., J. Biomechanics 26, 77-83 (1993). With permission.)
