**2.1. Introduction to elasticity and viscosity**

Elasticity is a material property that generates recovering force at an application of an external force to deform the material. When an external force is applied to a material and the material is in an equilibrium deformation, the external force is balanced by an inner force. The inner force is the recovering force. The recovering force divided by the cross sectional

area that the external force is working on is defined as stress, *σ*. Suppose the material is initially in a shape of rod of length *L*0 and cross-sectional area, *A*0. The force is applied to the length direction and the length after the deformation is *L.* The deformation is generally normalized as,

$$
\varepsilon = \frac{L - L\_0}{L\_0},
\tag{1}
$$

Viscoelastic Properties of Biological Materials 101

, also a materials constant of

. *<sup>d</sup><sup>γ</sup> <sup>σ</sup> dt* (5)

. *<sup>d</sup><sup>γ</sup> σ η dt* (6)

the flow velocity is zero. Every part of the fluid is sheared (Fig. 2). The shear is generated by an applied force causing the flow and an internal force against the former. The internal force is also transformed into a stress generated in the flow. Newton postulated the quantitative

Viscosity is a property of a fluid to resist the force for flow. The fluid described by the equation is classified as Newtonian. After stopping the flow, the fluid maintains its deformation strain at the time of stopping. There is a saying that water conforms to the shape

**Figure 2.** Shear strain in a fluid. A velocity gradient perpendicular to the bottom emerged.

When a step strain excitation is applied to an idealistically elastic material, the response of the material will be that shown in Fig. 3(a). This response can be easily expected from the Hookean constitutive equation. If the same strain excitation is applied to an idealistically viscous material, the response will be those as shown in the Fig. 3(b) and (c), which is also expectable from the Newtonian constitutive equation. Materials that exhibit viscoelastic properties can be considered as having both elastic and viscous components. Then, in the case of viscoelastic materials, the response is expected to have both characteristics of elasticity and viscosity. Schematic drawing of the stress response according to many experimental results is shown in Fig. 4. Here we define the stress relaxation as follows. Fig. 5 shows a step strain excitation and a stress response to it for a viscoelastic material. The instantaneous and equilibrium moduli, *m*g and *m*e, respectively, are material's constants. The

from the stand point of viscoelastic properties is to determine these material's constants and

(*t*) is a material's function. Phenomenological investigation of materials

relationship between the stress and the shear rate as,

The proportionality factor is defined as a viscosity coefficient,

a fluid. Then the constitutive relation for the viscosity is

of the vessel that contains it.

**2.2. Introduction to viscoelasticity** 

relaxation function

where is called the engineering strain or the Caushy strain. This is used for small deformations. For small strains, a Hookean relation,

$$\mathbf{Qx}\mathbf{x}\tag{2}$$

has been known to hold. The proportionality factor is defined as a modulus and the modulus is a material's constant. If the material was deformed by a tensile force, the modulus is defined as Young's modulus (Fig. 1 (a)). If the deformation was caused by a shear

**Figure 1.** Deformation modes and corresponding moduli. (a) Elongation and Young's modulus, and (b) shear deformation and the shear modulus.

shear force, the modulus is the shear modulus (Fig. 1(b)). In each case, the Hookean constitutive equations are written as,

$$
\sigma = E \cdot \varepsilon \quad \text{( $E$ : Young's modulus)}\tag{3}
$$

and

$$
\sigma = \mathbf{G} \cdot \boldsymbol{\gamma} \quad \text{( $\mathbf{G}$ : shear modulus)}\tag{4}
$$

where *γ* is the shear strain. After eliminating the external force, the recovering force and the deformation are completely diminished. Elasticity is a property of a material to resist the deformation by the external force.

Viscosity is a characteristic nature of a fluid. When a fluid is constantly flowing on an infinitively wide plain, the flow velocity is largest on the fluid surface. On the plain bottom the flow velocity is zero. Every part of the fluid is sheared (Fig. 2). The shear is generated by an applied force causing the flow and an internal force against the former. The internal force is also transformed into a stress generated in the flow. Newton postulated the quantitative relationship between the stress and the shear rate as,

$$
\sigma \propto \frac{d\underline{\nu}}{dt}.\tag{5}
$$

The proportionality factor is defined as a viscosity coefficient, , also a materials constant of a fluid. Then the constitutive relation for the viscosity is

$$
\sigma = \eta \frac{d\chi}{dt}.\tag{6}
$$

Viscosity is a property of a fluid to resist the force for flow. The fluid described by the equation is classified as Newtonian. After stopping the flow, the fluid maintains its deformation strain at the time of stopping. There is a saying that water conforms to the shape of the vessel that contains it.

**Figure 2.** Shear strain in a fluid. A velocity gradient perpendicular to the bottom emerged.

#### **2.2. Introduction to viscoelasticity**

100 Viscoelasticity – From Theory to Biological Applications

deformations. For small strains, a Hookean relation,

shear deformation and the shear modulus.

constitutive equations are written as,

deformation by the external force.

normalized as,

where

shear

and

area that the external force is working on is defined as stress, *σ*. Suppose the material is initially in a shape of rod of length *L*0 and cross-sectional area, *A*0. The force is applied to the length direction and the length after the deformation is *L.* The deformation is generally

> 0 0 , *L L <sup>ε</sup> <sup>L</sup>*

has been known to hold. The proportionality factor is defined as a modulus and the modulus is a material's constant. If the material was deformed by a tensile force, the modulus is defined as Young's modulus (Fig. 1 (a)). If the deformation was caused by a

**Figure 1.** Deformation modes and corresponding moduli. (a) Elongation and Young's modulus, and (b)

shear force, the modulus is the shear modulus (Fig. 1(b)). In each case, the Hookean

where *γ* is the shear strain. After eliminating the external force, the recovering force and the deformation are completely diminished. Elasticity is a property of a material to resist the

Viscosity is a characteristic nature of a fluid. When a fluid is constantly flowing on an infinitively wide plain, the flow velocity is largest on the fluid surface. On the plain bottom

*σ E ε* : Young's modulus *E* (3)

*σ G γ* : shear modulus *G* (4)

is called the engineering strain or the Caushy strain. This is used for small

(1)

*σ ε* (2)

When a step strain excitation is applied to an idealistically elastic material, the response of the material will be that shown in Fig. 3(a). This response can be easily expected from the Hookean constitutive equation. If the same strain excitation is applied to an idealistically viscous material, the response will be those as shown in the Fig. 3(b) and (c), which is also expectable from the Newtonian constitutive equation. Materials that exhibit viscoelastic properties can be considered as having both elastic and viscous components. Then, in the case of viscoelastic materials, the response is expected to have both characteristics of elasticity and viscosity. Schematic drawing of the stress response according to many experimental results is shown in Fig. 4. Here we define the stress relaxation as follows. Fig. 5 shows a step strain excitation and a stress response to it for a viscoelastic material. The instantaneous and equilibrium moduli, *m*g and *m*e, respectively, are material's constants. The relaxation function (*t*) is a material's function. Phenomenological investigation of materials from the stand point of viscoelastic properties is to determine these material's constants and

function and, then, deduce the molecular mechanism corresponding to the viscoelastic behavior. For this purpose, more generalized tool shall be introduced. Any given strain excitation can be constructed by a linear combination of many small step strains applied to the material at every Δt seconds. According to the linear response theory, stress response to the excitation also can be a linear combination of responses to small step strains applied to the material (Fig. 6). From any given strain excitation (*t*), stress response (*t*) would be obtained through the Boltzmann's superposition equation,

$$
\sigma(t) = \int\_{-\infty}^{t} \frac{d\varepsilon(u)}{du} m(t-u) du. \tag{7}
$$

Viscoelastic Properties of Biological Materials 103

**Figure 4.** Response of viscoelastic materials to a box-shaped strain excitation.

**Figure 5.** Definition of the response of a viscoelastic material to a step strain excitation.

**Figure 6.** Generalized strain excitation and the stress response of viscoelastic materials to the former. From the relationship between the excitation and the response, Boltzmann's equation, eq. 7, was

derived.

**Figure 3.** Responses of idealistically elastic materials (a) and that of idealistic viscous materials (b) and (c) to step strain and stress excitations, respectively.

**Figure 4.** Response of viscoelastic materials to a box-shaped strain excitation.

the material (Fig. 6). From any given strain excitation

obtained through the Boltzmann's superposition equation,

function and, then, deduce the molecular mechanism corresponding to the viscoelastic behavior. For this purpose, more generalized tool shall be introduced. Any given strain excitation can be constructed by a linear combination of many small step strains applied to the material at every Δt seconds. According to the linear response theory, stress response to the excitation also can be a linear combination of responses to small step strains applied to

> ( ) ( ). *<sup>t</sup> dε u σ t m t u du*

(a) (b)

**Figure 3.** Responses of idealistically elastic materials (a) and that of idealistic viscous materials (b) and

(c)

(c) to step strain and stress excitations, respectively.

(*t*), stress response

*du* (7)

(*t*) would be

**Figure 6.** Generalized strain excitation and the stress response of viscoelastic materials to the former. From the relationship between the excitation and the response, Boltzmann's equation, eq. 7, was derived.

For stress relaxation experiments, a step strain excitation is needed. But, actually, it is difficult to realize such an excitation strain. The realistic one should be a lamp followed by a plateau strain. An acceptable lamp width can be determined from the Boltzmann equation above. If the relaxation function is described as

$$\phi(t) = 1 - \exp(-t/\tau) \tag{8}$$

Viscoelastic Properties of Biological Materials 105

<sup>0</sup> ( 1) <sup>0</sup> *<sup>n</sup> <sup>e</sup> Et Et E n* (13)

<sup>0</sup> exp[ ( / ) ] ( 1). <sup>0</sup> *<sup>β</sup> Et E t <sup>τ</sup> <sup>b</sup>* (14)

For (C), it has been established that the non-single relaxation timed relaxation process can be well described by a power law relation or stretched exponential function, depending on the

One of the advantages of using these functions for describing the relaxation process is easily to establish the mechanistic model for the relaxation process. As these two functions have been found to describe well the mechanical or electric relaxation phenomena in many

When there is a dynamic structural inhomogeneity in glassy materials, Jonscher proposed that the power in the power law relation, eq. 13, indicates the correlation of molecular motions among motional groups and the variation in the number of motional unit in each molecular motional group (Jonscher, 1983). On the other hand, for explanation of the exponent value in the stretched exponential function, eq. 14, a diffusion-trap model is proposed (Klafter et al., 1986; Philips, 1996). When a step strain is applied to a material, many excited sites will emerge inside the material. In amorphous glassy materials, molecular motion of constituents could be regarded as a diffusion of a free volume of the constituents. When the free volume comes to an excited site, the site will be relaxed. This would be an elementary process of the relaxation. If the excited sites in the material is distributing on a fractal lattice, total relaxation of the material is expected to be described by the stretched exponential function, eq. 14. The exponent value, *β*, is strongly related the

In the introductory section, only the stress response of materials to strain excitation was explained. This is because our viscoelastic study of biological materials mainly has been employing the stress relaxation experiments. Therefore, in this chapter, the relaxation modulus of biological materials is discussed in conjunction with their structural information. In this section, the empirical determination of the relaxation modulus of

Figure 7 shows a schema for the stress relaxation experiments. By moving the load cell (LC), Kyowa Electric Works LTS series, strain is applied to the specimen through an equipped attachment mounted on a probe and the stress response to the excitation is detected by LC. LC is mounted on an auto micro-stage, Sigma Koki CTS-50X, derived by a stage controller Sigma Koki MARK-12. The sensitivity of LC and the attachments are chosen depending on

amorphous glassy materials, they are regarded as universal functions.

geometry of the exciting site distribution (Potuzak et al., 2011).

**3. Stress relaxation experiments** 

materials would be explained.

**3.1. Apparatus and equipments** 

materials and their state;

or

where *τ* is a characteristic relaxation time of a material, the stress response will be

$$
\sigma \sigma \left( t \right) = A \tau e^{-t/\tau} \left( e^{t\_1/\tau} - 1 \right). \tag{9}
$$

In this equation, *A* is a constant and *t*1 is the lamp width. If *t*1≪*τ*, it is rewritten as

$$
\sigma(t) = At\_1 \exp\left(-t/\tau\right) \tag{10}
$$

which is a usual relaxation modulus form of materials with single relaxation time *τ* as a response to the step strain excitation.

#### **2.3. Viscoelastic investigation**

As indicated above, the objective of the viscoelastic studies of materials is to understand the molecular mechanism corresponding to the viscoelastic behavior. For this purpose, to know the relaxation function of the material is needed. Almost all actual relaxation data, however, cannot be described by a relaxation function with single relaxation time as shown by eq. 8. In order to describe the relaxation modulus, then the relaxation function, the following methods have been traditionally employed: (A) multi relaxation time analysis, (B) relaxation time distribution analysis (Ferry, 1980), and (C) analysis using specific functions.

(A) and (B) are substantially similar analyses. For the analysis (A), experimental relaxation modulus, *E*(*t*), is fitted by

$$E(t) = E\_e + \sum\_{i=1}^{N} a\_i \exp\left(-t/\tau\_i\right) \tag{11}$$

where *E*e is an equilibrium modulus. If *N* is large and *ai* is a continual function of *τ*, *a*(*τ*), eq. 11 can be described as

$$E(t) = E\_e + \int\_{-\infty}^{t} a(\tau) \exp(-t/\tau) d\tau. \tag{12}$$

This description represents the relaxation time distribution analysis and *a*(*t*) is a relaxation time spectrum. By a numerical transformation of the experimental data, the relaxation spectrum will be obtained. At this point, relaxation spectrum does not have any more information than the original relaxation data. It is still difficult to deduce molecular events underlying the relaxation.

For (C), it has been established that the non-single relaxation timed relaxation process can be well described by a power law relation or stretched exponential function, depending on the materials and their state;

$$E(t) = E\_0 t^{-n} + E\_{\varepsilon} \quad \text{( $0 < n \le 1$ )}\tag{13}$$

or

104 Viscoelasticity – From Theory to Biological Applications

response to the step strain excitation.

**2.3. Viscoelastic investigation** 

modulus, *E*(*t*), is fitted by

11 can be described as

underlying the relaxation.

above. If the relaxation function is described as

For stress relaxation experiments, a step strain excitation is needed. But, actually, it is difficult to realize such an excitation strain. The realistic one should be a lamp followed by a plateau strain. An acceptable lamp width can be determined from the Boltzmann equation

<sup>1</sup> <sup>1</sup> / / . *<sup>t</sup> <sup>τ</sup> <sup>t</sup> <sup>τ</sup> <sup>σ</sup> t Aτe e*

which is a usual relaxation modulus form of materials with single relaxation time *τ* as a

As indicated above, the objective of the viscoelastic studies of materials is to understand the molecular mechanism corresponding to the viscoelastic behavior. For this purpose, to know the relaxation function of the material is needed. Almost all actual relaxation data, however, cannot be described by a relaxation function with single relaxation time as shown by eq. 8. In order to describe the relaxation modulus, then the relaxation function, the following methods have been traditionally employed: (A) multi relaxation time analysis, (B) relaxation

(A) and (B) are substantially similar analyses. For the analysis (A), experimental relaxation

 1

where *E*e is an equilibrium modulus. If *N* is large and *ai* is a continual function of *τ*, *a*(*τ*), eq.

 exp( / ) . *t <sup>e</sup> Et E a <sup>τ</sup> <sup>t</sup> <sup>τ</sup> <sup>d</sup><sup>τ</sup>*

This description represents the relaxation time distribution analysis and *a*(*t*) is a relaxation time spectrum. By a numerical transformation of the experimental data, the relaxation spectrum will be obtained. At this point, relaxation spectrum does not have any more information than the original relaxation data. It is still difficult to deduce molecular events

*ei i*

*N*

*i Et E a t τ* 

exp /

where *τ* is a characteristic relaxation time of a material, the stress response will be

In this equation, *A* is a constant and *t*1 is the lamp width. If *t*1≪*τ*, it is rewritten as

time distribution analysis (Ferry, 1980), and (C) analysis using specific functions.

*φt t* 1 exp( / ) *τ* (8)

(9)

<sup>1</sup> *σ t At t* exp / *τ* (10)

(11)

(12)

$$E\left(t\right) = E\_0 \exp[-\left(t \mid \tau \right)^{\beta}] \tag{14} \\
\leq b \leq 1 \\
\text{j.} \tag{14}$$

One of the advantages of using these functions for describing the relaxation process is easily to establish the mechanistic model for the relaxation process. As these two functions have been found to describe well the mechanical or electric relaxation phenomena in many amorphous glassy materials, they are regarded as universal functions.

When there is a dynamic structural inhomogeneity in glassy materials, Jonscher proposed that the power in the power law relation, eq. 13, indicates the correlation of molecular motions among motional groups and the variation in the number of motional unit in each molecular motional group (Jonscher, 1983). On the other hand, for explanation of the exponent value in the stretched exponential function, eq. 14, a diffusion-trap model is proposed (Klafter et al., 1986; Philips, 1996). When a step strain is applied to a material, many excited sites will emerge inside the material. In amorphous glassy materials, molecular motion of constituents could be regarded as a diffusion of a free volume of the constituents. When the free volume comes to an excited site, the site will be relaxed. This would be an elementary process of the relaxation. If the excited sites in the material is distributing on a fractal lattice, total relaxation of the material is expected to be described by the stretched exponential function, eq. 14. The exponent value, *β*, is strongly related the geometry of the exciting site distribution (Potuzak et al., 2011).

#### **3. Stress relaxation experiments**

In the introductory section, only the stress response of materials to strain excitation was explained. This is because our viscoelastic study of biological materials mainly has been employing the stress relaxation experiments. Therefore, in this chapter, the relaxation modulus of biological materials is discussed in conjunction with their structural information. In this section, the empirical determination of the relaxation modulus of materials would be explained.

#### **3.1. Apparatus and equipments**

Figure 7 shows a schema for the stress relaxation experiments. By moving the load cell (LC), Kyowa Electric Works LTS series, strain is applied to the specimen through an equipped attachment mounted on a probe and the stress response to the excitation is detected by LC. LC is mounted on an auto micro-stage, Sigma Koki CTS-50X, derived by a stage controller Sigma Koki MARK-12. The sensitivity of LC and the attachments are chosen depending on

the specimens. Soft materials are examined by a tension or indentation methods depending on the specimen. When the specimen can be shaped in a sheet, relaxation modulus was measured by the tensile strain application.

Viscoelastic Properties of Biological Materials 107

clump or indenter head were derived in 5 μm step. The judgment for an initial touch of the probe head on the specimen was made on the indication of signals from LC starting to change to more than a few με. The stress relaxation experiments were started one hour after the zero strain state detection, waiting for stress relaxations brought about by the zero point detecting procedures. For the stress relaxation measurement, strain values less than 1/2 of yield strain value were applied to the specimen at the deformation rate of 3.3 mm sec-1.

**4. Viscoelastic properties of agarose gel and cell-seeded agarose gel** 

purchased from Sigma.

estimated by using the Hayes equation,

incompressibility (Watase et al., 1983).

is a function of the ratio, *a*/*h*, and Poisson's ratio,

**4.1. Agarose gel** 

Agarose gel is a material of multi-purpose use. For example, it has been used widely as a cell culture matrix. In the following sections, how cultured cells in agarose gel change the mechanical properties of matrix agarose gel by precipitating extracellular matrix would be shown. For the precise estimation of the change in mechanical properties of the matrix, those of agarose gel itself must be quantitatively estimated. Here results obtained for pure agarose gel at first and then cell-seeded ones would be presented. Agarose used in this experiment and those in the following sections is type VII agarose for cell culture use

Recent investigation on the agarose gelation revealed that a phase separation takes place slightly below the gelation temperature, where the gelation process was expected to proceed competitively with the phase separation of the solution (Morita et al., 2008). It is expected that the mechanical properties of so prepared agarose gel depend remarkably on the thermal history during the preparation. For the reproducible data, gel preparation must be carefully conducted. Agarose type VII powder (Sigma Co. Ltd) was dissolved in distilled and deionized water and stirred for 12 hrs at room temperature. The aqueous solution was then incubated at 90oC for 5 hrs. The incubated solution was quenched from 90oC to 4oC, where the gelation temperature was determined to be 32oC. Before mechanical test, the gel specimens were swollen with the distilled and deionized water. Quench temperature 4oC is far below the spinodal point curve in the phase diagram. Fig. 9 shows the relaxation modulus of 2% (w/v) of Type VII agarose gel plotted against time. The modulus value was

> <sup>2</sup> <sup>1</sup> <sup>2</sup> , *F ν*

(15)

comes from the solution of the integral equation,

, of the gel. Following Watase *et al*.,

*daκ* 

where *F* is the recovering force of gel applied to the indenter surface, *a* is the radius of the indenter and *d* is the deformation depth (Hayes et al., 1972). For a specimen of thickness *h*,

Poisson's ratio of our agarose gel was considered to be very close to 0.5 because of its near

and its value is numerically calculated and tabulated (Hayes et al., 1972). The radius of the

*E*

**Figure 7.** Block diagram for stress relaxation measuring apparatus.

**Figure 8.** (a) shows the attachments for tension measurements. Cell-seeded gels are examined by the indentation. Fig. 8(b) shows the indenter for this measurement. Hard materials are shaped in beam and examined by a bending method. Fig. 8(c) shows three point bending equipment used for stress relaxation of bone specimens. In both measurements, specimens are soaked in Ringer's solution at 37oC during the relaxation experiments. Detected LC signals were passed to adata logger, Kyowa Electric Works PCD-300B, and finally stored into a PC.

#### **3.2. Measurements**

At each measuring mode, prior to the start of the stress relaxation experiment, an upper clump in the case of tensile strain application and an indenter head in the cases of indentation and beam bending must be placed at a zero strain point. For this procedure, the clump or indenter head were derived in 5 μm step. The judgment for an initial touch of the probe head on the specimen was made on the indication of signals from LC starting to change to more than a few με. The stress relaxation experiments were started one hour after the zero strain state detection, waiting for stress relaxations brought about by the zero point detecting procedures. For the stress relaxation measurement, strain values less than 1/2 of yield strain value were applied to the specimen at the deformation rate of 3.3 mm sec-1.

#### **4. Viscoelastic properties of agarose gel and cell-seeded agarose gel**

Agarose gel is a material of multi-purpose use. For example, it has been used widely as a cell culture matrix. In the following sections, how cultured cells in agarose gel change the mechanical properties of matrix agarose gel by precipitating extracellular matrix would be shown. For the precise estimation of the change in mechanical properties of the matrix, those of agarose gel itself must be quantitatively estimated. Here results obtained for pure agarose gel at first and then cell-seeded ones would be presented. Agarose used in this experiment and those in the following sections is type VII agarose for cell culture use purchased from Sigma.

#### **4.1. Agarose gel**

106 Viscoelasticity – From Theory to Biological Applications

measured by the tensile strain application.

**Figure 7.** Block diagram for stress relaxation measuring apparatus.

Works PCD-300B, and finally stored into a PC.

**3.2. Measurements** 

the specimens. Soft materials are examined by a tension or indentation methods depending on the specimen. When the specimen can be shaped in a sheet, relaxation modulus was

**Figure 8.** (a) shows the attachments for tension measurements. Cell-seeded gels are examined by the indentation. Fig. 8(b) shows the indenter for this measurement. Hard materials are shaped in beam and

(a) (b) (c)

relaxation of bone specimens. In both measurements, specimens are soaked in Ringer's solution at 37oC during the relaxation experiments. Detected LC signals were passed to adata logger, Kyowa Electric

At each measuring mode, prior to the start of the stress relaxation experiment, an upper clump in the case of tensile strain application and an indenter head in the cases of indentation and beam bending must be placed at a zero strain point. For this procedure, the

examined by a bending method. Fig. 8(c) shows three point bending equipment used for stress

Recent investigation on the agarose gelation revealed that a phase separation takes place slightly below the gelation temperature, where the gelation process was expected to proceed competitively with the phase separation of the solution (Morita et al., 2008). It is expected that the mechanical properties of so prepared agarose gel depend remarkably on the thermal history during the preparation. For the reproducible data, gel preparation must be carefully conducted. Agarose type VII powder (Sigma Co. Ltd) was dissolved in distilled and deionized water and stirred for 12 hrs at room temperature. The aqueous solution was then incubated at 90oC for 5 hrs. The incubated solution was quenched from 90oC to 4oC, where the gelation temperature was determined to be 32oC. Before mechanical test, the gel specimens were swollen with the distilled and deionized water. Quench temperature 4oC is far below the spinodal point curve in the phase diagram. Fig. 9 shows the relaxation modulus of 2% (w/v) of Type VII agarose gel plotted against time. The modulus value was estimated by using the Hayes equation,

$$E = \frac{F\left(1 - \nu^2\right)}{2da\kappa},\tag{15}$$

where *F* is the recovering force of gel applied to the indenter surface, *a* is the radius of the indenter and *d* is the deformation depth (Hayes et al., 1972). For a specimen of thickness *h*, is a function of the ratio, *a*/*h*, and Poisson's ratio, , of the gel. Following Watase *et al*., Poisson's ratio of our agarose gel was considered to be very close to 0.5 because of its near incompressibility (Watase et al., 1983). comes from the solution of the integral equation, and its value is numerically calculated and tabulated (Hayes et al., 1972). The radius of the

indenter head used in this work was *a* = 3 mm, and the average thickness of specimens was *h* = 3 mm. Since Poisson's ratio was assumed to be 0.5, of 3.609 was employed. There is a large relaxation up to 104 sec followed by a plateau which is characteristic of gels. As both axes are scaled in logarithm, it is clear that the relaxation modulus is not described by a power law relation. After examining the data, the empirical equation

$$E\left(t\right) = E\_0 \exp\left[-\left(t/\tau\right)^{\beta}\right] + E\_e \quad \text{(0 } \ll 1\text{)}\tag{16}$$

Viscoelastic Properties of Biological Materials 109

a control. All specimens showed the characteristic stress relaxation curve of agarose gel; a large relaxation up to 104 sec followed by a gel plateau where the former was attributed to molecular motions of polymer chains between two adjacent cross-links of the gel and the latter to the elasticity of the gel network. Each relaxation modulus curve was able to be described well by eq.16, and parameter determined by the fitting was listed in Table 1. Change in modulus value both *E*0 and *E*e as a function of culturing period was not so simple. Up to day 15, the change was not significant and at day 18, both modulus values significantly increased. The relaxation time *τ* and its distribution *β* did not change significantly through the cultivation period. Fig. 11 shows X-ray diffraction profiles of the cell-agarose gel composite specimens cultured for the indicated periods. The profile for a sintered HAp is also shown as a reference (Okazaki et al., 1997). There is a large peak at about 2θ20o, which was attributed to the diffraction peaks related to the agarose gel matrix (Foord & Atkins, 1989). The diffraction profile of the specimen after 9 days of culture showed diffraction peaks attributable to HAp crystal, where the peak at 2θ25o was indexed as (002) and those at 32o were indexed as (211), (112), and (300) (Okazaki et al., 1997). With further extension of culturing time, the intensities of peaks corresponding to HAp crystal gradually increased. In Figs. 12(a) and 12(b), *E0* and *Ee* values are respectively plotted against culture days. Relative integrated diffraction intensity in Fig. 11 is also plotted against days, defined as the ratio of intensity from 002 plan at each culture day against that at the day 18, *I*002/*I*002(18). The diffraction intensity is generally proportional to the amount of crystals, provided that the state of crystals in each specimen is not greatly different (Kakudo & Kasai, 1972). In Fig. 11, the half width of each (002) peak was almost the same throughout the culture time up to 18 days. In this case, the intensity can be considered to be proportional to the amount of HAp crystals in the specimen. The I002/I002(18) vs. culture time plot indicates that the amount of HAp crystals increases monotonically with culture period. The result indicates that even though HAp content proportionally increases with culture time, modulus values increment started only from the day 18. To explain this discrepancy between HAp increment and modulus values increment, the efficiency of the reinforcement of agarose gel matrix by precipitated mineral particles was appreciated. Fig. 13 shows a schema of the precipitation process of mineral particles into agarose gel matrix. In the early stage of HAp particle precipitation by MC3T3-E1 cells, the number of mineral particles increases inside a gel network [schema (a) in Fig. 13]. In this state of a cell-seeded agarose gel system, precipitated mineral particles do not contribute to the composite modulus. When a mineral particle cluster percolates inside the gel network, the modulus of the composite will increase [schema (b) in Fig. 13]. The percolation could occur almost simultaneously in every network around each cell. The effect of this percolation is considered to be similar to the increase in the crosslink density, which will lead to the increase in *Ee*. At the same time, the stiffness of network chains around the mineral percolation would be increased and their mobility would be decreased by the precipitated mineral particles. The empirical equation employed in this study, eq. 16, basically assumes that a relaxing entity can be described by a serial combination of a spring and a dashpot,

where the former represents an elasticity,

relaxation time, *τ*, is expressed as

, and the latter a viscosity,

, in a material. The

has been found suitable. In the Fig. 9, determined parameter values are listed. Recent investigation on the stretched exponential function revealed that *β* value suggests the relaxation mechanism at molecular level for homogenous systems: *β*=3/5 for relaxation arising from short-range forces and *β*=3/7 for relaxation dominated by long-range forces (Philips, 1996). According to *β*∼0.4 obtained here for agarose gel, the driving force for the relaxation would be long-range ones.

**Figure 9.** Relaxation modulus curve for agarose gel fabricated by quenching to 4oC.

#### **4.2. Osteoblast seeded agarose gel**

On the basis of the relaxation of agarose gel indicated above, how cells change their matrix by precipitating organic and inorganic materials has been investigated (Hanazaki et al., 2011). As seeded cells were osteoblast-like cells, MC3T3-E1, the system can be regarded as a model system of osteogenesis. MC3T3-E1 cells precipitate collagenous proteins and hydroxyl apatite (HAp)-like minerals. Fig.10 shows stress relaxation curves for osteoblasts seeded agarose gel cultured up to 18 days. Relaxation curve for agarose gel is also plotted as a control. All specimens showed the characteristic stress relaxation curve of agarose gel; a large relaxation up to 104 sec followed by a gel plateau where the former was attributed to molecular motions of polymer chains between two adjacent cross-links of the gel and the latter to the elasticity of the gel network. Each relaxation modulus curve was able to be described well by eq.16, and parameter determined by the fitting was listed in Table 1. Change in modulus value both *E*0 and *E*e as a function of culturing period was not so simple. Up to day 15, the change was not significant and at day 18, both modulus values significantly increased. The relaxation time *τ* and its distribution *β* did not change significantly through the cultivation period. Fig. 11 shows X-ray diffraction profiles of the cell-agarose gel composite specimens cultured for the indicated periods. The profile for a sintered HAp is also shown as a reference (Okazaki et al., 1997). There is a large peak at about 2θ20o, which was attributed to the diffraction peaks related to the agarose gel matrix (Foord & Atkins, 1989). The diffraction profile of the specimen after 9 days of culture showed diffraction peaks attributable to HAp crystal, where the peak at 2θ25o was indexed as (002) and those at 32o were indexed as (211), (112), and (300) (Okazaki et al., 1997). With further extension of culturing time, the intensities of peaks corresponding to HAp crystal gradually increased. In Figs. 12(a) and 12(b), *E0* and *Ee* values are respectively plotted against culture days. Relative integrated diffraction intensity in Fig. 11 is also plotted against days, defined as the ratio of intensity from 002 plan at each culture day against that at the day 18, *I*002/*I*002(18). The diffraction intensity is generally proportional to the amount of crystals, provided that the state of crystals in each specimen is not greatly different (Kakudo & Kasai, 1972). In Fig. 11, the half width of each (002) peak was almost the same throughout the culture time up to 18 days. In this case, the intensity can be considered to be proportional to the amount of HAp crystals in the specimen. The I002/I002(18) vs. culture time plot indicates that the amount of HAp crystals increases monotonically with culture period. The result indicates that even though HAp content proportionally increases with culture time, modulus values increment started only from the day 18. To explain this discrepancy between HAp increment and modulus values increment, the efficiency of the reinforcement of agarose gel matrix by precipitated mineral particles was appreciated. Fig. 13 shows a schema of the precipitation process of mineral particles into agarose gel matrix. In the early stage of HAp particle precipitation by MC3T3-E1 cells, the number of mineral particles increases inside a gel network [schema (a) in Fig. 13]. In this state of a cell-seeded agarose gel system, precipitated mineral particles do not contribute to the composite modulus. When a mineral particle cluster percolates inside the gel network, the modulus of the composite will increase [schema (b) in Fig. 13]. The percolation could occur almost simultaneously in every network around each cell. The effect of this percolation is considered to be similar to the increase in the crosslink density, which will lead to the increase in *Ee*. At the same time, the stiffness of network chains around the mineral percolation would be increased and their mobility would be decreased by the precipitated mineral particles. The empirical equation employed in this study, eq. 16, basically assumes that a relaxing entity can be described by a serial combination of a spring and a dashpot, where the former represents an elasticity, , and the latter a viscosity, , in a material. The relaxation time, *τ*, is expressed as

108 Viscoelasticity – From Theory to Biological Applications

relaxation would be long-range ones.

100000

**4.2. Osteoblast seeded agarose gel** 

1000

10000

E(t) (kPa)

*h* = 3 mm. Since Poisson's ratio was assumed to be 0.5,

power law relation. After examining the data, the empirical equation

**Figure 9.** Relaxation modulus curve for agarose gel fabricated by quenching to 4oC.

*E*0=19.7 kPa *E*e=3.2kPa

= 92.9 sec

= 0.44

On the basis of the relaxation of agarose gel indicated above, how cells change their matrix by precipitating organic and inorganic materials has been investigated (Hanazaki et al., 2011). As seeded cells were osteoblast-like cells, MC3T3-E1, the system can be regarded as a model system of osteogenesis. MC3T3-E1 cells precipitate collagenous proteins and hydroxyl apatite (HAp)-like minerals. Fig.10 shows stress relaxation curves for osteoblasts seeded agarose gel cultured up to 18 days. Relaxation curve for agarose gel is also plotted as

1 10 100 1000 10000 100000 time (sec)

indenter head used in this work was *a* = 3 mm, and the average thickness of specimens was

large relaxation up to 104 sec followed by a plateau which is characteristic of gels. As both axes are scaled in logarithm, it is clear that the relaxation modulus is not described by a

> <sup>0</sup> exp / ( 1) <sup>0</sup> *<sup>β</sup> <sup>e</sup> Et E t <sup>τ</sup> <sup>E</sup>*

has been found suitable. In the Fig. 9, determined parameter values are listed. Recent investigation on the stretched exponential function revealed that *β* value suggests the relaxation mechanism at molecular level for homogenous systems: *β*=3/5 for relaxation arising from short-range forces and *β*=3/7 for relaxation dominated by long-range forces (Philips, 1996). According to *β*∼0.4 obtained here for agarose gel, the driving force for the

of 3.609 was employed. There is a

(16)

agarose gel quenched to 4o

*Ee E t E t*

0

 exp /

C

$$
\pi = \frac{\eta}{\varepsilon}.\tag{17}
$$

Viscoelastic Properties of Biological Materials 111

**Figure 11.** X-ray diffraction profiles for cell-seeded agarose gel cultured for indicated days. That for HAp powder is also shown for reference (Okasaki at al., 1997). (From Hanazaki et al., J. Biorheology. In

**Figure 12.** Comparison of *E*0 (a) and *E*e (b) values with diffraction intensity for HAp 002 as a function of culture period. (From Hanazaki et al., J. Biorheology. In press. DOI 10.1007/s12573-011-0043-2. With

press. DOI 10.1007/s12573 -011- 0043-2. With permission.)

permission.)

An increase in the stiffness of the network chain implies an increase in and a decrease in their mobility, that is, an increase in , which would result in an increased *E*0 with relaxation time almost unchanged. Above the percolation threshold, the increase in the modulus of the system, (both *E*0 and *Ee* values) with culture time would become remarkable. In our case, the percolation was considered to take place between days 15 and 18 of culture.


(From Hanazaki et al., J. Biorheology. In press. DOI 10.1007/s12573-011-0043-2. With permission.)

\*All the parameter values are presented as the average value ± the standard error.

**Table 1.** Parameter values determined by fitting equation (16) to experimental data.

**Figure 10.** Relaxation modulus curves for MC3T3-E1 seeded agarose gels cultured for 1 day (○), 9 days (△), 12 days (□), 15 days (┼), and 18 days (х). Relaxation modulus for agarose gel (●) is also shown as a control. (From Hanazaki et al., J. Biorheology. In press. DOI 10.1007/s 12573 -011- 0043-2. With permission.)

their mobility, that is, an increase in

culture

. *<sup>η</sup> <sup>τ</sup>*

time almost unchanged. Above the percolation threshold, the increase in the modulus of the system, (both *E*0 and *Ee* values) with culture time would become remarkable. In our case, the

period (day) sample size *E*0 (kPa) τ (sec) <sup>β</sup> *Ee* (kPa)

**Figure 10.** Relaxation modulus curves for MC3T3-E1 seeded agarose gels cultured for 1 day (○), 9 days (△), 12 days (□), 15 days (┼), and 18 days (х). Relaxation modulus for agarose gel (●) is also shown as a control. (From Hanazaki et al., J. Biorheology. In press. DOI 10.1007/s 12573 -011- 0043-2. With permission.)

agarose 4 16.7 ± 0.6 162 ± 17 0.49 ± 0.01 2.19 ± 0.32 1 4 15.1 ± 2.4 178 ± 61 0.43 ± 0.03 2.36 ± 0.38 9 4 13.9 ± 3.0 144 ± 54 0.44 ± 0.04 2.56 ± 0.40 12 4 12.4 ± 2.0 168 ± 59 0.48 ± 0.03 2.21 ± 0.41 15 4 15.0 ± 2.0 136 ± 27 0.46 ± 0.01 2.22 ± 0.13 18 4 21.3 ± 3.6 173 ± 41 0.49 ± 0.02 3.38 ± 0.16

An increase in the stiffness of the network chain implies an increase in

percolation was considered to take place between days 15 and 18 of culture.

(From Hanazaki et al., J. Biorheology. In press. DOI 10.1007/s12573-011-0043-2. With permission.)

**Table 1.** Parameter values determined by fitting equation (16) to experimental data.

\*All the parameter values are presented as the average value ± the standard error.

*<sup>ε</sup>* (17)

, which would result in an increased *E*0 with relaxation

and a decrease in

**Figure 11.** X-ray diffraction profiles for cell-seeded agarose gel cultured for indicated days. That for HAp powder is also shown for reference (Okasaki at al., 1997). (From Hanazaki et al., J. Biorheology. In press. DOI 10.1007/s12573 -011- 0043-2. With permission.)

**Figure 12.** Comparison of *E*0 (a) and *E*e (b) values with diffraction intensity for HAp 002 as a function of culture period. (From Hanazaki et al., J. Biorheology. In press. DOI 10.1007/s12573-011-0043-2. With permission.)

Viscoelastic Properties of Biological Materials 113

modulus value of AGC0 significantly increased (\*p<0.05). In order to analyse the change in the viscoelastic properties of chondrocyte-agarose composite originated from the PCM-like material production by chondrocytes, the differences in the relaxation modulus values among three samples were calculated. Fig. 16 shows the difference in the relaxation

AGC AG AGC AG 0 0 *E EE* ,

AGC3 AGC AGC3 AGC 0 0 *E EE* ,

**Figure 14.** Microscopic images of chondrocyte-seeded agarose gel (a) before culture and (b) cultured for

**Figure 15.** Relaxation modulus curves for chondrocyte seeded agarose gel immediately after seeded (○) and that cultured for 3 weeks (△). Relaxation modulus curve for agarose gel is also shown as a reference

21 days.(From Sasaki et al., J. Biorheology 23, 95-101 (2009). With permission.)

(●). (From Sasaki et al., J. Biorheology 23, 95-101 (2009). With permission.)

modulus values between AG and AGC0 (○),

AGC3 and AGC0 (▲),

**Figure 13.** A model explaining the discrepancy of the mechanical behavior from the X-ray diffraction profiles as a function of culture period. (a) Precipitation of HAp in agarose gel up to day 15. (b) Partial percolation of HAp precipitates in a agarose net work area after day 18. (From Hanazaki et al., J. Biorheology. In press. DOI 10.1007/s12573-011-0043-2. With permission.)

#### **4.3. Chondrocyte seeded agarose gel**

Chondrocytes are responsible for the synthesis, maintenance, and gradual turnover of an extracellular matrix (ECM) composed principally of a hydrated collagen fibril network enmeshed in a gel of highly charged proteoglycan molecules. Each chondrocyte is surrounded by a narrow tissue region of pericellular matrix (PCM), the elastic modulus of which has been regarded to be larger than that of a chondrocyte and smaller than those of territorial matrix (TM) and inter territorial matrix (ITM). The unit consisting of a chondrocyte with PCM is generally termed a chondron. Since the volume fraction of chondrocytes in articular cartilage has been reported to be about 5-10%, chondrons will occupy more than this volume fraction. Even though the modulus of a chondron is much smaller than those of TM and ITM, the contribution of chondrons as mechanical elements to the mechanical function of articular cartilage is not negligible. To clarify the contribution of the viscoelastic nature of chondrons to that of articular cartilage tissue, relaxation modulus for chondrocyte-seeded agarose gel and that cultured for three weeks were measured as a model system of articular cartilage (Sasaki et al., 2009). The relaxation modulus curves for chondrocyte-seeded agarose gel were compared with that for agarose gel (AG). Fig. 14(a) shows a microscopic image of chondrocytes-seeded in agarose gel (AGC0) that was obtained immediately before mechanical measurements. The density of chondrocytes was 30 106 cell/m*l* and the average size of a chondrocyte was about 10 μm in diameter. Chondrocytes were shown to be dispersed almost homogeneously. Fig. 14(b) shows a microscopic image of chondrocyte-seeded agarose gel after 21 days of cultivation (AGC3). The magnification is the same as that in Fig. 14(a). Toluidine Blue staining was performed. Material was synthesized around each cell after 21 days of cultivation. Fig. 15 shows the relaxation modulus of AGC3 compared with those of AGC0 and AG. The relaxation modulus of AGC0 was increased by cultivation to be that of AGC3. In a short time region, up to 102 sec, the increment was not statistically significant. After103 sec, the relaxation modulus value of AGC0 significantly increased (\*p<0.05). In order to analyse the change in the viscoelastic properties of chondrocyte-agarose composite originated from the PCM-like material production by chondrocytes, the differences in the relaxation modulus values among three samples were calculated. Fig. 16 shows the difference in the relaxation modulus values between AG and AGC0 (○),

$$
\Delta E\_{\rm AGC0-AG} = E\_{\rm AGC0} - E\_{\rm AG'}
$$

AGC3 and AGC0 (▲),

112 Viscoelasticity – From Theory to Biological Applications

**4.3. Chondrocyte seeded agarose gel** 

**Figure 13.** A model explaining the discrepancy of the mechanical behavior from the X-ray diffraction profiles as a function of culture period. (a) Precipitation of HAp in agarose gel up to day 15. (b) Partial percolation of HAp precipitates in a agarose net work area after day 18. (From Hanazaki et al., J.

Chondrocytes are responsible for the synthesis, maintenance, and gradual turnover of an extracellular matrix (ECM) composed principally of a hydrated collagen fibril network enmeshed in a gel of highly charged proteoglycan molecules. Each chondrocyte is surrounded by a narrow tissue region of pericellular matrix (PCM), the elastic modulus of which has been regarded to be larger than that of a chondrocyte and smaller than those of territorial matrix (TM) and inter territorial matrix (ITM). The unit consisting of a chondrocyte with PCM is generally termed a chondron. Since the volume fraction of chondrocytes in articular cartilage has been reported to be about 5-10%, chondrons will occupy more than this volume fraction. Even though the modulus of a chondron is much smaller than those of TM and ITM, the contribution of chondrons as mechanical elements to the mechanical function of articular cartilage is not negligible. To clarify the contribution of the viscoelastic nature of chondrons to that of articular cartilage tissue, relaxation modulus for chondrocyte-seeded agarose gel and that cultured for three weeks were measured as a model system of articular cartilage (Sasaki et al., 2009). The relaxation modulus curves for chondrocyte-seeded agarose gel were compared with that for agarose gel (AG). Fig. 14(a) shows a microscopic image of chondrocytes-seeded in agarose gel (AGC0) that was obtained immediately before mechanical measurements. The density of chondrocytes was 30 106 cell/m*l* and the average size of a chondrocyte was about 10 μm in diameter. Chondrocytes were shown to be dispersed almost homogeneously. Fig. 14(b) shows a microscopic image of chondrocyte-seeded agarose gel after 21 days of cultivation (AGC3). The magnification is the same as that in Fig. 14(a). Toluidine Blue staining was performed. Material was synthesized around each cell after 21 days of cultivation. Fig. 15 shows the relaxation modulus of AGC3 compared with those of AGC0 and AG. The relaxation modulus of AGC0 was increased by cultivation to be that of AGC3. In a short time region, up to 102 sec, the increment was not statistically significant. After103 sec, the relaxation

Biorheology. In press. DOI 10.1007/s12573-011-0043-2. With permission.)

#### AGC3 AGC AGC3 AGC 0 0 *E EE* ,

**Figure 14.** Microscopic images of chondrocyte-seeded agarose gel (a) before culture and (b) cultured for 21 days.(From Sasaki et al., J. Biorheology 23, 95-101 (2009). With permission.)

**Figure 15.** Relaxation modulus curves for chondrocyte seeded agarose gel immediately after seeded (○) and that cultured for 3 weeks (△). Relaxation modulus curve for agarose gel is also shown as a reference (●). (From Sasaki et al., J. Biorheology 23, 95-101 (2009). With permission.)

Viscoelastic Properties of Biological Materials 115

change in the relaxation modulus in Δ*E*AGC3-AGC0 can be attributed to the material produced by chondrocytes after 21 days of cultivation. For the increase in gel modulus value in general, two possibilities would be responsible; (1) the increase in the modulus of gel network conforming polymer chain and (2) the increase in the cross-link density. The case (1) would result in a shortened relaxation time and the case (2) would provide a larger equilibrium modulus. In AGC3, the specimen seems to obtain a larger value of equilibrium modulus by the production of PCM-like material. According to Bushmann *et al*. (1995), the PCM-like material produced by chondrocytes was firstly deposited in the intra-network space of agarose gel in the vicinity of each chondrocyte surface in early cultures and with the cultivation period the material deposition area extended. From the stress relaxation results above, it is deduced that the PCM-like material in AGC3 could provide apparent cross-link points, which would contribute to the increase in the apparent cross-link density

In the previous section, viscoelastic properties of model systems of cartilaginous and osteogenesis tissues were shown. In this section, viscoelastic mechanical properties of actual

Bone has been often regarded mechanically as a composite material of hydrated organic matrix mainly composed of collagen and hydroxyapatite (HAp)-like mineral phase. It is thought that the pliant collagen is reinforced by stiff mineral particles, and, as a composite, the brittleness of the mineral is compensated for by the viscoelasticity of the collagen. Recently, the existence of non-collagenous glue proteins that connect mineralized collagen fibres has been revealed (Fantner *et al*., 2005). These organic phases have been suggested to be responsible for the toughness of bone. Because of this viscoelasticity of collagen fibres and non-fibrous proteins in the bone matrix, bone itself has noticeable viscoelasticity (Currey, 1965; Sasaki, 2000). Detailed experimental studies on the viscoelasticity of bone have been carried out only recently, despite the fact that it has been known for a long time that bone is viscoelastic (Currey, 1964; Lakes *et al*., 1979). According to the results of those experimental studies, the relaxation modulus of bone could not be adequately described by a single relaxation time and was found, unlike synthetic polymeric materials, to be thermo-

In our previous papers, as a new empirical equation for the description of stress relaxation of cortical bone, we proposed that stress relaxation of cortical bone could generally be described by a linear combination of two Kohlraush-Williams-Watts (KWW) functions (Iyo

<sup>0</sup> <sup>0</sup> 1 2 exp ( / ) 1 exp ( / ) , , , 1 , *<sup>β</sup> <sup>γ</sup> Et E A t <sup>τ</sup> A t <sup>τ</sup> <sup>A</sup> β γ* (18)

as compared with that in AGC0.

**5. Stress relaxation of bones** 

tissues, in particular bones, would be described.

**5.1. Relaxation modulus of cortical bone** 

rheologically complex (Vincent, 1982).

*et al*., 2004; Iyo *et al*., 2006),

**Figure 16.** Differences of relaxation modulus curves. EAGC0(t) EAG(t) (○), EAGC3(t) EAG(t) (●), and EAGC3(t) EAGC0(t) (┼). (From Sasaki et al., J. Biorheology 23, 95-101 (2009). With permission.)

and AGC3 and AG (◆),

$$
\Delta E\_{\rm AGC3-AG} = E\_{\rm AGC3} - E\_{\rm AG}
$$

where subscript letters AG, AGC0, and AGC3, indicate the relaxation modulus observed for AG, AGC0, and AGC3 specimens, respectively. The difference was remarkable in the gel viscoelasticity region. In Δ*E*AGC0-AG, as the modulus of AG is larger than that of AGC0, the difference was negative, showing the reduction of relaxation modulus by the seeding of chondrocytes. In the Figure, the modulus value difference looks to become levelled off in the short time region from 1 sec to more than 10 sec. The characteristic relaxation time of agarose gel has been reported to be around 102 sec (Watase et al., 1980), while that of a chondrocyte was determined to be 1 30 sec (Leipzig et al., 2005; Jones et al., 1999; Shieh et al., 2006). The behaviour of Δ*E*AGC0-AG in the short time region may be a reflection of the viscoelastic chondrocyte contribution as an elastic component in short time region of the relaxation process. Δ*E*AGC3-AGC0 represents the modulus increase by the PCM-like material production by chondrocytes. The modulus generally increased throughout the relaxation process. However, the increment also seems to be levelled off in the short time region as the reduction in Δ*E*AGC0-AG. Δ*E*AGC3-AG (◆) shows the re-increase in the relaxation modulus from AGC0 by the production of PCM-like material by chondrocytes. The plateau modulus value of AGC3 at times larger than 103 sec almost completely recovered the value of AG (the horizontal axis of Δ*E*=0), indicating that the gel modulus value once reduced by the chondrocyte seeding was increased with the PCM-like material production by chondrocytes after 21 days of cultivation. On the other hand, as Δ*E*AGC3-AG =Δ*E*AGC0-AG + Δ*E*AGC3-AGC0, the increment fromΔ*E*AGC0-AG was also shown to be significant in both the plateau (*t* larger than 103 sec) and the relaxation region (less than 10 sec) (\*p<0.05 and \*\*p<0.01, respectively). The change in the relaxation modulus in Δ*E*AGC3-AGC0 can be attributed to the material produced by chondrocytes after 21 days of cultivation. For the increase in gel modulus value in general, two possibilities would be responsible; (1) the increase in the modulus of gel network conforming polymer chain and (2) the increase in the cross-link density. The case (1) would result in a shortened relaxation time and the case (2) would provide a larger equilibrium modulus. In AGC3, the specimen seems to obtain a larger value of equilibrium modulus by the production of PCM-like material. According to Bushmann *et al*. (1995), the PCM-like material produced by chondrocytes was firstly deposited in the intra-network space of agarose gel in the vicinity of each chondrocyte surface in early cultures and with the cultivation period the material deposition area extended. From the stress relaxation results above, it is deduced that the PCM-like material in AGC3 could provide apparent cross-link points, which would contribute to the increase in the apparent cross-link density as compared with that in AGC0.

### **5. Stress relaxation of bones**

114 Viscoelasticity – From Theory to Biological Applications

and AGC3 and AG (◆),





0

(kPa)

5

10

15

20

**Figure 16.** Differences of relaxation modulus curves. EAGC0(t) EAG(t) (○), EAGC3(t) EAG(t) (●), and EAGC3(t) EAGC0(t) (┼). (From Sasaki et al., J. Biorheology 23, 95-101 (2009). With permission.)

0.1 1 10 100 1000 10000 100000 time (sec)

AGC3 AG AGC3 AG *E EE*

where subscript letters AG, AGC0, and AGC3, indicate the relaxation modulus observed for AG, AGC0, and AGC3 specimens, respectively. The difference was remarkable in the gel viscoelasticity region. In Δ*E*AGC0-AG, as the modulus of AG is larger than that of AGC0, the difference was negative, showing the reduction of relaxation modulus by the seeding of chondrocytes. In the Figure, the modulus value difference looks to become levelled off in the short time region from 1 sec to more than 10 sec. The characteristic relaxation time of agarose gel has been reported to be around 102 sec (Watase et al., 1980), while that of a chondrocyte was determined to be 1 30 sec (Leipzig et al., 2005; Jones et al., 1999; Shieh et al., 2006). The behaviour of Δ*E*AGC0-AG in the short time region may be a reflection of the viscoelastic chondrocyte contribution as an elastic component in short time region of the relaxation process. Δ*E*AGC3-AGC0 represents the modulus increase by the PCM-like material production by chondrocytes. The modulus generally increased throughout the relaxation process. However, the increment also seems to be levelled off in the short time region as the reduction in Δ*E*AGC0-AG. Δ*E*AGC3-AG (◆) shows the re-increase in the relaxation modulus from AGC0 by the production of PCM-like material by chondrocytes. The plateau modulus value of AGC3 at times larger than 103 sec almost completely recovered the value of AG (the horizontal axis of Δ*E*=0), indicating that the gel modulus value once reduced by the chondrocyte seeding was increased with the PCM-like material production by chondrocytes after 21 days of cultivation. On the other hand, as Δ*E*AGC3-AG =Δ*E*AGC0-AG + Δ*E*AGC3-AGC0, the increment fromΔ*E*AGC0-AG was also shown to be significant in both the plateau (*t* larger than 103 sec) and the relaxation region (less than 10 sec) (\*p<0.05 and \*\*p<0.01, respectively). The In the previous section, viscoelastic properties of model systems of cartilaginous and osteogenesis tissues were shown. In this section, viscoelastic mechanical properties of actual tissues, in particular bones, would be described.

#### **5.1. Relaxation modulus of cortical bone**

Bone has been often regarded mechanically as a composite material of hydrated organic matrix mainly composed of collagen and hydroxyapatite (HAp)-like mineral phase. It is thought that the pliant collagen is reinforced by stiff mineral particles, and, as a composite, the brittleness of the mineral is compensated for by the viscoelasticity of the collagen. Recently, the existence of non-collagenous glue proteins that connect mineralized collagen fibres has been revealed (Fantner *et al*., 2005). These organic phases have been suggested to be responsible for the toughness of bone. Because of this viscoelasticity of collagen fibres and non-fibrous proteins in the bone matrix, bone itself has noticeable viscoelasticity (Currey, 1965; Sasaki, 2000). Detailed experimental studies on the viscoelasticity of bone have been carried out only recently, despite the fact that it has been known for a long time that bone is viscoelastic (Currey, 1964; Lakes *et al*., 1979). According to the results of those experimental studies, the relaxation modulus of bone could not be adequately described by a single relaxation time and was found, unlike synthetic polymeric materials, to be thermorheologically complex (Vincent, 1982).

In our previous papers, as a new empirical equation for the description of stress relaxation of cortical bone, we proposed that stress relaxation of cortical bone could generally be described by a linear combination of two Kohlraush-Williams-Watts (KWW) functions (Iyo *et al*., 2004; Iyo *et al*., 2006),

$$E\{t\} \ = E\_0\left\{ A \exp\left[ - \left( t \,/\, \tau\_1\right)^{\beta} \right] + \left( 1 - A \right) \exp\left[ - \left( t \,/\, \tau\_2\right)^{\gamma} \right] \right\} \ , \; \left[ 0 < A , \beta , \gamma < 1 \right] \ \tag{18}$$

where *E*0 is the initial modulus value, *E*(0).*τ*1 and *τ*2 ( *τ*1) are characteristic times of the relaxation processes, *A* is the fractional contribution of the fast relaxation to the whole relaxation process, and *β* and *γ* are parameters describing the shape of the relaxation modulus. It has been revealed that the first term represents the relaxation in the collagen matrix in bone and the second term is related to the change in a higher-order structure of bone that is responsible for the anisotropic mechanical properties (Iyo *et al*., 2004). It seems to be possible to relate the viscoelastic properties and the hierarchical structure of bone by investigating these mechanical parameters. In this section, the effect of the structural anisotropy on the relaxation modulus of cortical bone would be presented, where relaxation parameters in eq. 18 will be analyzed in conjunction with the structural information of bone.

Viscoelastic Properties of Biological Materials 117

1, no more than 100 sec and a slow

2, in the order of 106 sec. In this

revealed to be expressed by a combination of two relaxation processes according to eq. 18: a

experiment, fitting of the average data to eq. 18 was performed. The relaxation modulus results obtained were described well by eq. 18. The relaxation parameters determined by the fitting, as well as the coefficient of determination, *R*2, and the mean square error, *s*, are listed in Table 2. The average initial value of the relaxation Young's modulus, *E*(0), of P was

**Figure 18.** Relaxation modulus curves for P (○) and N (●) bone specimens. (a) Decomposition into the

Magnification of decomposed fast relaxation modulus curves. (Iyo et al., J. Biomechanics 37, 1433-1437

fast relaxation and slow relaxation processes are shown in lines for each relaxation curve. (b)

(2004). With permission.)

fast process (KWW1 process) with a relaxation time,

process (KWW2 process) with a relaxation time,

significantly larger than that of N (p<0.05, ANOVA).

The bone samples used in this study were obtained from the mid-diaphysis of a 36-monthold bovine femur. Optical microscopic examination showed that all of the samples were generally plexiform but partly transformed into Haversian bone. The samples were cut using a band saw under tap water. In order to examine Young's moduli parallel and normal to the BA, we cut out specimen plates whose longer axes were parallel and normal to the BA, respectively. The cut sections were shaped into rectangular plates with approximate dimension of 0.5 cm (width), 3.2 cm (length), and 0.1 cm (thickness) by using emery paper under tap water. A plate with the longer edge parallel to the BA was coded specimen P, and a plate with the longer edge normal to the BA was coded specimen N. Fig. 17 shows the geometry of specimens.

**Figure 17.** Bone specimens cut parallel (P) and perpendicular (N) to the bone axis.(Iyo et al., J. Biomechanics 37, 1433-1437 (2004). With permission.)

Fig. 18 shows the average values of relaxation Young's modulus, *E*(*t*), plotted against time for specimens of P and N. At several points, standard errors are shown by vertical bars and are listed in Table 2. As mentioned above, stress relaxation of cortical bone has been revealed to be expressed by a combination of two relaxation processes according to eq. 18: a fast process (KWW1 process) with a relaxation time, 1, no more than 100 sec and a slow process (KWW2 process) with a relaxation time, 2, in the order of 106 sec. In this experiment, fitting of the average data to eq. 18 was performed. The relaxation modulus results obtained were described well by eq. 18. The relaxation parameters determined by the fitting, as well as the coefficient of determination, *R*2, and the mean square error, *s*, are listed in Table 2. The average initial value of the relaxation Young's modulus, *E*(0), of P was significantly larger than that of N (p<0.05, ANOVA).

116 Viscoelasticity – From Theory to Biological Applications

information of bone.

geometry of specimens.

where *E*0 is the initial modulus value, *E*(0).*τ*1 and *τ*2 ( *τ*1) are characteristic times of the relaxation processes, *A* is the fractional contribution of the fast relaxation to the whole relaxation process, and *β* and *γ* are parameters describing the shape of the relaxation modulus. It has been revealed that the first term represents the relaxation in the collagen matrix in bone and the second term is related to the change in a higher-order structure of bone that is responsible for the anisotropic mechanical properties (Iyo *et al*., 2004). It seems to be possible to relate the viscoelastic properties and the hierarchical structure of bone by investigating these mechanical parameters. In this section, the effect of the structural anisotropy on the relaxation modulus of cortical bone would be presented, where relaxation parameters in eq. 18 will be analyzed in conjunction with the structural

The bone samples used in this study were obtained from the mid-diaphysis of a 36-monthold bovine femur. Optical microscopic examination showed that all of the samples were generally plexiform but partly transformed into Haversian bone. The samples were cut using a band saw under tap water. In order to examine Young's moduli parallel and normal to the BA, we cut out specimen plates whose longer axes were parallel and normal to the BA, respectively. The cut sections were shaped into rectangular plates with approximate dimension of 0.5 cm (width), 3.2 cm (length), and 0.1 cm (thickness) by using emery paper under tap water. A plate with the longer edge parallel to the BA was coded specimen P, and a plate with the longer edge normal to the BA was coded specimen N. Fig. 17 shows the

**Figure 17.** Bone specimens cut parallel (P) and perpendicular (N) to the bone axis.(Iyo et al., J.

Fig. 18 shows the average values of relaxation Young's modulus, *E*(*t*), plotted against time for specimens of P and N. At several points, standard errors are shown by vertical bars and are listed in Table 2. As mentioned above, stress relaxation of cortical bone has been

Biomechanics 37, 1433-1437 (2004). With permission.)

**Figure 18.** Relaxation modulus curves for P (○) and N (●) bone specimens. (a) Decomposition into the fast relaxation and slow relaxation processes are shown in lines for each relaxation curve. (b) Magnification of decomposed fast relaxation modulus curves. (Iyo et al., J. Biomechanics 37, 1433-1437 (2004). With permission.)

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.

Viscoelastic Properties of Biological Materials 119

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

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.)

**mineral fraction** 


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

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


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

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

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 mechanically isotropic.

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 bone.
