A Continuous Wave Technique for the Measurement of the Elastic Properties of Cortical Bone

• Journal List
• HHS Author Manuscripts
• PMC7809538

J Biomech. Author manuscript; available in PMC 2021 Jan 15.

Published in final edited form as:

PMCID: PMC7809538

NIHMSID: NIHMS760669

Connection between elastic and electrical properties of cortical bone

X. Gao

¹Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Loughborough, UK.

I. Sevostianov

²Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, USA.

Abstract

The paper focuses on the connection between elastic and electrical properties of cortical bone. Both these properties are governed by microstructure that consists of several pore systems filled with mechanically soft and electrically conductive tissue. Microstructural changes induced by aging, various diseases, microgravity conditions etc. lead to variation in both properties. The paper address the problem of evaluation of the changes in mechanical performance (decrease in Young's moduli) via monitoring electrical conductivity. The theoretical results are verified experimentally.

Keywords: Cortical bone, elastic stiffness, electrical conductivity, cross-property connections

1. Introduction

We establish cross-property connections between mechanical and electrical properties of cortical bone. Such connections relate changes in different physical properties caused by the presence and/or development of certain microstructure. To the best of our knowledge, existence of the cross-property connections has been first recognized by Bristow (1960) who considered metals containing multiple randomly oriented microcracks. The existence of explicit quantitative cross-property connections between two physical properties depends on the possibility to express them in terms of the same or sufficiently similar microstructural parameter (Kachanov and Sevostianov, 2005). Critical review of the existing results on cross-property connections with detailed analysis of various applications is given by Sevostianov and Kachanov (2009).

In the context of properties of bone, the observations of cross-property connections are mostly of qualitative nature (see, for example, Sierpovska et al, 2006, and V. Sevostianov, 2014). The main challenge in developing the quantitative cross-property connections for cortical bone is related to the fact that the elastic properties are mostly determined by the porous dense tissue - effect of the presence of biological fluids and soft tissue in the porous space plays only a minor role. On the other hand, namely these constituents are responsible for bone's electrical conductivity. Cross-property connection for materials of this kind has been derived by Berrymann and Milton (1988) in the form of inequality for the case when both the constituents as well as the overall material are isotropic.

In the text to follow, we express the tensor of elastic compliances in terms of the overall conductivity tensor in the closed form using the similarity between the microstructural parameters governing the two said properties. The results are validated experimentally and significant correlation between experimental and theoretical results has been observed.

2. Modelling of the microstructure of cortical bone

Overall electrical and elastic properties of cortical bone are largely determined by their microstructure comprising a large number of interconnected diverse pores filled with electrically conductive biological liquids and soft tissue - blood, lymph, nerve tissue etc.

In our analysis, based on description given by Martin & Burr, (1989), Currey, (2002) and Fung, (1993) sketched in Figure 1, we model cortical bone as a porous elastically transversely isotropic material of low electrical conductivity comprised of three systems of pores filled with elastically soft and electrically highly conductive tissue. Haversian canals are modeled as a system of parallel cylindrical pores (strongly prolate spheroidal inhomogeneities, Mura, 1987), in which their axes of geometrical symmetry coincide with the axes of the material symmetry of the matrix (the axis of transverse-isotropy, x ₃ ). The osteocyte lacunae, modelled as oblate spheroidal cavities in a plane of transverse-isotropy (planes normal to Haversian canals). Canaliculi and Volkman's canals are treated as a set of thin cylindrical pores, with the axes of rotation perpendicular to the axis of transverse-isotropy of the matrix, which means they are lying in planes of transverse-isotropy and are randomly oriented in these planes.

Microstructure of cortical bone used in the present model: it is formed by osteons surrounding Haversian canals that contain blood and lymph vessels and nerves. Volkman's canals and canaliculi are randomly oriented in the planes orthogonal to the Haversian canals. The lamellae in osteons contain osteocytes located in oblate spheroidal pores (lacunae)

The aspect ratios of the spheroids have been estimated as follows:

• For Haversian canals we calculated the aspect ratio as γ = 3h/4R, where h is the height of the osteon (we used average value of 4 mm) and R is the radius of the Haversian canal (we used 25 μm). Thus calculated aspect ratio γ =120 allows one to preserve the volume of the inhomogeneity with fixed radius.

• Similarly, for Volkman's canals and canaliculi (accounted together) we used for h the difference between the radius of the osteon and the radius of the Haversian canal (125 μm) and for R, we used average radius of the canals (1.5 μm). The calculated aspect ratio is 80.

• The aspect ratio of the lacunae was taken as 0.2 in accordance with the observations of Currey (1962, 2002).

Although pores of different types have very different sizes, their partial porosities are comparable. Indeed, 1 mm³ of the bone typically contains about 25,000 of osteocyte lacunae with the total surface area 5 mm²/mm³, about 106 canaliculi with the total surface area of 160 mm²/mm³ and about 20 Haversian canals with the total surface area of 3 mm²/mm³ (Martin and Burr, 1989). These numbers imply partial porosities for each of these types in the range 0.075 – 0.120. Thus, it does not seem adequate to attribute entire porosity of the cortical bone to the Haversian canals as in the model proposed by Dong and Guo (2004, 2006).

As stated above, the channels and pores contain blood and lymph vessels, nerve fibers, and living cells. The influence of these fluids and soft tissues can be neglected in the context of the overall elastic response. Indeed, elastic stiffness of the mineralized tissue is of the order of several GPa, while Young's moduli of blood vessels are of the order of 10MPa at pressures of 100mm Hg, (Wesly et al., 1975). Young's moduli for the nerve tissue and for the cells are of the order of 4–10 MPa (Beel et al., 1984), and of 1kPa, (Theret et al., 1988), correspondingly. Thus, in the context of elastic properties, we treat the pores as empty ones embedded in the dense tissue that represents a combination of collagen fibers (protein), and hydroxyapatite Ca₁₀(PO₄)₆(OH)₂ crystals (mineral), (Katz, 1980). Mineralized tissue possesses transversally isotropic mechanical properties (Currey et al., 2001); in our calculations, we used the extrapolated data of Dong and Guo (2006) for dense tissue (Table 1) as well as our own measurements. In the context of electrical properties, we model the mineralized tissue as the isotropic background of very low conductivity, thus ignoring the bone matrix anisotropy since electrical conductivity of the matrix being different in different directions is still very small. The electrical conductivity of the bone matrix was extrapolated from the measurements to zero porosity and taken for calculation of cross-property coefficients as k ₀ = 3.841mS /m (see Section 6). The effect of the conductive soft tissue on the overall electrical properties of the bone is dominant; for the conductivity of the soft tissue we used k ₁ =1.5 S /m according to Hirsh et al (1950), Visser (1992), and Hoetink et al (2004).

Table 1.

The transversely isotropic elastic constants of cortical bone calculated based on mechanical testing done by Dong & Guo, (2004).

E 1(GPa)	E 3(GPa)	v 31	G 12(GPa)	G 13(GPa)
11.419	24.16	0.38	3.9	6.54

3. Background material: property contribution tensors

We consider a homogeneous elastic material (matrix), with the compliance tensor S ⁰ and electrical resistivity r ⁰ containing an inhomogeneity of volume V ₁ with the compliance tensor S ¹ and electrical resistivity r ¹. Compliance contribution tensors have been first introduced in the context of ellipsoidal pores and cracks in isotropic material by Horii and Nemat-Nasser (1983). For general case of ellipsoidal elastic inhomogeneities embedded in an isotropic matrix, these tensors were formally defined and calculated by Sevostianov and Kachanov (1999, 2002). Sevostianov et al (2005) calculated components of this tensor for a spheroidal inhomogeneity embedded in a transversely-isotropic material. The compliance contribution tensor of the inhomogeneity is a fourth-rank tensor H that gives the extra strain (per reference volume V ) due to its presence:

Δ ε = V 1 V H : σ ∞ , or , in components , Δ ε i j = V 1 V H i j k l σ k l ∞

(3.1)

where σ k l ∞ are remotely applied stresses that are assumed to be uniform within V in the absence of the inhomogeneity.

In the case of multiple inhomogeneities, the effective compliance, calculated in the non-interaction approximation (NIA) is given by

S i j k l = S i j k l 0 + 1 V ∑ m V m H i j k l ( m )

(3.2)

For an ellipsoidal inhomogeneity, its compliance contribution tensor is expressed in terms of Hill's tensor P _ijkl (Hill, 1963, Walpole, 1969) as

H = [ ( S 1 − S 0 ) − 1 + C 0 : ( J − P : C 0 ) ] − 1 ,

(3.3)

Hence, the problem of calculating the components of H _ijkl for an ellipsoidal inhomogeneity is reduced to the calculation of Hill's tensor. The expressions for the components P _ijkl of this tensor for a spheroidal inhomogeneity aligned with the axis of a transversely-isotropic material were derived by Sevostianov et al (2005) in the following form

P i j k l = p 1 T i j k l 1 + p 2 T i j k l 2 + p 3 T i j k l 3 + p 4 T i j k l 4 + p 5 T i j k l 5 + p 6 T i j k l 6

(3.4)

where basic tensors T i j k l m and coefficients p ₁ are given in the Appendix. Using formula (3.3) and the rule of multiplication for tensors represented in terms of standard tensor basis allow one to calculate components of the compliance contribution tensor. Figures 2a and ​ 2b illustrate dependence of the components of tensors P = ∑ k = 1 6 p k T ( k ) and H = ∑ k = 1 6 h k T ( k ) on the aspect ratio of the spheroid (with the elastic constants of the matrix given in Table 1).

Dependence on the aspect ratio of inhomogeneity of (a) Components of Hill's tensor; (b) components of the compliance contribution tensor (the elastic stiffnesses of the matrix material are given in Table 1); (c) components of the compliance contribution tensor of the pore embedded in the isotropic material constituting the best fit for transversely-isotropic cortical bone; (d) parameters A ₁ and A ₂ entering expression for the effective electrical conductivity.

For inhomogeneities not aligned with the symmetry axis of the matrix, like Volkman's canals and canaliculi, we use analytical approximation for components of tensor H _ijkl following approach proposed by Saadat et al. (2012). We first find the best isotropic approximation for a transversely isotropic tensor of elastic stiffness C _ijkl (see Fedorov, 1968) given by

λ 0 δ i j δ k l + G 0 ( δ i k δ l j + δ i l δ k j )

(3.5)

where

G 0 = ( 3 C i k i k − C i i k k ) / 30 , λ 0 = ( 2 C i i k k − C i k i k ) / 15

(3.6)

Using this best fit isotropy, we can calculate the components of the compliance contribution tensor for a spheroidal inclusion with semi axis a ₁=a ₂ =a, and a ₃ embedded in the matrix characterized by elastic constants G ₀ and λ ₀. For a pore or very soft inclusion coefficients in the representation H = ∑ k = 1 6 h k T ( k ) take the form (Sevostianov et al, 2006)

$$\begin{gathered} h 1 = κ ( f 0 − f 1 ) 2 G 0 ( 4 κ − 1 ) [ 2 κ ( f 0 − f 1 ) − ( 4 κ − 1 ) f 0 2 ] ; h 2 = 1 2 G 0 [ 1 − ( 2 − κ ) f 0 − κ f 1 ] ; \\ h 3 = h 4 = − ( 2 κ f 0 − f 0 + 2 κ f 1 ) 4 G 0 ( 4 κ − 1 ) [ 2 κ ( f 0 − f 1 ) − ( 4 κ − 1 ) f 0 2 ] ; \\ h 5 = 4 4 G 0 [ f 0 + 4 κ f 1 ] ; h 6 = 4 κ − 1 − 6 κ f 0 + 2 f 0 − 2 κ f 1 4 G 0 ( 4 κ − 1 ) [ 2 κ ( f 0 − f 1 ) − ( 4 κ − 1 ) f 0 2 ] \end{gathered}$$

(3.7)

where

κ = λ 0 + G 0 λ 0 + 2 G 0 ; f 0 = γ 2 ( 1 − g ) 2 ( γ 2 − 1 ) , f 1 = γ 2 4 ( γ 2 − 1 ) 2 [ ( 2 γ 2 + 1 ) g − 3 ]

(3.8)

and the shape factor g is expressed in terms of the aspect ratio γ = a ₃/a as follows

g ( γ ) = { 1 γ 1 − γ 2 arctan 1 − γ 2 γ , oblate shape ( γ < 1 ) 1 2 γ γ 2 − 1 ln γ + γ 2 − 1 γ − γ 2 − 1 , prolateshape ( γ > 1 )

(3.9)

The best isotropic approximation for the elastic constants in Table 2 is provided by G ₀ = 5.28GPa and λ ₀ =10.3GPa. The accuracy of the approximation is 12%. For these elastic constants, dependences of the components h _i on the aspect ratio of the spheroid is shown in Figure 2c.

Table 2.

Components of resistivity contribution tensor for different systems of pores (direction 3 is along the Haversian canals).

	1 k 11 0 ∑ K 11 = 1 k 22 0 ∑ K 22	1 k 33 0 ∑ K 33
Haversian canals	−2.001	−3.213×103
Osteocyte lacunae	−4.674	−8.016
Canaliculi and Volkman's canals	−7.86×102	−2.001

Similar approach can be applied for the electrical conductivity problem for a material containing an inhomogeneity. Assuming a linear relation between remotely applied electrical field E and the electrical current density J (electrical current per representative volume V ), the change in E required to maintain the same electrical current density if the inhomogeneity is introduced is:

where the symmetric second-rank tensor R is called the resistivity contribution tensor of an inhomogeneity. For inhomogeneity of spheroidal shape, tensor R has the following form:

R = − r 0 ⋅ [ A 1 ( I − nn ) + A 2 nn ]

(3.11)

where

and function f ₀ is given by (3.8). Dependences of these coefficients on the inhomogeneity aspect ratio are shown in Figure 2d.

In the case of multiple inhomogeneities, effective conductivity in the framework of NIA is

k i j = [ r i j 0 + 1 V ∑ m V m R i j ( m ) ] − 1

(3.13)

4. Overall properties of cortical bone and cross-property connections

4.1. Electrical resistivity

Geddes and Baker (1967) reported that the first experimental measurements of the electrical resistivity of cortical bone have been done by Osswald (1937). Considerable interest to the electrical properties of bone appeared after discovery of its piezoelectric properties by Fukada and Yasuda (1957).

At the present time the database of electrical properties of bone tissue is quite wide (see, for example, review of Gabriel et al, 1996 or book of Behari, 2009). However, to the best of our knowledge, the only quantitative micromechanical model connecting electrical properties of cortical bone with its microstructure has recently been proposed by Casas and Sevostianov (2013) using the model described in Section 2 and the concept of the resistivity contribution tensor.

To calculate effective electrical properties, we use non-interaction approximation (NIA). This approximation is accurate at low concentration of inhomogeneities (up to 15–20%, dependent on the properties contrast, Sevostianov and Sabina, 2008). Besides, the primary goal of the present work is to establish cross-property connections between elastic and conductive properties of cortical bone and, as hypothesized by Bristow (1960) and confirmed by a number of experiments (see review of Sevostianov and Kachanov, 2009), interaction affects both elastic and conductive properties of heterogeneous materials in a similar way. In the framework of NIA, each inhomogeneity is treated as one subjected to the same remotely applied electrical field. Contributions of the inhomogeneities into the change in the current density can be treated separately and the overall tensor of the effective conductivity of a material containing multiple inhomogeneities is given by (3.17). Table 2 provides the values of the components of ∑ R i j for three systems of pores filled with electrically conductive tissue that represent lymph and blood vessels, nerves, etc.

4.2. Elastic compliances

In contrast with the electrical conductivity, a number of analytical and semi-analytical models have been developed for overall elastic properties of cortical bone. To the best of our knowledge most of the existing analytical models of cortical bone either consider the dense bone matrix as isotropic, with any anisotropy due to oriented pores only, or they do not include a wide variety of pore type and size and account for Haversian canals only.

The first micromechanical model of bone has been proposed by Katz (1981). This model emphasizing the hierarchical structure of cortical bone accounts for the porous space modeled as a set of parallel Haversian canals and the microstructure of the dense mineralized tissue modeled as a fiber reinforced composite. Later, Sevostianov & Kachanov (1998, 2000) applied the concept of compliance contribution tensors to the microstructure accepted in the present paper (Section2) and estimated the impact of the complex porous microstructure on the overall elastic constants. The properties of the bone matrix were treated in this paper as isotropic. Dong & Guo (2006) modeled cortical bone as a set of parallel circular Haversian canals embedded in a transversely-isotropic matrix. Later, Martinez-Reina et al., (2010) proposed a numerical model that also treats cortical bone as a transversely isotropic material and expands the porosity to include canaliculi on the second level as well as takes into consideration the effects of water within the pores. The main advantage of this model that sets it apart is the ability to vary the mineral content of bone. Parnell et al. (2010) took the parallel fiber approach one step further by considering 3-phases - a matrix, pores, and reinforcing particles. Model of Nikolov et al. (2008) is similar to one of Dong & Guo (2006) - it accounts for Haversian canals only as the elements of the porous space within the transversely-isotropic bone matrix. We also have to mention several papers where hierarchical microstructure of cortical bone is considered (Hellmich et al., 2002, 2004; Fritsch et al. ,2008; Deuerling et al., 2009). In the present paper, we follow recent results of Salguero et al (2014) where three systems of pores described in Section 2 are embedded in a transversely-isotropic matrix. The results of the calculations of ∑ H i j k l for different pore systems are given in Table 3. Comparison of the model with experimental data of Dong & Guo (2006) is given by Salguero et al (2014) (in the framework of non-interaction approximation) and by Vilchevskaia and Sevostianov (2015) (in the framework of Maxwell homogenization scheme).

Table 3.

Components of the compliance contribution tensor (GPa ⁻¹) for different systems of pores (direction 3 is along the Haversian canals)

	Cylinders normal to the plane of isotropy (Haversian Canals)	Oblate spheroids (Lacunae)	Cylinders laying in the plane of isotropy (Volkman's Canals & Canaliculi)
ΣH 1111	0.251	0.108	0.143
ΣH 3333	0.041	0.324	0.179
ΣH 1122	−0.076	−0.043	−0.043
ΣH 1133	−0.016	−0.03	−0.05
ΣH 1212	0.163	0.075	0.093
ΣH 1313	0.076	0.163	0.115

4.3. Cross-property connections for cortical bone

In the case of the overall transverse isotropy, the change in the elastic compliance tensor has the following structure

S − S 0 = V * V [ W 1 II + W 2 t r J + W 3 ( Inn + nnI ) + W 4 ( J ⋅ nn + nn ⋅ J ) + W 5 nnnn ]

(4.1)

where coefficients W _i are expressed in terms of coefficients h _i as follows:

$$\begin{gathered} W 1 = h 1 − h 2 / 2 ; W 2 = h 2 ; W 3 = 2 h 3 + h 2 − 2 h 1 ; W 4 = h 5 − 2 h 2 \\ W 5 = h 6 − h 1 + h 2 / 2 − 2 h 3 − h 5 \end{gathered}$$

(4.2)

The key finding of Sevostianov and Kachanov (2002) is that tensor in the right hand side of (4.1) can be approximated with good accuracy by the following one:

V * V [ B 1 II + B 2 J ︸ i s o t r o p i c t e r m s + B 3 ( Ω I + I Ω ) + B 4 ( Ω ⋅ J + J ⋅ Ω ) ]

(4.3)

where B _i are scalar coefficients dependent on the inclusion shape and on the matrix-inclusion elastic contrast. To get this formula, one has to construct a fictitious compliance contribution tensor H ^ of an individual inhomogeneity, with coefficients h ^ i in the tensor basis that are obtained from h _i by multiplication of h _i by either (1+δ) or (1−δ), where

$$\begin{gathered} δ = h 6 + h 1 + h 2 / 2 − 2 h 3 − h 5 | h 6 | + | h 1 | + | h 2 | / 2 + 2 | h 3 | + | h | 5 : \\ h ^ 1 = h 1 ( 1 − δ sign h 1 ) h ^ 3 = h 3 ( 1 + δ sign h 3 ) \\ h ^ 2 = h 2 ( 1 − δ sign h 2 ) h ^ 5 = h 5 ( 1 + δ sign h 5 ) \\ h ^ 6 = h 6 ( 1 − δ sign h 6 ) \end{gathered}$$

(4.4)

The error of this approximation, as estimated by the norm max i j k l , H i j k l ≠ 0 | ( H i j k l − H ^ i j k l ) / H i j k l |, is equal to |δ|. Figure 3 provides an illustration of the accuracy of the representation (5.3) for a spheroidal inhomogeneity embedded in a transversely isotropic matrix in dependence on the spheroid aspect ratio: solid line corresponds to a spheroid aligned with the axis of symmetry of the matrix (Haversian canals and lacunae); dashed line corresponds to a spheroid embedded in a best fit isotropic matrix (our modelling of Volkman's canals and canaliculi). Coefficients B _i are expressed as

B 1 = h ^ 1 − h ^ 2 / 2 ; B 2 = h ^ 2 ; B 3 = 2 h ^ 3 + h ^ 2 − 2 h ^ 1 ; B 4 = h ^ 5 − 2 h ^ 2

(4.5)

Accuracy of the approximate representation of the pore compliance tensor H as a function of pore aspect ratio for transversely isotropic material given in Table 1 (solid line) and its best fit isotropic approximation (dashed line).

Sevostianov and Kachanov (2002) shown that, if inhomogeneity' aspect ratios are not correlated with either orientations of the inclusions or their volumes (volumes and orientations may be correlated), tensor S can be expressed in terms of the second rank symmetric tensor (they called it "inclusions' concentration tensor") ω = 1 V ∑ k ( V * nn ) ( k ) as

S = S 0 + c ( b 1 II + b 2 J ) + [ b 3 ( ω I + I ω ) + b 4 ( ω ⋅ J + J ⋅ ω ) ] ; b i = ∫ 0 ∞ B i ( γ ) f ( γ ) d γ

(4.6)

where f(γ) is the shape distribution density. The trace of inclusions' concentration tensor t r ω = ( 1 / V ) ∑ V * is the volume fraction of inhomogeneities c. Taking into account that overall electrical conductivity can be expressed in terms of the same tensor:

k − 1 ⋅ k 0 − I = a 1 c I + a 2 ω ; a i = ∫ 0 ∞ A i ( γ ) f ( γ ) d γ

(4.7)

the following cross-property connection can be explicitly obtained between elastic compliances and electrical conductivities of cortical bone by elimination of ω :

$$\begin{gathered} ( S − S 0 ) = [ b 1 a 2 − b 3 a 1 a 2 ( a 2 + 3 a 1 ) II + b 2 a 2 − b 4 a 1 a 2 ( a 2 + 3 a 1 ) J ] [ t r ( k 0 K − 1 ) − 3 ] \\ + b 3 k 0 a 2 [ ( K − 1 − I ) I + I ( K − 1 − I ) ] + b 4 k 0 a 2 [ ( K − 1 − I ) ⋅ J + J ⋅ ( K − 1 − I ) ] \end{gathered}$$

(4.8)

In particular we can write the following relation between Young's modulus and conductivity in the direction of Haversian canals:

( E 3 0 − E 3 k 0 E 3 0 ) = [ ( b 1 + b 2 ) a 2 − ( b 3 + b 4 ) a 1 a 2 ( a 2 + 3 a 1 ) ] [ t r ( K − 1 ) − 3 k 0 ] + 2 ( b 3 + b 4 ) a 2 ( 1 k 33 − 1 k 0 )

(4.9)

Note that for specimens with unknown concentration of inhomogeneities trace of the resistivity tensor is required to evaluate Young's modulus in any direction. If volume fraction c of the inhomogeneities is known independently, (5.14) is simplified as

$$\begin{gathered} ( E 3 0 − E 3 E 3 ) = c [ ( b 1 + b 2 ) a 2 − ( b 3 + b 4 ) a 1 a 2 ( a 2 + 3 a 1 ) ] + 2 ( b 3 + b 4 ) a 2 ( k 0 − k 33 k 33 ) \\ = M c + N ( k 0 − k 33 k 33 ) \end{gathered}$$

(4.10)

In the next section we validate expression (4.10) by comparison with experimental data.

5. Experimental verification

5.1. Specimens preparation

Specimens were taken from two bovine femurs belonging to the same animal (two years old female cow with live weight approximately 500 kg). Twenty rectangular sections with the length of more than 60 mm were cut from the middle part of the femurs by diamond saw at low speed. Then, the sections were cut into two main pieces – the thin pieces for microscopy and the long specimens of the length of 55 mm for electrical and mechanical testing, as shown in the upper image in Fig. 4a. Specimens were polished with, in turn, 120 and 600 grid sand paper until the shape of rectangular parallelograms is reached (lower image in Fig. 4a). Finally, specimens were immersed into physiological solution (0.9% NaCl solution) at room temperature for 24 hrs (Fig. 4b).

(a) Cortical bone specimens;(b) storage of polished specimens (c) schematic diagram of rectangular parallelograms specimen;(d) TCS SP5 II Broadband confocal microscope to obtain morphology of cortical bone; (e) HP 4338B milliohmmeter to measure electrical resistance of wet cortical bone (f) three-point bending testing to measure Young's modulus of wet cortical bone.

5.2. Porosity evaluation

The porosity of specimens was evaluated by the ratio between the volume of liquid and total volume. The dimensions of the specimens were measured by caliper with accuracy of ± 0.001 mm (Fig. 4c). The volume of liquid was calculated by the ratio between the difference of wet mass m _w and dry mass m _d of specimens, and the density of water. The porosity c is:

c = 4 ( m w − m d ) / ρ ( W 1 + W 2 ) ( H 1 + H 2 ) L

(5.1)

5.3. Microscopy

Microstructure of the cortical bone was studied using confocal microscope (TCS SP5 II Broadband confocal microscope, Leica, Germany, Fig. 4d). The thin pieces were polished and macerated by boiling to remove bone marrow and sonicating in a 350°C water bath to remove polishing grit. Then, before the microscopic study, the polished specimens were soaked for a few hours in glutaraldehyde solution, to increase the florescence of the bone required for the confocal microscopy.

5.4. Electrical resistivity measurements

The electrical resistance R of specimens in longitudinal direction was measured by four-point-resistance-measurement (HP 4338B milliohmmeter, HP, USA, Fig. 4e) immediately after taken out of the saline solution. Two alligator chips griped at each edge of the specimens, respectively. The other two clips griped at the specimen to measure the electrical resistance of gauge length between them. To reduce the error in measurement, the electrical resistance of each wet specimen was averaged over three results obtained from three gauge length l ₁, l ₂, and l ₃, as shown in Fig. 4e. The conductivity k in longitudinal direction was calculated by:

where A is the cross-sectional area of the specimen.

5.5. Young's modulus measurements

The Young's modulus in the longitudinal direction of the specimens was measured by three-point-bending test. The specimens were placed onto a custom-made platform with support span a of 35.64 mm, as shown in Fig. 4f. The indenter targeted to the middle point of the support span. A universal testing machine (Instron 5882, Instron, USA) provided precise displacement and force measurement. A pre-load of 20 N was set to ensure that specimens were fully loaded at the beginning of the tests. Specimens were loaded under a load rate of 0.04 mm/min till failure. Spraying was carried out during the tests to keep the specimens wet. The Young's modulus E in longitudinal direction was calculated by:

E 3 = 4 a 3 ( W 1 + W 2 ) ( H 1 + H 2 ) 3 m

(5.3)

where m is slope of the force-displacement curve, found by linear curve fitting.

Results of the Young's modulus and electrical conductivity measurements are presented in Figure 5 a, ​ b and in Table 4. Figure 5c shows dependence of the Young's modulus on the electrical conductivity (experimental cross-property connection).

(a) Dependence of the Young's modulus E ₃ of the cortical bone on the porosity; (b) Dependence of the longitudinal conductivity k ₃₃ on the porosity; (c) Young's modulus E ₃ as function of the longitudinal conductivity k ₃₃ of the cortical bone.

Table 4.

Measured values of the electrical conductivity and Young's modulus of cortical bone and comparison of measured ( E 3 0 − E 3 ) / E 3 with calculated according cross-property connection (5.15)

Porosity	k 3 (mS/m)	E 3 (GPa))	(k 0 - k 3)/ k 3	( E 3 0 − E 3 ) / E 3 measured	( E 3 0 − E 3 ) / E 3 calculated
6.452423	11.93181	13.96869	−0.67809	0.839829	0.827265
7.08277	14.28688	13.99619	−0.73115	0.836214	0.972262
7.201798	13.17711	11.8434	−0.70851	1.169985	1.158708
7.283548	14.74553	13.39724	−0.73951	0.918305	1.060036
7.370781	15.13435	12.43235	−0.74621	1.067187	1.083274
7.470491	14.79972	13.77267	−0.74047	0.866014	1.175035
7.493851	13.86830	11.85249	−0.72304	1.16832	1.274873
7.606253	15.82473	10.98545	−0.75728	1.339459	1.180055
7.724463	15.78201	12.11348	−0.75662	1.121603	1.258909
7.777196	14.89987	12.03763	−0.74221	1.134972	1.36283
8.065902	13.60026	10.45751	−0.71758	1.457565	1.667568
8.238407	15.75324	9.271674	−0.75618	1.771883	1.589998
8.732316	16.30992	9.72629	−0.7645	1.642323	1.865571
8.838663	16.11964	11.11373	−0.76172	1.312455	1.947171
9.063449	15.45601	8.56843	−0.75149	1.999383	2.140859
9.071876	16.75523	9.355984	−0.77076	1.746905	2.052408
9.402075	17.17566	7.865861	−0.77637	2.267284	2.236408
9.441977	17.39841	9.282241	−0.77923	1.768728	2.248002
10.31277	18.93474	7.730121	−0.79715	2.324657	2.718077
10.40556	18.41383	7.202872	−0.79141	2.568021	2.805404

We can now compare the theoretically derived cross-property connection (4.10) with experimentally obtained results. The values of E 3 0 and k ₀ have been taken from the extrapolation of data in Fig. 5a and ​ b to zero porosity as

E 3 0 = 25.68 G P a ; k 0 = 3.841 m S / m

(5.4)

For these constants, assuming equal partial porosities of three types of pores, coefficients M and N, entering (4.10) are

Figure 6 provides comparison between experimentally measured Young's modulus and one calculated by (4.10) using measurements of electrical conductivity and porosity. For readers' convenience, the results of these calculations are also given in the last column of Table 4. The covariance cov(X,Y) between experimentally measured (E ₀ − E ₃)/E ₃ (set X) and calculated according to (5.15) (set Y) is 0.275624 while the standard deviations (sd) for two data sets are 0.511142 and 0.573636, respectively. It leads to the correlation coefficient

cor ( X , Y ) = cov ( X , Y ) sd ( X ) sd ( Y ) = 0.94

(5.6)

Comparison of the analytically derived cross-property connection (5.15) with experimental data.

This correlation is significant and we can conclude that analytically derived cross-property connection (4.10) is experimentally verified.

Supplementary Material

Acknowledgement

Financial support from the FP7 Project TAMER IRSES-GA-2013-610547, NIH grant number R25GM061222, and New Mexico Space Grant Consortium contained in the NASA Cooperative Agreement NNX13AB19A to New Mexico State University are gratefully acknowledged.

Footnotes

Conflict of interests statement

The authors of this manuscript have no conflict of interest with the presented work.

Publisher's Disclaimer: This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

References

• Barthélémy J-F (2009) Effective Permeability of Media with a Dense Network of Long and Micro Fractures. Transport in Porous Media 76, 153–178. [Google Scholar]
• Beel JA, Groswald DE, Luttges MW, (1984) Alterations in the mechanical properties of peripheral nerve following crush injury. Journal of Biomechanics 17, 185–193. [PubMed] [Google Scholar]
• Behari J (2009) Biophysical Bone Behaviour: Principles and Applications. Wiley. [Google Scholar]
• Berryman JG and Milton GW (1988) Microgeometry of random composites and porous media. Journal of Physics D 21, 87–94. [Google Scholar]
• Bristow JR (1960) Microcracks, and the static and dynamic elastic constants of annealed heavily cold-worked metals. British Journal of Applied Physics 11, 81–85. [Google Scholar]
• Casas R and Sevostianov I (2013) Electrical resistivity of cortical bone: micromechanical modeling and experimental verification, International Journal of Engineering Science 62, 106–112. [Google Scholar]
• Currey JD (1962) Stress concentrations in bone. Quarterly Journal of Microscopical Science 103, 111–133. [Google Scholar]
• Currey JD (2002) Bones: Structure and Mechanics. Princeton, NJ: Princeton University Press. [Google Scholar]
• Currey JD, Zioupos P (2001) The Effect of Porous Microstructure on the Anisotropy of Bone-Like Tissue: A Counterexample. Journal of Biomechanics 34, 707–710. [PubMed] [Google Scholar]
• Dong XN, Guo XE (2004) The Dependence of Transversely Isotropic Elasticity of Human Femoral Cortical Bone on Porosity. Journal of Biomechanics 37, 1281–1287. [PubMed] [Google Scholar]
• Dong XN, Guo XE (2006) Prediction of Cortical Bone Elastic Constants by a Two-Level Micromechanical Model Using a Generalized Self-Consistent Method. Journal of Biomechanical Engineering 128, 309–316. [PubMed] [Google Scholar]
• Deuerling Justin M., Yue Weimin, Orias Espinoza, Alejandro A., Roeder, Ryan K. (2009) Specimen-specific multi-scale model for the anisotropic elastic constants of human cortical bone. Journal of Biomechanics 42, 2061–2067. [PMC free article] [PubMed] [Google Scholar]
• Fedorov F (1968) Theory of elastic waves in crystals. New York: Plenum Press. [Google Scholar]
• Fritsch A, Dormieux L, Hellmich C, Sanahuja J (2008) Mechanical behavior of hydroxyapatite biomaterials: An experimentally validated micromechanical model for elasticity and strength. Wiley Inter Science. [PubMed] [Google Scholar]
• Fukada E and Yasuda I (1957) On the piezoelectrical effect of bone. Journal of the Physical Society of Japan 120, 1158–1162. [Google Scholar]
• Fung YC (1993) Biomechanics: Mechanical Properties of Living Tissues, Springer Verlag, New York. [Google Scholar]
• Gabriel C, Gabriel S, and Corthout E (1996) The dielectrical properties of biological tissues: I. Literature survey. Physics in Medicine and Biology 41, 2231–2249. [PubMed] [Google Scholar]
• Geddes LA, and Baker LE (1967) The specific resistance of biological material-A compendium of data for the biomedical engineer and physiologist. Medical and Biological Engineering 5, 271–293. [PubMed] [Google Scholar]
• Helmich C, Ulm FJ (2002). Micromechanical model for ultrastructural stiffness of mineralized tissues. Journal of Engineering Mechanics 128, 898–908. [Google Scholar]
• Hellmich C, Ulm FJ, Dormieux L (2004) Can the diverse elastic properties of trabecular and Cortical bone be attributed to only a few tissue-dependent phase properties and their interactions? Arguments from a multiscale approach. Biomechanical Modeling and Mechanobiology 2, 219–238. [PubMed] [Google Scholar]
• Hill R (1963). Elastic properties of reinforced solids: some theoretical principles. Journal of the Mechanics and Physics of Solids 11, 357–372. [Google Scholar]
• Hirsh FG, Texter EC Jr., Wood LA, Ballard WC, Horan FE, and Wright IS. (1950) The electrical conductivity of blood, Blood, 5, 1017–1035. [PubMed] [Google Scholar]
• Hoetink AE, Faes Th.J.C., Visser KR, and Heethaar RM (2004) On the Flow Dependency of the Electrical Conductivity of Blood, IEEE Transactions on Biomedical Engineering 51, 1251–1261. [PubMed] [Google Scholar]
• Horii H, Nemat-Nesser S (1983) Overall moduli of solids with microcracks: load-induced anisotropy. Journal of the Mechanics and Physics of Solids, 31, 155–171. [Google Scholar]
• Kachanov M and Sevostianov I (2005) On quantitative characterization of microstructures and effective properties, International Journal of Solids and Structures, 42, 309–336. [Google Scholar]
• Kanaun SK and Levin VM (2008) Self-Consistent Methods for Composites, Vol.1: Static Problems. Springer. [Google Scholar]
• Kunin IA (1983), Elastic media with microstructure, Springer Verlag, Berlin. [Google Scholar]
• Katz JL (1980) Anisotropy of Young's modulus of bone. Nature 283, 106–107. [PubMed] [Google Scholar]
• Katz JL (1981) Composite material models for cortical bone In: Cowin SC (Ed.), Mechanical Properties of Bone, ASME, New York, Vol. 45, pp. 171–184 [Google Scholar]
• Martin RB and Burr DB (1989) Structure, Function and Adaptation of Compact Bone, Raven Press, New York. [Google Scholar]
• Martínez-Reina J, Dominguez J, Garcia-Aznar JM (2010) Effect of porosity and mineral content on the elastic constants of cortical bone: a multiscale approach. Biomechanical Modeling and Mechanobiology 10, 309–322. [PubMed] [Google Scholar]
• Mura T (1987) Micromechanics of Defects in Solids, Martinus Nijhoff Publ, Dordrecht. [Google Scholar]
• Nye JF (1985) Physical Properties of Crystals. Oxford University Press. [Google Scholar]
• Nikolov S, Raabe D (2008) Hierarchical modeling of the elastic properties of bone at submicron scales. The role of extrafibrillar mineralization. Biophysical Journal 94, 4220–4232. [PMC free article] [PubMed] [Google Scholar]
• Osswald K (1937). Messung der Leitfahigkeit und Dielektrizitatkonstante biologischer Gewebe und Flussigkeiten bei kurzen Wellen, Hochfrequenz Technishe Elektroakustik 49, 40–50. [Google Scholar]
• Parnell WJ, Vu MB, Grimal Q, Naili S (2011) Analytical methods to determine the effective mesoscopic and macroscopic elastic properties of cortical bone. Biomechanical Modeling and Mechanobiology 11, 883–901. [PubMed] [Google Scholar]
• Saadat F, Sevostianov I, and Giraud A (2012).Approximate representation of a compliance contribution tensor for a cylindrical inhomogeneity normal to the axis of symmetry of transversely isotropic material, International journal of Fracture 174, 237–244. [Google Scholar]
• Salguero L, Saadat F and Sevostianov I (2014) Micromechanical modeling of elastic properties of cortical bone accounting for anisotropy of dense tissue. Journal of Biomechanics 47, 3279–3287. [PMC free article] [PubMed] [Google Scholar]
• Sevostianov I and Kachanov M (1998) On the relationship between microstructure of the cortical bone and its overall elastic properties, International Journal of Fracture 92, L3–L8. [Google Scholar]
• Sevostianov I and Kachanov M (1999) Compliance tensors of ellipsoidal inclusions, International Journal of Fracture 96, L3–L7. [Google Scholar]
• Sevostianov I, Kachanov M (2000) Impact of the porous microstructure on the overall elastic properties of the osteonal cortical bone. Journal of Biomechanics 33, 881–888. [PubMed] [Google Scholar]
• Sevostianov I and Kachanov M (2002) Explicit cross-property correlations for anisotropic two-phase composite materials Journal of the Mechanics and Physics of Solids, 50, 253–282. [Google Scholar]
• Sevostianov I and Sabina F (2007) Cross-property connections for fiber reinforced piezoelectrical materials, International Journal of Engineering Sciences 45, 719–735. [Google Scholar]
• Sevostianov I, Yilmaz N, Kushch V, Levin V (2005) Effective elastic properties of matrix composites with transversely-isotropic phases. International Journal of Solids and Structures 42, 455–476. [Google Scholar]
• Sevostianov I, Kováčik J and Simančík F (2006) Elastic and electrical properties of closed-cell aluminum foams. Cross-property connection. Materials Science and Engineering, A-420, 87–99. [Google Scholar]
• Sevostianov I and Kachanov M (2009) Connections between elastic and conductive properties of heterogeneous materials, In: Advances in Applied Mechanics, 42, (van der Giessen E and Aref H, Eds.), Academic Press, pp.69–252. [Google Scholar]
• Sevostianov V (2014) Evaluation of Decalcification Induced Changes in Bone Strength Using Electrical Conductivity Measurements. In Proceedings of ASME 2014 International Mechanical Engineering Congress and Exposition, Montreal, Quebec, Canada, November 14–20, 2014 Volume 3: Biomedical and Biotechnology Engineering, Paper No. IMECE2014–38638, pp. V003T03A071. doi: 10.1115/IMECE2014-38638 [CrossRef] [Google Scholar]
• Sierpowska J, Hakulinen MA, Töyräs J, Day JS, Weinans H, Kiviranta I, Jurvelin JS, and Lappalainen R (2006) Interrelationships between Electrical Properties and Microstructure of Human Trabecular Bone. Physics in Medicine and Biology 51, 5289–303. [PubMed] [Google Scholar]
• Visser KR (1992) Electrical Conductivity of Stationary and Flowing Human Blood at Low Frequencies. Medical & Biological Engineering & Computing 30, 636–40. [PubMed] [Google Scholar]
• Walpole LJ (1969). On the overall elastic moduli of composite materials, Journal of the Mechanics and Physics of Solids 17, 235–251. [Google Scholar]
• Walpole LJ (1984) Fourth-Rank Tensors of the Thirty-Two Crystal Classes: Multiplication Tables. Proceedings of the Royal Society of London A-391, 149–179. [Google Scholar]

theisbefer1966.blogspot.com

Source: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7809538/

Share this post