International Journal of Engineering Science 62 (2013) 106–112

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International Journal of Engineering Science

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Electrical resistivity of cortical bone: Micromechanical modeling andexperimental verification

Rafael Casas, Igor Sevostianov ⇑Department of Mechanical and Aerospace Engineering, New Mexico State University, P.O. Box 30001, Las Cruces, NM 88003, USA

a r t i c l e i n f o

Article history:Received 29 August 2012Accepted 2 September 2012Available online 16 October 2012

Keywords:Cortical boneAnisotropyMicrostructureElectrical resistivity

0020-7225/$ - see front matter � 2012 Elsevier Ltdhttp://dx.doi.org/10.1016/j.ijengsci.2012.09.001

⇑ Corresponding author. Tel.: +1 575 6463322; faE-mail address: Igor@nmsu.edu (I. Sevostianov).

a b s t r a c t

Electrical resistivity of bone tissue is largely determined by its microstructure. The lattercomprises a large number of pores filled with electrically conductive material – blood,lymph, nerve tissue, etc. The present paper analyzes a connection between morphologyof the osteonal cortical bone and its overall anisotropic electrical resistivity. Bone’s micro-structure is modeled using available micrographs. The calculated anisotropic electric resis-tivity is in agreement with available experimental data and with our measurements. Theinfluence of each of the pore types on the overall electric properties is examined.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

In the present paper, we develop a micromechanical model for electrical conductivity of cortical bone and verify it exper-imentally. Geddes and Baker (1967) reported that the first experimental measurements of the electrical resistivity of corticalbone have been done by Osswald (1937). Considerable interest to the electric properties of bone appeared after discovery ofits piezoelectric properties by Fukada and Yasuda (1957). Cochran, Pauluk, and Bassett (1968) measured electromechanicalconstants of canine femoral bone in vitro at physiological moisture conditions. For experiments, they used thin strips of cor-tical bone and whole bone specimens.

Swanson and Lafferty (1972) studied changes in electrical characteristics of rat tibiae as a function of age, immobilization,and vibration and related them to compositional and structural changes in the femurs. They reported that the increase incortical bone conductivity corresponds to the age-dependent increase in the inorganic portion of bone and bone density.Behari, Guha, and Agarwal (1974) measured electrical properties of human cortical bone in dependence on temperatureand observed increase in conductivity with temperature in the interval 30–60 �C. Liboff, Rinaldi, Lavine, and Shamos(1975) measured electrical conductivity in rabbit femur and human tibia in vivo. For these tissues, they obtained the valuesof 2–5 � 105 X/cm and 0.7–1.0 � 105 X/cm, respectively. Sansen, Dicker, Stan, and Mulier (1978) first proposed to use four-point method to measure electrical properties of bone in vivo. This method was used by Durand, Christel, and Assailly (1978)who measured electric impedance of whole sheep bones. Detailed review of various approaches to measurement of electricalconductivity of bone tissue and the results obtained in 1960s–1970s is given by Singh and Saha (1984).

Substantial progress in understanding the nature of the electrical properties of bone has been achieved in 1980s. Chak-kalakal, Johnson, Harper, and Katz (1980) measured dielectric relaxation of bovine femoral cortical bone in vitro saturatedwith physiological solution (0.9% NaCl solution). They showed that the dielectric behavior of fluid saturated compact boneis determined mostly by fluid-filled pores. They also observed substantial anisotropy of electric properties: the long-time

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R. Casas, I. Sevostianov / International Journal of Engineering Science 62 (2013) 106–112 107

resistivity of longitudinal specimens was obtained to be 45–48 X m while for the radial specimens it was 3–4 times higher(resistivity of the fluid was 0.72 X m). Kosterich, Foster, and Pollack (1983, 1984) studied dielectric permittivity and electri-cal conductivity of fluid saturated specimens of rat femoral bone (in vitro). The conductivity of the fresh bone was found to betwo to three times higher than those of the bone fixed in formalin. It is probably related to the fact that electrical conduc-tivity of formalin is much lower than those of biofluids. Indeed, in their second paper (Kosterich, Foster, & Pollack, 1984) thespecimens were immersed in several solutions of various sodium chloride concentrations with the conductivity varying from0.13 to 3.55 S/m. Their results are quite close to the in vivo measurements. Saha and Williams (1995) compared electrical anddielectric properties of wet human cortical and trabecular bone tissue from the dial tibia. Specimens were measured at fre-quencies of 120 Hz, 100 kHz, and 1 MHz in various orientations. Both of the bone tissues displayed differences in theirdependency on frequency because of the variances in their microstructure. Conductivity of cortical bone tissue was foundto be 66.2 ± 15.3 mS/cm. Connection between electrical properties and microstructure of human trabecular bone was dis-cussed by Sierpowska et al. (2006). Electrical properties of the specimens have been measured at frequencies varying from50 Hz to 5 MHz and related to trabecular bone volume fraction (BV/TV). An interesting observation on linear dependence ofrelative permittivity on BV/TV at high and low frequencies has been made.

At the present time the database of electrical properties of bone tissue is quite expansive (see, for example, review ofGabriel, Gabriel, and Corthout (1996) or book of Behari (2009)). However, to the best of our knowledge, no quantitativemicromechanical model has been developed to connect electrical properties of cortical bone with its microstructure. We fillthis gap in the present paper.

2. Experimental procedure

Twenty specimens have been cut from two bovine femur bones taken from the same animal (29 month old female animalwith live weight � 497 kg). The specimens were cut in the shape of approximately rectangular parallelograms using dia-mond saw at low speed keeping bone tissue wet with water during cutting. Mass of the specimens varied from 14 to22 g. The specimens were polished with 600 grid sand paper, washed and put into containers filled with physiological solu-tion (0.9% NaCl solution, with electrical resistivity 1.5 X m). After approximately 24 h, they were taken out and their resis-tance in longitudinal direction was measured by four-point method using HP milliohmmeter, model 4338B. Two alligatorclips were attached to the edges of the specimen, and a plastic clip with two probes attached 20 mm apart was placed ontothe bone. The clip was placed into the center, on the left side of the specimen, and on the right side of the specimen. Theresults for each specimen were averaged over these three measurements. From the data on resistance and geometry ofthe specimens, the conductivity in longitudinal direction was calculated. Density of each specimen was estimated by weight-ing them in the physiological solution. Fig. 1a illustrates thus obtained dependence of the electrical conductivity on densityof the bone specimens. Assuming density of the dense bone tissue being equal to 1.25 g/cc (Lees & Heeley, 1981), we alsoestimated porosity of each specimen.

Microstructure of the cortical bone was studied using SP5II confocal microscope. The thin sections have been cut from thespecimens, polished and macerated by boiling to remove bone marrow and sonicating in a 350 �C water bath to remove pol-ishing grit. Then, before the microscopic study, the polished specimens were soaked for a few hours in glutaraldehyde solu-tion, to increase the florescence of the bone, which is necessary for the confocal microscopy. Based on the obtained pictures(Fig. 1b) and the reviewed literature we model the microstructure of cortical bone as discussed in Section 3.2.

3. Micromechanical modeling

The proposed micromechanical model is based on the concept of resistivity contribution tensor, first introduced by Sevos-tianov and Kachanov (2002). The detailed description of the method is given by Sevostianov, Kovácik, and Simancík (2006).

Fig. 1. (a) Dependence of measured longitudinal electrical conductivity of saturated cortical bone on specimens density; and (b) confocal image of corticalbone microstructure.

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3.1. Electrical resistivity contribution tensor

We consider the electrical conductivity problem for a material with isotropic electrical resistivity q0 containing an inho-mogeneity of volume V⁄with the isotropic resistivity q1. Assuming a linear relation between remotely applied electric field Eand the electric current density J (electric current per representative volume V), the change in E required to maintain thesame electric current density if the inhomogeneity is introduced is:

DE ¼ V�

V0R � J ð3:1Þ

where the symmetric second-rank tensor R can be called the resistivity contribution tensor of an inhomogeneity. Treating theinhomogeneity as an isolated one allows one to calculate R-tensors for certain shapes (see, for example, review of Sevostia-nov and Kachanov, 2008).

Alternatively, relation (3.1) can be written in a dual form

DJ ¼ V�

V0K � E ð3:2Þ

where K is the conductivity contribution tensor of an inhomogeneity. Tensors R and K are simply proportional to each other. Inthe text to follow, we will use conductivity contribution tensor. For inhomogeneity of ellipsoidal shape, tensor K is expressedin terms of second-rank Eshelby’s tensor for conductivity sC as

K ¼ k0 sC � k0

k1 � k0I

� ��1

ð3:3Þ

where k0 and k1 is the electrical conductivities of the matrix material and inhomogeneity, correspondingly.For spheroidal inhomogeneity of aspect ratio c, relevant results were given by Carslaw and Jaeger (1959). These results

lead to the following expression for the Eshelby’s tensor:

sC ¼ f0ðI � nnÞ þ ð1� 2f 0Þnn ð3:4Þ

where n is the unit vector along the spheroid’s axis of symmetry,

f0 ¼c2ð1� gÞ2ðc2 � 1Þ ð3:5Þ

and the shape factor g is expressed in terms of the aspect ratio c as follows

g ¼1

cffiffiffiffiffiffiffiffi1�c2p arctan

ffiffiffiffiffiffiffiffi1�c2p

c ; oblate shape ðc < 1Þ

1cffiffiffiffiffiffiffiffic2�1p ln cþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 � 1

p� �; prolate shape ðc > 1Þ

8>><>>:

ð3:6Þ

In particular, for a sphere, c = 1 and f0 = 1/3; for a cylinder, c ?1 and f0 = 1/2.Then,

K ¼ �k0ðA1I þ A2nnÞ ð3:7Þ

where dimensionless factors A1 and A2 are as follows:

A1 ¼k0 � k1

k0 þ ðk1 � k0Þf0; A2 ¼

ðk0 � k1Þ2ð1� 3f 0Þ½k1 � 2ðk1 � k0Þf0�½k0 þ ðk1 � k0Þf0�

ð3:8Þ

when k1 >> k0 (as in the case of cortical bone when all the pores are filled with soft conductive tissue), these relations arereduced to

A1 ¼�1f0; A2 ¼

1� 3f 0

f0ð1� 2f 0Þð3:9Þ

Fig. 2 illustrates behavior of factors A1 and A2 for this case.

3.2. Microstructure of cortical bone

Microstructure of cortical bone (see Figs. 1b and 3) is formed by osteons embedded into extra-osteonal bone matrix. Anosteon measures approximately 300 lm in diameter and 3–5 mm in length. Each osteon has a central vascular channel –Haversian canal (containing electrically conductive soft tissue – blood, lymph and nerve fibers) of average diameter of theorder of 50 lm surrounded by concentric circular lamellae. Much smaller Volkman’s canals of diameters 2–10 lm linkHaversian canals by a network of blood and lymph vessels. The lamellae contain pores of oblate shapes of diameter ofthe order of 5 lm, called lacunae, with bone cells – osteocytes. These cells receive nutrients via channels (canaliculi) that

Fig. 2. Dependence of the parameters A1 and A2 entering expression for the effective electrical conductivity on the spheroid aspect ratio c.

Fig. 3. Microstructure of cortical bone used in the present model.

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emanate from Haversian canals and have diameters of the order of 1 lm. Both Volkman’s canals and canaliculi tend to lie inplanes perpendicular to Haversian canals and are randomly oriented in these planes.

For the analysis of the overall electric properties, we model the rather complex morphology of the osteonal cortical bone,according to Martin and Burr (1989), Currey (2002) and Fung (1993), as a solid of high electric resistivity containing threesets of highly conductive inhomogeneities: prolate spheroids oriented in x3 direction (Haversian canals), prolate spheroidstrandomly oriented in the x1x2 plane (canliculi and Volksman’s canals) and oblate spheroids (lacunae with osteocytes). Thesymmetry axes of lacunae tend to lie in planes perpendicular to Haversian canals and do not appear to have any preferentialorientations in these planes (Martin & Burr, 1989).

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

� For Haversian canals we used c = 3h/4R, where h is the height of the osteon (we used average value of 4 mm) and R is theradius of the Haversian canal (we used 25 lm). Thus calculated aspect ratio c = 120 allows one to preserve the volume ofthe 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 lm) and for R, we used average radius of the canals (1.5 lm). The cal-culated aspect ratio is 80.� The aspect ratio of the lacunae was taken as 0.2 (Sevostianov & Kachanov, 1998, 2000) in accordance with the observa-

tions of Currey (1962, 2002).

Table 1Factors Ai for pores filled with highly conductive liquid and components of resistivitycontribution tensor for different systems of pores (direction 3 is along the Haversiancanals).

A1 A2

Single prolate spheroid c = 120 �2.001 �3.211 � 103

Single prolate spheroid c = 80 �2.001 �1.568 � 103

Single oblate spheroid c = 0.2 �8.016 6.683PR11 =

PR22

PR33

Haversian canals �2.001 �3.213 � 103

Osteocyte lacunae �4.674 �8.016Canaliculi and Volkman’s canals �7.86 � 102 �2.001

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

We model the mineralized tissue (bone matrix) as the isotropic ‘‘background’’, thus ignoring the bone matrix anisotropysince electrical conductivity of the matrix being different in different directions is still very small. So that effect of the con-ductive soft tissue on the overall properties is dominant. The electrical resistivity of the bone matrix was taken in our cal-culations as q0 = 70 kX m, while for the conductivity of the soft tissue we used q1 = 1.5 X m according to Hirsh et al. (1950),Visser (1992), and Hoetink, Faes, Visser, and Heethaar (2004).

4. Results

Results of calculations in the framework of our model are presented and discussed in this section. Table 1 gives values ofAi and A2 for the shapes corresponding to canals and osteocyte lacunae in our model as well as components of

PRij; calcu-

lated for systems of pores filled with electrically conductive tissue that represent lymph and blood vessels, nerves, etc. It isseen that the main contribution to the conductivity in longitudinal direction comes from Haversian canals. The main contri-bution to the electrical conductivity in radial direction comes from canaliculi and Volkman’s canals. The separate contribu-tions of each of the three pore types to the overall resistivity are illustrated in Fig. 4.

To calculate effective electrical properties, we use non-interaction approximation. This approximation is reasonably accu-rate at low concentration of inhomogeneities. However, as shown by Sevostianov and Sabina (2007) this ‘‘low’’ may be ashigh as 20% of volume fraction of inhomogeneities. If interaction between the inhomogeneities is neglected, each inhomo-geneity can be assumed to be subjected to the same remotely applied electric field. Contribution of the inhomogeneity intothe change in the current density can be treated separately and the overall tensor of the effective resistivity qij of a materialcontaining multiple inhomogeneities is a sum

qij ¼ q0ij þ

1V0

XV�ðpÞRðpÞij

� ��1

ð4:1Þ

Fig. 4. Contribution of different pore systems into overall (a) longitudinal and (b) transverse electrical conductivity of cortical bone.

Fig. 5. Dependence of overall electrical conductivities of cortical bone on total porosity according to (4.1). It is assumed that porosity produced byHaversian canals is 0.5 of total porosity and three other types of pores (Volkman’s canals, canaliculi, and osteocyte lacunae) have equal partial porosities.

Fig. 6. Comparison of the predicted longitudinal electrical conductivity of cortical bone with experimental measurements.

R. Casas, I. Sevostianov / International Journal of Engineering Science 62 (2013) 106–112 111

Assuming that the porosity produced by Haversian canals is 0.5 of total porosity of the cortical bone and three other typesof pores (Volkman’s canals, canaliculi, and osteocyte lacunae) have approximately equal partial porosities, the anisotropicelectric resistivities are obtained as functions of the overall porosity and presented in Fig. 5. Unfortunately, it was impossibleto make quantitative comparison of our results with the data available in literature since porosity of the bone specimens isusually not reported. Due to that, we provide comparison with our own measurements that are in qualitative agreementwith the data given in the book of Behari (2009). We compared the model predictions according to (4.1) with conductivitymeasurements in longitudinal direction of the specimens discussed in Section 2. Fig. 6 illustrates normalized resistivity ofthe cortical bone as function of total porosity. The value of the resistivity of the dense bone tissue used in the calculationsis q0 = 70 kX m. It is seen that the agreement between analytical prediction and experimental measurements is better than12%.

5. Concluding remarks

In the present work, we developed a micromechanical model for electrical properties of cortical bone and verified itexperimentally. The proposed model accounts for realistic complex microstructure of the bone that is formed by Haversiancanals, Volkman canals, system of canaliculi and osteocyte lacunae. Anisotropy of the dense bone tissue formed by collagenand hydroxyapatite is ignored since its electrical conductivity is negligible compared to those of the soft tissue. The predic-tion of the model is in a good agreement with experimental measurements of the longitudinal electrical conductivity ofbovine cortical bone.

Acknowledgement

The financial support of New Mexico Space Grant Consortium is gratefully acknowledged.

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