ELECTRICAL PROPERTIES OF BONE
a review

Overview
    Fukada and Yasuda (5.1) first demonstrated that dry bone is piezoelectric in the classic sense, i.e. mechanical stress results in electric polarization, the indirect effect; and an applied electric field causes strain, the converse effect. The piezoelectric properties of bone are of interest in view of their hypothesized role in bone remodelling (4.2.1). Wet collagen (the protein in bone, tendon and ligament), however, does not exhibit piezoelectric response. Studies of the dielectric and piezoelectric properties of fully hydrated bone raise some doubt as to whether wet bone is piezoelectric at all at physiological frequencies (5.2). Piezoelectric effects occur in the kilohertz range, well above the range of physiologically significant frequencies (5.2). Both the dielectric properties (5.3) and the piezoelectric properties of bone (5.4) depend strongly upon frequency. The magnitude of the piezoelectric sensitivity coefficients of bone depends on frequency, on direction of load, and on relative humidity. Values up to 0.7 pC/N have been observed (5.4), to be compared with 0.7 and 2.3 pC/N for different directions in quartz, and 600 pC/N in some piezoelectric ceramics. It is, however, uncertain whether bone is piezoelectric in the classic sense at the relatively low frequencies which dominate in the normal loading of bone. The streaming potentials examined originally by Anderson and Eriksson (5.5,5.6) can result in stress generated potentials at relatively low frequencies even in the presence of dielectric relaxation or electrical conductivity but this process is as yet poorly understood.
    Becker and co-workers (5.19-5.22) have also explored tissue electrical properties in connection with growth, repair and regeneration. For example, (5.22, 5.23) partial limb regeneration in rats was stimulated by application of weak electrical signals. Electrical signals in amphibians (5.24), which can naturally regenerate lost limbs, differ from those in mammals, which ordinarily do not regenerate lost limbs. Cartilage (5.2) exhibits electrical response to applied force.

Piezoelectricity equations
    Piezoelectricity is a coupled field effect as is thermoelasticity (which governs thermal expansion). In piezoelectric materials stress and strain are coupled to electrical field and polarization. Not all materials are piezoelectric; only those materials lacking a center of symmetry on the atomic or molecular scale can be piezoelectric: these are chiral materials. Examples of piezoelectric materials include quartz, Rochelle salt, barium titanate, and lead titanate zirconate ceramics. Piezoelectric materials are used as vibrating elements for time keeping, as emitters of sound (speakers) or ultrasound, and as microphones, or other sensors and actuators.

    For a piezoelectric material, in the linear domain, the constitutive equations are (8.3.1):

D_i =[d_ijk]_T s_jk+[K_ij]_s,TE_j + [p_i]_σDT        (8.10.1)

e_ij= [S_ijkl ]_E,Ts_kl+[d_kij]_TE_k + [a_ij]_EDT        (8.10.2)

Here D is the electric displacement vector, d is the piezoelectric modulus tensor at constant temperature T, K is the dielectric tensor at constant stress s and temperature T, E is the electric field vector, p is the pyroelectric coefficient at constant stress, eis the strain, S is the elastic compliance tensor at constant electric field,and ais the thermal expansion tensor. The usual Einstein summation convention over repeated subscripts is used. In a material described by these equations, the isothermal compliance measured at constant electric displacement [S_ijkl]_D differs [8.3.1] from the compliance measured at constant field [S_ijkl]_E:

[S_ijkl]_D -[S_ijkl]_E = -d_mijd_nkl[K_mn]_σ^-1.        (8.10.3)

    Piezoelectric relaxation refers to dependence of the piezoelectric coefficients on time or frequency. Such relaxation has been observed in many materials, including ceramics [8.10.2],composites [8.10.3], and bone [8.10.4]. Such relaxation can be represented with complex piezoelectric coefficients or by a piezoelectric loss tangent [8.10.2], [8.10.5]. Mechanical relaxation also occurs in piezoelectric materials and is important in applications: large damping is considered desirable in materials used to generate short acoustic pulses for flaw detection [8.10.6]; small damping (high mechanical Q) is desirable in stable resonators and high power transducers.

    For materials which do relax, the coefficients in Eq. 8.10.1 and Eq. 8.10.2 are complex. The sign is negative since these are compliances.

d*_ijk= d'_ijk - id"_ijk

K*_ij= K'_ij - iK"_ij

S*_ijkl= S'_ijkl - iS"_ijkl.        (8.10.4)

Bone electricity: wet and dry
    Potentials observed in bent bone differ from predictions based on the results of experiments performed in compression (5.7). The piezoelectric polarization may consequently depend on the strain gradient (5.7) as well as on the strain. This piezoelectric theory has been criticized as ad hoc by some authors, however, the idea has some appeal in view of Frost's modeling (4.1.3, 4.1.4) and Currey's suggestion (4) that strain gradients may be significant in this regard. The gradient theory is not ad hoc but can be obtained theoretically from general nonlocality considerations (5.8). Moreover, gradient effects are known in ferroelectric ceramics as originally reported in the Russian literature and recently referred to as "flexo electric". The physical mechanism for such effects is hypothesized to lie in the fibrous architecture of bone (2.10.7,2.10.10, 2.10.11). Theoretical analyses of bone piezoelectricity (5.9-5.12) maybe relevant to the issue of bone remodeling. Recent, thorough, studies have explored electromechanical effects in wet and dry bone. They suggest that two different mechanisms are responsible for these effects: classical piezoelectricity due to the molecular asymmetry of collagen in dry bone, and fluid flow effects, possibly streaming potentials in wet bone (5.13).

    Bone exhibits additional electrical properties which are of interest. For example, the dielectric behavior (e.g. the dynamic complex permittivity) governs the relationship between the applied electric field and the resulting electric polarization and current. Dielectric permittivity of bone has been found to increase dramatically with increasing humidity and decreasing frequency (5.2,5.3). For bone under partial hydration conditions, the dielectric permittivity(which determines the capacitance) can exceed 1,000 and the dielectric loss tangent (which determines the ratio of conductivity to capacitance) can exceed unity. Both the permittivity and the loss are greater if the electric field is aligned parallel to the bone axis. Bone under conditions of full hydration in saline behaves differently: the behavior of bovine femoral bone is essentially resistive, with very little relaxation (5.14). The resistivity is about 45-48 ohm-m for the longitudinal direction, and three to four times greater in the radial direction. These values are to be compared with a resistivity of 0.72 ohm-m for physiological saline alone. Since the resistivity of fully hydrated bone is about 100 times greater than that of bone under 98% relative humidity, it is suggested that at 98% humidity the larger pores are not fully filled with fluid (5.14).

Compact bone also exhibits a permanent electric polarization as well as pyroelectricity, which is a change of polarization with temperature (5.15,5.16). These phenomena are attributed to the polar structure of the collagen molecule; these molecules are oriented in bone. The orientation of permanent polarization has been mapped in various bones and has been correlated with developmental events.

References

5.1 Fukada, E. and Yasuda, I. On the piezoelectric effect of bone, J. Phys. Soc. Japan,12, 1158-1162 , 1957.

5.2. Reinish, G., A. S. Nowick, Piezoelectric properties of bone as functions of moisture content, Nature, 253, 626-627, 1975.

5.3. Lakes, R. S. and Katz, J. L., Dielectric relaxation in cortical bone, J. Appl. Phys.,48, 808-811, 1977.

5.4. Bur, A., Measurements of the dynamic piezoelectric properties of bone as a function of temperature and humidity, J. Biomech. 1, 495-507, 1976.

5.5. Anderson, J. C. and Eriksson, C., Electrical properties of wet collagen, Nature, 218, 167-169, 1968.

5.6. Anderson, J.C. and Eriksson, C., Piezoelectric properties of dry and wet bone, Nature, 227, 491-492, 1970.

5.7. Williams, W. S., Sources of piezoelectricity in tendon and bone, CRC Crit. Rev. in Bioengrg.2, 95-117, 1974.

5.8. Lakes, R. S. The role of gradient effects in the piezoelectricity of bone, IEEE Trans. Biomedical Engineering, BME27, 282-283, 1980.

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5.12. Guzelsu, N., A piezoelectric model for dry bone and tissue, J. Biomech. 11, 257-267, 1978.

5.13. Johnson, M., Chakkalakal, D., Harper, R. A., and Katz, J. L., Comparison of the electromechanical effects in wet and dry bone, J. Biomech. 13, 437-442, 1980.

5.14 Chakkalakal, D. A., Johnson, M. W., Harper, R. A., and Katz, J. L., Dielectric properties of fluid saturated bone, IEEE Trans. Biomed. Engng. BME-27, 95-100, 1980.

5.15 Athenstaedt, H., Permanent electric polarization and pyroelectric behavior of the vertebrate skeleton. VI, the appendicular skeleton of man. Z. Anat. Entwickl. Gesch. 131, 21-30, 1970.

5.16 Lang, S. B., Pyroelectric effect in bone and tendon, Nature 212, 704-705, 1966.

5.17 Bassett, C. A. L., Pilla, A. A., and Pawluk, R. J., A non-operative salvage of surgically resistant pseudarthrosis and non-unions by pulsing electromagnetic fields: a preliminary report, Clinical Orthop. 124, 128-143, 1977.

5.18 Brighton, C. T., Friedenberg, Z. B., Mitchell, E. I., and Booth, R. E., Treatment of non-union with constant direct current, Clinical Orthop. 124, 106-123, 1977.

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8.10.5 Holland, R. "Representation of dielectric, elastic, and piezoelectric losses by complex coefficients," IEEE Trans. Sonics Ultrason., SU-14, 18-20, Jan. 1967.

8.10.6 Jaffe, H. and Berlincourt, D. A., "Piezoelectric transducer materials," Proc. IEEE,53, 1372-1386, 1965.

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8.10.9 "IRE standards on piezoelectric crystals: Measurements of piezoelectric ceramics,"Proc. IRE, 49, 1161-1169, July 1961.

8.10.10 IEEE Standard on Piezoelectricity, IEEE 176-1978; Inst. Electrical, Electronics Engineers, New York, 1978.