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A linear piezoelectric model for characterizing stress generated potentials in bone
Vol. 12, pp. 335-347. PergamonPress. 1979. Printedin Great Britain.
J. Biomechsnics
A LINEAR PIEZOELECTRIC MODEL FOR CHARACTERIZING STRESS GENERATED POTENTIALS IN BONE* E.
KOROSTOFF
Restorative Dentistry, School of Dental Medicine; Metallurgy and Materials Science, and Bioengineering, College of Engineering and Applied Science; Orthopaedic Surgery, School of Medicine, University of Pennsylvania, Philadelphia, PA 19104, U.S.A. Abstract - A comprehensive model is presented for the characterization of stress generated potentials (SGP) in bone, based entirely upon the linear piezoelectricity (PZE) of single crystal collagen. The model calls for a theoretical antiparallel polarization configuration in adjacent lamellae that yields to parallel polarization for minimization of free energy. The direction of reorientation is guided by a stress-gradient-induced trigger polarization that specifically determines the polarity in bending and artifactually determines polarity under normal stress. It describes a decoupling of the signs of the PZE tensor of single crystal collagen from the coordinates of the bone : hence, no assumption of the validity of a PZE matrix applied directly to bone can lead to correct results. The model is consistent with all major experimental SGP observations and provides a fresh viewpoint for externally applied electricity for b&e st&uIation.
ment of those osteons. Note that this tensor states that
INTRODUCTION
The literature shows that a satisfactory characterization of stress generated potentials (SGP) in bone does not exist. We know that bone polarizes electrically when placed under mechanical stress (Gayda, 1912; Fukada et al., 1957) but no one has been able to quantitatively define the relationship between stress and polarization. True, we can write: Pi = dijpj~.
(1)
which gives that relationship in a purely formal sense, but we have not been able to quantitatively characterize a tensor d, that is compatible with experimental data. Its lack has led authors in this field (most recently Pfeiffer, 1977; Jendrucko, 1977) to accept the piezoelectric tensor (PZE) for single crystal collagen (C, or Dh) as being applicable to bone. It is not ; and for two reasons. The first reason is that bone is polycrystalline in collagen and it is therefore impossible to represent its directional relationships with the tensor for a single orientation of collagen. Recognizing this, Gundjian et al. (1974) attempted a statistical correlation of collagen orientations, but the resulting tensor retained the same form as that forthe single crystal and did not correlate with experiment. Near success was obtained by Korostoff (1977) who derived a parametric SGP tensor for osteonal bone whose development started with the D, tensor for single crystal collagen, which was successively modified by transformations for the elliptical osteonal structure and the geometric interarrange___ * Received 3 January 1978. t Throughout this paper, bent bone refers to four-point bending. B.“. 1213% A
each bone specimen has its own unique numerical tensor which is based upon the specimen’s individual structure. The parametric tensor was highly consistent with the relative magnitudes of a set of experimental data, but was ambiguous as to the signs of the magnitudes. Its failure with respect to polarity was a necessary consequence of the second reason for the inability of any PZE or SGP tensor to be wholly applicable for bone. That second reason is that the signs ofany tensor are rigid with respect to its coordinates, whereas the polarity for bone has been experimentally shown to be determined not by its spatial coordinates, but by the sign of the stress (Cochran, 1966). Thus, at the very most, an SGP tensor for bone may be capable of characterizing magnitudes, but not the signs of those magnitudes. It is important to note the qualifier in the last sentence, i.e. “. . at the very most .“, since even the magnitudes are often not characterized by the tensor. This was shown by Ritter (1976)and Ritter et al. (1976) where the measured SGP differed significantly for ordinary vs lubricated compression. This was confirmed by Iannocone et al. (1977). If, as is currently accepted, bone is considered to behave as though governed by a single PZE matrix (Fukada, 1968) referenced to its macroscopic coordinates, then it is difficult to see how any SGP at all can develop in bent bone.7 This is because any polarization would have to be oppositely directed in the compressed and tensed halves of the specimen, resulting in the same sign of charge on both convex and concave surfaces. This is schematically demonstrated in Fig. 1. A further inconsistency, as mentioned above, is that the PZE tensor representation predicts sign 335
336
E.
KOROSTOFF
(a)
(b)
Fig. 1. Schematic representation of normal components of polarization due to bending stress for a material governed by a single PZE tensor: (a) and (b) are for tensors of opposite sign.
+
@$
(a)
@
lb) -
-
@J
(4
The above observations have cast doubt on the ability of SGP matrix formulations to characterize experimental data. For this reason, hypotheses not based upon linear piezoelectricity have been proposed. One is based on streaming potentials (Anderson et al., 1968), but this has to be regarded as a special case hypothesis for wet bone since it cannot explain the essentially similar SGP behavior of dry bone. Another hypothesis, inspired by the apparent failure of linear PZE theory to explain SGP of bone in bending, is based upon a proposed generation of electric potential by stress gradients (Breger, 1973 ; Williams et al., 1976) that requires a fourth order tensor for its characterization. Its relationship to the current model will be discussed. This paper continues, from the paper of Korostoff (1977), to develop a model grounded in linear piezoelectricity that satisfactorily accounts for all major experimental SGP observations, including the anomalies outlined here. THE MODEL
If PZE theory, in which signs are fixed by specimen orientation, is to explain the experimentally observed Fig. 2. Schematic representation as in Fig. 1: (a) specimen signs of potential in bending, then there must be some with positive charge on convex surface; (b) same specimen aspect ofbone that decouples the sign of the SGP in the oppositely bent; (c) end-for-end reversal of (b) showing PZE or SGP matrix from the macroscopic specimen negative charge on convex surfaces. (The solid circle, 0, is a coordinates and leaves it free to be oriented by some reference point to identify orientation of specimen.) property or consequence of the prevailing stress. We propose that the decoupling mechanism resides reversal of surface charge when the specimen is rebent in the microstructure of lamellar bone.* To develop after physically reversing it end-for-end. This is illus- this, we start with idealized parallel plate lamellae for a trated in Fig. 2, where comparison of the convex (and specimen in bending (Fig. 3), in which the prevailing concave) surfaces of (a) and (c) show just such a sign collagen orientations in a local region are alternately reversal. But, in actual fact, the signs of potential at approximate right angle to each other.? Each of generated for normal bone in bending are always these lamellar regions is composed of unidirectional determined by the sign of the stress: concave (comg collagen and may therefore be considered to be ression) always negative; convex (tension) always described by the same PZE matrix. With the justifipositive. cation previously given (Korostoff, 1977), the D, PZE matrix is selected (equation 2). (The alternate choice of the Cb matrix would not affect the essentials of the * Considering all bone to be either lamellar or woven. hypothesis.) t This model is convenient for discussion. The hypothesis Acting upon these crossed (90”) orientations of being developed here does not depend upon rigid 90” alternation or upon each lamella having a single collagen collagen are the tensile (or compressive) stresses orientation throughout its length and breadth. The hy- parallel to the plane but at some angle to the collagen pothesis is consistent with the structure of real bone : domains axes. From equations (1) and (2) it can be shown that of orientation within given lamella and varied orientations (all parallel to the lamellar planes, Boyde, 1969 ; Black, 1974). the resulting polarizations are perpendicular to the from plane to plane. lamellar plane and that they are antiparallel for
A linear oiezoelectric
model for characterizing
Fig. 3. Schematic representation of idealized parallel lamellae in a bent specimen of lamellar bone, showing collagen orientations (C).
(a)
(b)
Fig. 4. Bent bone, as in Fig. 1, showing: (a) antiparallel polarization predicted by the PZE matrix; (b) net polarization obtained from experiment (schematic only: no correlation intended between direction or magnitude of polarization with stress).
adjacent planes (Fig. 4a). Note that these polarizations are derived purely from the applied stress, and there is no implication of any pre-existing polarizations as would exist for a ferroelectric material. 0 d,,=O
0
0
d,,
0
0
0
0
0
-d,h
0.
0
0
0000
(2)
Such an antiparallel arrangement would result in cancellation of polarization to give a zero net polarization perpendicular to the lamellar plane. We know from experiment that this is not so; indeed, that a net unidirectional polarization exists, as in Fig. 4(b), to provide negative and positive voltage on the concave and convex surfaces, respectively. One cannot explain away the experimental fact that a strong polarization exists. Yet, the analysis leading to an antiparallel arrangement must also be correct since it involves no seriously challengeable assumptions. The only possible reconciliation of the apparent contradiction lies in some mechanism for the reorientation of the antiparallel polarization configuration to a parallel configuration. Such an orientation change would occur if the relative energies so dictated, i.e. if the antiparallel configuration energy is high, and therefore unstable, relative to the parallel configuration energy. In Note 1 (Loughin, 1977) these energies are approximated to be +36 J/m’ and - 36 J/m’, respectively, for a polarization strength of 2 x 10e6 C/m*. No calculation for the alternate state of zero polarization was made because that possibility is not indicated by experimental results. The higher energy of the antiparallel configuration derives from the interaction of the electric field due to the polari-
stress generated
potentials
in bone
337
zation in a given lamella with the oppositely directed polarization of the adjacent lamella. The concurrent energy increase for strain and demagnetizing field which accompany the change from antiparallel to parallel alignment are very small, as shown in Note I. Thus, the antiparallel state predicted by the PZE tensor is unstable and yields to the equilibrium configuration in which the polarizations are parallel. The process of redirection of polarization from the unstable antiparallel configuration thus decouples the macroscopic bone specimen from the rigid polarity imposed by the PZE matrix. What remains to be answered is how the system decides which set of antiparallel polarizations to reverse, i.e. what is the trigger that determines the dominant polarization direction? We now postulate that the trigger is based upon the net molecular surface charge of bone which is normally negative (Nieders, 1970; Eriksson, 1976). We next idealize a specimen of bone as composed of surfaces with net negative charge. In a bent configuration the charges will be compressed and dilated, respectively, at the concave and convex surfaces, resulting in a polarization vector as shown in Fig. 5. This polarization can be thought of as adding vectorially to the antiparallel piezoelectric polarizations of Fig. 4(a) to enhance that direction. The magnitude of the trigger need not be large in order to determine the resultant orientation. An order of magnitude estimation is given in Note 2 where it is seen that the trigger polarization is about 0.1% of the PZE polarization. In summary, our model is as follows: the PZE matrix for single crystal collagen interacting with the structure of lamellar bone requires that the PZE polarization be antiparallel. However, the high energy of the antiparallel configuration makes it unstable relative to a parallel configuration. The choice of orientation is determined by a trigger polarization that arises from stress induced geometric dilation and compression of surface charge density of the bone. What happens in the absence of a trigger polarization is discussed subsequently. It is important to note that no attempt is made here to present a detailed mechanism for the system’s movement to the parallel configuration. That will be the work of subsequent experimentation from which the various energy dependencies may be determined. Specifically, we do not now have sufficient data to know whether there is a polarization switching from antiparallel to parallel, or whether the parallel configuration is the only one physically to take place. Now that the hypothesis has been presented, let us
Fig. 5. Shape polarization due to differential density of surface charge on concave and convex surfaces of bent bone with negative zeta potential.
E. KOROSTOFF
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go on to see how consistent are its predictions with the major experimental observations. THE MODEL APPLIED TO PLANE LAMELLAR BONE
There is no experimental SGP data for plane lamellar bone. However, since it is simpler to illustrate the consequences of the model for such bone, we do so here. After doing so, we will show that the consequences are substantially the same for Haversian osteonal bone. Bending
The case of plane lamellar bone in bending was developed with the model in the previous section. It states that the direction of parallel orientation of the transverse PZE polarization is guided by the weak shape-induced polarization, i.e. the magnitude of transverse polarization is determined by the appropriate PZE tensor components, and the polarity of polarization is determined by the weak shape-induced polarization. Compression
Consider a compression specimen, under uniform stress, whose structure is oriented as shown in Fig. 6(a).
Application of the matrix of equation (2) gives the antiparallel polarizations shown in Fig. 6(b). In contrhst with the.case of the bent specimen of Fig. 5, there is no triggering mechanism present to guide the reorientation that must occur to reduce the system energy. Depending upon the specifics of the energy balance among antiparallel energy, depolarizing field energy and strain energy, it is expected that the system will decompose into polarization domains, each containing a number of lamellae, as indicated schematically in Fig. 6(c). For an idealized case of perfectly uniform internal stress in the specimen, the polarization domain directions will be random and result in statistical cancellation of polarization. Under such conditions, the experimentally measured voltage should be zero. On the other hand, consider a real compression experiment in which there is frictional constraint of the specimen surface in contact with the testing machine. The well known result is “barreling”, or bending of the specimen walls. Again, for a real specimen the bending will be unequal at opposite faces, and this will lead to a favored polarization direction by the mechanism of Fig. 5. The favored polarization direction will largely depend on accidents of the precision of the geometry of the specimen, on how precisely it is clamped each time it is placed in the testing machine, and on the regularity of the microscopic structure. Polarity determination
by the zeta potential*
The sign of the trigger polarization, induced by bending as shown in Fig. 5, depends on the sign of the net molecular surface charge of the bone. If the sign of the zeta potential of bone is reversed, then the polarization of Fig. 5 will also reverse to the direction of the compressed, concave surface. Thus, the hypothesis, applied to changes in zeta potential of bone predicts : a change in the bone structure that produces a reversal sign of the zeta potential will cause a polarity reversal in the measured SGP. This prediction is precisely the same for osteonal bone, as is seen in a following section which also presents some experimental verification for osteonal bone and tendon.
THE MODEL APPLIED TO HAVERSIANBONE
DOI;;LIN-
DDRWN
Fig. 6. (a) Schematic representation of idealized parallel lamellae in a compression bone specimen, showing stress and collagen orientations; (b) directions of polarization as predicted by the PZE matrix; (c) association into domains in absence of a triggering polarization. l This is the electrical potential across the diffuse portion of the electrical double layer that forms upon contact of an electrolyte with a solid surface.
Consider a single osteon under a uniform cylindrical stress. The tangential (circumferential) stress in a hollow cylinder under uniform radial stress is inversely proportional to the radial position of the circumference. Thus, the tangential stress is greatest at the Haversian canal perimeter (to) and least at the osteon cement line as indicated in Figs. 7(a) and (b) for uniform radial tension. A charge density modulation takes place within the osteon, which results in the geometric trigger polarization, P,, that points radially
A linear piezoelectric model for characterizing stress generated potentials in bone
339
+Por+o t
(b)
(a)
Fig. 7. (a) Schematic single osteon embedded in interstitial lamellae (not shown) under applied radial tensile stress, 0, with resulting tangential tensile stress, o,, which decreases with distance from Haversian canal according to f(r,,/r). (b) Magnitude of stress or of radial polarization resulting from P, = d-u, ; P, is + for tension and - for compression. (P, is radial polarization; d, is the PZE matrix in polar coordinates ; CT, is the stress in polar coordinates.) inward for tension (Fig. 8a) and radially outward for compression (Fig. 8b). As noted in the description for plane lamellar bone, this trigger polarization is also small compared with the radially antiparallel PZE polarizations within the cylindrical lamellae of the osteon. As in the case for plane lamellar bone, this trigger is postulated to determine the directionality of the PZE polarizations as they take up a parallel orientation. Since the PZE polarization is linearly related to the stress, Fig. 7(b) also represents a two dimensional plot of polarization vs cross sectional position in the osteon. Note in Fig. 7 that all values of P are positive because of the tensile stress. An osteon in compression would be deuicted bv the mirror image of Fig. 7(b).
Fig. 8. Schematic of single osteons, demonstrating dependence of geometric polarization, P,, on sign of applied radial stress : (a) Tensile strain greater at canal wall provides a lower surface density of negative charge than osteon exterior and, hence, an inward pointing P,; (b) Compressive strain greater at canal wall yields a higher surface density of negative charge than osteon exterior for an outward pointing P,.
Compression of Haversiarl bow
Haversian bone can be idealized as a collection of parallel osteons to which the descriptions of Fig. 7 may be applied. To do this, we must decide on a state of microscopic stress that will act upon the piezoelectric collagen. This requires a knowledge that we do not possess, about the microscopic mechanical parameters for bone, e.g. the viscoelastic moduli for the material at the osteon cement line. The simplifying assumption made here is that an osteon in a region of bone under uniaxial stress can be treated as being in radial stress. This is the approximation for the case where the cement line material behaves as a viscous fluid. With the above assumption and the information in Fig. 7(b), a bone specimen in uniaxial tension follows directly as represented in Fig. 9. It gives a detailed correspondence between bone architecture and polarization (or SGP) on a microscale, but indicates a zero macroscopic polarization (or SGP) across a row of osteons. This pattern within the osteon of increasing potential (absolute value) as the Haversian canal is
Fig. 9. (a) Schematic row ofosteons embedded in interstitial lamellae (not shown) for bone under tension. (b) The SGP for each osteon is as shown in Fig. 7. Note that the two sides of the specimen are at equipotential. hence a zero SGP.
E. KOR~STOFF
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approached, and zero macroscopic SGP, is precisely in accord with recent experimental results (Iannacone et al., 1977).
Bmdirlg of Hauersim borte
We use the same scheme as above for representing Haversian osteonal bone and note that the stress varies linearly from one side of the bent specimen to the other, from maximum compression at one surface, through zero at the midplane, to maximum tension at the opposite surface. Since our model requires a linearity between the magnitudes of stress and voltage, there will be a scaled congruence of stress and voltage as illustrated in Fig. 10. Note its difference from Fig. 9
----
TRACE
OF P IN
for uniform stress in respect to: (1) the difference in cusp sign as the stress changes sign; (2) the superimposed voltagegradient. Here, in distinction from the case of uniform stress, a macroscopic SGP prevails. A one dimensional representation is given in Fig. 11 for comparison with the experimental results of Starkebaum et al. (1977) given in Fig. 12. The voltageposition curve predicted by the model clearly reproduces the essential features of the experimental curve. These features are: (1) a net voltage difference between the tension and compression surfaces; (2) a superposed amplification of voltage as the Haversian wall is approached ; (3) dependence of total amplitude on position of the osteon; (4) opposite directions of SGP for osteons under opposite sign of stress.
y DIRECTION
Fig. 10. Similar to Fig. 9 except there is a constant stress gradient due to specimen bending. Note that there is a net SGP between the two sides of the specimen.
(b)
(a)
Fig. 11. (a) A row of idealized osteonal cross sections in a bent specimen of bone; (b) calculated voltage (or stress, in arbitrary units) vs position.
A linear piezoelectric model for characterizing stress generated potentials in bone
341
3.c
il
HAVERSIAN
CANAL
COMPRESSION
2.C -
SIDE d”
5 E
0 I
ti i-Q 1.0
A bOO/
6 >
0, 00 \
TENSION -
0
d
“I 0 0
SIDE
0
- I .o
Fig. 12. Typical microelectrode experimental results (Starkebaum, 1977) for bent bone for comparison with Fig. 11.
A linear piezoelectric model for characterizing stress generated potentials in bone Polarity determination G.ul bow
by the zeta potential
in Haver-
As previously indicated, the polarization dependence on zeta potential discussed for plane lamellar bone is also true for Haversian osteonal bone. Here, the directional dependence manifests itself within each osteon, and is radial. It arises because the deformational strain is greatest at the perimeter of the Haversian canal, which provides either greater compression of net surface charge or greater dilation of that charge for compressive vs tensile stresses, respectively (compared with the strain at the outer perimeter of the osteon). Thus, it is clear that a change In the sign of the zeta potential of a bone specimen would lead to a reversal of the geometric trigger polarization. This is the same as the previously given prediction for plane lamellar bone. It is not clear whether this prediction has been experimentally tested, since in the three experiments cited below, the measured potentials may have rel-, evance only to electrokinetic phenomena. However, if these potentials are interpreted to be relevant to SGP, then the experiments below confirm the prediction of the model. Cignitti et al. (1970/71) found a reversed polarity for bent bone which had been steeped in dilute solutions of Th(NO,), and ThCl,. Because of the relationship of the zeta potential to streaming potentials theseauthors and Eriksson (1976) inferred that the sign reversal was evidence for streaming potentials as the origin of the SGP. However, reversal of the polarity of SGP in bending due to reversal of sign of the zeta potential is also consistent with our PZE generated SGP. Likewise, Anderson et a/. (1968) found, for tendon under impact tensile load, that the pH of its contained electrolyte controlled the magnitude and sign of the SGP at low pH (negative zeta potential) which decreased in magnitude to the isoelectric point (pH = 4.7) of tendon collagen, and then became positive, increasing in magnitude as the pH was progressively raised (positive zeta potential). Anomalous results were obtained (Anderson et al., 1968) for bent bone in a range of pH, but considering the imperviousness of bone and its high mineral content, it is not clear to this author that the zeta potential of the specimen was fully determined by the bathing electrolyte.
343
compression. These two facts are consistent with the proposed model because it states that in bending there is always that stress gradient which firmly guides the osteonal polarity,whereas there are varying moderate to zero stress gradients in compression. An entirely different matter than predicting scatter, however, is the prediction of sign reversal. It is the author’s experience that specimens occasionally show nonconservation of polarity upon retesting. This observation has probably been observed elsewhere but not reported because of the difficulty of placing it within some known context. It has been reported (Ritter, 1976; Ritter et al., 1976), and confirmed (Iannacone et al., 1977) that for specimens of human tibia and bovine femur, the uniformity of stress in a specimen has an important influence on the magnitude of the SGP developed in compression and tension. In brief, these studies showed that tensile specimens, when compared with compression specimens from the same bone region, had potentials significantly small and approaching zero. To test the hypothesis that uniform vs nonuniform strain was responsible for the different behavior, the same compression specimens were tested under lubricated compression (which freed the contacting surfaces from frictional constraint). This hypothesis was confirmed by the similar behavior (very low SGP) in tensile and lubricated compression tests. There was a marked difference between human and bovine specimens in that the more microscopically regular human specimens showed very much greater reduction in SGP in the lubricated tests. Thus, both internal and external geometric factors affect the SGP magnitude through their influence on stress uniformity. These observations, and the expression of the model for the cases of uniform stress and bending, can be approached in a quantitative fashion from its basis in linear PZE. A purely formalistic relation between the macroscopic polarization and the gradient of the stress can be derived by differentiating the expression of linear PZE of equation (1) (where now the polarization, Pi, is the macroscopic polarization and where d,, is the macroscopic SGP tensor, Korostoff, 1977) with respect to X, (distance in the gradient direction) and then integrating with respect to Xi. This gives, for constant gradient
(3) A CONSTITUTIVE EQUATION FOR THE MODEL
The SGP data obtained by a number of investigators leads the author to a couple of general statements. One is that bending of bone always develops a voltage whereas compression or tension of bone does not always develop a voltage. Second, at least in the author’s experience, is that the magnitude ofvoltage in bending is far more reproducible (specimen removed from apparatus and then restressed) than it is in
This expression does not provide the polarity because ofthe sign indeterminacy of the dijk that was previously described, and because there is no demonstrated explicit phenomenological relation between the sign of the gradient and the polarity of the resulting polarization. However, the implicit relation between the two provides a means for removing the sign indeterminacy of the above expression. That is, since the sign of P, determines the polarity of P,, and the sign of P, is explicitly determined by the gradient, then the polarity
344
E. KOROSTOFF
of Pi is implicitly ,determined by the gradient. Thus, equation (3) can be rewritten as
with the understanding that the gradient determines the sign of Pi only in an implicit manner. This is phenomenologically different from the stress gradient hypothesis of Williams and Breger (1976) since they proposed the SGP to arise from a postulated fourth rank gradient tensor. An alternative hypothesis by Williams (1978) proposed a third rank tensor where the PZE moduli are functions of position. This is also phenomenologically different from the model proposed in this paper, which requires an invariant third rank PZE tensor. In any case, while these two hypotheses are, in principle, phenomenologically consistent with experimentally determined behavior of bone, they have not been explicitly related to the known structure of bone. Equation (4) is the constitutive equation for the model presented in this paper and is consistent with the prior qualitative description of the model and discussions of its consequences. Thus, under uniform stress, the stress gradient is zero and the macroscopic polarization is likewise zero. On the other hand, for nonuniform stress, there exists by definition a stress gradient which provides a macroscopic polarization. Further, the sign of the polarization is implicitly determined by the sign of the stress gradient, and is independent of the signs of the third rank PZE tensor. By dividing by the product of permittivity and resistivity of bone, one can use equation (4) to approximate the current density associated with the gradient-induced voltage. One can show that the current densities predicted by this equation compare well with the range of current densities that prevail for externally powered bone stimulation electrodes within a radius of 1 mm. However, such calculations are not presented here because of their unreliability with respect to the appropriate magnitudes of the resistivity and particularly the permittivity. If reliable calculations should show such a coincidence of current densities between those prevailing in stimulation experiments and those calculated from the theoretical model, it would constitute simultaneous support for their commonality and for the validity of the presented model. It would further imply that the effect of applied stimulation current does not depend upon local electrode reactions, but is rather more direct in its impact on the cellular milieu.
DISCUSSION
A. In vivo implications Aside from hard experimental facts that the hypothesis must fit, are the philosophical expectations, based upon physiology, with which the hypothesis must lie comfortably. If electrical signals mediate the
bone morphology, then their generation should take place only as required by physiological need. Thus, the existence of stress within physiological limits should not, in itself, be an automatic provider of an electrical signal for bone remodelhng. The signal should occur only when the stress is excessive, or more particularly, unbalanced. The hypothesis presented in this paper fulfills this requirement through the disappearance of the PZE polarization in Haversian osteonal bone when the stress is uniform. Only when there is sufficient imbalance of stress, e.g. so as to cause bending, is there an overall directionality imparted to the polarization that will lead to a macroscopic potential. This suggests a fundamental criterion for the mechanical design of intra-osseous implants, i.e. that the functioning implant shall not produce in the bone significant tensile stress associated with stress gradients.
The above raises the question of bone maintenance in the absence of this macroscopic potential when normal balanced stresses prevail. Here, however, the microscopic potentials continue to exist within each osteon, and these may play the maintenance role. For zero gravity or bedrest, these microscopic potentials will decrease or disappear, and the loss of bone mass (osteoporosis) may be related to that decline. It has been shown that the SGP magnitude is a function of the age of bone during animal maturation (rat femur: Pollack et al., 1977 ; Koh et al., 1975) and that extrapolation indicates zero SGP at 5 weeks or younger. This has been attributed to the develop ment of intermolecular collagen cross linking with age, and it is suggested here that an additional influence may be the concurrent continuing development of lamellar bone from the immature woven bone. It is interesting to consider a possible role of the interaction energy on the transformation of young woven bone into mature lamellar bone with its parallel collagen orientation. The woven bone with its random collagen orientations is at a distinct energy disadvantage under stress, since the statistical net angle among the collagen polarizations leads to a system energy above the equilibrium energy for parallel polarizations. In the prenatal infant, the bone is directed genetically, and the question of stress induced polarization energy is not moot. However, as the developing musculature and functional use result in mechanical stresses in the bone, the interaction energy may provide the impetus for reorientation of a labile woven structure into the mature lamellar one. The fresh view of the detailed polarization configuration, opened up by the hypothesis, should provide insights for in t&o bone stimulation experiments in which external electric and electromagnetic fields and currents are applied. For instance, the external fields may, be seen as triggers for determining the directionality ofantiparallel polarizations due to existing stresses. Thus, there could be some advantage in the simultaneous application of external stress and
A linear piezoelectric model for characterizing stress generated potentials in bone
external field, or crossed d.c. and r.f. fields, to provide a known polarization configuration in the living bone. In this context, the kinetics of polarization reorientation may be important for applied r.f. fields, just as it may play a role in the (in vitro) study by Bur (1976) and Pfeiffer (1977a) of frequency dependency in the measurement of SGP. B. 7he model and competing hypotheses The model presented in this paper is consistent with the major SGP experimental observations. These are : (1) the existence of SGP and the consistency of its polarity in bent bonespecimens; (2) the large scatter in magnitude and the sometimes observed inconsistency of polarity of specimens in compression; (3) the smaller magnitude of SGP for tension and lubricated compression compared with normal compression; (4) the reversal of sign and magnitude of SGP where the zeta potential is reversed and changed in magnitude; (5) the detailed dependence of potential vs position of micropotentials in bent Haversian bone; (6) the essential similarity of SGP in both wet and dry bone; (7) the relative magnitudes of the components of the SGP experimental matrix. The last point, (7), was developed in a previous paper (Korostoff, 1977) which was a first step in the formulation of this paper. The earlier paper* did not recognize the non-significance for bone of the polarity of the matrix components and therefore agonized lightly over three polarities where theory and experiment disagreed. This (current) paper takes the position that all polarities in compression and tension are artifactual (determined by stress gradient), and that only in bending is there a predictable polarity. Some specimens will have reproducible signs (of voltage) in compression while others will show a sign reversal capability. According to our model, the most nearly perfect specimens in regard to their external geometry and internal arrangement of osteons or lamellae, will be the most likely to show instability of sign in compression. This follows since the only possibility for a triggering polarization is in the manner of positioning of such specimens for compression. On the other hand, a specimen with external geometric bias or one in which the internal arrangement of osteons or lamella provide a pronounced nonuniformstrain will have a built-in trigger and hence have a reproducible polarity under stress. These same comments apply to the observation of reduced SGP for lubricated compression, i.e. the more nearly perfect are the internal and external specimen geometries, the greater will be _ * There is an error on p, 42 of that paper in that “the other is an equal, but opposite matrix” is true, not of sign, but in interchange of p and o. Thus, the sum of the two matrices, with equal weighting, leads to -d;s = ai4 = (p’ + oz)d/4 = d/4. The same is true for the elliptical osteon, for which c4 = (2p - l)d/4, & = -pd/4, and & = (1 - p)d/4. The significance of this correction is that the SGP matrix is independent of p and W.The numerical results, e.g. Table 1, of that paper are unchanged.
345
the difference in measured SGP between lubricated and unlubricated compression. With regard to thesign ofpotential in bending, it has been reported (Steinberg et al., 1976) that some unfixed specimens of whole rat femur show polarity reversal after approximately 6 days from excision. The sign reversal was a continuation of a steady decrease in SGP magnitude as a function of time. The observation was not then explicable, and it is suggested here that structural changes affecting the trigger polarity may be responsible. The mimicry by our hypothesis of the detailed microelectrode experimental behavior of Haversian bone in bending, item (S), is a powerful inferential argument in favor of linear PZE as the property responsible for the observed SGP in bending. The stress gradient hypothesis of Williams et al. (1976) is, in principle, consistent with the major SGP experimental observations, (l)-(6), noted above. There is, however, no evidence that it could be made consistent with item (7) i.e. the relative magnitudes of an experimentally determined SGP matrix. Further, the lack of an explicit relationship between the phenomenology of this hypothesis with the known structure of bone must be recognized. Additionally, the provision by the postulated fourth rank gradient tensor of a polarization magnitude equivalent to that provided by the third rank PZE tensor has no counterpart for other materials in nature, and must be regarded in a tentative manner. Streaming potentials, on the other hand, is a demonstrable property of appropriate systems. However, no detailed model based upon specific bone structure has yet been advanced to explain the strongly anisotropic SGP behavior of bone. With respect to the major experimental SGP observations, this hypothesis is consistent in principle with (1) and especially (4), but is strongly contra-indicated for item (6). Also, there is no evidence that it could be made consistent with item (7). CONCLUSIONS
The model developed in this paper states the following : (1) Themterplay of lamellar bone structure and the PZE tensor for single crystal collagen results in an unstable antiparallel PZE polarization configuration in bone. This results in decoupling the signs of the PZE tensor from the bone coordinates. (2) The unstable, antiparallel configuration reorients to the lower energy parallel configuration whose direction is guided by a small, shape-induced polarization. The consequences of these statements are the following : (3) Neither PZE nor SGP tensors are capable of predicting macroscopic SGP polarity in bone under tension or compression. (4) The macroscopic SGP polarity ofa bone specimen
346
E.
KOR~XTOFF
in bending, tension or compression is determined by the sign of the stress gradient. (5) The macroscopic SGP polarity ofa bone specimen in bending, tension or compression can be reversed by reversal of the zeta potential of the bone. (6) The macroscopic SGP magnitude of a bone specimen in bending is based upon the SGP parametric third rank tensor. (7) The macroscopic SGP magnitude of a bone speci-
men in tension or compression may be in the range of zero up to the magnitude predicted by the SGP parametric tensor, depending upon the magnitude of the stress gradient, which in turn depends upon the degree of structural irregularity in the specimen and upon the non-uniformity of the applied stress. (8) The model presented is consistent with all major experimental observations of SGP. Acknowledgements - The author wishes to thank : Dr. Takeshi Egami of the Department of Metallurgy and Materials Science, University of Pennsylvania and Dr. R. Bruce Martin of the Division of Orthopaedic Surgery of the University of West Virginia for their critical reading of the manuscript and for their valuable suggestions. This work was supported by the National Science Foundation under Grant No. GH39170X and by the National Institutes of Health (NIDR) under Grant No. DEO0104. NOMENCLATURE
E 1 M P r arr aik 0”
d a, V
lamellar width, m specimen thickness, m component of PZE matrix, C/N component of PZE tensor, C/N viscosity, Ns/m’ permittivity, F/m energy density, J/m3 electric field, V/m strain, m/m modulus, N/m2 polarization, C/m2 distance, m stress, N/m’ surface charge density, C/m2 volume charge density, C/m3 electric potential, V electrophoretic mobility, m’/Vsec.
REFERENCES
Anderson, J. C. and Eriksson, C. (1968) Electrical properties of wet collagen. Nature, Lond. 218, 166-167. Black, J., Mattson, R. and Korostoff, E. (1974) Haversian osteons : size, distribution, internal structure and orientation. J. Biomed. Mater. Res. 8, 299-319. Boyde, A. and Hobdell, M. H. (1969) Scanning electron microscopy of lamellar bone. Z. Zellfi mikrosk. Anat. 93, 213-231. Breger, L. (1973) Piezoelectricity in bone and tendon. Ph.D. Thesis, University of Illinois. Bur, A. J. (1976) Measurement of the dynamic piezoelectric properties of bone as a function of temperature and humidity. J. Biomech. 9, 495-507. Cignitti, M., Figura, F., Marchetti, M. and Sallas, A. (1970/71) Electrokinetic effects in mechano-electrical phenomenology of the bone. Arch. Fisol. 68, 232.
Cochran, G. V. B. (1966) Electromechanical properties of moist bone. Sc.D. Thesis, Columbia University. Eriksson, C. (1976) The Biochemistry and Physiology of Bone (Edited by Bourne, G. H.), Vol. 4, pp. 340 et seq. Academic Press, New York. Fukada, E. (1968) Mechanical deformation and electrical polarization in biological substances. Biorheology 5, 199-208.
Fukada, E. and Yasuda, I. (1957) On the piezoelectric effect of bone. J. phys. Sot. Japan 12, 1158-1162. Gayda, T. (1912) Recherches d’electrophysiologie sur lea tissue de soutien. Archs ital. Biol. S8, 417432. Gundjian, A. A. and Chen, H. L. (1974) Standardization and interpretation of the electromechanical properties of bone. I.E.E.E. Trans. Biomed. Engng 21, 177-182. Iannocone, W., Korostoff, E. and Pollack, S. R. (1977)Stress generated potentials of bone at the osteon level. Abstract, Ann. Conf. Engng Med. Biol., Los Angeles. Jendrucko, R. J., Cheng, C. J. and Hyman, W. A. (1977) The distribution of induced electrical activity in bent long bone. J. Biomech. 10,493-503.
Koh, J. K., Steinberg, M. E., Korostoff, E. and Pollack, S. R. (1975) Changes in electrical and mechanical properties of whole bone related to maturation. Surg. Forum 26, 515-516. Korostoff, E. (1977) Stress generated potentials in bone: relationship to piaoelectricity of collagen. J. Biomech. 10, 41-44. Lakes, R. S., Harper, P. A. and Katz, J. L. (1977) Dielectric relaxation in cortical bone. J. appl. Phys. 48, 808-811. Liboff, A. R. and Furst, M. (1974) Pyroelectric effect in collagenous structures. Ann. N.Y. Acad. Sci. 238, 26-35. Loughin, S. (1977) Burroughs Corp., Private communication. Nieders, M. E., Weiss, L. and Cudney, T. L. (1970) An electrokinetic characterization of human tooth surfaces. Archs oral Biol. 15, 135. Pfeiffer, B. H. (1977) A model to estimate the piezoelectric polarization in the osteon system. J. Biomech. 10,487492. Pfeiffer, B. H. (1977a) Local piezoelectric polarization of human cortical bone as a function of stress frequency. J. Biomech. 10, 53-57. Pollack, S. R., Korostoff, E., Steinberg, M. E. and Koh, J. K. (1977) Stress generated potentials in bone: effects of collagen modification. J. Biomed. Mater. Res. l&677-700. Ritter, T. A. (1976) Strain-related potentials in human and bovine bone. Ph.D. Thesis, University of Pennsylvania. Ritter,T. A., Korostoff, E., Pollack, S. R. and Steinberg, M. E. (1976) Strain-related potentials in human cortical bone. Abstract, Orthopaedic Res. Sot., New Orleans. Starkebaum, W. (1977) Stress generated potentials in bone on a microscale. Ph.D. Thesis, University of Pennsylvania. Starkebaum, W., Pollack, S. R. and Korostoff, E. (1977) Stress generated potentials in bone at the cell level, Sot. Biomat., Abstract No. 111. Steinberg, M. E., Finegan, W. J., Labosky, D. A. and Black, J. (1976) Temporal and thermal effects ofdeformation potentials in bone. Calc. Tissue Res. 21, 135-144. Williams, W. S. and Breger, L. (1976) Piezoelectricity in tendon and bone. J. Biomech. 8, 407-413. Williams, W. S. (1978) Piezoelectric description of strain related potentials from whole bone and single osteons. Conf. Elect. and Magn. Control Musculoskeletal Growth and Repair. University of Pennsylvania.
APPENDIX
1
Approximation of Energies Involved in Polarization Reorientation
There are three energy terms involved in polarization reorientation : (a) energy reduction in going from antiparallel to parallel configuration; (b) energy increase due to
347
A linear piezoelectric model for characterizing stress generated potentials in bone mechanical strain which accompanies the reorientation; (c) energy increase due to the emergence of a depolarizing field which accompanies the parallel configuration. The sum of these terms must be negative if reorientation is to take place. Following are order of magnitude numerical calculations of these three energies.
E = g = 6.4 x lo3 v/m, i; thus :
E.= d,,E = -1.29 x lo-‘. and the strain energy is:
(a) Lower limit approximation of the anriparallel polarization interaction energ)
The electric field due to polarization P is :
< = fM[(?., + 6)2 - ill, where i., is the strain due to stress (u) of direct PZE effect 5 = &r
3(P.r)r-rZP E(r) =: 4nfr4 Figure 13 designates two lamellae with their polarization as though acting from their midplanes at a distance a apart. This approximation yields a lower energy than an exact solution which would sum the interactions between closely spaced opposite polarizations; hence, this is a lower limit approximation. The energy of interaction between the field from one lamella with the oppositely directed polarization of the other Iamella is given by: 5 = j(--P).dE
= 0.0129 J/m’. (c)
Depolarizingfield energy
The depolarizing field energy is given approximately by D‘?
where 6 is the specimen thickness that we take to be 10e3 m, and r = 6.5 J/m’.
where dE = E(r)pdpd+.
APPENDIX
Integration of this expression over 0 through 2s and p from 0 to y. gives: <=-,
2P2 ta
where E is the permittivity of bone, P is the magnitude of P, and the factor 2 reflects inclusion of both oppositely polarized adjacent lamellae. As is seen from the field equation above, E falls off rapidly with r and only the two adjacent lamellae are important. The same calculation for parallel polarizations on adjacent lamellae gives the same answer with opposite sign. For wet bone permittivity of 4.4 x lo-* F/m,* lamellar thickness of 5 x 1O-6 m and a polarization of2 x 10e6 C/m2 calculated from Liboff and Furst (1974) for a stress of IO’ N/m2 acting upon d,, = -0.203 x 10-i’ C/N.7 Srntipar.,llel = +36 J/m3, iparirlls,= -36 J/m’.
Order of Magnirude Estimation Trigger Polarization
of the
If we know the differential charge density per unit volume (‘a,) of the two surface regions of maximum (and opposite) strain during bending, then with the specimen thickness (6) we can calculate the polarization due to bending as : P, = *UC.6.
Consider a maximum strain, I,. Neglecting the Poisson strain, the change in surface charge density is a,(1 + A,,,)for the convex surface and a,(1 - I,) for the concave surface. Subtracting the surface charge density for the unstressed condition (u,), we have the differential charge densities + l,u, and -&,,u. for the convex and concave surfaces, respectively. The magnitude of u, can be obtained from electrokinetic experiments in which the electrophoretic mobility (V) and viscosity (n) are measured :
uu = Kqv,
(b) Strain energy The strain that accompanies a field reversal of polarization can be calculated from the converse piezoelectric effect. For proper comparison of the strain energy with the antiparallel interaction energy, the electric field, E, should be that required to produce twice the polarization as in (a) above.
2
where K is the Debye-Huckel parameter. The use of u, to calculate u, requires a knowledge of the volume associated with Us. We assume for this calculation that the depth from the surface is approximately the collagen molecular diameter, 2r,. Thus : P, = i,KqV6/2r,
From electrophoretic experiments on dentin (Nieders et al., 1970): n = 0.8177 x IO-’ Pa. set; F = 0.54 x lo-‘msec~’
V-l;
we take Fig. 13. Geometric relations for two opposite polarizations idealized as acting from their respective lamellar midplanes.
.-__ * Lakes et al. (1977). t This value of dt5 for bone is smaller than the d,, of the piezoelectric matrix for single crystal collagen.
Ke
I.
Let I, = 10M3, 6 = 10m3m
and 2r, = 20 x 10-‘om
Pb = 2.2 x 10mgC/m’.
The result is 0.1% of the piezoelectric polarization used in Appendix 1.
J. Biomechsnics
A LINEAR PIEZOELECTRIC MODEL FOR CHARACTERIZING STRESS GENERATED POTENTIALS IN BONE* E.
KOROSTOFF
Restorative Dentistry, School of Dental Medicine; Metallurgy and Materials Science, and Bioengineering, College of Engineering and Applied Science; Orthopaedic Surgery, School of Medicine, University of Pennsylvania, Philadelphia, PA 19104, U.S.A. Abstract - A comprehensive model is presented for the characterization of stress generated potentials (SGP) in bone, based entirely upon the linear piezoelectricity (PZE) of single crystal collagen. The model calls for a theoretical antiparallel polarization configuration in adjacent lamellae that yields to parallel polarization for minimization of free energy. The direction of reorientation is guided by a stress-gradient-induced trigger polarization that specifically determines the polarity in bending and artifactually determines polarity under normal stress. It describes a decoupling of the signs of the PZE tensor of single crystal collagen from the coordinates of the bone : hence, no assumption of the validity of a PZE matrix applied directly to bone can lead to correct results. The model is consistent with all major experimental SGP observations and provides a fresh viewpoint for externally applied electricity for b&e st&uIation.
ment of those osteons. Note that this tensor states that
INTRODUCTION
The literature shows that a satisfactory characterization of stress generated potentials (SGP) in bone does not exist. We know that bone polarizes electrically when placed under mechanical stress (Gayda, 1912; Fukada et al., 1957) but no one has been able to quantitatively define the relationship between stress and polarization. True, we can write: Pi = dijpj~.
(1)
which gives that relationship in a purely formal sense, but we have not been able to quantitatively characterize a tensor d, that is compatible with experimental data. Its lack has led authors in this field (most recently Pfeiffer, 1977; Jendrucko, 1977) to accept the piezoelectric tensor (PZE) for single crystal collagen (C, or Dh) as being applicable to bone. It is not ; and for two reasons. The first reason is that bone is polycrystalline in collagen and it is therefore impossible to represent its directional relationships with the tensor for a single orientation of collagen. Recognizing this, Gundjian et al. (1974) attempted a statistical correlation of collagen orientations, but the resulting tensor retained the same form as that forthe single crystal and did not correlate with experiment. Near success was obtained by Korostoff (1977) who derived a parametric SGP tensor for osteonal bone whose development started with the D, tensor for single crystal collagen, which was successively modified by transformations for the elliptical osteonal structure and the geometric interarrange___ * Received 3 January 1978. t Throughout this paper, bent bone refers to four-point bending. B.“. 1213% A
each bone specimen has its own unique numerical tensor which is based upon the specimen’s individual structure. The parametric tensor was highly consistent with the relative magnitudes of a set of experimental data, but was ambiguous as to the signs of the magnitudes. Its failure with respect to polarity was a necessary consequence of the second reason for the inability of any PZE or SGP tensor to be wholly applicable for bone. That second reason is that the signs ofany tensor are rigid with respect to its coordinates, whereas the polarity for bone has been experimentally shown to be determined not by its spatial coordinates, but by the sign of the stress (Cochran, 1966). Thus, at the very most, an SGP tensor for bone may be capable of characterizing magnitudes, but not the signs of those magnitudes. It is important to note the qualifier in the last sentence, i.e. “. . at the very most .“, since even the magnitudes are often not characterized by the tensor. This was shown by Ritter (1976)and Ritter et al. (1976) where the measured SGP differed significantly for ordinary vs lubricated compression. This was confirmed by Iannocone et al. (1977). If, as is currently accepted, bone is considered to behave as though governed by a single PZE matrix (Fukada, 1968) referenced to its macroscopic coordinates, then it is difficult to see how any SGP at all can develop in bent bone.7 This is because any polarization would have to be oppositely directed in the compressed and tensed halves of the specimen, resulting in the same sign of charge on both convex and concave surfaces. This is schematically demonstrated in Fig. 1. A further inconsistency, as mentioned above, is that the PZE tensor representation predicts sign 335
336
E.
KOROSTOFF
(a)
(b)
Fig. 1. Schematic representation of normal components of polarization due to bending stress for a material governed by a single PZE tensor: (a) and (b) are for tensors of opposite sign.
+
@$
(a)
@
lb) -
-
@J
(4
The above observations have cast doubt on the ability of SGP matrix formulations to characterize experimental data. For this reason, hypotheses not based upon linear piezoelectricity have been proposed. One is based on streaming potentials (Anderson et al., 1968), but this has to be regarded as a special case hypothesis for wet bone since it cannot explain the essentially similar SGP behavior of dry bone. Another hypothesis, inspired by the apparent failure of linear PZE theory to explain SGP of bone in bending, is based upon a proposed generation of electric potential by stress gradients (Breger, 1973 ; Williams et al., 1976) that requires a fourth order tensor for its characterization. Its relationship to the current model will be discussed. This paper continues, from the paper of Korostoff (1977), to develop a model grounded in linear piezoelectricity that satisfactorily accounts for all major experimental SGP observations, including the anomalies outlined here. THE MODEL
If PZE theory, in which signs are fixed by specimen orientation, is to explain the experimentally observed Fig. 2. Schematic representation as in Fig. 1: (a) specimen signs of potential in bending, then there must be some with positive charge on convex surface; (b) same specimen aspect ofbone that decouples the sign of the SGP in the oppositely bent; (c) end-for-end reversal of (b) showing PZE or SGP matrix from the macroscopic specimen negative charge on convex surfaces. (The solid circle, 0, is a coordinates and leaves it free to be oriented by some reference point to identify orientation of specimen.) property or consequence of the prevailing stress. We propose that the decoupling mechanism resides reversal of surface charge when the specimen is rebent in the microstructure of lamellar bone.* To develop after physically reversing it end-for-end. This is illus- this, we start with idealized parallel plate lamellae for a trated in Fig. 2, where comparison of the convex (and specimen in bending (Fig. 3), in which the prevailing concave) surfaces of (a) and (c) show just such a sign collagen orientations in a local region are alternately reversal. But, in actual fact, the signs of potential at approximate right angle to each other.? Each of generated for normal bone in bending are always these lamellar regions is composed of unidirectional determined by the sign of the stress: concave (comg collagen and may therefore be considered to be ression) always negative; convex (tension) always described by the same PZE matrix. With the justifipositive. cation previously given (Korostoff, 1977), the D, PZE matrix is selected (equation 2). (The alternate choice of the Cb matrix would not affect the essentials of the * Considering all bone to be either lamellar or woven. hypothesis.) t This model is convenient for discussion. The hypothesis Acting upon these crossed (90”) orientations of being developed here does not depend upon rigid 90” alternation or upon each lamella having a single collagen collagen are the tensile (or compressive) stresses orientation throughout its length and breadth. The hy- parallel to the plane but at some angle to the collagen pothesis is consistent with the structure of real bone : domains axes. From equations (1) and (2) it can be shown that of orientation within given lamella and varied orientations (all parallel to the lamellar planes, Boyde, 1969 ; Black, 1974). the resulting polarizations are perpendicular to the from plane to plane. lamellar plane and that they are antiparallel for
A linear oiezoelectric
model for characterizing
Fig. 3. Schematic representation of idealized parallel lamellae in a bent specimen of lamellar bone, showing collagen orientations (C).
(a)
(b)
Fig. 4. Bent bone, as in Fig. 1, showing: (a) antiparallel polarization predicted by the PZE matrix; (b) net polarization obtained from experiment (schematic only: no correlation intended between direction or magnitude of polarization with stress).
adjacent planes (Fig. 4a). Note that these polarizations are derived purely from the applied stress, and there is no implication of any pre-existing polarizations as would exist for a ferroelectric material. 0 d,,=O
0
0
d,,
0
0
0
0
0
-d,h
0.
0
0
0000
(2)
Such an antiparallel arrangement would result in cancellation of polarization to give a zero net polarization perpendicular to the lamellar plane. We know from experiment that this is not so; indeed, that a net unidirectional polarization exists, as in Fig. 4(b), to provide negative and positive voltage on the concave and convex surfaces, respectively. One cannot explain away the experimental fact that a strong polarization exists. Yet, the analysis leading to an antiparallel arrangement must also be correct since it involves no seriously challengeable assumptions. The only possible reconciliation of the apparent contradiction lies in some mechanism for the reorientation of the antiparallel polarization configuration to a parallel configuration. Such an orientation change would occur if the relative energies so dictated, i.e. if the antiparallel configuration energy is high, and therefore unstable, relative to the parallel configuration energy. In Note 1 (Loughin, 1977) these energies are approximated to be +36 J/m’ and - 36 J/m’, respectively, for a polarization strength of 2 x 10e6 C/m*. No calculation for the alternate state of zero polarization was made because that possibility is not indicated by experimental results. The higher energy of the antiparallel configuration derives from the interaction of the electric field due to the polari-
stress generated
potentials
in bone
337
zation in a given lamella with the oppositely directed polarization of the adjacent lamella. The concurrent energy increase for strain and demagnetizing field which accompany the change from antiparallel to parallel alignment are very small, as shown in Note I. Thus, the antiparallel state predicted by the PZE tensor is unstable and yields to the equilibrium configuration in which the polarizations are parallel. The process of redirection of polarization from the unstable antiparallel configuration thus decouples the macroscopic bone specimen from the rigid polarity imposed by the PZE matrix. What remains to be answered is how the system decides which set of antiparallel polarizations to reverse, i.e. what is the trigger that determines the dominant polarization direction? We now postulate that the trigger is based upon the net molecular surface charge of bone which is normally negative (Nieders, 1970; Eriksson, 1976). We next idealize a specimen of bone as composed of surfaces with net negative charge. In a bent configuration the charges will be compressed and dilated, respectively, at the concave and convex surfaces, resulting in a polarization vector as shown in Fig. 5. This polarization can be thought of as adding vectorially to the antiparallel piezoelectric polarizations of Fig. 4(a) to enhance that direction. The magnitude of the trigger need not be large in order to determine the resultant orientation. An order of magnitude estimation is given in Note 2 where it is seen that the trigger polarization is about 0.1% of the PZE polarization. In summary, our model is as follows: the PZE matrix for single crystal collagen interacting with the structure of lamellar bone requires that the PZE polarization be antiparallel. However, the high energy of the antiparallel configuration makes it unstable relative to a parallel configuration. The choice of orientation is determined by a trigger polarization that arises from stress induced geometric dilation and compression of surface charge density of the bone. What happens in the absence of a trigger polarization is discussed subsequently. It is important to note that no attempt is made here to present a detailed mechanism for the system’s movement to the parallel configuration. That will be the work of subsequent experimentation from which the various energy dependencies may be determined. Specifically, we do not now have sufficient data to know whether there is a polarization switching from antiparallel to parallel, or whether the parallel configuration is the only one physically to take place. Now that the hypothesis has been presented, let us
Fig. 5. Shape polarization due to differential density of surface charge on concave and convex surfaces of bent bone with negative zeta potential.
E. KOROSTOFF
338
go on to see how consistent are its predictions with the major experimental observations. THE MODEL APPLIED TO PLANE LAMELLAR BONE
There is no experimental SGP data for plane lamellar bone. However, since it is simpler to illustrate the consequences of the model for such bone, we do so here. After doing so, we will show that the consequences are substantially the same for Haversian osteonal bone. Bending
The case of plane lamellar bone in bending was developed with the model in the previous section. It states that the direction of parallel orientation of the transverse PZE polarization is guided by the weak shape-induced polarization, i.e. the magnitude of transverse polarization is determined by the appropriate PZE tensor components, and the polarity of polarization is determined by the weak shape-induced polarization. Compression
Consider a compression specimen, under uniform stress, whose structure is oriented as shown in Fig. 6(a).
Application of the matrix of equation (2) gives the antiparallel polarizations shown in Fig. 6(b). In contrhst with the.case of the bent specimen of Fig. 5, there is no triggering mechanism present to guide the reorientation that must occur to reduce the system energy. Depending upon the specifics of the energy balance among antiparallel energy, depolarizing field energy and strain energy, it is expected that the system will decompose into polarization domains, each containing a number of lamellae, as indicated schematically in Fig. 6(c). For an idealized case of perfectly uniform internal stress in the specimen, the polarization domain directions will be random and result in statistical cancellation of polarization. Under such conditions, the experimentally measured voltage should be zero. On the other hand, consider a real compression experiment in which there is frictional constraint of the specimen surface in contact with the testing machine. The well known result is “barreling”, or bending of the specimen walls. Again, for a real specimen the bending will be unequal at opposite faces, and this will lead to a favored polarization direction by the mechanism of Fig. 5. The favored polarization direction will largely depend on accidents of the precision of the geometry of the specimen, on how precisely it is clamped each time it is placed in the testing machine, and on the regularity of the microscopic structure. Polarity determination
by the zeta potential*
The sign of the trigger polarization, induced by bending as shown in Fig. 5, depends on the sign of the net molecular surface charge of the bone. If the sign of the zeta potential of bone is reversed, then the polarization of Fig. 5 will also reverse to the direction of the compressed, concave surface. Thus, the hypothesis, applied to changes in zeta potential of bone predicts : a change in the bone structure that produces a reversal sign of the zeta potential will cause a polarity reversal in the measured SGP. This prediction is precisely the same for osteonal bone, as is seen in a following section which also presents some experimental verification for osteonal bone and tendon.
THE MODEL APPLIED TO HAVERSIANBONE
DOI;;LIN-
DDRWN
Fig. 6. (a) Schematic representation of idealized parallel lamellae in a compression bone specimen, showing stress and collagen orientations; (b) directions of polarization as predicted by the PZE matrix; (c) association into domains in absence of a triggering polarization. l This is the electrical potential across the diffuse portion of the electrical double layer that forms upon contact of an electrolyte with a solid surface.
Consider a single osteon under a uniform cylindrical stress. The tangential (circumferential) stress in a hollow cylinder under uniform radial stress is inversely proportional to the radial position of the circumference. Thus, the tangential stress is greatest at the Haversian canal perimeter (to) and least at the osteon cement line as indicated in Figs. 7(a) and (b) for uniform radial tension. A charge density modulation takes place within the osteon, which results in the geometric trigger polarization, P,, that points radially
A linear piezoelectric model for characterizing stress generated potentials in bone
339
+Por+o t
(b)
(a)
Fig. 7. (a) Schematic single osteon embedded in interstitial lamellae (not shown) under applied radial tensile stress, 0, with resulting tangential tensile stress, o,, which decreases with distance from Haversian canal according to f(r,,/r). (b) Magnitude of stress or of radial polarization resulting from P, = d-u, ; P, is + for tension and - for compression. (P, is radial polarization; d, is the PZE matrix in polar coordinates ; CT, is the stress in polar coordinates.) inward for tension (Fig. 8a) and radially outward for compression (Fig. 8b). As noted in the description for plane lamellar bone, this trigger polarization is also small compared with the radially antiparallel PZE polarizations within the cylindrical lamellae of the osteon. As in the case for plane lamellar bone, this trigger is postulated to determine the directionality of the PZE polarizations as they take up a parallel orientation. Since the PZE polarization is linearly related to the stress, Fig. 7(b) also represents a two dimensional plot of polarization vs cross sectional position in the osteon. Note in Fig. 7 that all values of P are positive because of the tensile stress. An osteon in compression would be deuicted bv the mirror image of Fig. 7(b).
Fig. 8. Schematic of single osteons, demonstrating dependence of geometric polarization, P,, on sign of applied radial stress : (a) Tensile strain greater at canal wall provides a lower surface density of negative charge than osteon exterior and, hence, an inward pointing P,; (b) Compressive strain greater at canal wall yields a higher surface density of negative charge than osteon exterior for an outward pointing P,.
Compression of Haversiarl bow
Haversian bone can be idealized as a collection of parallel osteons to which the descriptions of Fig. 7 may be applied. To do this, we must decide on a state of microscopic stress that will act upon the piezoelectric collagen. This requires a knowledge that we do not possess, about the microscopic mechanical parameters for bone, e.g. the viscoelastic moduli for the material at the osteon cement line. The simplifying assumption made here is that an osteon in a region of bone under uniaxial stress can be treated as being in radial stress. This is the approximation for the case where the cement line material behaves as a viscous fluid. With the above assumption and the information in Fig. 7(b), a bone specimen in uniaxial tension follows directly as represented in Fig. 9. It gives a detailed correspondence between bone architecture and polarization (or SGP) on a microscale, but indicates a zero macroscopic polarization (or SGP) across a row of osteons. This pattern within the osteon of increasing potential (absolute value) as the Haversian canal is
Fig. 9. (a) Schematic row ofosteons embedded in interstitial lamellae (not shown) for bone under tension. (b) The SGP for each osteon is as shown in Fig. 7. Note that the two sides of the specimen are at equipotential. hence a zero SGP.
E. KOR~STOFF
340
approached, and zero macroscopic SGP, is precisely in accord with recent experimental results (Iannacone et al., 1977).
Bmdirlg of Hauersim borte
We use the same scheme as above for representing Haversian osteonal bone and note that the stress varies linearly from one side of the bent specimen to the other, from maximum compression at one surface, through zero at the midplane, to maximum tension at the opposite surface. Since our model requires a linearity between the magnitudes of stress and voltage, there will be a scaled congruence of stress and voltage as illustrated in Fig. 10. Note its difference from Fig. 9
----
TRACE
OF P IN
for uniform stress in respect to: (1) the difference in cusp sign as the stress changes sign; (2) the superimposed voltagegradient. Here, in distinction from the case of uniform stress, a macroscopic SGP prevails. A one dimensional representation is given in Fig. 11 for comparison with the experimental results of Starkebaum et al. (1977) given in Fig. 12. The voltageposition curve predicted by the model clearly reproduces the essential features of the experimental curve. These features are: (1) a net voltage difference between the tension and compression surfaces; (2) a superposed amplification of voltage as the Haversian wall is approached ; (3) dependence of total amplitude on position of the osteon; (4) opposite directions of SGP for osteons under opposite sign of stress.
y DIRECTION
Fig. 10. Similar to Fig. 9 except there is a constant stress gradient due to specimen bending. Note that there is a net SGP between the two sides of the specimen.
(b)
(a)
Fig. 11. (a) A row of idealized osteonal cross sections in a bent specimen of bone; (b) calculated voltage (or stress, in arbitrary units) vs position.
A linear piezoelectric model for characterizing stress generated potentials in bone
341
3.c
il
HAVERSIAN
CANAL
COMPRESSION
2.C -
SIDE d”
5 E
0 I
ti i-Q 1.0
A bOO/
6 >
0, 00 \
TENSION -
0
d
“I 0 0
SIDE
0
- I .o
Fig. 12. Typical microelectrode experimental results (Starkebaum, 1977) for bent bone for comparison with Fig. 11.
A linear piezoelectric model for characterizing stress generated potentials in bone Polarity determination G.ul bow
by the zeta potential
in Haver-
As previously indicated, the polarization dependence on zeta potential discussed for plane lamellar bone is also true for Haversian osteonal bone. Here, the directional dependence manifests itself within each osteon, and is radial. It arises because the deformational strain is greatest at the perimeter of the Haversian canal, which provides either greater compression of net surface charge or greater dilation of that charge for compressive vs tensile stresses, respectively (compared with the strain at the outer perimeter of the osteon). Thus, it is clear that a change In the sign of the zeta potential of a bone specimen would lead to a reversal of the geometric trigger polarization. This is the same as the previously given prediction for plane lamellar bone. It is not clear whether this prediction has been experimentally tested, since in the three experiments cited below, the measured potentials may have rel-, evance only to electrokinetic phenomena. However, if these potentials are interpreted to be relevant to SGP, then the experiments below confirm the prediction of the model. Cignitti et al. (1970/71) found a reversed polarity for bent bone which had been steeped in dilute solutions of Th(NO,), and ThCl,. Because of the relationship of the zeta potential to streaming potentials theseauthors and Eriksson (1976) inferred that the sign reversal was evidence for streaming potentials as the origin of the SGP. However, reversal of the polarity of SGP in bending due to reversal of sign of the zeta potential is also consistent with our PZE generated SGP. Likewise, Anderson et a/. (1968) found, for tendon under impact tensile load, that the pH of its contained electrolyte controlled the magnitude and sign of the SGP at low pH (negative zeta potential) which decreased in magnitude to the isoelectric point (pH = 4.7) of tendon collagen, and then became positive, increasing in magnitude as the pH was progressively raised (positive zeta potential). Anomalous results were obtained (Anderson et al., 1968) for bent bone in a range of pH, but considering the imperviousness of bone and its high mineral content, it is not clear to this author that the zeta potential of the specimen was fully determined by the bathing electrolyte.
343
compression. These two facts are consistent with the proposed model because it states that in bending there is always that stress gradient which firmly guides the osteonal polarity,whereas there are varying moderate to zero stress gradients in compression. An entirely different matter than predicting scatter, however, is the prediction of sign reversal. It is the author’s experience that specimens occasionally show nonconservation of polarity upon retesting. This observation has probably been observed elsewhere but not reported because of the difficulty of placing it within some known context. It has been reported (Ritter, 1976; Ritter et al., 1976), and confirmed (Iannacone et al., 1977) that for specimens of human tibia and bovine femur, the uniformity of stress in a specimen has an important influence on the magnitude of the SGP developed in compression and tension. In brief, these studies showed that tensile specimens, when compared with compression specimens from the same bone region, had potentials significantly small and approaching zero. To test the hypothesis that uniform vs nonuniform strain was responsible for the different behavior, the same compression specimens were tested under lubricated compression (which freed the contacting surfaces from frictional constraint). This hypothesis was confirmed by the similar behavior (very low SGP) in tensile and lubricated compression tests. There was a marked difference between human and bovine specimens in that the more microscopically regular human specimens showed very much greater reduction in SGP in the lubricated tests. Thus, both internal and external geometric factors affect the SGP magnitude through their influence on stress uniformity. These observations, and the expression of the model for the cases of uniform stress and bending, can be approached in a quantitative fashion from its basis in linear PZE. A purely formalistic relation between the macroscopic polarization and the gradient of the stress can be derived by differentiating the expression of linear PZE of equation (1) (where now the polarization, Pi, is the macroscopic polarization and where d,, is the macroscopic SGP tensor, Korostoff, 1977) with respect to X, (distance in the gradient direction) and then integrating with respect to Xi. This gives, for constant gradient
(3) A CONSTITUTIVE EQUATION FOR THE MODEL
The SGP data obtained by a number of investigators leads the author to a couple of general statements. One is that bending of bone always develops a voltage whereas compression or tension of bone does not always develop a voltage. Second, at least in the author’s experience, is that the magnitude ofvoltage in bending is far more reproducible (specimen removed from apparatus and then restressed) than it is in
This expression does not provide the polarity because ofthe sign indeterminacy of the dijk that was previously described, and because there is no demonstrated explicit phenomenological relation between the sign of the gradient and the polarity of the resulting polarization. However, the implicit relation between the two provides a means for removing the sign indeterminacy of the above expression. That is, since the sign of P, determines the polarity of P,, and the sign of P, is explicitly determined by the gradient, then the polarity
344
E. KOROSTOFF
of Pi is implicitly ,determined by the gradient. Thus, equation (3) can be rewritten as
with the understanding that the gradient determines the sign of Pi only in an implicit manner. This is phenomenologically different from the stress gradient hypothesis of Williams and Breger (1976) since they proposed the SGP to arise from a postulated fourth rank gradient tensor. An alternative hypothesis by Williams (1978) proposed a third rank tensor where the PZE moduli are functions of position. This is also phenomenologically different from the model proposed in this paper, which requires an invariant third rank PZE tensor. In any case, while these two hypotheses are, in principle, phenomenologically consistent with experimentally determined behavior of bone, they have not been explicitly related to the known structure of bone. Equation (4) is the constitutive equation for the model presented in this paper and is consistent with the prior qualitative description of the model and discussions of its consequences. Thus, under uniform stress, the stress gradient is zero and the macroscopic polarization is likewise zero. On the other hand, for nonuniform stress, there exists by definition a stress gradient which provides a macroscopic polarization. Further, the sign of the polarization is implicitly determined by the sign of the stress gradient, and is independent of the signs of the third rank PZE tensor. By dividing by the product of permittivity and resistivity of bone, one can use equation (4) to approximate the current density associated with the gradient-induced voltage. One can show that the current densities predicted by this equation compare well with the range of current densities that prevail for externally powered bone stimulation electrodes within a radius of 1 mm. However, such calculations are not presented here because of their unreliability with respect to the appropriate magnitudes of the resistivity and particularly the permittivity. If reliable calculations should show such a coincidence of current densities between those prevailing in stimulation experiments and those calculated from the theoretical model, it would constitute simultaneous support for their commonality and for the validity of the presented model. It would further imply that the effect of applied stimulation current does not depend upon local electrode reactions, but is rather more direct in its impact on the cellular milieu.
DISCUSSION
A. In vivo implications Aside from hard experimental facts that the hypothesis must fit, are the philosophical expectations, based upon physiology, with which the hypothesis must lie comfortably. If electrical signals mediate the
bone morphology, then their generation should take place only as required by physiological need. Thus, the existence of stress within physiological limits should not, in itself, be an automatic provider of an electrical signal for bone remodelhng. The signal should occur only when the stress is excessive, or more particularly, unbalanced. The hypothesis presented in this paper fulfills this requirement through the disappearance of the PZE polarization in Haversian osteonal bone when the stress is uniform. Only when there is sufficient imbalance of stress, e.g. so as to cause bending, is there an overall directionality imparted to the polarization that will lead to a macroscopic potential. This suggests a fundamental criterion for the mechanical design of intra-osseous implants, i.e. that the functioning implant shall not produce in the bone significant tensile stress associated with stress gradients.
The above raises the question of bone maintenance in the absence of this macroscopic potential when normal balanced stresses prevail. Here, however, the microscopic potentials continue to exist within each osteon, and these may play the maintenance role. For zero gravity or bedrest, these microscopic potentials will decrease or disappear, and the loss of bone mass (osteoporosis) may be related to that decline. It has been shown that the SGP magnitude is a function of the age of bone during animal maturation (rat femur: Pollack et al., 1977 ; Koh et al., 1975) and that extrapolation indicates zero SGP at 5 weeks or younger. This has been attributed to the develop ment of intermolecular collagen cross linking with age, and it is suggested here that an additional influence may be the concurrent continuing development of lamellar bone from the immature woven bone. It is interesting to consider a possible role of the interaction energy on the transformation of young woven bone into mature lamellar bone with its parallel collagen orientation. The woven bone with its random collagen orientations is at a distinct energy disadvantage under stress, since the statistical net angle among the collagen polarizations leads to a system energy above the equilibrium energy for parallel polarizations. In the prenatal infant, the bone is directed genetically, and the question of stress induced polarization energy is not moot. However, as the developing musculature and functional use result in mechanical stresses in the bone, the interaction energy may provide the impetus for reorientation of a labile woven structure into the mature lamellar one. The fresh view of the detailed polarization configuration, opened up by the hypothesis, should provide insights for in t&o bone stimulation experiments in which external electric and electromagnetic fields and currents are applied. For instance, the external fields may, be seen as triggers for determining the directionality ofantiparallel polarizations due to existing stresses. Thus, there could be some advantage in the simultaneous application of external stress and
A linear piezoelectric model for characterizing stress generated potentials in bone
external field, or crossed d.c. and r.f. fields, to provide a known polarization configuration in the living bone. In this context, the kinetics of polarization reorientation may be important for applied r.f. fields, just as it may play a role in the (in vitro) study by Bur (1976) and Pfeiffer (1977a) of frequency dependency in the measurement of SGP. B. 7he model and competing hypotheses The model presented in this paper is consistent with the major SGP experimental observations. These are : (1) the existence of SGP and the consistency of its polarity in bent bonespecimens; (2) the large scatter in magnitude and the sometimes observed inconsistency of polarity of specimens in compression; (3) the smaller magnitude of SGP for tension and lubricated compression compared with normal compression; (4) the reversal of sign and magnitude of SGP where the zeta potential is reversed and changed in magnitude; (5) the detailed dependence of potential vs position of micropotentials in bent Haversian bone; (6) the essential similarity of SGP in both wet and dry bone; (7) the relative magnitudes of the components of the SGP experimental matrix. The last point, (7), was developed in a previous paper (Korostoff, 1977) which was a first step in the formulation of this paper. The earlier paper* did not recognize the non-significance for bone of the polarity of the matrix components and therefore agonized lightly over three polarities where theory and experiment disagreed. This (current) paper takes the position that all polarities in compression and tension are artifactual (determined by stress gradient), and that only in bending is there a predictable polarity. Some specimens will have reproducible signs (of voltage) in compression while others will show a sign reversal capability. According to our model, the most nearly perfect specimens in regard to their external geometry and internal arrangement of osteons or lamellae, will be the most likely to show instability of sign in compression. This follows since the only possibility for a triggering polarization is in the manner of positioning of such specimens for compression. On the other hand, a specimen with external geometric bias or one in which the internal arrangement of osteons or lamella provide a pronounced nonuniformstrain will have a built-in trigger and hence have a reproducible polarity under stress. These same comments apply to the observation of reduced SGP for lubricated compression, i.e. the more nearly perfect are the internal and external specimen geometries, the greater will be _ * There is an error on p, 42 of that paper in that “the other is an equal, but opposite matrix” is true, not of sign, but in interchange of p and o. Thus, the sum of the two matrices, with equal weighting, leads to -d;s = ai4 = (p’ + oz)d/4 = d/4. The same is true for the elliptical osteon, for which c4 = (2p - l)d/4, & = -pd/4, and & = (1 - p)d/4. The significance of this correction is that the SGP matrix is independent of p and W.The numerical results, e.g. Table 1, of that paper are unchanged.
345
the difference in measured SGP between lubricated and unlubricated compression. With regard to thesign ofpotential in bending, it has been reported (Steinberg et al., 1976) that some unfixed specimens of whole rat femur show polarity reversal after approximately 6 days from excision. The sign reversal was a continuation of a steady decrease in SGP magnitude as a function of time. The observation was not then explicable, and it is suggested here that structural changes affecting the trigger polarity may be responsible. The mimicry by our hypothesis of the detailed microelectrode experimental behavior of Haversian bone in bending, item (S), is a powerful inferential argument in favor of linear PZE as the property responsible for the observed SGP in bending. The stress gradient hypothesis of Williams et al. (1976) is, in principle, consistent with the major SGP experimental observations, (l)-(6), noted above. There is, however, no evidence that it could be made consistent with item (7) i.e. the relative magnitudes of an experimentally determined SGP matrix. Further, the lack of an explicit relationship between the phenomenology of this hypothesis with the known structure of bone must be recognized. Additionally, the provision by the postulated fourth rank gradient tensor of a polarization magnitude equivalent to that provided by the third rank PZE tensor has no counterpart for other materials in nature, and must be regarded in a tentative manner. Streaming potentials, on the other hand, is a demonstrable property of appropriate systems. However, no detailed model based upon specific bone structure has yet been advanced to explain the strongly anisotropic SGP behavior of bone. With respect to the major experimental SGP observations, this hypothesis is consistent in principle with (1) and especially (4), but is strongly contra-indicated for item (6). Also, there is no evidence that it could be made consistent with item (7). CONCLUSIONS
The model developed in this paper states the following : (1) Themterplay of lamellar bone structure and the PZE tensor for single crystal collagen results in an unstable antiparallel PZE polarization configuration in bone. This results in decoupling the signs of the PZE tensor from the bone coordinates. (2) The unstable, antiparallel configuration reorients to the lower energy parallel configuration whose direction is guided by a small, shape-induced polarization. The consequences of these statements are the following : (3) Neither PZE nor SGP tensors are capable of predicting macroscopic SGP polarity in bone under tension or compression. (4) The macroscopic SGP polarity ofa bone specimen
346
E.
KOR~XTOFF
in bending, tension or compression is determined by the sign of the stress gradient. (5) The macroscopic SGP polarity ofa bone specimen in bending, tension or compression can be reversed by reversal of the zeta potential of the bone. (6) The macroscopic SGP magnitude of a bone specimen in bending is based upon the SGP parametric third rank tensor. (7) The macroscopic SGP magnitude of a bone speci-
men in tension or compression may be in the range of zero up to the magnitude predicted by the SGP parametric tensor, depending upon the magnitude of the stress gradient, which in turn depends upon the degree of structural irregularity in the specimen and upon the non-uniformity of the applied stress. (8) The model presented is consistent with all major experimental observations of SGP. Acknowledgements - The author wishes to thank : Dr. Takeshi Egami of the Department of Metallurgy and Materials Science, University of Pennsylvania and Dr. R. Bruce Martin of the Division of Orthopaedic Surgery of the University of West Virginia for their critical reading of the manuscript and for their valuable suggestions. This work was supported by the National Science Foundation under Grant No. GH39170X and by the National Institutes of Health (NIDR) under Grant No. DEO0104. NOMENCLATURE
E 1 M P r arr aik 0”
d a, V
lamellar width, m specimen thickness, m component of PZE matrix, C/N component of PZE tensor, C/N viscosity, Ns/m’ permittivity, F/m energy density, J/m3 electric field, V/m strain, m/m modulus, N/m2 polarization, C/m2 distance, m stress, N/m’ surface charge density, C/m2 volume charge density, C/m3 electric potential, V electrophoretic mobility, m’/Vsec.
REFERENCES
Anderson, J. C. and Eriksson, C. (1968) Electrical properties of wet collagen. Nature, Lond. 218, 166-167. Black, J., Mattson, R. and Korostoff, E. (1974) Haversian osteons : size, distribution, internal structure and orientation. J. Biomed. Mater. Res. 8, 299-319. Boyde, A. and Hobdell, M. H. (1969) Scanning electron microscopy of lamellar bone. Z. Zellfi mikrosk. Anat. 93, 213-231. Breger, L. (1973) Piezoelectricity in bone and tendon. Ph.D. Thesis, University of Illinois. Bur, A. J. (1976) Measurement of the dynamic piezoelectric properties of bone as a function of temperature and humidity. J. Biomech. 9, 495-507. Cignitti, M., Figura, F., Marchetti, M. and Sallas, A. (1970/71) Electrokinetic effects in mechano-electrical phenomenology of the bone. Arch. Fisol. 68, 232.
Cochran, G. V. B. (1966) Electromechanical properties of moist bone. Sc.D. Thesis, Columbia University. Eriksson, C. (1976) The Biochemistry and Physiology of Bone (Edited by Bourne, G. H.), Vol. 4, pp. 340 et seq. Academic Press, New York. Fukada, E. (1968) Mechanical deformation and electrical polarization in biological substances. Biorheology 5, 199-208.
Fukada, E. and Yasuda, I. (1957) On the piezoelectric effect of bone. J. phys. Sot. Japan 12, 1158-1162. Gayda, T. (1912) Recherches d’electrophysiologie sur lea tissue de soutien. Archs ital. Biol. S8, 417432. Gundjian, A. A. and Chen, H. L. (1974) Standardization and interpretation of the electromechanical properties of bone. I.E.E.E. Trans. Biomed. Engng 21, 177-182. Iannocone, W., Korostoff, E. and Pollack, S. R. (1977)Stress generated potentials of bone at the osteon level. Abstract, Ann. Conf. Engng Med. Biol., Los Angeles. Jendrucko, R. J., Cheng, C. J. and Hyman, W. A. (1977) The distribution of induced electrical activity in bent long bone. J. Biomech. 10,493-503.
Koh, J. K., Steinberg, M. E., Korostoff, E. and Pollack, S. R. (1975) Changes in electrical and mechanical properties of whole bone related to maturation. Surg. Forum 26, 515-516. Korostoff, E. (1977) Stress generated potentials in bone: relationship to piaoelectricity of collagen. J. Biomech. 10, 41-44. Lakes, R. S., Harper, P. A. and Katz, J. L. (1977) Dielectric relaxation in cortical bone. J. appl. Phys. 48, 808-811. Liboff, A. R. and Furst, M. (1974) Pyroelectric effect in collagenous structures. Ann. N.Y. Acad. Sci. 238, 26-35. Loughin, S. (1977) Burroughs Corp., Private communication. Nieders, M. E., Weiss, L. and Cudney, T. L. (1970) An electrokinetic characterization of human tooth surfaces. Archs oral Biol. 15, 135. Pfeiffer, B. H. (1977) A model to estimate the piezoelectric polarization in the osteon system. J. Biomech. 10,487492. Pfeiffer, B. H. (1977a) Local piezoelectric polarization of human cortical bone as a function of stress frequency. J. Biomech. 10, 53-57. Pollack, S. R., Korostoff, E., Steinberg, M. E. and Koh, J. K. (1977) Stress generated potentials in bone: effects of collagen modification. J. Biomed. Mater. Res. l&677-700. Ritter, T. A. (1976) Strain-related potentials in human and bovine bone. Ph.D. Thesis, University of Pennsylvania. Ritter,T. A., Korostoff, E., Pollack, S. R. and Steinberg, M. E. (1976) Strain-related potentials in human cortical bone. Abstract, Orthopaedic Res. Sot., New Orleans. Starkebaum, W. (1977) Stress generated potentials in bone on a microscale. Ph.D. Thesis, University of Pennsylvania. Starkebaum, W., Pollack, S. R. and Korostoff, E. (1977) Stress generated potentials in bone at the cell level, Sot. Biomat., Abstract No. 111. Steinberg, M. E., Finegan, W. J., Labosky, D. A. and Black, J. (1976) Temporal and thermal effects ofdeformation potentials in bone. Calc. Tissue Res. 21, 135-144. Williams, W. S. and Breger, L. (1976) Piezoelectricity in tendon and bone. J. Biomech. 8, 407-413. Williams, W. S. (1978) Piezoelectric description of strain related potentials from whole bone and single osteons. Conf. Elect. and Magn. Control Musculoskeletal Growth and Repair. University of Pennsylvania.
APPENDIX
1
Approximation of Energies Involved in Polarization Reorientation
There are three energy terms involved in polarization reorientation : (a) energy reduction in going from antiparallel to parallel configuration; (b) energy increase due to
347
A linear piezoelectric model for characterizing stress generated potentials in bone mechanical strain which accompanies the reorientation; (c) energy increase due to the emergence of a depolarizing field which accompanies the parallel configuration. The sum of these terms must be negative if reorientation is to take place. Following are order of magnitude numerical calculations of these three energies.
E = g = 6.4 x lo3 v/m, i; thus :
E.= d,,E = -1.29 x lo-‘. and the strain energy is:
(a) Lower limit approximation of the anriparallel polarization interaction energ)
The electric field due to polarization P is :
< = fM[(?., + 6)2 - ill, where i., is the strain due to stress (u) of direct PZE effect 5 = &r
3(P.r)r-rZP E(r) =: 4nfr4 Figure 13 designates two lamellae with their polarization as though acting from their midplanes at a distance a apart. This approximation yields a lower energy than an exact solution which would sum the interactions between closely spaced opposite polarizations; hence, this is a lower limit approximation. The energy of interaction between the field from one lamella with the oppositely directed polarization of the other Iamella is given by: 5 = j(--P).dE
= 0.0129 J/m’. (c)
Depolarizingfield energy
The depolarizing field energy is given approximately by D‘?
where 6 is the specimen thickness that we take to be 10e3 m, and r = 6.5 J/m’.
where dE = E(r)pdpd+.
APPENDIX
Integration of this expression over 0 through 2s and p from 0 to y. gives: <=-,
2P2 ta
where E is the permittivity of bone, P is the magnitude of P, and the factor 2 reflects inclusion of both oppositely polarized adjacent lamellae. As is seen from the field equation above, E falls off rapidly with r and only the two adjacent lamellae are important. The same calculation for parallel polarizations on adjacent lamellae gives the same answer with opposite sign. For wet bone permittivity of 4.4 x lo-* F/m,* lamellar thickness of 5 x 1O-6 m and a polarization of2 x 10e6 C/m2 calculated from Liboff and Furst (1974) for a stress of IO’ N/m2 acting upon d,, = -0.203 x 10-i’ C/N.7 Srntipar.,llel = +36 J/m3, iparirlls,= -36 J/m’.
Order of Magnirude Estimation Trigger Polarization
of the
If we know the differential charge density per unit volume (‘a,) of the two surface regions of maximum (and opposite) strain during bending, then with the specimen thickness (6) we can calculate the polarization due to bending as : P, = *UC.6.
Consider a maximum strain, I,. Neglecting the Poisson strain, the change in surface charge density is a,(1 + A,,,)for the convex surface and a,(1 - I,) for the concave surface. Subtracting the surface charge density for the unstressed condition (u,), we have the differential charge densities + l,u, and -&,,u. for the convex and concave surfaces, respectively. The magnitude of u, can be obtained from electrokinetic experiments in which the electrophoretic mobility (V) and viscosity (n) are measured :
uu = Kqv,
(b) Strain energy The strain that accompanies a field reversal of polarization can be calculated from the converse piezoelectric effect. For proper comparison of the strain energy with the antiparallel interaction energy, the electric field, E, should be that required to produce twice the polarization as in (a) above.
2
where K is the Debye-Huckel parameter. The use of u, to calculate u, requires a knowledge of the volume associated with Us. We assume for this calculation that the depth from the surface is approximately the collagen molecular diameter, 2r,. Thus : P, = i,KqV6/2r,
From electrophoretic experiments on dentin (Nieders et al., 1970): n = 0.8177 x IO-’ Pa. set; F = 0.54 x lo-‘msec~’
V-l;
we take Fig. 13. Geometric relations for two opposite polarizations idealized as acting from their respective lamellar midplanes.
.-__ * Lakes et al. (1977). t This value of dt5 for bone is smaller than the d,, of the piezoelectric matrix for single crystal collagen.
Ke
I.
Let I, = 10M3, 6 = 10m3m
and 2r, = 20 x 10-‘om
Pb = 2.2 x 10mgC/m’.
The result is 0.1% of the piezoelectric polarization used in Appendix 1.