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A piezoelectric model for dry bone tissue
A PIEZOELECTRIC
MODEL FOR DRY BONE TISSUE* NEJAT GUZELSU
Applied Mathematics Division, Marmara Research Institute, P.K. 21, Gebze, Kocaeli, Turkey Abstract - A linear theory of piezoelectric bars is used for analyzing a dry femur under constant load and a cantilever dry bone beam subjected to a vertical end load. The resuhs obtained from this theoretical study are: (a) The faster healing of the fractured bone tissue by the aid of negative electrical surface charge; @IThe causes of the allular level electrical stimulation due to the alternating structure of the collagen fibers ih the osteom ; (c) The faster rate of remodelling of the bone tissue due to the shear force.
In this study the bone is assumed to be a linear anisotropic piezoelectric material because of the constant temperature assumption. Although the piezoelectric properties are measured mostly on the dry bone samples, the theory stili gives some conclusions for wet bone. In the first part of this work a feedback mechanism for a bone tissue is given. In the second part, a compact portion of a diaphysis of a human femur is investigated under the axial uniform pressure. The last section contains a solution of a cantilever beam, made of compact bone, which is subjected to end-point loading.
m’-nIODUCTION The objective of this study is to investigate in a theoretical way some of the properties of bone tissue.
Among its properties, bone tissue has regeneration and remodelling capacity (Bassett, 1971). These two characteristics have been investigated by many authors (Athenstadt, 1970, 1974; Bassett et a/., 1964; Bassett, 1965; Connoll et a(., 1974; F&de&erg et al., 1970, 1971, 1974; Lavine et al., 1971, 1974). In recent years active research in the area of some tissues such as bone and collagen has shown these materials to be pyroelectric (Anderson er al., 1970; Fukada er al., 1957, 1964; Liboff et al., 1974; Reinish, 1974). The bones of the skeleton, which are mostly subjected to mechanical forces, are made of a tissue exhibiting pyroelectric behaviour, thus suggesting that the remodelling and the regeneration of bone are due to the interaction of both the mechanical and electrical fields. The oriented dipoles existing in a body cause an internal polarization, but these polarization charges are neutralized by leakage of the material (Cady, 1964, Vol. 2). Under constant temperature a pyroelectric material does exhibit a piezoelectricity property (Cady, 1964, Vol. 1). The physical description of the remodelling of bone tissue, in terms of a very simplified form of the linear theory of piezoelectricity, was given by Gjelsvik (Gjelsvik, 1973). As was suggested by Fukada (Fukada, 1968a) the stress gradient theory can be applied for explaining the experimental data for a cantilever bone beam subjected to constant end load (Williams and Breger, 1974). It is shown that the approximate gradient theory is in good agreement with the experimental data, though the treatment is formal only. According to Athenstaedt (Athenstaedt, 1970, 1974), the existing pyroelectric axis in a tissue usually shows the direction of growth of the organ made of this tissue. The remodelling and the regeneration of the bone tissue, exhibiting a pyroelectric property, can be explained by this property (Bassett, 1971; Athenstaedt, 1970, 1974; Bassett, 1964, 1968).
* Receioed 2 August 1976; receiwd January
1978.
for
publication3
THE FEEDBACK MECHANISM IN BONE
Bone tissue has a porous structure containing mineral salts at fluctuating levels. The elasticity, growth, regeneration and remodelling properties are some of its bioiogical and physical characteristics. In general the bone tissue consists of an amorphous and oriented fiber-based matrix, carrying mineral salts and the cells. The microscopic structure of the bone changes according to age, external effects and its place in the body (Warwick, 1973). Collagen molecules (Herring, 1971) make the tropocollagen macromolecules. The macromolecules consist of three a-polypeptide chains, each chain being twisted into a left hand helix, and the three chains being wound around a common axis in the form of right-handed superhelix. It forms a long thin rod of CQ. 2800 A in length and 14 A in diameter. The organization of tropocollagen molecules into fibrils takes place extracellularly. The osteons which are usually parallel to the femur axis are made of lamellas. Each osteon lamella contains collagen fibrils oriented in a helical way, and this orientation changes at each consecutive lamella, causing 90” angle between the helix lines. This particular structure creates a complex composite material made of collagen fiber layers with the direction of the fibrils in the neighbouring layers oriented at a 90” angle to the other. One of the main properties of collagen fib& is that of pyroelectricity
257
258
N. G&ELS.U
with the helix axis of the tropocollagen molecule as the pyroelectric axis (Fukada et al., 1964). Sixty-five per cent of the bone weight is due to the hydroxyapatite crystals (Bourne, 1971; Cameron, 1971). This material generally can be found around the collagen fibers in a crystalline structure. The organic matrix of the bone and the apatite crystals make a coherent unit vital to the bone tissue function. The tropocollagen molecules create heterogeneous centers for crystallization of the bone minerals (SchifTman, 1970; Kummer, 1972). These centers are usually located at the points where the more polar heads of collagen fibrils are located, Bones adjust their densities and orientations according to their functions. This response is known as Wolffs law. The generated electrical signals, due to the internal forces of the bone caused by the external effects, carry the necessary information to the bone cells, regulating their biological functions and controlling the orientation of their secreted macromolecules (Bassett et al., 1962). Wolff s law is summarized as a feedback mechanism (Bassett, 1971) in Fig. 1. Bone tissue is generated or resorbed by the osteoblast and osteoclast cells, respectively. The deposited calcium in bones can easily go into the blood with the aid of enzymes and hormones. Normally the exchange of calcium in bone is balanced and a steady state is maintained. For example: in the absence of the gravity force, the astronauts demonstrated that the process of bone decalcification was so accelerated that it was
INITIATES AND CONTROLS MINERALIZATION C ____-__--___THE ORIENTATION OF THE EXTRACELLULAR FXBRILS
BALANCED STEADY STATE EN
BONE TISSUE
obviously due to a loss of mineral salts rather than the remodelling (resorption) of bone tissue (Kummer, 1972). Consequently recalcification occurs within a short time. But the remodelling of bone tissue is made by the cells and it requires a more extended period. One can summarize the reactions of bone tissue to the external effects in the following way: a rapid one with intake and loss of mineral salts causing density variations, and a slow one with generation or resorbtion of bone tissue, namely remodellittg. AN INVESITCATION
OF A DRY FEMUR THE AXIAL FORCE
LPJDER
A femur located between a hip and a knee is subjected to rather big axial forces, especially during exercises in the human (Warwick, 1973). In this part of the study, the effect of the axial force on a femur is analyzed. A compact part of the diaphysis of a femur is approximated by a 2 f. long hollow cylinder Fig. 2. The coordinate z coincides with the femur axis. The xt, x2 plane which contains the cross section of a femur is bounded by r = a and r = b radii. The polar coordinates are used instead of x1 and x2 x1 = rcos0 x2 = rsin6.
The bone tissue has a hexagonal polar structure (Anderson et al., 1970; Fukada et al., 1957.1964; Lang,
I
I. L.-cr--_
STRUCTURAL RESPONSE TO RESIST FORCE
-
STRAIN
_-__-__----_ INCREASE IN THE DENSITY AND THE ORIENTATION CHANGING THE MECHANICAL PROPERTIES OF BONE
-
NOR!'X=; n ib
>
G?;
I
1 ELASTIC DEFORMATION 1 STIMLiUTES CELL REACTIONS -
CELL&B AND EKTRACELLLIMR TIWJSDUCER
I
PROPORTIONAL ELECTRICAL CO!DI.fAND S1GK.U
TRANSDUCER
,-_----_-__ PIEZOELECTRIC PROPERTIES IS EMRACELLULAR
NUTRITION ENZYBES CELL-CELL COMWXICATION
cN=NOR14ALSTRAIN FEEDBACK CONTROL SYSTE?fIX BONES
Fig: 1.
39
A piezoelectric model for dry bone tissue E, = -
s r"q$", m=O
(11)
where u, L;,w, cp,sij and Ei, are, respectively, m&hanical displacement in the radial direction, mechanical displacement in the 0 direction, mechanical displacement in the I direction , electrical potential, components of the strain tensor and the components of the electrical field. The field equations are: -,,,rr’
- T(;‘+
PZ;;L’ + T!m”’ = 0
(12)
- &-I;’ + FZ;,;” + T:;‘ ” = 0
(13)
7-L;!, = 0
(14)
Fig. 2. A model for a diaphysis of a femur
bone tissue is not homogeneous, and its properties change from point to point. The following analysis will be correct only in so far as the assumed piezoelectric texture correctly describes a real specimen. Imposing the rotational symmetry condition (independent of 0) and the hexagonal polar crystal structure (Cady, 1964, Vol. l), on the theory of linear piezoelectric bars (Dakmeci, 1974; Mindlin, 1975) one can obtain the following set of equations for a femur under the constant load u(r.2)
=
x
rmufm’(z)
(1)
r”w(“‘(z)
(2)
rmt’(m’(z) s wcl)(z)
(3)
“=I3 I
x
w(r,z) =
m-0
2
v(r,z) =
Inso Z (t.2’ = cp mZ, rm4+m’(z) Z
(4)
d;yZ+” + d'"'" = 0.
-mdj”‘+
1966, 1970; Liboff, 1974; Reinish, 1974).* However,
(15)
These equations are respectively, the balance of moments in the directions of r, t and 0 and Gauss’s law. For example p,F’ and d!“’ are TbyJ = Irmtr,dA
d’“’
=
I
= IO* Pr,,.krdr,
s
rmD,dA.
(16)
A
where t,j, d,, Ti,, T,, d,, d and A are, respectively, components of the stress tensor, components of the dielectric displacement, a component of the resultant force which is obtained by the integration of the corresponding stress over the cross section of the bar, the resultant value of the stresses at the boundary, a component of the resultant dielectric displacement, resultant dielectric displacement on the boundary and the cross section ofthe femur. Because ofthe absence of tractions on r = a and r = b surfaces give
cc s,
=
C (m + l)Fu’“+”
TW+ 1’
(5)
=
()t
m=CJ
&+
1) =
0.
1
sBe
=
1
pUh+l)
(6)
“=O
s
zi
= f
r=m,y:
(7)
m=o
s
Bx
=
s,, =
rc!,”
i
(8) F(u$)(m
+ l)kmC1’
m=o
E, =
-“$, (m +
In the calculations the electric field existing outside the piezoelectric material is ignored because the dielectric constant of vacuum (E = 1 in CGS units) is much smaller than the piezoelectric body (Mindlin, 1961, 1972; Tiersten, 1969): Therefore, the boundary condition for the dielectric displacement is reduced to the following form: -D, = 4nw/(z).
(9)
Hence, the term in (15) can be calculated from ~)r=@‘+~’
(10)
2x _d’“‘=
l
(17)
In the Appendix, the physical constants are given.
t Thebold&phab&c character indicates that this value is defined on the boundary. : A dielectric permitivity of a bone specimen with 40% relative humidity is EII = 40 (Reinkh, 1974).
-4~
Q. .I”
o,(z)b”bdO
= -2Zxbm+’
47=J/(Z), (18)
where o/(z) is the free surface charge on surface r = b. The physical discussion of o/(z) is given at the end of this section.
260
N. GijZELW
The other boundary conditions FZ’ =
)
-
Tj:‘=
-
pm’ = *I
0
-d’“’
=
dA
porn
are
following set of equations for the unknowns u(“, w(O), q(O) and #‘j.
3= +e
(1%
z=
(20)
+p,FdA
-t
i=
L
d”“’ t = 0
+ c13w!z’
= 0
(29)
J(l + O)[Zc,,u[:) ‘r c3Jw!$j + e3+#I”:] = 0
(30)
(22)
where p,, stands for the uniform pressure which is distributed over the surfaces I = & f. T,, = 0 at 2 = k e implies that T,, I 0 along the cylinder. The eiements of the stress tensor and the dielectric displacement vector are given
- 2nb24no,(t)
(31)
+ O)[(c,,+ c,~)u(~)+ c13w!Q) + e31q!9’]
-J(l
CL&(*’
+ O)[(c,, 4
-J(l
+ cIJw!~’ + e31@j’rj = 0 J(2 + O)[Zc,,u!:’ + -J(l
(23)
c33w!2
+
e,,cp!2]
=
0
(32) (33)
+ O)[e,,(u!C$ + w(l)) - .sII$‘)]
+.I(2 + O)[Ze,,@ = c J(m + P)([C,,(P
= 0
m=l
+ 1) + C&(~+‘l
+ c,~w(~) .: + e,,cp”‘}* .J ;
+ e,,qv$‘]
I:(
(21)
+f
z=-E,
7-F) = c J(m + p){[c,,(p P
-J(O + O)[(c,2 + Cll)U”
+e
z=
0
m=O
+ e3,w!$i
- E&;;]
f 1) + Cl ,l@+l’
- 2nb34xo,(t)
= 0. (34)
P +
rgy’
=
1
J(m
p)([c,,(p
+
c,3w!:’
+
(24)
e,,cp!?I
+ 1) + c&(p+l)
p,;’
=
1
J(m
c,pY
p){[c&Z’
+
+
, w(O),c#” and $I’ are solved from the following equations ; Tit) = 0, integrated forms of (30), (31) and (2%
P
f
u(l)
(25)
e33cP!?f
#)
=
Cnffz
-k
’ H,H,
+ (P + l)w’p+ 9 e15(p
+
(26)
l)cp’p+l’)
=
C
J(m
l){c,,u~
+
+ e,,(p + lWp’ ‘)I
e3,H, + @II + cd3
+ dcrn’ : 1 J(m + ,
p){el,$’
+
H,H,
elj
(H,Hz-tH,H,)
P x
[u?’
+
(p
+
-+ H,Hh
(35) 1
1 1
(cl1 + ci2W2- H4e3, H,H,+ H,H, ’
w(O)= k I 7-g
-
o,ti)dl + k,
P +
H,c,3
+ H,H,
l)w(p+ I)]
- EII(P + &#p+ ‘q
(27) (36)
d’“’ r = 1 J(m + p){2~~(p
-t l)dp+”
P
(0) cp
(Pb +
e33w!:)
-
~33(P.~
I
(28)
=
k,c,,H, H,H, + Hdf4
HICI~ H,H, + HaH,
z -
b J(m
+
p)
f’
= I
+“~HTdr,
a
where ci,, et,, cjk are, respectively, the elastic coefficients, the piezoelectric strain constants and the dielectric permitivities. If one’ substitutes the equations (23-28) into the equations (12-15) form = 0, 1 and p = 0 will give the
p
=
(38) J
[b-$]=[: - i (b=+:;+o’)] > 0,
where l
The summation
sign C 3 ; . I
p=O
H, =
c3,@,1
H,
e33e31
=
+ +
~12) ~13~33
-
2c:,
261
A piezoelectric model for dry bone tissue
ff,
=
c13e33
H,
=
e33(cll
From
==
+
CIA
-
2e31c13
dr
139)
2nb34~o,(z) b - Jo,
(43)
p,=
(01
2
c33e31
b
kl=_.$ k
-
p,,Zzr
I0
0.
the above values one can obtain f,, = 0
t zz -4 D,=
--
J(l)
D
[
Jw
1
staedt, 1970, Fig. 6). the electric polarization pattern in the lower extremity of new-born infant is shown. A human femur starts to ossify from its ossification center located in the middle of the femur. The bone grows and calcifies in the longitudinal and radial direction. When a human reaches a certain age the radial polarization vector disappears but the longitudinal polarization vector remains. This longitudinal polarization vector is balanced out with the opposite polarization vector of metaphysis. From the above experimental results one can conclude that, if one creates an artificial radiaI polarization vector in an area then the bone growth and the regeneration can be induced locally in the radial direction. P, and P, components of the polarization vector are computed by the following formula in terms of (43) and (44) Di = Et + 4~Pi
~fKW
The physical explanation of these results : As it was mentioned at the beginning of this study, a tissue has a permanent polarization vector in the direction of its growth (Athenstaedt, 1970, 1974). In Fig. 3 (Athen-
P,=
-
(45j
(46) I--
(bl
(cl
Fig. 3. Reference (Athenstaedt, 1970, Fig. 6).
Electric polarization pattern in the lower extremity of newborn infants. (a) Total view of the extremity; (b) distal femur head, enlarged, with the epiphysis as it normally exists at birth. (1) Femur; (2) tibia ; (3) fibula ; ossification centers of the metatarsal bone (4), the phalanges (5). the tales (6), the calcaneus (7). and the cuboid bone (8); (9) distal femur epiphysis : (10) a pan of the periosteum showing the direction of radial polarization in it (arrow); (c) A ring of compact substance cut out from the femur diaphysis, showing its direction of radial polarization.
HlCl3 b-0 A
A = H,H,
+ H,H,
The polarization vector P, solely depends upon the free surface charge o/(z) of the surface r = b. If w,(s) < 0 then P, is positive. In an adult, applying free negative surface charge from a negative pole of a battery to the fracture area will induce a radial polarization vector locally at that region a condition similar to that existing in the infant where the regeneration and healing capacity for fractures and wounds is quite extended. The experimental observations (Bassett et of., 1964; Bassett 1965 ; Connoll et al., 1974; Friedenberg et al., 1970, 197 1, 1974; Lavine et al., 1971, 1974) of bones show that a negative pole of a battery, which is applied to the fractured sideof a bone, accelerates the ossification process in the bone. Such treatment may cause two things : (a) First, the deposition ofpositive electrical charge carrying Ca ions to the negative pole (faster ossification); (b) Second, a constant radial polarization vector near the fracture area similar to that existing in infants (increased regeneration capacity). The P, polarization vector, given by the equation (46), consists of two terms. The first term on the right hand side of this equation shows the effect of the surface charge on P,. The second term shows the effect of the external force. At each step of a human the second term changes sign causing an ahernating
262
N. G~~ZEISJ
electrical stimulation in the bone tissue because of the leakage in living tissue. The final implication of the above calculation is a proposed explanation of cellular-level electrical stimulation. There are three kinds ofcollagen fiber arrangements in osteons (Ascenzi and Bonucci, 1968): the first kind, in which the fibres in all lamellae are roughly parallel to the axis of the osteone, makes a 90” helix angle ; the second kind, in which the fibres in alternate lamellae are longitudinal and circumferential gives a helix angle between 90” and 0” ; the third kind, in which the fibres in all lamellae are inclined but in opposite directions and therefore the successive lamellaes have an alternating helix angle between plus and minus 45”. In this study, the collagen structure of an osteon of the third kind which is shown in Fig. 4 is investigated. The osteon axis is taken parallel to the z axis. The osteon lamellas (I) and (II) are located between the curves 1 and 2 and (2 and 3), respectively. The osteocytes are shown with dark ellipsoidical points. The curved lines starting from curve 1 show the direction of the collagen fibrils in the first osteon lamella (I) and the curved lines starting from 2 shown the orientation of the collagen fibrils in the second osteon lamella (II).
Ostefytes
collagen
(al
(47)
djm= eJ&-,l ‘iJk =
cjk -
hJk -
(48) dJ&krw
dJmstands for the pietoelectric stress constants. If there
Of the
L___:_________J
Of the fibers
The structure
of an OsteOn.
Section (bl
The position ,”
0”
Osteon lomello
A -A
of the
OsteOCyteS
osteon.
I Osteon lame1 la IT
(c)
The
section
(49)
a,, = {i = k, 1, j # k, 01.
I*
Direction
Direction
An angle between the tangents of the two different layers of collagen fibrils is cu. 90”. In Fig. 4(c) a very small element, which is taken around the M-M line in Fig. 4(a), is shown. It is referred to a system of righthanded Cartesian coordinates where 2 axis is parallel to the femur axis, x axis is perpendicular toy and 2. The surfaces passing through 1,2,3 curves defining the boundaries of the element are approximated by planes. In Fig. 4(d) the direction of the collagen fibrils are shown. If this element is subjected to the uniform pressure p. parallel to the z axis, then the state ofstress on a square element, located with its edge parallel to the collagen fibrils is shown in Fig. 5. The relation between the stress tensor and the polarization vector is given in the following way.
The direction of the collagen fibers
M-M.
Cd) I -rT 0steon lamellos and the direction of the coiloqen fibers.
Fig. 4.
A piezoelectric model for dry bone tissue
for the second layer. d,, = - +
8 = 135’.
When the bone tissue is considered as a whole then this alternating structure of the osteon lamellae are not observable in the experiments. The shear stress (p,/2) (this stress (tY.:. = - pJ2)) is defined with respect to the x y’ z’ coordinate system which is obtained with the rotation of Y: coordinates 45” about the x axis) in Fig. 5(a) gives negative polarization for the first lamella 8 = 45’ la) The state of stress of an OSteOn element which has on angle 45' with the 2 axis.
P, = d;.&
= % ( -P&9
d,,i+,
= - 4
In the second osteon lamella it gives Osteacytes d - 2
P, = d14tyr =
(
dwpo (- p,/Z) = -
>
4
e = 135’
Iv=
Osteon lamellas (b) The localpolarization in the osteon lamelias.
Fig. 5.
is no electrical external effects only mechanical then equation (47) can be approximated
ones
This approximate relation has been used for the calculations of the piezoelectric constants (Fukada et al., 1957, 1964). Fukada (Fukada, 1968) gave the piezoelectric stress constants d,, for fibrous material (wood)* in terms of the angle between the direction of fibers and the direction of applied stresses.
In a similar manner in each osteon lamella direction of the fibers change, so one can give (d,,)t terms of 0, which is the angle between the direction the applied stresses to the bone and the direction fibers. For example, for the first layer, 0 = 45’ d,, positive
9
the in of of is
8 = 45’
The wood tissue fits the hexagonal holoaxiai crystal structure l
. . . .
.
.
d
14 .
The magnitude of thkse local electrical charges are the function of the external load P,. This electrical stimulations may be the trigger of the cellular response to the external effects which is described in the feedback mechanism. From this, one can conclude that, if a broken bone can be exposed to mechanical external effects :hen the healing would be faster than a broken bone which is casted.
A?4 INVESTIGATION OF A CANTILEVER BEAM MADE A COMPACT BONE
(d,,) = % sin 20.
d,, = +
positive polarization. These opposite polarization vectors in alternating lamellas give a total zero polarization along the x axis, but in a cellular level these polarization vectors cause local electrical surface charges on the largest surface side of the osteocytes Fig. 5(b).
.
.
-d,,
.
t d,, is the biggest piezoelectric stress constant in the bone tissue see Appendix.
OF
In this section, a cantilever beam having a hexagonal polar structure is analyzed. In Fig. 6, the dimensions of the beam and the referred Cartesian coordinates are shown. The boundary conditions are summarized in the following way: P, is the external end load at z = 0 in they direction which is assumed to be distributed on the xy plane as a shear stress. The normal stress t,, = 0 at z = 0. The displacements and the derivative of the vertical displacement are zero at -_ = e. The side surfaces of the beam (0 < -_< E) are free ofstresses. The electrical displacement is assumed to be zero on the all surfaces of the beam, because of the absence of free electrical surface charges and the fact that the dielectric permitivity of the bone is much bigger than the vacuum (Mindlin. 1961, 1972; Tiersten ; 1969). Shortly, in these calculations the assumption of plane stress is applied, consequently the stress components in the x direction and the dielectric
N. GOZELSU
264
displacement component D, are assumed to be zero. The series form of the electrical potential cp used in calculations is :
The procedure of the solution for the cantilever beam is similar to the solution of the femur under the constant load which is given in the previous section. Because of the length of the expressions and the straightforward nature of the calculations, the steps of the computations are not given here. After the calculations the final form of the procedure is written in the following way : P&H,
cp=-
coshrz + P,a,h’ -sinh f.r T&1+&)
Io,H,ra’,
Y3 P&l, + Z I,,,(ef, + ~4~~)
where P, is the external end load, H,, H,, r, a;. a3, M, K2v
e15,
c44.
are the material
cl1
combinations,
constants
or their
of the beam, 2h is the
F is the length
depth of the beam, lo, is the moment of inertia, a’, c 0. k
=
-
Cl1
Cl2
K
c13
Cl1
Cl1
H L=
c33k11
+
Cl21
-
263
H4
e33kl,
+
C12)
-
2e3lc,3
=
-
c33c11
=
2
e14
M-l+ 4
H, =
+ 4 lc44
e33c1
I -
H 1 = 8.86 x 10” (dyn/cm’)’
H3
H&f4
---HH,cO H
a’, = ke31 -H
1
r2=-K
Hy’
= 1.56 x 1013 (stat C/~rn’)~ H:dr)r)= 0.13 x 10”
1
H, = -4.50
45 >
+ c,,
2b = 0.2 cm.
2h = 0.8 cm
The values of the necessary constants are*
e31c13
Cl1
K2 = ill
of the cantilever beam are
e=7cm
Cl1
H, =
In order to evaluate these results one must refer to the related experiments. Some of the conclusions drawn from the experiments conducted with a cantilever beam bone can be summarized (Bassett et al., 1962) as follows : 1. Piezoelectricity is generated by the extracellular matrix and does not require the presence of living cells for its generation. 2. The observed electrical potential following a load application between points A and B in Fig. 6. is neutralized by the leakage of the bone tissue in a short period of time. When the load is removed, then an opposite electrical potential is generated in the beam. If the bone sample is dry then the observed potential remains constant until the load is removed. 3. The amplitude of the electric voltage is proportional to the applied load. 4. The concave surface under the maxima1 compression has a negative electric potential and the convex surface under the maximal tension has a positive electrical potential. This last observation is quite important in the remodelling process of the bones. The bone formation occurs on the concave surface where electronegativity is dominant and the resorption of the bone tissue takes place on this convex side which is electropositive. The expression of cpverifies both conclusion 3 and 4. The numerical calculation for this theoretical result is given in the following example: The dimensions
e33~ll +eL
H,
ZMe,, a3=31,,,c,,-H,lo,.
0,
H, = c13e3) - cj3h
x 10” (dyn/cm’)(stat
H, = 1.33 x 1013 (dyn/cm’)(stat
C/cm’) C/cm’)
H$“c’)= 1.4 x lo2 stat C/cm2 dyn
~MK, 1
H:dry’= 0.12 x 10’ H, = 0.74 x 10’ stat C/cm2 K = 2.81 x 10” dyn/cm’ k = 0.57 K$““)
=
1.2 x 10’ (stat C/dyn cm’) Kyry)
r(we’)
=
3.275
&
=
0.1
X
10’
redry)= 3.227
+eo = - 1.4 x lO’stat C/cm2 al Fig. 6. The dimensions of the cantilever beam.
* Values computed from the Appendix.
a’!dv) = -0.12 a3 = 9 x lo-’
stat C/dyn cm3, M = 1
X
10’
265
+. piezoelectric model for dry bone tissue f
Zf
=
c,,ks,,
+
KS,, - H,E:
(50)
I 12j= 8 x 10v3 cm’. It is assumed that
at an electrode by the passage of current 1 through a solution for a time f. The passing current for the wet bone beam with one square centimeter cross section is
P, = lo6 dyn = 10N = 1.019 kg. A vertical load is applied at the end of the cantilever beam. The normal stress t,, = +3.5 x lo’* dyn or + 350 N/cm1 at y = + h, z = E. The maximum shear stress at y = 0 is f,, = 9.375 x lo6 dyn/cm’ = 93.75 N/C&.’ The vertical displacement of the point c in Fig. 6 is 0.5 mm. These stresses in the cantilever beam are in the same order as in the living femur. The potential difference between the points A(: = e/2, y = + h) and B(z = e/2, y = -h) is Aq, = (Pi*)- p(B) = 300 x 1.73 x 10W4 = 0.051V = 51mV. In the caSe of the wet bone
and this value is in the range of the experimentat results (l-5 mv). The variation of the Aq along the beam is given in the following formula.
AiPU3 I=-.-.-= R
where R = 100 ohm/cm (Gremy et al., 1966). Thus the m gram Ca to be liberated under the electrode requires t = m x 1.045 x IO’days M = 40.08
m=MIr
HF
where f is Faraday’s constant (F=96479 C). Here m is the mass of an element of atomic weight M liberated * From the equation (50). t &ivalent Ca concentration is approximately 4.8% of the total cations in the plasma.
and
1.~1= 2 for the calcium ions.
The average thickness of the osteon lamella is co. 1Opm = 10 -3 cm. The amount of Ca required to achieve a thickness equal to that of osteon lamella on the concave side of the beam can be found in the following way. 1 cm3 bone tissue contains 60% inorganic material (Herring, 1971) with a density of ca. 2.40 gjcm3. 1 cm3 bone tissue has the following inorganic elements.
A9 =9(+h)--9(-h)= 17.379 x lo-’ + 57.41 x lo-5 e-22.589+3.227:. If one compares the expression of Ap with the expression which is given by Williams and Breger (1974) for the potential difference one can see that the approximate solution of the stress gradient theory which gives a linear relation between the potential difference and the distance for the cantilever beam solution, fits the experimental data better than the above exact solution. This suggests that even the exact solution for the case of classical piezoekctricity is a very crude physical model for bone tissue. To conclude one must evaluate the amount of calcium ions i&erated at the electronegative.side of the wet bone beam (Bassett, 19681971). Although the theory is given for dry bone some of the results computed for the wet bone give some conclusive results. Let us assume that the potential difference cpAe is neutralized by the Ieakage of the blood conduction in the wet bone. As is suggested by experiments (Bassett er al., 1964; Bassett, 1965; Control1 et al., 1974; Friedenberg er nl., 1970. 1971,1974; Lavine et al., 1971,1974) with bone tissues, the calcium ions liberated from the electropositive side are deposited on the electronegative side. Faraday’s laws of electroiysis can be summarized in the equation (Moore, 1962)
15.94 x 104 e.s.u./sec,
Element
Atomic weight
Ca P 0 Ca3(POJ2
40.08 30.9 16.0 3 10.0
The amount of Calcium in 1 cm3 bone is K,
120.24 = 2.40 x 0.60 x - 310 = 0.558 g/cm’.
Therefore the time required for the deposition of a layer having the same thickness as that of an osteon lamella is, 1
t--x0.558x 0.048
1O-3 x 1.045 x IO3 2 12 days?
Bones remodel themselves to make the stresses equal (Kummer, 1972; Pauwels, 1965). A combination of both the muscles and the bones contribute to minimize the bending moments of bones. But these moments do not vanish totally because of variations in the external effects. In the bones the bending moments and the shear stresses are induced because of the external effects. Shear stresses cause (big)electrical potentials in the bones, which in turn cause remodelling in the bones in such a way that the bone tissue removed from the tension side (electropositive) is laid again on the compression side (electronegative) in order to minimize the bending effects. Pauwels (1965) demonstrated how a bar, stressed like the human ulna, can be adapted by a bending stress to a special curved axis form to minimize the bending effect over the length of the bone. Also the tension and the compression stresses and the variation from one to another
266
N. Giizusu
may also be an important factor in the remodelling of bones (Williams and Breger, 1974). The ossification time for an osteon lamella is quite prolonged (Lacroix, 1971) under compression force. The time which will be needed by an osteon to reach its full load of calcium varies and is a matter of several months. The above numerical example shows that under shear stress, a layer of osteon requires a short period of time for ossification with respect to the compression force. Therefore one can conclude that a bone which is subjected to the shear stress adopts its functional shape in a short time, in other words, the remodelling process is faster under shear stress than under normal stress condition with no stress gradients.
REFERENCES
Anderson,
J. C. and Ericksson,
C. (1970) Piezoelectric
properties Of dry and wet bone. Nature 227, 491-492. Asanzi, A. and Bonucci, E. (1968) The compressive properties of single osteons. Anat. Rec. 161. 377-392. Athenstaedt, H. (1970) Permanent electric polarization and pyroelectric behaviour of the vertebrate skeleton-VI. The
appendicular skeleton of man. 2. Anat. Enrw. Gesch. 131, 21-30. Athenstaedt, H. (1974) Pyroelectric and piezoelectric properties of vertebrates. Ann. N.Y. Acad. Sci. 238, 68-94. Bassett, C. A. L. and Becker, R. 0. (1962) Generation of electric potentials by bone in response to mechanical stress. Science 137, 1063-1064. Bassett, C. A. L. (1964) Bone B.iodynamics (Edited by H. M. Frost). Little % Brown, Boston, MA. Bassett, C. A. L., Pawluk, R. J. and Becker, R.O. (1964) Effects of electric currents on bone in uico. Nature 204, 652-654. Bassett, C. A. L. (1965) Electrical effects in bone. S&n?. Am.
213, 18-25. CONCLUSION
Experiments reported in the literature were conducted to discover the physical properties of bone tissue and were concerned mostly with mechanical and pyroelectrical properties. These experiments show that the dry bone tissue has an hexagonal polar structure and that this physical structure reflects approximately the properties of the wet bone. The source of this pyroelectric property of the bone is its collagen fibers. The results obtained from this theoretical study can be summarized as belows : (a) The healing of fractured bone tissue by negative electrical surface charge method as applied in clinics by experimental means is rationalized theoretically. The ossification process is accelerated in the treated area by creating an artificial piezoelectric axis which shows the direction of growth in the living tissues, thus increasing artificially the regeneration process and accelerating deposition on the fracture side. (b) The electrical stimulation necessary to the feedback mechanism can be generated at a cellular level under compression force ‘because of the particular orientation of alternating collagen fibers in the osteons. (c) The bones in the body change their geometric shape in an accelerated way under shear stress or under the shear stress and the normal stress gradients, due to external forces. In this remodelling process the bone becomes thicker on the compression side, in order to reduce the effect of the bending moments. The electrical potential caused by the shear stress may trigger the feedback mechanism for the remodelling and also’facilitate calcium deposition. (d) Even the exact theory of linear piezoelectricity cannot explain some of the experimental data properly which are observed. (Fukada, 1968a; Williams and Breger, 1974). Therefore, the theoretical models for bones must be changed in such a way that the experimental results must be explained more correctly bv these models.
Bassett, C. A. L. (1968) Biological significance of pieZ*leCtricity. Colt. T&e Res. 1.252-272. Bassett. C. A. L. (19711 The Biochemisrrv and Phvsioloav of So4 (Edited b; G. I-i. Bourne) 2nd Ed&on. Vol 3,pp. i-76. Academic Press, New York. Boume, G. H. (1971) The Biochemistry and Physiology of Bone. 2nd Edition Academic Press. New York. Vol. 2. on . .” 79-120. Cady, W. G. (1964) Pieroelecrricicy. Dover. New York, Vol. 1, p. 2. Cameron, D. A. (1971) The Biochemistry and Physiology of Bone. 2nd Edition. Academic Press, New York, Vol. 1, pp. 9 l-236. Connoll, J. F., Ortiz, J., Price, R. R. and Baytick, R. J. (1974) The effect of electrical stimulation on the biophysical properties of fracture healing. Ann. N.Y. Acad. Sci. 238, 519-529.
DBkmeci, M. C. (1974) A theory of high frequency vibrations of piezoelectric crystal bars. Inr. J. Solids Struct. 10, 401-409. F&de&erg, 2. B., Andrews, E. T., Smolenski. B. I., Pearl, B. W. and Brighton, C. T. (1970) Bone reaction to varying amounts of direct current. Surg. Gynecol. Obstet. 131,
894-899. Friedenberg, 2. B., Roberts, R. G. Diolizian, N. H. and Brighton, C. T. (1971) Stimulation of fracture healing by direct current in the rabbit fibula. J. Bone Jnt Surg. 53A, 1400-1408. Friendenberg, 2. B. and Brighton, C. T. (1974) Electrical fracture healing. Ann. N.Y. Acad. Sci. 238, 564-574. Fukada, E. and Yasuda, I. (1957) On the piezcelectric effect of bone, J. Phys. Sot. Japan 12, 1158-1162. Fukada E. and Yasuda. I. (19641 Piezoelcctric effects in.. collagen, Jup. J. Appl. ‘Phyi. 3, i17-121. Fukada, E. (1968) Piezoeiectricity as a fundamental property of wood. Food Sci. Tech. 2,299-306. Fukada. E. (l%Sa) Piezoelectricity in polymers and biological materials. CJIrrasonics6.229-234. Gjelsvik, A. (1973) Bone remodelling and piezoelectritity - 1. J. Biomechanics 6, 69-77. Gremy. F. and Pages, J. C. (1966) Elemenrs de Biophysique, Tome 1. Editions Medicales Flammarion. Herring, G. M. (1971) The Biochemistry and Physiology of Bone. 2nd Edition Academic Press, New York, Vol. 1, pp. 127-189. Kummer, B. K. F. (1972) Biomechanics (Y. C. Fung, N. Perrone and M. Anliker, ed.). Prentice-Ha& New York, pp. 237-271. LaCroix, P. (1971) The Biochemistry and Physiology of Bone. f;z_Ftion Academic Press, New York, Vol. 3, pp.
367
A piezoelectric model for dry bone tissue Lang, S. B. (1966) Pyroelectric effect in bone and tendon. Nature 212, 12, 704-705. Lang, S. 8. (1970) Ultrasonic method for measuring elastic coefficients of bone and results on fresh and dried bones. IEEE Trans. Biomed. Engng 17, 101-105. Lavine, L. S., Lustrin. I., Shamos, M. H. and Moss, M. L. (1971) The influence of electric current on bone regeneration in oico. Acta orthop. stand. 42, 305-314. Lavine, L., Lustrin, I., Rinaldi, R. and Shamos, M. (1974) Clinical and ultrastructural investigations of electrical enhancement of bone healing, Ann. N.Y. Acad. Sci. 238, 552-563. L&off, A. R. and Furst, M. (1974) Pyroelectric
effect in
collagenous structures. Ann. N.Y. Acad. Sci. 238, 26-35. Mindlin, R. D. (1961) High frequency vibrations of crystal plates. Quarr. Appl. Math. 19, 51-61. Mindlin, R. D. (1972) High frequency vibrations ofpiezoelectric crystal plates, Inc. J. Solids Strucr. 8, 895-906, Mindlin, R. D. (1976) Low frequency vibrations of elastic bars. Int. 1. Solids Struct. 12, 27-49. Moore, W. J. (1972) Physical Chemistry. Prentice-Hall, New York. Pauwels, F. (1965) Gesammelte Abhandlungen zurfunktioneilen Anatomie des Bewegunqsapparates. Springer. Berlin. Reinish, G. B. (1974) Dielectric and piezoefectric properties of bone as functions of moisture content. Ph.D. Thesis, Columbia University. Schiffman, E., Martin, G. R. and Miller, E. J. (1970) Biological calcification (H. Schraer, ed.). Appeleton, New York, p. 727. Tiersten, H. F. (1969) Linear Piezoelectric Plate Vibrations. Plenum Press, New York. Warwick, R. and Williams, P. L. (1973) Gray’s AMromy, 35th Edition. Longman. London. Williams, S. W. and Breger, L. (1974) Analysis of stress distribution and piezoelectric response in cantilever bending of bone and tendon. Ann. N. Y. Acad. Sci. U&121-130.
These equations can be written in the following way because of the symmetry properties of some coefficient tensors t, = c_sl - e,E,
+ ctiEi
1 < m, n f 6. The elastic stiffness coefficients c,, (10” dyn/cm’) (Lang, 1970).
ct1 cl? c13 c33 c44
Dry Phalanx
Dry femur
Wet Phalanx
2.12 (0.07)* 0.95 (0.03) 1.02 (0.14) 3.74 (0.16) 0.75 (0.02)
2.38 (0.14) 1.02 (0.06) 1.12 (0.21) 3.34 (0.12) 0.82 (0.02)
1.97 (0.05) 1.21 (0.04) 1.26 (0.12) 3.20 (0.11) 0.54 (0.01)
Piezoeiectric strain constants e,t
(LibotT~eId~ femur e3t e33 et4 ets
stat C (e.s.u.) 10’ cm2 [
I
(Reinish, 1974) Human femur dry
(Reinish, 1974) Human femur wet
4.10
2.378
1.902
1.643 4.305 0.271
1
stat C Dielectric permitivity e,, ___ (Reinish, 1974). [ cm. stat V
El1 833
APPENDIX
D, = e,p,
The indices take the following values 1 < i, j, k. f < 3 and
Human femur dry
Human femur wet
10 12
120 140
The material constants of bone The structure
of the bone tissue fits the hexagonal polar group. The constitutive equations for the bone tissue are: ‘ij
=
CC!ske
-
eki$k
DJ
=
e,kfskf
+
E,&,.
* The values in the parenthesis deviations.
show the standard
MODEL FOR DRY BONE TISSUE* NEJAT GUZELSU
Applied Mathematics Division, Marmara Research Institute, P.K. 21, Gebze, Kocaeli, Turkey Abstract - A linear theory of piezoelectric bars is used for analyzing a dry femur under constant load and a cantilever dry bone beam subjected to a vertical end load. The resuhs obtained from this theoretical study are: (a) The faster healing of the fractured bone tissue by the aid of negative electrical surface charge; @IThe causes of the allular level electrical stimulation due to the alternating structure of the collagen fibers ih the osteom ; (c) The faster rate of remodelling of the bone tissue due to the shear force.
In this study the bone is assumed to be a linear anisotropic piezoelectric material because of the constant temperature assumption. Although the piezoelectric properties are measured mostly on the dry bone samples, the theory stili gives some conclusions for wet bone. In the first part of this work a feedback mechanism for a bone tissue is given. In the second part, a compact portion of a diaphysis of a human femur is investigated under the axial uniform pressure. The last section contains a solution of a cantilever beam, made of compact bone, which is subjected to end-point loading.
m’-nIODUCTION The objective of this study is to investigate in a theoretical way some of the properties of bone tissue.
Among its properties, bone tissue has regeneration and remodelling capacity (Bassett, 1971). These two characteristics have been investigated by many authors (Athenstadt, 1970, 1974; Bassett et a/., 1964; Bassett, 1965; Connoll et a(., 1974; F&de&erg et al., 1970, 1971, 1974; Lavine et al., 1971, 1974). In recent years active research in the area of some tissues such as bone and collagen has shown these materials to be pyroelectric (Anderson er al., 1970; Fukada er al., 1957, 1964; Liboff et al., 1974; Reinish, 1974). The bones of the skeleton, which are mostly subjected to mechanical forces, are made of a tissue exhibiting pyroelectric behaviour, thus suggesting that the remodelling and the regeneration of bone are due to the interaction of both the mechanical and electrical fields. The oriented dipoles existing in a body cause an internal polarization, but these polarization charges are neutralized by leakage of the material (Cady, 1964, Vol. 2). Under constant temperature a pyroelectric material does exhibit a piezoelectricity property (Cady, 1964, Vol. 1). The physical description of the remodelling of bone tissue, in terms of a very simplified form of the linear theory of piezoelectricity, was given by Gjelsvik (Gjelsvik, 1973). As was suggested by Fukada (Fukada, 1968a) the stress gradient theory can be applied for explaining the experimental data for a cantilever bone beam subjected to constant end load (Williams and Breger, 1974). It is shown that the approximate gradient theory is in good agreement with the experimental data, though the treatment is formal only. According to Athenstaedt (Athenstaedt, 1970, 1974), the existing pyroelectric axis in a tissue usually shows the direction of growth of the organ made of this tissue. The remodelling and the regeneration of the bone tissue, exhibiting a pyroelectric property, can be explained by this property (Bassett, 1971; Athenstaedt, 1970, 1974; Bassett, 1964, 1968).
* Receioed 2 August 1976; receiwd January
1978.
for
publication3
THE FEEDBACK MECHANISM IN BONE
Bone tissue has a porous structure containing mineral salts at fluctuating levels. The elasticity, growth, regeneration and remodelling properties are some of its bioiogical and physical characteristics. In general the bone tissue consists of an amorphous and oriented fiber-based matrix, carrying mineral salts and the cells. The microscopic structure of the bone changes according to age, external effects and its place in the body (Warwick, 1973). Collagen molecules (Herring, 1971) make the tropocollagen macromolecules. The macromolecules consist of three a-polypeptide chains, each chain being twisted into a left hand helix, and the three chains being wound around a common axis in the form of right-handed superhelix. It forms a long thin rod of CQ. 2800 A in length and 14 A in diameter. The organization of tropocollagen molecules into fibrils takes place extracellularly. The osteons which are usually parallel to the femur axis are made of lamellas. Each osteon lamella contains collagen fibrils oriented in a helical way, and this orientation changes at each consecutive lamella, causing 90” angle between the helix lines. This particular structure creates a complex composite material made of collagen fiber layers with the direction of the fibrils in the neighbouring layers oriented at a 90” angle to the other. One of the main properties of collagen fib& is that of pyroelectricity
257
258
N. G&ELS.U
with the helix axis of the tropocollagen molecule as the pyroelectric axis (Fukada et al., 1964). Sixty-five per cent of the bone weight is due to the hydroxyapatite crystals (Bourne, 1971; Cameron, 1971). This material generally can be found around the collagen fibers in a crystalline structure. The organic matrix of the bone and the apatite crystals make a coherent unit vital to the bone tissue function. The tropocollagen molecules create heterogeneous centers for crystallization of the bone minerals (SchifTman, 1970; Kummer, 1972). These centers are usually located at the points where the more polar heads of collagen fibrils are located, Bones adjust their densities and orientations according to their functions. This response is known as Wolffs law. The generated electrical signals, due to the internal forces of the bone caused by the external effects, carry the necessary information to the bone cells, regulating their biological functions and controlling the orientation of their secreted macromolecules (Bassett et al., 1962). Wolff s law is summarized as a feedback mechanism (Bassett, 1971) in Fig. 1. Bone tissue is generated or resorbed by the osteoblast and osteoclast cells, respectively. The deposited calcium in bones can easily go into the blood with the aid of enzymes and hormones. Normally the exchange of calcium in bone is balanced and a steady state is maintained. For example: in the absence of the gravity force, the astronauts demonstrated that the process of bone decalcification was so accelerated that it was
INITIATES AND CONTROLS MINERALIZATION C ____-__--___THE ORIENTATION OF THE EXTRACELLULAR FXBRILS
BALANCED STEADY STATE EN
BONE TISSUE
obviously due to a loss of mineral salts rather than the remodelling (resorption) of bone tissue (Kummer, 1972). Consequently recalcification occurs within a short time. But the remodelling of bone tissue is made by the cells and it requires a more extended period. One can summarize the reactions of bone tissue to the external effects in the following way: a rapid one with intake and loss of mineral salts causing density variations, and a slow one with generation or resorbtion of bone tissue, namely remodellittg. AN INVESITCATION
OF A DRY FEMUR THE AXIAL FORCE
LPJDER
A femur located between a hip and a knee is subjected to rather big axial forces, especially during exercises in the human (Warwick, 1973). In this part of the study, the effect of the axial force on a femur is analyzed. A compact part of the diaphysis of a femur is approximated by a 2 f. long hollow cylinder Fig. 2. The coordinate z coincides with the femur axis. The xt, x2 plane which contains the cross section of a femur is bounded by r = a and r = b radii. The polar coordinates are used instead of x1 and x2 x1 = rcos0 x2 = rsin6.
The bone tissue has a hexagonal polar structure (Anderson et al., 1970; Fukada et al., 1957.1964; Lang,
I
I. L.-cr--_
STRUCTURAL RESPONSE TO RESIST FORCE
-
STRAIN
_-__-__----_ INCREASE IN THE DENSITY AND THE ORIENTATION CHANGING THE MECHANICAL PROPERTIES OF BONE
-
NOR!'X=; n ib
>
G?;
I
1 ELASTIC DEFORMATION 1 STIMLiUTES CELL REACTIONS -
CELL&B AND EKTRACELLLIMR TIWJSDUCER
I
PROPORTIONAL ELECTRICAL CO!DI.fAND S1GK.U
TRANSDUCER
,-_----_-__ PIEZOELECTRIC PROPERTIES IS EMRACELLULAR
NUTRITION ENZYBES CELL-CELL COMWXICATION
cN=NOR14ALSTRAIN FEEDBACK CONTROL SYSTE?fIX BONES
Fig: 1.
39
A piezoelectric model for dry bone tissue E, = -
s r"q$", m=O
(11)
where u, L;,w, cp,sij and Ei, are, respectively, m&hanical displacement in the radial direction, mechanical displacement in the 0 direction, mechanical displacement in the I direction , electrical potential, components of the strain tensor and the components of the electrical field. The field equations are: -,,,rr’
- T(;‘+
PZ;;L’ + T!m”’ = 0
(12)
- &-I;’ + FZ;,;” + T:;‘ ” = 0
(13)
7-L;!, = 0
(14)
Fig. 2. A model for a diaphysis of a femur
bone tissue is not homogeneous, and its properties change from point to point. The following analysis will be correct only in so far as the assumed piezoelectric texture correctly describes a real specimen. Imposing the rotational symmetry condition (independent of 0) and the hexagonal polar crystal structure (Cady, 1964, Vol. l), on the theory of linear piezoelectric bars (Dakmeci, 1974; Mindlin, 1975) one can obtain the following set of equations for a femur under the constant load u(r.2)
=
x
rmufm’(z)
(1)
r”w(“‘(z)
(2)
rmt’(m’(z) s wcl)(z)
(3)
“=I3 I
x
w(r,z) =
m-0
2
v(r,z) =
Inso Z (t.2’ = cp mZ, rm4+m’(z) Z
(4)
d;yZ+” + d'"'" = 0.
-mdj”‘+
1966, 1970; Liboff, 1974; Reinish, 1974).* However,
(15)
These equations are respectively, the balance of moments in the directions of r, t and 0 and Gauss’s law. For example p,F’ and d!“’ are TbyJ = Irmtr,dA
d’“’
=
I
= IO* Pr,,.krdr,
s
rmD,dA.
(16)
A
where t,j, d,, Ti,, T,, d,, d and A are, respectively, components of the stress tensor, components of the dielectric displacement, a component of the resultant force which is obtained by the integration of the corresponding stress over the cross section of the bar, the resultant value of the stresses at the boundary, a component of the resultant dielectric displacement, resultant dielectric displacement on the boundary and the cross section ofthe femur. Because ofthe absence of tractions on r = a and r = b surfaces give
cc s,
=
C (m + l)Fu’“+”
TW+ 1’
(5)
=
()t
m=CJ
&+
1) =
0.
1
sBe
=
1
pUh+l)
(6)
“=O
s
zi
= f
r=m,y:
(7)
m=o
s
Bx
=
s,, =
rc!,”
i
(8) F(u$)(m
+ l)kmC1’
m=o
E, =
-“$, (m +
In the calculations the electric field existing outside the piezoelectric material is ignored because the dielectric constant of vacuum (E = 1 in CGS units) is much smaller than the piezoelectric body (Mindlin, 1961, 1972; Tiersten, 1969): Therefore, the boundary condition for the dielectric displacement is reduced to the following form: -D, = 4nw/(z).
(9)
Hence, the term in (15) can be calculated from ~)r=@‘+~’
(10)
2x _d’“‘=
l
(17)
In the Appendix, the physical constants are given.
t Thebold&phab&c character indicates that this value is defined on the boundary. : A dielectric permitivity of a bone specimen with 40% relative humidity is EII = 40 (Reinkh, 1974).
-4~
Q. .I”
o,(z)b”bdO
= -2Zxbm+’
47=J/(Z), (18)
where o/(z) is the free surface charge on surface r = b. The physical discussion of o/(z) is given at the end of this section.
260
N. GijZELW
The other boundary conditions FZ’ =
)
-
Tj:‘=
-
pm’ = *I
0
-d’“’
=
dA
porn
are
following set of equations for the unknowns u(“, w(O), q(O) and #‘j.
3= +e
(1%
z=
(20)
+p,FdA
-t
i=
L
d”“’ t = 0
+ c13w!z’
= 0
(29)
J(l + O)[Zc,,u[:) ‘r c3Jw!$j + e3+#I”:] = 0
(30)
(22)
where p,, stands for the uniform pressure which is distributed over the surfaces I = & f. T,, = 0 at 2 = k e implies that T,, I 0 along the cylinder. The eiements of the stress tensor and the dielectric displacement vector are given
- 2nb24no,(t)
(31)
+ O)[(c,,+ c,~)u(~)+ c13w!Q) + e31q!9’]
-J(l
CL&(*’
+ O)[(c,, 4
-J(l
+ cIJw!~’ + e31@j’rj = 0 J(2 + O)[Zc,,u!:’ + -J(l
(23)
c33w!2
+
e,,cp!2]
=
0
(32) (33)
+ O)[e,,(u!C$ + w(l)) - .sII$‘)]
+.I(2 + O)[Ze,,@ = c J(m + P)([C,,(P
= 0
m=l
+ 1) + C&(~+‘l
+ c,~w(~) .: + e,,cp”‘}* .J ;
+ e,,qv$‘]
I:(
(21)
+f
z=-E,
7-F) = c J(m + p){[c,,(p P
-J(O + O)[(c,2 + Cll)U”
+e
z=
0
m=O
+ e3,w!$i
- E&;;]
f 1) + Cl ,l@+l’
- 2nb34xo,(t)
= 0. (34)
P +
rgy’
=
1
J(m
p)([c,,(p
+
c,3w!:’
+
(24)
e,,cp!?I
+ 1) + c&(p+l)
p,;’
=
1
J(m
c,pY
p){[c&Z’
+
+
, w(O),c#” and $I’ are solved from the following equations ; Tit) = 0, integrated forms of (30), (31) and (2%
P
f
u(l)
(25)
e33cP!?f
#)
=
Cnffz
-k
’ H,H,
+ (P + l)w’p+ 9 e15(p
+
(26)
l)cp’p+l’)
=
C
J(m
l){c,,u~
+
+ e,,(p + lWp’ ‘)I
e3,H, + @II + cd3
+ dcrn’ : 1 J(m + ,
p){el,$’
+
H,H,
elj
(H,Hz-tH,H,)
P x
[u?’
+
(p
+
-+ H,Hh
(35) 1
1 1
(cl1 + ci2W2- H4e3, H,H,+ H,H, ’
w(O)= k I 7-g
-
o,ti)dl + k,
P +
H,c,3
+ H,H,
l)w(p+ I)]
- EII(P + &#p+ ‘q
(27) (36)
d’“’ r = 1 J(m + p){2~~(p
-t l)dp+”
P
(0) cp
(Pb +
e33w!:)
-
~33(P.~
I
(28)
=
k,c,,H, H,H, + Hdf4
HICI~ H,H, + HaH,
z -
b J(m
+
p)
f’
= I
+“~HTdr,
a
where ci,, et,, cjk are, respectively, the elastic coefficients, the piezoelectric strain constants and the dielectric permitivities. If one’ substitutes the equations (23-28) into the equations (12-15) form = 0, 1 and p = 0 will give the
p
=
(38) J
[b-$]=[: - i (b=+:;+o’)] > 0,
where l
The summation
sign C 3 ; . I
p=O
H, =
c3,@,1
H,
e33e31
=
+ +
~12) ~13~33
-
2c:,
261
A piezoelectric model for dry bone tissue
ff,
=
c13e33
H,
=
e33(cll
From
==
+
CIA
-
2e31c13
dr
139)
2nb34~o,(z) b - Jo,
(43)
p,=
(01
2
c33e31
b
kl=_.$ k
-
p,,Zzr
I0
0.
the above values one can obtain f,, = 0
t zz -4 D,=
--
J(l)
D
[
Jw
1
staedt, 1970, Fig. 6). the electric polarization pattern in the lower extremity of new-born infant is shown. A human femur starts to ossify from its ossification center located in the middle of the femur. The bone grows and calcifies in the longitudinal and radial direction. When a human reaches a certain age the radial polarization vector disappears but the longitudinal polarization vector remains. This longitudinal polarization vector is balanced out with the opposite polarization vector of metaphysis. From the above experimental results one can conclude that, if one creates an artificial radiaI polarization vector in an area then the bone growth and the regeneration can be induced locally in the radial direction. P, and P, components of the polarization vector are computed by the following formula in terms of (43) and (44) Di = Et + 4~Pi
~fKW
The physical explanation of these results : As it was mentioned at the beginning of this study, a tissue has a permanent polarization vector in the direction of its growth (Athenstaedt, 1970, 1974). In Fig. 3 (Athen-
P,=
-
(45j
(46) I--
(bl
(cl
Fig. 3. Reference (Athenstaedt, 1970, Fig. 6).
Electric polarization pattern in the lower extremity of newborn infants. (a) Total view of the extremity; (b) distal femur head, enlarged, with the epiphysis as it normally exists at birth. (1) Femur; (2) tibia ; (3) fibula ; ossification centers of the metatarsal bone (4), the phalanges (5). the tales (6), the calcaneus (7). and the cuboid bone (8); (9) distal femur epiphysis : (10) a pan of the periosteum showing the direction of radial polarization in it (arrow); (c) A ring of compact substance cut out from the femur diaphysis, showing its direction of radial polarization.
HlCl3 b-0 A
A = H,H,
+ H,H,
The polarization vector P, solely depends upon the free surface charge o/(z) of the surface r = b. If w,(s) < 0 then P, is positive. In an adult, applying free negative surface charge from a negative pole of a battery to the fracture area will induce a radial polarization vector locally at that region a condition similar to that existing in the infant where the regeneration and healing capacity for fractures and wounds is quite extended. The experimental observations (Bassett et of., 1964; Bassett 1965 ; Connoll et al., 1974; Friedenberg et al., 1970, 197 1, 1974; Lavine et al., 1971, 1974) of bones show that a negative pole of a battery, which is applied to the fractured sideof a bone, accelerates the ossification process in the bone. Such treatment may cause two things : (a) First, the deposition ofpositive electrical charge carrying Ca ions to the negative pole (faster ossification); (b) Second, a constant radial polarization vector near the fracture area similar to that existing in infants (increased regeneration capacity). The P, polarization vector, given by the equation (46), consists of two terms. The first term on the right hand side of this equation shows the effect of the surface charge on P,. The second term shows the effect of the external force. At each step of a human the second term changes sign causing an ahernating
262
N. G~~ZEISJ
electrical stimulation in the bone tissue because of the leakage in living tissue. The final implication of the above calculation is a proposed explanation of cellular-level electrical stimulation. There are three kinds ofcollagen fiber arrangements in osteons (Ascenzi and Bonucci, 1968): the first kind, in which the fibres in all lamellae are roughly parallel to the axis of the osteone, makes a 90” helix angle ; the second kind, in which the fibres in alternate lamellae are longitudinal and circumferential gives a helix angle between 90” and 0” ; the third kind, in which the fibres in all lamellae are inclined but in opposite directions and therefore the successive lamellaes have an alternating helix angle between plus and minus 45”. In this study, the collagen structure of an osteon of the third kind which is shown in Fig. 4 is investigated. The osteon axis is taken parallel to the z axis. The osteon lamellas (I) and (II) are located between the curves 1 and 2 and (2 and 3), respectively. The osteocytes are shown with dark ellipsoidical points. The curved lines starting from curve 1 show the direction of the collagen fibrils in the first osteon lamella (I) and the curved lines starting from 2 shown the orientation of the collagen fibrils in the second osteon lamella (II).
Ostefytes
collagen
(al
(47)
djm= eJ&-,l ‘iJk =
cjk -
hJk -
(48) dJ&krw
dJmstands for the pietoelectric stress constants. If there
Of the
L___:_________J
Of the fibers
The structure
of an OsteOn.
Section (bl
The position ,”
0”
Osteon lomello
A -A
of the
OsteOCyteS
osteon.
I Osteon lame1 la IT
(c)
The
section
(49)
a,, = {i = k, 1, j # k, 01.
I*
Direction
Direction
An angle between the tangents of the two different layers of collagen fibrils is cu. 90”. In Fig. 4(c) a very small element, which is taken around the M-M line in Fig. 4(a), is shown. It is referred to a system of righthanded Cartesian coordinates where 2 axis is parallel to the femur axis, x axis is perpendicular toy and 2. The surfaces passing through 1,2,3 curves defining the boundaries of the element are approximated by planes. In Fig. 4(d) the direction of the collagen fibrils are shown. If this element is subjected to the uniform pressure p. parallel to the z axis, then the state ofstress on a square element, located with its edge parallel to the collagen fibrils is shown in Fig. 5. The relation between the stress tensor and the polarization vector is given in the following way.
The direction of the collagen fibers
M-M.
Cd) I -rT 0steon lamellos and the direction of the coiloqen fibers.
Fig. 4.
A piezoelectric model for dry bone tissue
for the second layer. d,, = - +
8 = 135’.
When the bone tissue is considered as a whole then this alternating structure of the osteon lamellae are not observable in the experiments. The shear stress (p,/2) (this stress (tY.:. = - pJ2)) is defined with respect to the x y’ z’ coordinate system which is obtained with the rotation of Y: coordinates 45” about the x axis) in Fig. 5(a) gives negative polarization for the first lamella 8 = 45’ la) The state of stress of an OSteOn element which has on angle 45' with the 2 axis.
P, = d;.&
= % ( -P&9
d,,i+,
= - 4
In the second osteon lamella it gives Osteacytes d - 2
P, = d14tyr =
(
dwpo (- p,/Z) = -
>
4
e = 135’
Iv=
Osteon lamellas (b) The localpolarization in the osteon lamelias.
Fig. 5.
is no electrical external effects only mechanical then equation (47) can be approximated
ones
This approximate relation has been used for the calculations of the piezoelectric constants (Fukada et al., 1957, 1964). Fukada (Fukada, 1968) gave the piezoelectric stress constants d,, for fibrous material (wood)* in terms of the angle between the direction of fibers and the direction of applied stresses.
In a similar manner in each osteon lamella direction of the fibers change, so one can give (d,,)t terms of 0, which is the angle between the direction the applied stresses to the bone and the direction fibers. For example, for the first layer, 0 = 45’ d,, positive
9
the in of of is
8 = 45’
The wood tissue fits the hexagonal holoaxiai crystal structure l
. . . .
.
.
d
14 .
The magnitude of thkse local electrical charges are the function of the external load P,. This electrical stimulations may be the trigger of the cellular response to the external effects which is described in the feedback mechanism. From this, one can conclude that, if a broken bone can be exposed to mechanical external effects :hen the healing would be faster than a broken bone which is casted.
A?4 INVESTIGATION OF A CANTILEVER BEAM MADE A COMPACT BONE
(d,,) = % sin 20.
d,, = +
positive polarization. These opposite polarization vectors in alternating lamellas give a total zero polarization along the x axis, but in a cellular level these polarization vectors cause local electrical surface charges on the largest surface side of the osteocytes Fig. 5(b).
.
.
-d,,
.
t d,, is the biggest piezoelectric stress constant in the bone tissue see Appendix.
OF
In this section, a cantilever beam having a hexagonal polar structure is analyzed. In Fig. 6, the dimensions of the beam and the referred Cartesian coordinates are shown. The boundary conditions are summarized in the following way: P, is the external end load at z = 0 in they direction which is assumed to be distributed on the xy plane as a shear stress. The normal stress t,, = 0 at z = 0. The displacements and the derivative of the vertical displacement are zero at -_ = e. The side surfaces of the beam (0 < -_< E) are free ofstresses. The electrical displacement is assumed to be zero on the all surfaces of the beam, because of the absence of free electrical surface charges and the fact that the dielectric permitivity of the bone is much bigger than the vacuum (Mindlin. 1961, 1972; Tiersten ; 1969). Shortly, in these calculations the assumption of plane stress is applied, consequently the stress components in the x direction and the dielectric
N. GOZELSU
264
displacement component D, are assumed to be zero. The series form of the electrical potential cp used in calculations is :
The procedure of the solution for the cantilever beam is similar to the solution of the femur under the constant load which is given in the previous section. Because of the length of the expressions and the straightforward nature of the calculations, the steps of the computations are not given here. After the calculations the final form of the procedure is written in the following way : P&H,
cp=-
coshrz + P,a,h’ -sinh f.r T&1+&)
Io,H,ra’,
Y3 P&l, + Z I,,,(ef, + ~4~~)
where P, is the external end load, H,, H,, r, a;. a3, M, K2v
e15,
c44.
are the material
cl1
combinations,
constants
or their
of the beam, 2h is the
F is the length
depth of the beam, lo, is the moment of inertia, a’, c 0. k
=
-
Cl1
Cl2
K
c13
Cl1
Cl1
H L=
c33k11
+
Cl21
-
263
H4
e33kl,
+
C12)
-
2e3lc,3
=
-
c33c11
=
2
e14
M-l+ 4
H, =
+ 4 lc44
e33c1
I -
H 1 = 8.86 x 10” (dyn/cm’)’
H3
H&f4
---HH,cO H
a’, = ke31 -H
1
r2=-K
Hy’
= 1.56 x 1013 (stat C/~rn’)~ H:dr)r)= 0.13 x 10”
1
H, = -4.50
45 >
+ c,,
2b = 0.2 cm.
2h = 0.8 cm
The values of the necessary constants are*
e31c13
Cl1
K2 = ill
of the cantilever beam are
e=7cm
Cl1
H, =
In order to evaluate these results one must refer to the related experiments. Some of the conclusions drawn from the experiments conducted with a cantilever beam bone can be summarized (Bassett et al., 1962) as follows : 1. Piezoelectricity is generated by the extracellular matrix and does not require the presence of living cells for its generation. 2. The observed electrical potential following a load application between points A and B in Fig. 6. is neutralized by the leakage of the bone tissue in a short period of time. When the load is removed, then an opposite electrical potential is generated in the beam. If the bone sample is dry then the observed potential remains constant until the load is removed. 3. The amplitude of the electric voltage is proportional to the applied load. 4. The concave surface under the maxima1 compression has a negative electric potential and the convex surface under the maximal tension has a positive electrical potential. This last observation is quite important in the remodelling process of the bones. The bone formation occurs on the concave surface where electronegativity is dominant and the resorption of the bone tissue takes place on this convex side which is electropositive. The expression of cpverifies both conclusion 3 and 4. The numerical calculation for this theoretical result is given in the following example: The dimensions
e33~ll +eL
H,
ZMe,, a3=31,,,c,,-H,lo,.
0,
H, = c13e3) - cj3h
x 10” (dyn/cm’)(stat
H, = 1.33 x 1013 (dyn/cm’)(stat
C/cm’) C/cm’)
H$“c’)= 1.4 x lo2 stat C/cm2 dyn
~MK, 1
H:dry’= 0.12 x 10’ H, = 0.74 x 10’ stat C/cm2 K = 2.81 x 10” dyn/cm’ k = 0.57 K$““)
=
1.2 x 10’ (stat C/dyn cm’) Kyry)
r(we’)
=
3.275
&
=
0.1
X
10’
redry)= 3.227
+eo = - 1.4 x lO’stat C/cm2 al Fig. 6. The dimensions of the cantilever beam.
* Values computed from the Appendix.
a’!dv) = -0.12 a3 = 9 x lo-’
stat C/dyn cm3, M = 1
X
10’
265
+. piezoelectric model for dry bone tissue f
Zf
=
c,,ks,,
+
KS,, - H,E:
(50)
I 12j= 8 x 10v3 cm’. It is assumed that
at an electrode by the passage of current 1 through a solution for a time f. The passing current for the wet bone beam with one square centimeter cross section is
P, = lo6 dyn = 10N = 1.019 kg. A vertical load is applied at the end of the cantilever beam. The normal stress t,, = +3.5 x lo’* dyn or + 350 N/cm1 at y = + h, z = E. The maximum shear stress at y = 0 is f,, = 9.375 x lo6 dyn/cm’ = 93.75 N/C&.’ The vertical displacement of the point c in Fig. 6 is 0.5 mm. These stresses in the cantilever beam are in the same order as in the living femur. The potential difference between the points A(: = e/2, y = + h) and B(z = e/2, y = -h) is Aq, = (Pi*)- p(B) = 300 x 1.73 x 10W4 = 0.051V = 51mV. In the caSe of the wet bone
and this value is in the range of the experimentat results (l-5 mv). The variation of the Aq along the beam is given in the following formula.
AiPU3 I=-.-.-= R
where R = 100 ohm/cm (Gremy et al., 1966). Thus the m gram Ca to be liberated under the electrode requires t = m x 1.045 x IO’days M = 40.08
m=MIr
HF
where f is Faraday’s constant (F=96479 C). Here m is the mass of an element of atomic weight M liberated * From the equation (50). t &ivalent Ca concentration is approximately 4.8% of the total cations in the plasma.
and
1.~1= 2 for the calcium ions.
The average thickness of the osteon lamella is co. 1Opm = 10 -3 cm. The amount of Ca required to achieve a thickness equal to that of osteon lamella on the concave side of the beam can be found in the following way. 1 cm3 bone tissue contains 60% inorganic material (Herring, 1971) with a density of ca. 2.40 gjcm3. 1 cm3 bone tissue has the following inorganic elements.
A9 =9(+h)--9(-h)= 17.379 x lo-’ + 57.41 x lo-5 e-22.589+3.227:. If one compares the expression of Ap with the expression which is given by Williams and Breger (1974) for the potential difference one can see that the approximate solution of the stress gradient theory which gives a linear relation between the potential difference and the distance for the cantilever beam solution, fits the experimental data better than the above exact solution. This suggests that even the exact solution for the case of classical piezoekctricity is a very crude physical model for bone tissue. To conclude one must evaluate the amount of calcium ions i&erated at the electronegative.side of the wet bone beam (Bassett, 19681971). Although the theory is given for dry bone some of the results computed for the wet bone give some conclusive results. Let us assume that the potential difference cpAe is neutralized by the Ieakage of the blood conduction in the wet bone. As is suggested by experiments (Bassett er al., 1964; Bassett, 1965; Control1 et al., 1974; Friedenberg er nl., 1970. 1971,1974; Lavine et al., 1971,1974) with bone tissues, the calcium ions liberated from the electropositive side are deposited on the electronegative side. Faraday’s laws of electroiysis can be summarized in the equation (Moore, 1962)
15.94 x 104 e.s.u./sec,
Element
Atomic weight
Ca P 0 Ca3(POJ2
40.08 30.9 16.0 3 10.0
The amount of Calcium in 1 cm3 bone is K,
120.24 = 2.40 x 0.60 x - 310 = 0.558 g/cm’.
Therefore the time required for the deposition of a layer having the same thickness as that of an osteon lamella is, 1
t--x0.558x 0.048
1O-3 x 1.045 x IO3 2 12 days?
Bones remodel themselves to make the stresses equal (Kummer, 1972; Pauwels, 1965). A combination of both the muscles and the bones contribute to minimize the bending moments of bones. But these moments do not vanish totally because of variations in the external effects. In the bones the bending moments and the shear stresses are induced because of the external effects. Shear stresses cause (big)electrical potentials in the bones, which in turn cause remodelling in the bones in such a way that the bone tissue removed from the tension side (electropositive) is laid again on the compression side (electronegative) in order to minimize the bending effects. Pauwels (1965) demonstrated how a bar, stressed like the human ulna, can be adapted by a bending stress to a special curved axis form to minimize the bending effect over the length of the bone. Also the tension and the compression stresses and the variation from one to another
266
N. Giizusu
may also be an important factor in the remodelling of bones (Williams and Breger, 1974). The ossification time for an osteon lamella is quite prolonged (Lacroix, 1971) under compression force. The time which will be needed by an osteon to reach its full load of calcium varies and is a matter of several months. The above numerical example shows that under shear stress, a layer of osteon requires a short period of time for ossification with respect to the compression force. Therefore one can conclude that a bone which is subjected to the shear stress adopts its functional shape in a short time, in other words, the remodelling process is faster under shear stress than under normal stress condition with no stress gradients.
REFERENCES
Anderson,
J. C. and Ericksson,
C. (1970) Piezoelectric
properties Of dry and wet bone. Nature 227, 491-492. Asanzi, A. and Bonucci, E. (1968) The compressive properties of single osteons. Anat. Rec. 161. 377-392. Athenstaedt, H. (1970) Permanent electric polarization and pyroelectric behaviour of the vertebrate skeleton-VI. The
appendicular skeleton of man. 2. Anat. Enrw. Gesch. 131, 21-30. Athenstaedt, H. (1974) Pyroelectric and piezoelectric properties of vertebrates. Ann. N.Y. Acad. Sci. 238, 68-94. Bassett, C. A. L. and Becker, R. 0. (1962) Generation of electric potentials by bone in response to mechanical stress. Science 137, 1063-1064. Bassett, C. A. L. (1964) Bone B.iodynamics (Edited by H. M. Frost). Little % Brown, Boston, MA. Bassett, C. A. L., Pawluk, R. J. and Becker, R.O. (1964) Effects of electric currents on bone in uico. Nature 204, 652-654. Bassett, C. A. L. (1965) Electrical effects in bone. S&n?. Am.
213, 18-25. CONCLUSION
Experiments reported in the literature were conducted to discover the physical properties of bone tissue and were concerned mostly with mechanical and pyroelectrical properties. These experiments show that the dry bone tissue has an hexagonal polar structure and that this physical structure reflects approximately the properties of the wet bone. The source of this pyroelectric property of the bone is its collagen fibers. The results obtained from this theoretical study can be summarized as belows : (a) The healing of fractured bone tissue by negative electrical surface charge method as applied in clinics by experimental means is rationalized theoretically. The ossification process is accelerated in the treated area by creating an artificial piezoelectric axis which shows the direction of growth in the living tissues, thus increasing artificially the regeneration process and accelerating deposition on the fracture side. (b) The electrical stimulation necessary to the feedback mechanism can be generated at a cellular level under compression force ‘because of the particular orientation of alternating collagen fibers in the osteons. (c) The bones in the body change their geometric shape in an accelerated way under shear stress or under the shear stress and the normal stress gradients, due to external forces. In this remodelling process the bone becomes thicker on the compression side, in order to reduce the effect of the bending moments. The electrical potential caused by the shear stress may trigger the feedback mechanism for the remodelling and also’facilitate calcium deposition. (d) Even the exact theory of linear piezoelectricity cannot explain some of the experimental data properly which are observed. (Fukada, 1968a; Williams and Breger, 1974). Therefore, the theoretical models for bones must be changed in such a way that the experimental results must be explained more correctly bv these models.
Bassett, C. A. L. (1968) Biological significance of pieZ*leCtricity. Colt. T&e Res. 1.252-272. Bassett. C. A. L. (19711 The Biochemisrrv and Phvsioloav of So4 (Edited b; G. I-i. Bourne) 2nd Ed&on. Vol 3,pp. i-76. Academic Press, New York. Boume, G. H. (1971) The Biochemistry and Physiology of Bone. 2nd Edition Academic Press. New York. Vol. 2. on . .” 79-120. Cady, W. G. (1964) Pieroelecrricicy. Dover. New York, Vol. 1, p. 2. Cameron, D. A. (1971) The Biochemistry and Physiology of Bone. 2nd Edition. Academic Press, New York, Vol. 1, pp. 9 l-236. Connoll, J. F., Ortiz, J., Price, R. R. and Baytick, R. J. (1974) The effect of electrical stimulation on the biophysical properties of fracture healing. Ann. N.Y. Acad. Sci. 238, 519-529.
DBkmeci, M. C. (1974) A theory of high frequency vibrations of piezoelectric crystal bars. Inr. J. Solids Struct. 10, 401-409. F&de&erg, 2. B., Andrews, E. T., Smolenski. B. I., Pearl, B. W. and Brighton, C. T. (1970) Bone reaction to varying amounts of direct current. Surg. Gynecol. Obstet. 131,
894-899. Friedenberg, 2. B., Roberts, R. G. Diolizian, N. H. and Brighton, C. T. (1971) Stimulation of fracture healing by direct current in the rabbit fibula. J. Bone Jnt Surg. 53A, 1400-1408. Friendenberg, 2. B. and Brighton, C. T. (1974) Electrical fracture healing. Ann. N.Y. Acad. Sci. 238, 564-574. Fukada, E. and Yasuda, I. (1957) On the piezcelectric effect of bone, J. Phys. Sot. Japan 12, 1158-1162. Fukada E. and Yasuda. I. (19641 Piezoelcctric effects in.. collagen, Jup. J. Appl. ‘Phyi. 3, i17-121. Fukada, E. (1968) Piezoeiectricity as a fundamental property of wood. Food Sci. Tech. 2,299-306. Fukada. E. (l%Sa) Piezoelectricity in polymers and biological materials. CJIrrasonics6.229-234. Gjelsvik, A. (1973) Bone remodelling and piezoelectritity - 1. J. Biomechanics 6, 69-77. Gremy. F. and Pages, J. C. (1966) Elemenrs de Biophysique, Tome 1. Editions Medicales Flammarion. Herring, G. M. (1971) The Biochemistry and Physiology of Bone. 2nd Edition Academic Press, New York, Vol. 1, pp. 127-189. Kummer, B. K. F. (1972) Biomechanics (Y. C. Fung, N. Perrone and M. Anliker, ed.). Prentice-Ha& New York, pp. 237-271. LaCroix, P. (1971) The Biochemistry and Physiology of Bone. f;z_Ftion Academic Press, New York, Vol. 3, pp.
367
A piezoelectric model for dry bone tissue Lang, S. B. (1966) Pyroelectric effect in bone and tendon. Nature 212, 12, 704-705. Lang, S. 8. (1970) Ultrasonic method for measuring elastic coefficients of bone and results on fresh and dried bones. IEEE Trans. Biomed. Engng 17, 101-105. Lavine, L. S., Lustrin. I., Shamos, M. H. and Moss, M. L. (1971) The influence of electric current on bone regeneration in oico. Acta orthop. stand. 42, 305-314. Lavine, L., Lustrin, I., Rinaldi, R. and Shamos, M. (1974) Clinical and ultrastructural investigations of electrical enhancement of bone healing, Ann. N.Y. Acad. Sci. 238, 552-563. L&off, A. R. and Furst, M. (1974) Pyroelectric
effect in
collagenous structures. Ann. N.Y. Acad. Sci. 238, 26-35. Mindlin, R. D. (1961) High frequency vibrations of crystal plates. Quarr. Appl. Math. 19, 51-61. Mindlin, R. D. (1972) High frequency vibrations ofpiezoelectric crystal plates, Inc. J. Solids Strucr. 8, 895-906, Mindlin, R. D. (1976) Low frequency vibrations of elastic bars. Int. 1. Solids Struct. 12, 27-49. Moore, W. J. (1972) Physical Chemistry. Prentice-Hall, New York. Pauwels, F. (1965) Gesammelte Abhandlungen zurfunktioneilen Anatomie des Bewegunqsapparates. Springer. Berlin. Reinish, G. B. (1974) Dielectric and piezoefectric properties of bone as functions of moisture content. Ph.D. Thesis, Columbia University. Schiffman, E., Martin, G. R. and Miller, E. J. (1970) Biological calcification (H. Schraer, ed.). Appeleton, New York, p. 727. Tiersten, H. F. (1969) Linear Piezoelectric Plate Vibrations. Plenum Press, New York. Warwick, R. and Williams, P. L. (1973) Gray’s AMromy, 35th Edition. Longman. London. Williams, S. W. and Breger, L. (1974) Analysis of stress distribution and piezoelectric response in cantilever bending of bone and tendon. Ann. N. Y. Acad. Sci. U&121-130.
These equations can be written in the following way because of the symmetry properties of some coefficient tensors t, = c_sl - e,E,
+ ctiEi
1 < m, n f 6. The elastic stiffness coefficients c,, (10” dyn/cm’) (Lang, 1970).
ct1 cl? c13 c33 c44
Dry Phalanx
Dry femur
Wet Phalanx
2.12 (0.07)* 0.95 (0.03) 1.02 (0.14) 3.74 (0.16) 0.75 (0.02)
2.38 (0.14) 1.02 (0.06) 1.12 (0.21) 3.34 (0.12) 0.82 (0.02)
1.97 (0.05) 1.21 (0.04) 1.26 (0.12) 3.20 (0.11) 0.54 (0.01)
Piezoeiectric strain constants e,t
(LibotT~eId~ femur e3t e33 et4 ets
stat C (e.s.u.) 10’ cm2 [
I
(Reinish, 1974) Human femur dry
(Reinish, 1974) Human femur wet
4.10
2.378
1.902
1.643 4.305 0.271
1
stat C Dielectric permitivity e,, ___ (Reinish, 1974). [ cm. stat V
El1 833
APPENDIX
D, = e,p,
The indices take the following values 1 < i, j, k. f < 3 and
Human femur dry
Human femur wet
10 12
120 140
The material constants of bone The structure
of the bone tissue fits the hexagonal polar group. The constitutive equations for the bone tissue are: ‘ij
=
CC!ske
-
eki$k
DJ
=
e,kfskf
+
E,&,.
* The values in the parenthesis deviations.
show the standard