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Ultrasonic wave propagation in human cortical bone—I. Theoretical considerations for hexagonal symmetry
J. &om&mnics.
1976. Vol. 9. pp. 407412.
Pergamon PXSS.
Printed I”
GreatBrlta~n
ULTRASONIC WAVE PROPAGATION IN HUMAN CORTICAL BONE-I. THEORETICAL CONSIDERATIONS FOR HEXAGONAL SYMMETRY*+ HYO SUB Yoo~f Laboratory for Crystallographic
and J. LAWRENCE KATZ
Biophysics, Department of Physics, Rensselaer Polytechntc Instttute. Troy, NY 12181. U.S.A.
the theory of wave propagation in an anisotropic elastic medium are derived the basic equations relating the five independent second-order elastic stiffness constants (fourth-rank tensor) to the ultrasonic wave speeds in a hexagonal medium, with special emphasis on determining the microtextural symmetry of human cortical bone. In addition. the three pure mode directions of high symmetry in a hexagonal medium are explicitly shown. Finally. expressions relating the ‘technical moduli’ such as Young’s modulus, shear modulus and bulk modulus to the elastic compliances are presented for the most general case (triclinic symmetry) and then are specialized for the hexagonal system. Abstract-From
INTRODUCTION Although Wertheim (1847) made the observation that bone is viscoelastic, this aspect of its behavior appears to have been neglected experimentally for over a century. It was not until the late 1950’s and mid 1960’s that measurements of the viscoelastic properties of bone began to appear in the literature. In 1958, Craig and Peyton reported on creep studies on dentin, while Smith and Walmsley (1959) performed similar experiments on human bone. Sedlin (1965) and Lugassy and Korostoff (1969) reported on creep and relaxation experiments on human and bovine bone respectively. Smith and Keiper (1965) determined the dynamic elastic and viscoelastic moduli of human and canine bones using an electromagnetic transducer and were unable to find a frequency dependence of the elastic constants within the audio frequency range of 0.553.5 kHz. McElhaney (1966) reported on the strain rate dependence of both the compressive strength and the Young’s modulus of bovine and human bone in the strain rate range from 1 x 10-3/sec to 1.5 x 103/sec, using an air gun type compression testing machine. Lang (1970) measured the elastic constants of bovine cortical bone employing an ultrasonic technique at two different frequencies, a dilatational mode at 5 MHz and an equivoluminal mode at 2.25 MHz. Lang assigned hexagonal symmetry a priori to the bone based mainly on the symmetries of the major constituents, collagen and crystalline hydroxyapatite. in addition to information derived from measure* Received 25 July 1975. t Contribution No. 78 from the Laboratorv
for Crvstallographic Biophysics; supported by USPHS through NIDR Grant Number STl-DE-117-11. $ NIDR Postdoctoral Trainee. This paper is the first in a series of three, the second and third parts will be published in Vol. 9 No. 7 and 8 respectively.
ments of the pyroelectric and piezoelectric properties and morphological structure of bone. However. this is misleading since: bone collagen is not a well-crystallized phase in the sense of three-dimensional translational symmetry: the mineral phase includes both crystalline and amorphous components; and the wavelengths of the ultrasonic waves involved are of the order of magnitude longer than the dimensions of the individual collagen and apatitic units which exhibit 6-fold crystalline symmetry. According to Currey (1970). (secondary) osteons are a prominent feature of the bones of man and some large carnivores while they are less plentiful in herbivores. Previous calculations by Katz (1971) and Katz and Ukraincik (1971) indicated the dependence of the anisotropic elastic behavior of human cortical bone on the nature of the microstructure of the osteons. Of special importance in this modeling, Katz and Ukraincik (1971) suggested that the arrangements of osteons and interstitial lamellae in transverse cross section could be considered to be pseudo-hexagonal in nature (Fig. la). This is quite distinct from the elongated, elliptically shaped osteons of young bovine compact bone which. for the most part, are arranged concentrically around the medullary cavity (Fig. tb). In addition, preliminary experiments by the present investigators have shown that bovine bone is more compact, dense and brittle than human bone. Therefore, the present four part study was undertaken in order to obtain independent experimental data concerning the dependence of the dynamic elastic properties of human cortical bone on the morphology of osteons and interstitial lamellae at the microstructural level of organization. Firstly, an ultrasonic pulse transmission method at a fixed frequency of 5 MHz was used to measure anisotropic velocities in dry bone; the conventional metallographic techniques for preparing the specimens were employed in this portion of the study.
407
408
HYO SUB YOONand J. LAWRENCEKATZ
Since the point of interest here was to assess the effect of bone symmetry on the anisotropy of wave propagation and not to obtain exact numbers for. the in uivo elastic stiffnesses (or elastic moduli), which do depend upon the wetness of bone (e.g. Evans and Lebow, 1951; Sedlin, 1965), this was deemed an acceptable procedure. Secondly, a review of the theory of ultrasonic wave propagation in hexagonally anisotropic materials was undertaken in order to ensure that a full description of the elastic anisotropy of bone was available and indeed’justified the theoretical assumptions and experimental procedures. Thirdly, since there is significant evidence that bone is piezoelectric (Yasuda et al., 1954; Fukada and Yasuda, 1957; Bassett and Becker, 1962; Shamos et at., 1963), the piezoelectric contribution to the elastic stiffnesses of human bone was calculated from the ultrasonic measurements. Finally, a frequency and temperature dependence study of both wet and dry human cortical bone was undertaken in the frequency range 0.5-1OMHz. This was done in order to determine whether there were any changes in the ultrasonic properties which might be interpreted as viscoelastic behavior even at these frequencies. The results of these studies are reported in a series of four papers. For the purpose of pedagogy the theoretical analysis is presented here in the first paper. Although the choice of hexagonal symmetry did depend upon the analysis of the 5 MHz measurements on dry bone reported in the second paper, once accepted as verified, it does make it easier to see why the measurements were made the way they were. The third paper shows that the piezoelectric contribution to the elastic stiffnesses of bone are negligibly small; the fourth paper in this series provides new frequency and temperature dependent data in a range not covered by previous dynamic measurements. DETERMINATION OF THE ELASTIC STIFFNESSRS OF A HEXAGONAL.MEDIUM*
Plane waves of small amplitude can propagate in any direction in an infinitely extended elastic medium. Associated with each direction there are three independent waves whose directions of particle displacement form a mutually orthogonal set. In general these * Viscoelastic effects are neglected in this discussion even though bone is viscoefastic at low frequencies. This is justitied because the frequency employed was 5 MHz while the temperature was reasonably constant (room temperature). t The plane wave solutions or displacements are of the form uj = Li,exp[ik(N,x, - Vt] or uj = Ujexp[i(ot - k’x)], where k is the wave propagation vector (k//k1 = N), w the angular frequency, t the time, x the position vector, i the imaginary unit, and the rest are the same as in the text. $The matrix notation (or Voigt notation) of indices is related to the tensor notation as follows: Tensor notation 11 22 33 23,32 31,13 12,21 Matrix notation 1 2 3 4 5 6
displacement directions or polarization directions are neither parallel to nor perpendicular to the direction of wave propagation, i.e. the waves are neither purely longitudinal nor purely transverse. However, there exist special directions of high symmetry in the medium, along which pure modes of propagation occur. The first kind allows propagation of one longitudinal and two transverse modes, whereas along directions of the second kind one transverse and two mixed modes (one quasilongitudinal and one quasitransverse) exist. Sakadi (1941) and Borgnis (1955) have determined the pure mode directions of the first kind for several crystal systems of higher symmetry. Brugger (1965) applied Borgnis’ method to most of the Laue groups for the pure mode directions of the first kind while Chang (1968) did the same for those of the second kind. In studying the elastic properties of a material by ultrasonic techniques, pure modes of propagation are utilized whenever possible because there is no ambiguity in results and the analyses are less involved. Since the strains normally encountered in ultrasonic measurements are of the order of 10e6 to 10m7 (West and Einspruch, 1960; Claiborne and Einspruch, 1963) the assumptions of linear elasticity (e.g. Love, 1927; Nye, 1957) are well justified. In general, combining the generalized Hooke’s law with the Newton’s second law of motion and substituting plane wave solutionst into these equations of motion for an anisotropic medium give the wellknown relations pV2U,
= cm,,,N,NJJ,,
(1)
where summation over repeated indices is implied. p is the density of the medium and V the wave speed. U and N are unit vectors along the directions of particle displacement and wave propagation, respectively, and hence their components U,, N,, etc. are the direction cosines. The c,,,,,, are the second-order adiabatic elastic stiffness constants in tensor notation (Brugger, 1965). Since the symmetry of the human cortical bone turns out to be hexagonal (as will be discussed in detail later) the relations (1) will be specialized for the hexagonal system. In the hexagonal system, there are five independent second-order elastic stiffnesses (fourth-rank tensor) which relate stress to strain; these c,, in matrix form are given by$
where ch6 = acll - ~~2). Hexagonal materials transversely isotropic (or degenerate) so that directions in a conical surface at the same angle to X3 axis (the bone axis) are equivalent. The relations
are all the (1)
Fig. 1. Microstructures of femora, 75 x.
Facing ". 408)
Ultrasonic wave propagation-I for the hexagonal explicitly:
system
are,
when
written
out
In order that not all components of amplitude vanish, the determinant of the coefficients must vanish. Thus, we arrive at a secular equation which restricts the possible values of pVz :
There are three possible values of pV2, all of which are real; they are the roots of the cubic equation equation (4). For the wave propagating along a direction in the X,X3 plane, N2 = 0 and equation (4) assumes a block diagonal form: cb6N: + COIN: - pV2 0 0
pV&,
= ce6Nf + c,,N:
Fig. 2. Pure mode
given by the relations N, = 1, m
(6)
where C#Iis the polar angle of N from the X3 axis. The polarization vector or eigenvector corresponding to this eigenvalue is [0 - lo]*. As explained previously, this wave is of a pure mode of the second kind. For the two mixed modes (or coupled modes), one quasi-longitudinal (&) and one quasitransverse (bT,), one obtains, by solving the 2 x 2 determinant,
system
(see also Fig. 3): N, = N, = 0.
N: + IV: = I.
N, = 0.
19H-r) l9H-11) (9H-7,)
0 = 0.
(~13 + CU)NINS cd4N; + cj3N:
(~13 + (.dN,N3
= ce6 sin24 + c4., cos24,
for hexagonal
c,i - ci3 - 2C,, = ~-:~,--~-2~’
N:
2 + ~z,z,N3- pV2
to the element
directions
For the hexagonal system there are three pure mode directions of the first kind. _*. /L 7. which are
0 c,,N:
The eigenvalue, pV&,, corresponding (1,l) is known immediately:
4OY
(5)
- /IV’
Note that the j direction is given in terms of as yet unknown second-order elastic stiffnesses. The relations in a hexagonal Cl,,.= pv2 along given directions medium are shown in Table 1. In order to obtain c,~, an arbitrary direction
which is in general a pure mode direction second kind. was chosen in this work. RELATIONSHIPS MODULI’
of the
BETWEEN THE ‘TECHNIC’AL.
AND ELASTIC CONSTANTS
Since the static measurements
of the elastic proper-
-__ 2
Pl/f$,
U,,
= Ic, 1 + c441 N+ + k33
= [sin c#J~,0, cos
+ cd
N:
-y
+
N: (Cl 1 - c44) -
2
N= 2 tc44
-
c33)
-?
2J
+ (c,~ + c,,)‘N:N;.
(7)
+ (c,,+i,,i’N:N:.
(8)
d,l,
N:.
PVC&,,= (Cl1 + La) -y + k33 + c44) N’ y - d-t,, u ,+,, = r-cos
+
cx3) $I’
.~ ties of a material give the elastic compliances
$jr_ 0. sin b,].
*The expression [Y,.Y~,Y~] was written simply [r, y2 ys] without commas throughout this paper.
as
(isother-
ma1) which are simply related to the ‘technical elastic moduli’, i.e. Young’s modulus (El, shear modulus (G).
I
i
HYO SUB
410
YOON
and J. LAWRENCEKATZ
Table 1. Relations between elastic stiffnesses and wave speeds for hexagonal system Mode
PV2
aL aT
C33 C4.4
YL YTh
Cl1 C66
~7-3
c44
where ys = cos 4. (B) Shear modulus (rigidity modulus or torsional modulus for a circular cylinder), G,,, j.2y31.
45L*
&II
45 T,,
f(c.44 + C6.5)
4ST:
$(C,, + C33+ 2C44) --bh, - Cd2
+ c33 + 2c4‘4 + ml,
- d + WI3 +
+
*c13 =
c44wz
1 -= 3bk4 + 65) GIYIY2Y31 =
4(c,, + cJ1”2
[(Cll + Cd4 - 2pVZ)(c33 + c44 - 2pV)]“2
- cqq
fY:@55
+
+
w44
+
Y%G22
+
4x56 -
St%) + +
2s33
s44)
+
is44
-
-
4s12 +
+
Y:YrY3(2S24
+
s44)
%5) +
+ $66 +
bulk modulus (K), and others, their expressions along a given direction will be described. Following Voigt (1910) (see also Cady, 1964; Nye, 19_57),the general expression (i.e. for the triclinic system) for each of the ‘technical moduli’ is first written out in full along a given direction whose direction cosines with respect to the X1,X2,X3 axes are y1,y2,y3, and then is simplified for the hexagonal system where the only one angle, 4, from the X3 axis (the unique axis) can specify any orientation.
fY%%
4s23
Y:Y:(fs,,
+
fS55
+ 2Sll - 4%1
Y:Y:(2sII +
s55)
+
fS55
-
%6)
2334
-
431, - 3%)
‘ts44 + +
-
+ Y:YJY1(2s35+ 2sl5 - 4s,5 -
3s64)
4S36 -
3S45)
+
&Y2(2SM
+
+ Y:[Y3@,5
2%6
-
2s22
- 2S3s -t %4)
+ YA2Si6
-
2S26 +
s4s)l
+ YzJ1G%6 - 2SIL5 + s45)
(A) Young’s modulus, E,;,, i12)13,.
f Y3b4
1 ___ = $3 -J%172Yll
+ S&1
+ ?':b'd2%4- 2s2.2 + %6)
= Yhl
+ Yh,
+ Y1@35 -
+ Y&3
+ Y:Y:Gk
+ s44)+ Y:Y:(%
+ Y:Y:(%
+ 566) + G6)
+ %Y,Y*(%s
+ s4.5)
+ ~Y:Y,YA%, + s45) + mY2%5
+
Y3S15)
2Y:(Y,%, + Y1s26)
+ &:(Y,s,,
+
2s15
+
s64)1t
+ %s) =
+ 2Y:YzY&c
+
- &4
YzS34).
* For example, c33 is not equal to ErOOll(= 1/s3& even for an isotropic material. Some writers tend to list c33 next to EIool, without specifying each of them separately (Liboff and Shamos, 1973). In addition, the term modulus of elasticity (or elastic modulus), which was in most cases referred to as Young’s modulus, is becoming more and more ambiguous since more people tend to adopt the convention recommended in the IRE Standards on Piezoelectric Crystals (1949) (or Nye, 1957), and should therefore be replaced by the other more specific terms, such as, at least, Young’s modulus, shear modulus, etc. t CL,(matrix notation) or c& (tensor notation) and s;. or sLqrsare obtained by the transformation law for a fourthrank tensor, e.g. %W = a,ia,ja,ka.tcijk~~ where api, etc. are the direction cosines between the “new” X; axes and the “old” Xj axes.
s44 +
-
(s11
+ 26, I +
s12 -
S33 -
+44x1
2S,3
-
-
r:,
S44h’%l
-
Y:,>
where Y3 = cos I#_ (C) Bulk modulus (the reciprocal of the volume compressibility), K. 1 - = siiti = S11 + S22 + S33 + 2(S,2 + S23 + S31) K
1 = K(hex) =
s33 +
2(s,, + s12 + 2~~~)
Cl1 +c12 -t 2c33 -4c13 C33k11
+ c12) -
263
(12)
Note that ci3 is not equal to ECYIYIYll because the matrices (c,,) and (s,,), not the corresponding elements in pairs, are inverse to each other, i.e. (c,,) x (s,,“)= l.* This is also clear from the fact that ci3 is the bulk (infinitely extended medium) elastic stiffness with the Xi axis along the direction (yl Y2Y3), which, however, does not correspond to the longitudinal stiffness along this direction (i.e. not one of the three roots)t. On the other hand EtY,Y2.j3j is the exten-
Ultrasonic wave propagation--I sional elastic stiffness of a thin rod whose long axis coincides with the direction (y1yZy3), only along which the force is applied. Also note that ck4 is in general not equal to G,,,,,, %,.K(hex) happens to be equal to the Reuss quasi-isotropic average of bulk modulus (Hill, 1952) because of the assumption that the stress in a medium is uniform. Since all the static and dynamic measurements to give the ‘technical moduli’ involve the surface of a material in a certain manner, these moduli can be considered to represent the stiffness of its bulk and surface. Therefore, such a diagram as Fig. 6 in Part II (Yoon and Katz, 1976) is a more meaningful representation of the bulk (intrinsic) elastic properties than that for the ‘technical moduli’, and the elastic constants, ci,,, or sp,, are more fundamental than the ‘technical moduli’.
REFERENCES Bassett. C. A. L.
and Becker. R. 0. (1962) Generation of
electric potentials by bone in response stress. Science 137, 1063-1064.
to mechanical
Borgnis. F. E. (1955) Specific directions of longitudinal wave propagation in anisotropic media. Phcs. Rev. 98, 1000~1005.
Brugger. K. (1965) Pure modes for elastic waves in crystals. J. rrppl. Phv. 36. 7.59 768. (‘ad!. W. G. (1964) Picxwlectricity. 822 pp. Dover, New York. Chang. Z. P. (1968) Pure transverse modes for elastic waves in crystals. J. uppl. P/xl,.\.39, 5669-5681. Claibornc. L. T. and Einspruch. N. G. (1963) Measurement of ultrasonically produced strains in solids by a calorimetric technique. Pitya. Lett. 7, 301-302. Craig. R. G. and Peyton. F. A. (1958) Elastic and mechanical nronerties of human dentin. J. Dent. Rrs. 37, 7l(i~?lx: Currey. J. (1970) +lrlimr/ Skeletons, 52 pp. St. Martin’s Press. New York. Evans. F. G. and Lebow. M. (1951) Regional differences in some of the physical properties of the human femur. J. trppl. PIlkol.
3, 563. j72.
Fukada. E. and Yasuda. I. (1957) On the oiezoelectric effect of bone. .1. Ph!,s. SW. J;pm 12, 1158-1162. Hill. R. (1951) The elastic behaviour of a crystalline aggregate. Proc. P/I,IX SIX. (Lond.) A65. 349-354. IRE Stundards on Pirzoelectric Crystals. 1~4~ (1949) Proc. IRE (now IEEE) 37, 1378-1395; IEEE Std 176-1949 Katy, J. L. (lY71 ) Anisotropic elastic properties of calcified tissues. 1.-IDR Progrctm crnri Ahstrucrs of Papers (49th General Session. Intern. Assoc. Dent. Res.. Chicago. Ill.). p. 71. Abstract No. 73. Kat/. J. L. and Ukraincik. K. (1971) On the anisotropic elastic properties of hydroxyapatite. J. Bionwchanics 4, 221 ?37. Lang. S. B. (1970) Ultrasonic method for measuring elastic coefficients of bone and results on fresh and dried bovine bones. IEEE fitins. Bio-Med. Engng. 17. 101-105: (1969) Elastic coefficients of animal bone. Science 165, 287-288. Lihoff. A. R. and Shamos. M. H. (1973) Solid state physics of bone. In Biological Mineralization (Edited by Zipkin. 1.1. Chao. 14. on. 335%395. Wilev. New York. Lobe. A. ‘E. H.’ il927) A 7kztik on the Mathemtical Thror~~ of Elaticitl. 4th Edn.. 643 pp. Cambridge Univ. Press; Reprinted by Dover, New York, 1944. Lugassy, A. A. and Korostoff, E. (1969) Viscoelastic behavlor of bovine femoral cortical bone and sperm whale dentin. In RYSC~W/I in Drntal und Medical Muterials (I ditsd h) KorostoH: E.). pp. 1 17. Plenum Press. New
411
York; Lugassy. A. A. (1968) Mechanical and viscoelastic properties of cow bone and sperm whale dentin studied under compression. Ph.D. Thesis, Univ. of Pennsylvania, Philadelphia. PA. 246 pp. (Unpublished). McElhaney. J. H. (1966) Dynamic response of bone and muscle tissue. J. nppl. Physiol. 21, 1231-1136. Nye. J. F. (1957) P/1j,tical Properties of Crv.Qul.\. 322 pp. Clarendon Press. Oxford. Sakadi. Z. (1941) Elastic waves in crystals. Proc. Phys.Math. Sot. Jupun (3rd Ser.) 23, 539-547. Sedlin. E. D. (1965) A rheologic model for cortical bone. Acta Orrhop, Stand. Suppl. 83, 1-77. Shamos. M. H.. Lavine. L. S. and Shamos. M. 1. (1963) Piezoelectric effect in bone. Nature 197. 81. Smith. J. W. and Walmsley. R. (1959) Factors affecting the elasticity of bone. J. Amt. (Land.) 93. 503-523. Smith. Jr.. R. W. and Keiper. D. A. (1965) Dynamic measurement of viscoelastic properties of bone. .4n1. J. Mt,tl. Elccrron. 4, 156 -I 60. Voiet. 964 pp.. y W. (19101 Lehrhuch der Kristclllphvsik, Teubner. Leipiig; Reprinted with an ahditional appeidix (1928) 978pp.. Teubner. Leipzig: (1966) Johnson Reprint. New York. Wertheim, G. (1847) M&moire sur I’Clasticite et la cohCsion des principaux tissus du corps humain. Aim. Chrtn. ~‘1 Phjx (3rd Su.) 21, 385 414; (1846) Co!npt. Rmtf. 23. 1151~ 1154. West, F. G. and Emspruch. N. G. (1960) Calorimelrlc measurement of ultrasonically produced strains in solids J. ~rt’ousi. Sot. Ant. 32, I 160. Yasuda. I.. Noguchi, K. and Sata. T. (1954) Dynamic callus and electric callus. Nippon Sei!kgrkct Gukkui Zasshi (1. Jap. Orthop. .4~soc.) 28, 267-268 (in Japanese): (1955) J. Borw .J,tt Swq. 37A. 1292~ 1293. Yoon. H. S. and Katz, J. L. (1976) Ultrasonic wave propagation in human cortical bone--II. Measurement of elastic properties and microhardness. J. Biomrchctnics 9.
NO.MENCLAT<‘RE
direction cosine elastic stiffness in matrix notation. N/m’ transformed elastic stiffness in matrix notation. N/m’ elastic stiffness in tensor notation. N/m’ transformed elastic stiffness in tensor notation. Njm’ Young’s modulus, N/m” Young’s modulus along the directlon b,.;‘2;f31, N/m’ Young‘s modulus along the direction pi,] in a hexagonal medium. which is independent of ;‘, and y2? N/m’ shear or rigidity modulus. N/m’ modulus along shear the direction [;‘I.yz.j)3], N/m’ shear modulus along the dlrection [y,] in a hexagonal medium. which I\ indepcntlentf ;‘, and ;‘z. N/m’ \ - 1. the imaginary unit wave-number vector along the directlon of wave propagation /k\ = (X2). the wave numher. radim kilohertz. kc/set megahertz. Mc.‘sec unit propagation vector. k//k component of the unit propagation vector elastic compliance in matrlr notation. m’N transformed elastic compliance in matrix notation. m’:N
412
V
HYO SUB YCQNand J. LAWRENCEKATZ elastic comphance 111 tensor notation. m2/N transformed elastic compliance in tensor notation, m’/N time, set component of the particle displacement vector, m unit particle displacement vector, m component of the unit particle displacement vector, m unit “longitudinal” displacement vector of the wave along the direction 4 in a hexagonal material, 4L mode, m unit “horizontal” displacement vector of the wave along the direction 4 in a hexagonal material. 4Th mode, m unit “vertical” displacement vector of the wave along the direction C#Iin a hexagonal material, 47;. mode, m wave speed or phase velocity, m/set or km/set
CYIJLY31
P
4 % 0
position vector. m coordinates of the oosition. m right-handed rectangular axes, “old” axes “new” right-handed rectangular axes r mode. see Fig. 2 /I mode. see Fig. 2 y mode. see Fig. 2 direction cosines of the direction along which the ‘technical moduli’ are defined wavelength. m density of the medium, kg/m3 or g/cm3 polar angle of the wave propagation direction from the X3 axis, rad polar angle of the particle displacement vector UdL, in general 4, # 4, rad angular frequency, rad/sec
Roman indices 1.2,3. Greek indicrs 1-6.
1976. Vol. 9. pp. 407412.
Pergamon PXSS.
Printed I”
GreatBrlta~n
ULTRASONIC WAVE PROPAGATION IN HUMAN CORTICAL BONE-I. THEORETICAL CONSIDERATIONS FOR HEXAGONAL SYMMETRY*+ HYO SUB Yoo~f Laboratory for Crystallographic
and J. LAWRENCE KATZ
Biophysics, Department of Physics, Rensselaer Polytechntc Instttute. Troy, NY 12181. U.S.A.
the theory of wave propagation in an anisotropic elastic medium are derived the basic equations relating the five independent second-order elastic stiffness constants (fourth-rank tensor) to the ultrasonic wave speeds in a hexagonal medium, with special emphasis on determining the microtextural symmetry of human cortical bone. In addition. the three pure mode directions of high symmetry in a hexagonal medium are explicitly shown. Finally. expressions relating the ‘technical moduli’ such as Young’s modulus, shear modulus and bulk modulus to the elastic compliances are presented for the most general case (triclinic symmetry) and then are specialized for the hexagonal system. Abstract-From
INTRODUCTION Although Wertheim (1847) made the observation that bone is viscoelastic, this aspect of its behavior appears to have been neglected experimentally for over a century. It was not until the late 1950’s and mid 1960’s that measurements of the viscoelastic properties of bone began to appear in the literature. In 1958, Craig and Peyton reported on creep studies on dentin, while Smith and Walmsley (1959) performed similar experiments on human bone. Sedlin (1965) and Lugassy and Korostoff (1969) reported on creep and relaxation experiments on human and bovine bone respectively. Smith and Keiper (1965) determined the dynamic elastic and viscoelastic moduli of human and canine bones using an electromagnetic transducer and were unable to find a frequency dependence of the elastic constants within the audio frequency range of 0.553.5 kHz. McElhaney (1966) reported on the strain rate dependence of both the compressive strength and the Young’s modulus of bovine and human bone in the strain rate range from 1 x 10-3/sec to 1.5 x 103/sec, using an air gun type compression testing machine. Lang (1970) measured the elastic constants of bovine cortical bone employing an ultrasonic technique at two different frequencies, a dilatational mode at 5 MHz and an equivoluminal mode at 2.25 MHz. Lang assigned hexagonal symmetry a priori to the bone based mainly on the symmetries of the major constituents, collagen and crystalline hydroxyapatite. in addition to information derived from measure* Received 25 July 1975. t Contribution No. 78 from the Laboratorv
for Crvstallographic Biophysics; supported by USPHS through NIDR Grant Number STl-DE-117-11. $ NIDR Postdoctoral Trainee. This paper is the first in a series of three, the second and third parts will be published in Vol. 9 No. 7 and 8 respectively.
ments of the pyroelectric and piezoelectric properties and morphological structure of bone. However. this is misleading since: bone collagen is not a well-crystallized phase in the sense of three-dimensional translational symmetry: the mineral phase includes both crystalline and amorphous components; and the wavelengths of the ultrasonic waves involved are of the order of magnitude longer than the dimensions of the individual collagen and apatitic units which exhibit 6-fold crystalline symmetry. According to Currey (1970). (secondary) osteons are a prominent feature of the bones of man and some large carnivores while they are less plentiful in herbivores. Previous calculations by Katz (1971) and Katz and Ukraincik (1971) indicated the dependence of the anisotropic elastic behavior of human cortical bone on the nature of the microstructure of the osteons. Of special importance in this modeling, Katz and Ukraincik (1971) suggested that the arrangements of osteons and interstitial lamellae in transverse cross section could be considered to be pseudo-hexagonal in nature (Fig. la). This is quite distinct from the elongated, elliptically shaped osteons of young bovine compact bone which. for the most part, are arranged concentrically around the medullary cavity (Fig. tb). In addition, preliminary experiments by the present investigators have shown that bovine bone is more compact, dense and brittle than human bone. Therefore, the present four part study was undertaken in order to obtain independent experimental data concerning the dependence of the dynamic elastic properties of human cortical bone on the morphology of osteons and interstitial lamellae at the microstructural level of organization. Firstly, an ultrasonic pulse transmission method at a fixed frequency of 5 MHz was used to measure anisotropic velocities in dry bone; the conventional metallographic techniques for preparing the specimens were employed in this portion of the study.
407
408
HYO SUB YOONand J. LAWRENCEKATZ
Since the point of interest here was to assess the effect of bone symmetry on the anisotropy of wave propagation and not to obtain exact numbers for. the in uivo elastic stiffnesses (or elastic moduli), which do depend upon the wetness of bone (e.g. Evans and Lebow, 1951; Sedlin, 1965), this was deemed an acceptable procedure. Secondly, a review of the theory of ultrasonic wave propagation in hexagonally anisotropic materials was undertaken in order to ensure that a full description of the elastic anisotropy of bone was available and indeed’justified the theoretical assumptions and experimental procedures. Thirdly, since there is significant evidence that bone is piezoelectric (Yasuda et al., 1954; Fukada and Yasuda, 1957; Bassett and Becker, 1962; Shamos et at., 1963), the piezoelectric contribution to the elastic stiffnesses of human bone was calculated from the ultrasonic measurements. Finally, a frequency and temperature dependence study of both wet and dry human cortical bone was undertaken in the frequency range 0.5-1OMHz. This was done in order to determine whether there were any changes in the ultrasonic properties which might be interpreted as viscoelastic behavior even at these frequencies. The results of these studies are reported in a series of four papers. For the purpose of pedagogy the theoretical analysis is presented here in the first paper. Although the choice of hexagonal symmetry did depend upon the analysis of the 5 MHz measurements on dry bone reported in the second paper, once accepted as verified, it does make it easier to see why the measurements were made the way they were. The third paper shows that the piezoelectric contribution to the elastic stiffnesses of bone are negligibly small; the fourth paper in this series provides new frequency and temperature dependent data in a range not covered by previous dynamic measurements. DETERMINATION OF THE ELASTIC STIFFNESSRS OF A HEXAGONAL.MEDIUM*
Plane waves of small amplitude can propagate in any direction in an infinitely extended elastic medium. Associated with each direction there are three independent waves whose directions of particle displacement form a mutually orthogonal set. In general these * Viscoelastic effects are neglected in this discussion even though bone is viscoefastic at low frequencies. This is justitied because the frequency employed was 5 MHz while the temperature was reasonably constant (room temperature). t The plane wave solutions or displacements are of the form uj = Li,exp[ik(N,x, - Vt] or uj = Ujexp[i(ot - k’x)], where k is the wave propagation vector (k//k1 = N), w the angular frequency, t the time, x the position vector, i the imaginary unit, and the rest are the same as in the text. $The matrix notation (or Voigt notation) of indices is related to the tensor notation as follows: Tensor notation 11 22 33 23,32 31,13 12,21 Matrix notation 1 2 3 4 5 6
displacement directions or polarization directions are neither parallel to nor perpendicular to the direction of wave propagation, i.e. the waves are neither purely longitudinal nor purely transverse. However, there exist special directions of high symmetry in the medium, along which pure modes of propagation occur. The first kind allows propagation of one longitudinal and two transverse modes, whereas along directions of the second kind one transverse and two mixed modes (one quasilongitudinal and one quasitransverse) exist. Sakadi (1941) and Borgnis (1955) have determined the pure mode directions of the first kind for several crystal systems of higher symmetry. Brugger (1965) applied Borgnis’ method to most of the Laue groups for the pure mode directions of the first kind while Chang (1968) did the same for those of the second kind. In studying the elastic properties of a material by ultrasonic techniques, pure modes of propagation are utilized whenever possible because there is no ambiguity in results and the analyses are less involved. Since the strains normally encountered in ultrasonic measurements are of the order of 10e6 to 10m7 (West and Einspruch, 1960; Claiborne and Einspruch, 1963) the assumptions of linear elasticity (e.g. Love, 1927; Nye, 1957) are well justified. In general, combining the generalized Hooke’s law with the Newton’s second law of motion and substituting plane wave solutionst into these equations of motion for an anisotropic medium give the wellknown relations pV2U,
= cm,,,N,NJJ,,
(1)
where summation over repeated indices is implied. p is the density of the medium and V the wave speed. U and N are unit vectors along the directions of particle displacement and wave propagation, respectively, and hence their components U,, N,, etc. are the direction cosines. The c,,,,,, are the second-order adiabatic elastic stiffness constants in tensor notation (Brugger, 1965). Since the symmetry of the human cortical bone turns out to be hexagonal (as will be discussed in detail later) the relations (1) will be specialized for the hexagonal system. In the hexagonal system, there are five independent second-order elastic stiffnesses (fourth-rank tensor) which relate stress to strain; these c,, in matrix form are given by$
where ch6 = acll - ~~2). Hexagonal materials transversely isotropic (or degenerate) so that directions in a conical surface at the same angle to X3 axis (the bone axis) are equivalent. The relations
are all the (1)
Fig. 1. Microstructures of femora, 75 x.
Facing ". 408)
Ultrasonic wave propagation-I for the hexagonal explicitly:
system
are,
when
written
out
In order that not all components of amplitude vanish, the determinant of the coefficients must vanish. Thus, we arrive at a secular equation which restricts the possible values of pVz :
There are three possible values of pV2, all of which are real; they are the roots of the cubic equation equation (4). For the wave propagating along a direction in the X,X3 plane, N2 = 0 and equation (4) assumes a block diagonal form: cb6N: + COIN: - pV2 0 0
pV&,
= ce6Nf + c,,N:
Fig. 2. Pure mode
given by the relations N, = 1, m
(6)
where C#Iis the polar angle of N from the X3 axis. The polarization vector or eigenvector corresponding to this eigenvalue is [0 - lo]*. As explained previously, this wave is of a pure mode of the second kind. For the two mixed modes (or coupled modes), one quasi-longitudinal (&) and one quasitransverse (bT,), one obtains, by solving the 2 x 2 determinant,
system
(see also Fig. 3): N, = N, = 0.
N: + IV: = I.
N, = 0.
19H-r) l9H-11) (9H-7,)
0 = 0.
(~13 + CU)NINS cd4N; + cj3N:
(~13 + (.dN,N3
= ce6 sin24 + c4., cos24,
for hexagonal
c,i - ci3 - 2C,, = ~-:~,--~-2~’
N:
2 + ~z,z,N3- pV2
to the element
directions
For the hexagonal system there are three pure mode directions of the first kind. _*. /L 7. which are
0 c,,N:
The eigenvalue, pV&,, corresponding (1,l) is known immediately:
4OY
(5)
- /IV’
Note that the j direction is given in terms of as yet unknown second-order elastic stiffnesses. The relations in a hexagonal Cl,,.= pv2 along given directions medium are shown in Table 1. In order to obtain c,~, an arbitrary direction
which is in general a pure mode direction second kind. was chosen in this work. RELATIONSHIPS MODULI’
of the
BETWEEN THE ‘TECHNIC’AL.
AND ELASTIC CONSTANTS
Since the static measurements
of the elastic proper-
-__ 2
Pl/f$,
U,,
= Ic, 1 + c441 N+ + k33
= [sin c#J~,0, cos
+ cd
N:
-y
+
N: (Cl 1 - c44) -
2
N= 2 tc44
-
c33)
-?
2J
+ (c,~ + c,,)‘N:N;.
(7)
+ (c,,+i,,i’N:N:.
(8)
d,l,
N:.
PVC&,,= (Cl1 + La) -y + k33 + c44) N’ y - d-t,, u ,+,, = r-cos
+
cx3) $I’
.~ ties of a material give the elastic compliances
$jr_ 0. sin b,].
*The expression [Y,.Y~,Y~] was written simply [r, y2 ys] without commas throughout this paper.
as
(isother-
ma1) which are simply related to the ‘technical elastic moduli’, i.e. Young’s modulus (El, shear modulus (G).
I
i
HYO SUB
410
YOON
and J. LAWRENCEKATZ
Table 1. Relations between elastic stiffnesses and wave speeds for hexagonal system Mode
PV2
aL aT
C33 C4.4
YL YTh
Cl1 C66
~7-3
c44
where ys = cos 4. (B) Shear modulus (rigidity modulus or torsional modulus for a circular cylinder), G,,, j.2y31.
45L*
&II
45 T,,
f(c.44 + C6.5)
4ST:
$(C,, + C33+ 2C44) --bh, - Cd2
+ c33 + 2c4‘4 + ml,
- d + WI3 +
+
*c13 =
c44wz
1 -= 3bk4 + 65) GIYIY2Y31 =
4(c,, + cJ1”2
[(Cll + Cd4 - 2pVZ)(c33 + c44 - 2pV)]“2
- cqq
fY:@55
+
+
w44
+
Y%G22
+
4x56 -
St%) + +
2s33
s44)
+
is44
-
-
4s12 +
+
Y:YrY3(2S24
+
s44)
%5) +
+ $66 +
bulk modulus (K), and others, their expressions along a given direction will be described. Following Voigt (1910) (see also Cady, 1964; Nye, 19_57),the general expression (i.e. for the triclinic system) for each of the ‘technical moduli’ is first written out in full along a given direction whose direction cosines with respect to the X1,X2,X3 axes are y1,y2,y3, and then is simplified for the hexagonal system where the only one angle, 4, from the X3 axis (the unique axis) can specify any orientation.
fY%%
4s23
Y:Y:(fs,,
+
fS55
+ 2Sll - 4%1
Y:Y:(2sII +
s55)
+
fS55
-
%6)
2334
-
431, - 3%)
‘ts44 + +
-
+ Y:YJY1(2s35+ 2sl5 - 4s,5 -
3s64)
4S36 -
3S45)
+
&Y2(2SM
+
+ Y:[Y3@,5
2%6
-
2s22
- 2S3s -t %4)
+ YA2Si6
-
2S26 +
s4s)l
+ YzJ1G%6 - 2SIL5 + s45)
(A) Young’s modulus, E,;,, i12)13,.
f Y3b4
1 ___ = $3 -J%172Yll
+ S&1
+ ?':b'd2%4- 2s2.2 + %6)
= Yhl
+ Yh,
+ Y1@35 -
+ Y&3
+ Y:Y:Gk
+ s44)+ Y:Y:(%
+ Y:Y:(%
+ 566) + G6)
+ %Y,Y*(%s
+ s4.5)
+ ~Y:Y,YA%, + s45) + mY2%5
+
Y3S15)
2Y:(Y,%, + Y1s26)
+ &:(Y,s,,
+
2s15
+
s64)1t
+ %s) =
+ 2Y:YzY&c
+
- &4
YzS34).
* For example, c33 is not equal to ErOOll(= 1/s3& even for an isotropic material. Some writers tend to list c33 next to EIool, without specifying each of them separately (Liboff and Shamos, 1973). In addition, the term modulus of elasticity (or elastic modulus), which was in most cases referred to as Young’s modulus, is becoming more and more ambiguous since more people tend to adopt the convention recommended in the IRE Standards on Piezoelectric Crystals (1949) (or Nye, 1957), and should therefore be replaced by the other more specific terms, such as, at least, Young’s modulus, shear modulus, etc. t CL,(matrix notation) or c& (tensor notation) and s;. or sLqrsare obtained by the transformation law for a fourthrank tensor, e.g. %W = a,ia,ja,ka.tcijk~~ where api, etc. are the direction cosines between the “new” X; axes and the “old” Xj axes.
s44 +
-
(s11
+ 26, I +
s12 -
S33 -
+44x1
2S,3
-
-
r:,
S44h’%l
-
Y:,>
where Y3 = cos I#_ (C) Bulk modulus (the reciprocal of the volume compressibility), K. 1 - = siiti = S11 + S22 + S33 + 2(S,2 + S23 + S31) K
1 = K(hex) =
s33 +
2(s,, + s12 + 2~~~)
Cl1 +c12 -t 2c33 -4c13 C33k11
+ c12) -
263
(12)
Note that ci3 is not equal to ECYIYIYll because the matrices (c,,) and (s,,), not the corresponding elements in pairs, are inverse to each other, i.e. (c,,) x (s,,“)= l.* This is also clear from the fact that ci3 is the bulk (infinitely extended medium) elastic stiffness with the Xi axis along the direction (yl Y2Y3), which, however, does not correspond to the longitudinal stiffness along this direction (i.e. not one of the three roots)t. On the other hand EtY,Y2.j3j is the exten-
Ultrasonic wave propagation--I sional elastic stiffness of a thin rod whose long axis coincides with the direction (y1yZy3), only along which the force is applied. Also note that ck4 is in general not equal to G,,,,,, %,.K(hex) happens to be equal to the Reuss quasi-isotropic average of bulk modulus (Hill, 1952) because of the assumption that the stress in a medium is uniform. Since all the static and dynamic measurements to give the ‘technical moduli’ involve the surface of a material in a certain manner, these moduli can be considered to represent the stiffness of its bulk and surface. Therefore, such a diagram as Fig. 6 in Part II (Yoon and Katz, 1976) is a more meaningful representation of the bulk (intrinsic) elastic properties than that for the ‘technical moduli’, and the elastic constants, ci,,, or sp,, are more fundamental than the ‘technical moduli’.
REFERENCES Bassett. C. A. L.
and Becker. R. 0. (1962) Generation of
electric potentials by bone in response stress. Science 137, 1063-1064.
to mechanical
Borgnis. F. E. (1955) Specific directions of longitudinal wave propagation in anisotropic media. Phcs. Rev. 98, 1000~1005.
Brugger. K. (1965) Pure modes for elastic waves in crystals. J. rrppl. Phv. 36. 7.59 768. (‘ad!. W. G. (1964) Picxwlectricity. 822 pp. Dover, New York. Chang. Z. P. (1968) Pure transverse modes for elastic waves in crystals. J. uppl. P/xl,.\.39, 5669-5681. Claibornc. L. T. and Einspruch. N. G. (1963) Measurement of ultrasonically produced strains in solids by a calorimetric technique. Pitya. Lett. 7, 301-302. Craig. R. G. and Peyton. F. A. (1958) Elastic and mechanical nronerties of human dentin. J. Dent. Rrs. 37, 7l(i~?lx: Currey. J. (1970) +lrlimr/ Skeletons, 52 pp. St. Martin’s Press. New York. Evans. F. G. and Lebow. M. (1951) Regional differences in some of the physical properties of the human femur. J. trppl. PIlkol.
3, 563. j72.
Fukada. E. and Yasuda. I. (1957) On the oiezoelectric effect of bone. .1. Ph!,s. SW. J;pm 12, 1158-1162. Hill. R. (1951) The elastic behaviour of a crystalline aggregate. Proc. P/I,IX SIX. (Lond.) A65. 349-354. IRE Stundards on Pirzoelectric Crystals. 1~4~ (1949) Proc. IRE (now IEEE) 37, 1378-1395; IEEE Std 176-1949 Katy, J. L. (lY71 ) Anisotropic elastic properties of calcified tissues. 1.-IDR Progrctm crnri Ahstrucrs of Papers (49th General Session. Intern. Assoc. Dent. Res.. Chicago. Ill.). p. 71. Abstract No. 73. Kat/. J. L. and Ukraincik. K. (1971) On the anisotropic elastic properties of hydroxyapatite. J. Bionwchanics 4, 221 ?37. Lang. S. B. (1970) Ultrasonic method for measuring elastic coefficients of bone and results on fresh and dried bovine bones. IEEE fitins. Bio-Med. Engng. 17. 101-105: (1969) Elastic coefficients of animal bone. Science 165, 287-288. Lihoff. A. R. and Shamos. M. H. (1973) Solid state physics of bone. In Biological Mineralization (Edited by Zipkin. 1.1. Chao. 14. on. 335%395. Wilev. New York. Lobe. A. ‘E. H.’ il927) A 7kztik on the Mathemtical Thror~~ of Elaticitl. 4th Edn.. 643 pp. Cambridge Univ. Press; Reprinted by Dover, New York, 1944. Lugassy, A. A. and Korostoff, E. (1969) Viscoelastic behavlor of bovine femoral cortical bone and sperm whale dentin. In RYSC~W/I in Drntal und Medical Muterials (I ditsd h) KorostoH: E.). pp. 1 17. Plenum Press. New
411
York; Lugassy. A. A. (1968) Mechanical and viscoelastic properties of cow bone and sperm whale dentin studied under compression. Ph.D. Thesis, Univ. of Pennsylvania, Philadelphia. PA. 246 pp. (Unpublished). McElhaney. J. H. (1966) Dynamic response of bone and muscle tissue. J. nppl. Physiol. 21, 1231-1136. Nye. J. F. (1957) P/1j,tical Properties of Crv.Qul.\. 322 pp. Clarendon Press. Oxford. Sakadi. Z. (1941) Elastic waves in crystals. Proc. Phys.Math. Sot. Jupun (3rd Ser.) 23, 539-547. Sedlin. E. D. (1965) A rheologic model for cortical bone. Acta Orrhop, Stand. Suppl. 83, 1-77. Shamos. M. H.. Lavine. L. S. and Shamos. M. 1. (1963) Piezoelectric effect in bone. Nature 197. 81. Smith. J. W. and Walmsley. R. (1959) Factors affecting the elasticity of bone. J. Amt. (Land.) 93. 503-523. Smith. Jr.. R. W. and Keiper. D. A. (1965) Dynamic measurement of viscoelastic properties of bone. .4n1. J. Mt,tl. Elccrron. 4, 156 -I 60. Voiet. 964 pp.. y W. (19101 Lehrhuch der Kristclllphvsik, Teubner. Leipiig; Reprinted with an ahditional appeidix (1928) 978pp.. Teubner. Leipzig: (1966) Johnson Reprint. New York. Wertheim, G. (1847) M&moire sur I’Clasticite et la cohCsion des principaux tissus du corps humain. Aim. Chrtn. ~‘1 Phjx (3rd Su.) 21, 385 414; (1846) Co!npt. Rmtf. 23. 1151~ 1154. West, F. G. and Emspruch. N. G. (1960) Calorimelrlc measurement of ultrasonically produced strains in solids J. ~rt’ousi. Sot. Ant. 32, I 160. Yasuda. I.. Noguchi, K. and Sata. T. (1954) Dynamic callus and electric callus. Nippon Sei!kgrkct Gukkui Zasshi (1. Jap. Orthop. .4~soc.) 28, 267-268 (in Japanese): (1955) J. Borw .J,tt Swq. 37A. 1292~ 1293. Yoon. H. S. and Katz, J. L. (1976) Ultrasonic wave propagation in human cortical bone--II. Measurement of elastic properties and microhardness. J. Biomrchctnics 9.
NO.MENCLAT<‘RE
direction cosine elastic stiffness in matrix notation. N/m’ transformed elastic stiffness in matrix notation. N/m’ elastic stiffness in tensor notation. N/m’ transformed elastic stiffness in tensor notation. Njm’ Young’s modulus, N/m” Young’s modulus along the directlon b,.;‘2;f31, N/m’ Young‘s modulus along the direction pi,] in a hexagonal medium. which is independent of ;‘, and y2? N/m’ shear or rigidity modulus. N/m’ modulus along shear the direction [;‘I.yz.j)3], N/m’ shear modulus along the dlrection [y,] in a hexagonal medium. which I\ indepcntlentf ;‘, and ;‘z. N/m’ \ - 1. the imaginary unit wave-number vector along the directlon of wave propagation /k\ = (X2). the wave numher. radim kilohertz. kc/set megahertz. Mc.‘sec unit propagation vector. k//k component of the unit propagation vector elastic compliance in matrlr notation. m’N transformed elastic compliance in matrix notation. m’:N
412
V
HYO SUB YCQNand J. LAWRENCEKATZ elastic comphance 111 tensor notation. m2/N transformed elastic compliance in tensor notation, m’/N time, set component of the particle displacement vector, m unit particle displacement vector, m component of the unit particle displacement vector, m unit “longitudinal” displacement vector of the wave along the direction 4 in a hexagonal material, 4L mode, m unit “horizontal” displacement vector of the wave along the direction 4 in a hexagonal material. 4Th mode, m unit “vertical” displacement vector of the wave along the direction C#Iin a hexagonal material, 47;. mode, m wave speed or phase velocity, m/set or km/set
CYIJLY31
P
4 % 0
position vector. m coordinates of the oosition. m right-handed rectangular axes, “old” axes “new” right-handed rectangular axes r mode. see Fig. 2 /I mode. see Fig. 2 y mode. see Fig. 2 direction cosines of the direction along which the ‘technical moduli’ are defined wavelength. m density of the medium, kg/m3 or g/cm3 polar angle of the wave propagation direction from the X3 axis, rad polar angle of the particle displacement vector UdL, in general 4, # 4, rad angular frequency, rad/sec
Roman indices 1.2,3. Greek indicrs 1-6.