J. Biomechanrcs.
1976. Vol.
9. pp. 703-710.
Pergamon
Press
Printed
m Great
Bntain
ELASTIC PROPERTIES OF CORTICAL BONE IN FEMALE HUMAN FEMURS* D. VIANO?, U. HELFENSTEIN, M. ANLIKERand P. R~~EGSEGGER Institut ftir Biomedizinische Technik der Universitlt Ziirich und der Eidgeniissischen Technischen Hochschule, Moussonstrasse 18, 8044 Ziirich. Switzerland
Abstract-The evaluation of the physiological and pathophysiological changes in the status of human bones is incomplete without information on the elastic properties of the bone material. To assess the merits of a corresponding noninvasive technique for cortical bone the mechanical parameters of macerated human female femur shafts are evaluated by way of measuring the resonance frequencies and associated mode shapes for the bending, axial and torsional motions. The shafts are considered to behave as hollow cylinders of variable cross section and the bone material is assumed to be isotropic and linearly elastic. It is characterized by its density, Young’s modulus and shear modulus. The density and the geometry of the cross sections are determined by digital tomography. Young’s modulus and the shear modulus are considered constant along the shaft and calculated on the basis of classical theories by matching iterative mathematical predictions with experimental data. Flexural, longitudinal and torsional motions are measured by means of a stabilized Michelson interferometer. Up to 16 eigenfrequencies in the range of 1 and 20 kHz are examined. Results for the age range of 24-85 years show that Young’s modulus of cortical bone decreases with age by approximately 10% whereas the compacta density diminishes by ca. 8%. In view of the uncertainty involved in determining Young’s modulus of compact bone and the relative change it exhibits with age the efforts of developing a noninvasive technique for its quantification may not be warranted.
INTRODUCTION The mechanical integrity of bones is defined by the bone mineral mass, its geometric distribution and the elastic properties of the bone material. Whereas noninvasive clinical methods of determining the mineral mass of certain bones and its geometric distribution have become available (Cameron and Sorensen, 1963; Riiegsegger et al., 1974), we are still lacking the corresponding validated techniques of measuring the elastic properties of the bone material. An evaluation of the physiological and pathophysiological changes in the status of human bones should include information on their elastic properties because they may not only depend on the amount of bone mineral present but also on the bonding between the constituents and the lamellar structure of bone. Estimates for the strength of the human ulna on the basis of resonance measurements and bone mineral mass determinations have been reported in the literature (Jurist and Kianian, 1973; Spiegl and Jurist, 1975). However, the method applied does not permit a separation of the effects of a variation of the cross-section along the bone, of the boundary conditions at the two ends of the ulna and of the elastic properties of the bone material on changes in the resonance frequency. Thompson et a[. (1970) found in their experiments on dog radii that variations in geometry along the length of the bone may significantly influence the resonance frequency. * Received 6 May 1976. t Now at: General Motors Warren, MI, U.S.A.
Research
Laboratories,
With this investigation we aim to assess the merit of a noninvasive method of measuring changes in the elastic properties of bone material. To this end we evaluate the elastic parameters of excised human femurs from their resonance frequencies for flexural, longitudinal, and torsional vibrations. We thereby make use of classical linear elastic beam theory and take into account the geometric variations of the cross section along the femur. These variations are determined nondestructively by the method described by Riiegsegger et al. (1974). As the ends of the femur are composed of both, cortical and spongy bone, we restrict our study for the sake of simplicity to the femur shaft which is primarily made up of compact bone. EXPERIMENTALPROCEDURE A schematic drawing of the experimental arrangement used to detect the resonance frequencies and the associated mode shapes of human femur shafts is given in Fig. 1. The principal component of the measuring system is a stabilized linear Michelson interferometer. An optical detection technique has the advantage that no contact is required with the moving structure. The interferometer is based on a design reported by Sizgoric and Gundjian (1969). With a refined stabilization against drift and low frequency vibrations described by Goodbread et al. (1975) it is ossible to resolve surface displacements as small as 0.01 1 Sinusoidal stress waves are generated by means of a piezoelectric transducer glued to one of the end sections of the femur shaft. The transducer is driven by a high frequency sweep generator (Wavetek, Model 144) with which the frequency is increased from 1 to 20 kHz in approx. 5 min. To measure the surface displacements, twenty equally spaced mirrors attached along the length of the bone (Fig. 2) are successively used as reflectors for one leg of the interferometer. A He-Ne Laser (Spectra
703
704
D. VIANO,U. HELFJZNSTEIN, M. ANLIKER and P. R~EGSEGGER
SHAFT DRIVER
0 STABkIZATlON
Fig. 1. Schematic drawing of the experimentul arrangement. A sweep generator excites stress waves in the femur shaft by means of a piezoelectric transducer. The lateral displacement of the bone surface is detected optically by a stabilized laser interferometer. The amplified detector voltage is fed in two lock-in amplifiers. The resulting phase-locked displacement signal is recorded together with the sweep voltage which is proportional to the frequency of the excitation signal.
longitudinal
Physics, Mod. 120; I = 6328 A) serves as a light source. A beam splitter sends 50% of the light to the mirror sensing the shaft displacement, the other 50% to the reference mirror. The reflected beams are recombined on a silicon pin photodiode (Monsanto MDl). While the low frequency component of its output signal is fed back to the stabilization circuit, the high frequency component is analyzed by two lock-in amplifiers (Brookdeal, Model 401) with a phase difference of 0” and 90” in relation to the driving signal. With these amplifiers the femur shaft movement is detected by multiplying the amplified photodiode voltage with the driver voltage and then filtering the product with a lowpass filter. This arrangement allows for the improvement of the signal to noise ratio up to 100 dB. Therefore surface movements on the order of 0.01 A are still measurable. A typical recording from one measuring site is given in Fig. 3. The surface displacements at resonance are on the order of several Angstroms. The accuracy of the resonance frequency determination depends on the peak separation. Good resolution is obtained especially for the lower modes for which the error is less than 1%. To minimize alterations of the bone resonances due to the effects of mirror and transducer masses, their total weight was kept below 1% of that of the femur shaft. THEORETICAL
ASPECTS
ELASTIC
Fig. 2. Mirror orientation for resonance and mode shape determination. Twenty mirrors are equally spaced at intervals of approx. 1.25 cm.
AND
EVALUATION
OF
PARAMETERS
For the purpose of mathematically analyzing the mechanical behavior of the femur shaft, we model it as a hollow cylinder composed of uniform compact
705
Elastic properties of cortical bone steady displacement (90)
phase.deg.
camp (0)
frequency (khz)
order.type
motion we can introduce a matrix w (x; j) such that II~ (x; j) = W$(X;j)bi (i).
16.43
L-th
torsion
15.12
2-nd
long
14 53
5-th
flexw
13 79
5-th
flexw
12 ?I
3-rd
torsion
As described in the appendix for the case of flexural vibrations, the matrix K (x; j) contains the eigenfunctions for the harmonic vibrations (Fltigge, 1962) and the components of the vector bi (j) are the unknown coefficients in the expression for the general mode shape for segment j. The vectors bi(~1are determined for each frequency of the enforced oscillatory motion by satisfying the boundary conditions and by requiring continuity between the segments: ui (xj; j) = ui (xj; ,j + I).
The vector bi (i) for a general segment is related to the vector bi (j + 1) of the adjacent segment by hi 0’ + 1) = fJi(.Hk where 2-nd
torsion
3-rd
flexw
1-st
torsion
U,(j) = Wi-‘(Xj; j + l)w(Xj, j = 1,2. . . . . (j,,, - 1).
Therefore the vectors b,(j) can all be expressed in
i. Fig. 3. Typktrl wcordiq of’ a displacrr~~olf~f~ryuolc! SMWI). Resonance frequencies and corresponding modes are indicated.
bone. The latter is taken as a linearly elastic, isotropic and homogeneous material, characterized by its density. p, Young’s modulus, E, and shear modulus, G. Variations in the geometric cross sectional properties along the length of the shaft are taken into account by subdividing the shaft into segments of uniform cross sectional properties. These are determined by a linear interpolation of the measured data as shown in Fig. 4. By requiring continuity of the displacements and stress resultants at the segment boundaries, a continuous femur shaft model is established. Four types of steady state forced harmonic vibrational motions of the femur shaft are considered: flexure in the direction of minimum stiffness (i = 1) and maximum stiffness (i = 2), longitudinal extension (i = 3) and torsion (i = 4). The corresponding mode shapes and stresses for each femur shaft segment j are defined by a vector ui (x; j) where x is the axial coordinate as shown in Fig. 2. For each type of
j)
:.
compacta
,\
area
(cm21
3. I.
moment
of inertia (cm‘)
1. ‘2 !.
/--‘I
1.0
radius
of gyration
(cm)
0.9
//./
0.6
,0
,5
,I0
position 115
km) ,20
Fig. 4. Sample
cross-sections and related geometrical parameters of a human femur shaft. Locations of the illustrated
cross-sections are indicated on the femur drawn in a reduced scale. Subscripts 1 and 2 refer to the principal directions of the moment of inertia.
106
D. VIANO,U.
HELFENSTEIN, M. ANLIKER
terms of the vector bi (1) of the first segment. Accordingly we have ui(O,l) = Il$(O;l)bi(l)
and P.
R~~EGSEGGER
mechanical parameters may be compared against additional experimental data and thereby serve as a check of the consistency and accuracy of the theory.
and udkjrnax) = WW,,,)~(j,,,,, - l)bi(l) whereF(.L, - 1)= ~i(jmax- l)K(j,,, - 2) and G(l) = U,(l). Finally, the boundary conditions at x = 0,l yield the frequency equation for resonance. To describe the flexural motions we make use of Timoshenko’s beam theory, whereby the geometric shear stiffness parameters k; and k; for the two principal directions are assumed constant over the entire length of the femur shaft (Fltigge, 1962). For longitudinal vibration Love’s theory is applied and for the torsional vibration the classical bar theory (Fltigge (1962)). In the latter case the geometric torsional stiffness parameter k is considered to be the same for all segments. With the aid of standard matrix techniques a computer algorithm is developed to determine the root of the frequency equations and the corresponding mode shapes. To evaluate the elastic parameters from the four types of motion, an iterative procedure is applied. As an initial step we assume a reasonable set of values for the parameters and determine the corresponding resonance frequencies as the roots of the appropriate frequency equations. The lowest theoretical and experimental resonance frequencies for each type of motion are compared. On the basis of the discrepancies we choose a new value for the wave speed ci = JE/p (i = 1,2,3) in each of the lowest bending and axial vibrations and also a new value for the torsional wave speed cq = ,/kGlp in case of the lowest torsional mode. The procedure is continued iteratively until the absolute difference between the theoretical and experimental resonance frequency is less than 1%. For the two bending directions the shear correction terms E/k’G are then adjusted by a similar iterative procedure which reduces the difference between the 4th theoretical and experimental resonance frequencies to 1% retaining the previously determined value for E/p. These values of E/p and E/k’G are used as new guesses, and the previous procedure involving the lowest and fourth modes is repeated until both E/p and E/k’G change less than 1%. In the case of longitudinal vibrations, Poisson’s ratio is given a fixed value of 0.32 (Lang, 1969). This means that cg is the only parameter determined iteratively from the lowest resonance frequency for longitudinal vibrations. Likewise, for torsional vibrations it is cq which is evaluated iteratively in a similar manner. After the wave speeds ci to cq and the two values for EIk’G have been determined as outlined, all other resonances and mode shapes in the frequency range 1-17 kHz are determined. Those theoretical predictions which have not been used to evaluate the
RESULTS
To determine the feasibility of our procedure, a macerated femur shaft of a 24 year old female was analyzed in detail. The cross-sectional data at ten stations equally spaced by 2.5 cm were obtained by densitometry (Rtiegsegger et al., 1974). The cross sectional image provided by this method allows for the determination of the compacta area, moments of inertia and radii of gyration (Fig. 4). These geometrical features are used for the calculation of the resonance frequencies, mode shapes and material properties (cl to cd, EIk;G and EIk;G) as described in the previous section. The 16 computed resonance frequencies are listed in Table 1 together with those obtained from measurements. The 10 resonance frequencies which are not used to determine the mechanical parameters of the models, as well as the entire set of mode shapes constitute a basis for assessing the validity of the theoretical models employed in describing the various motions of the femur shaft. The maximum difference between the theoretical and experimental resonance frequencies is 3%. From the corresponding mode shapes given in Fig. 5 it also follows that the measured and calculated amplitude patterns are in good agreement. The discrepancy observed for the fifth bending mode may be explained by the difficulty in separating closely spaced resonances at higher frequencies. Also included in Table 1 are the material properties deduced by the iterative procedure. The uncertainty of the wave speeds is between 2 and 3% and can be estimated from the uncertainties of the geometrical parameters and the precision of the resonance frequency determination. The mechanical properties for a set of female femurs are compiled in Table 2. Of these properties the density of the compacta, the longitudinal Young’s modulus E, = pci and the modified shear modulus G’ = kG are plotted against age in Fig. 6. Included in the table are also weight and length of the whole femur. DISCUSSION
AND CONCLUSIONS
The wave speeds ci, c1 and cj derived from the resonances for bending and longitudinal vibrations do not exhibit a significant age dependence. The values for c1 and c2 are between 3000 and 3400 m/s whereas cg ranges from 3200 to 34OOmls. These results are in general agreement with the average wave speed of 32OOm/s found by Craven and Costantini (1973) for the proximal radius. As evident from Fig. 6 the compacta density decreases by ca. 8% between the ages 24 and 85 whereas the Young’s modulus El = peg determined from the speed of longitudinal waves and the density decreases by approx. 10% over
.
707
Elastic properties of cortical bone Table 1. Resonance frequencies and material parameters of a 24 year old female femur shaft. The theoretical and experimental resonance frequencies are given for all four types of vibrations together with the corresponding material properties Vibration Order Type Flexure min.
Flexure max.
Longitudinal Torsion
Resonance frequencies Experimental Theoretical &Hz) NW
Material properties
I .67
1 2 3 4
1.67 4.26 1.29 10.44
4.17 1.22 10.45
5
13.79
13.80
I 2 3 4 5 1 2 1 2 3 4
1.73 4.32 7.51 10.88 14.53 7.78 15.12 3.91 8.14 12.37 16.43
1.73 4.3 1 1.47 10.89 14.39 7.78 15.15 4.03 8.14 12.30 16.45
c1 = 3.32 x IO’ cmjsec E/k,‘G = 6.07
c2 = 3.23 x 10’ cm/set E/k2’G = 5.17 cj = 3.29 x 10’ cmjsec
cq = 1.8 x 10’ cmjsec
this age range and thus diminishes slightly more than measurements diminish with decreasing density. does the density p. This observation appears to be However, the primary cause for a decrease in the in contradiction with the results obtained by Abendeffective density of their bone samples is an increase schein and Hyatt (1970) for fresh human tibia speci- in the volume fraction of pores filled with fluid, which mens. According to their data, both the static Young’s in effect means a loss in stress carrying material. Conmodulus and that derived from ultrasonic wave speed sequently the data given by Abendschein and Hyatt
flrxural
-----
mode shapes
tor5~onal
mode
shapes
throretical
I
longlfudinal
mode shapes
Fig. 5. Comparison of experimental and theoretical mode shapes. Solid lines represent experimentally determined modes, whereas the dashed lines are those predicted by theory.
I
708
D. VIANO,U. HEDENSTEIN,M. ANLIKERand P. RCJEGSEGGER
Table 2. Material properties of 10 macerated female femur shafts. The density p is a mean value for the compact bone. Also listed are weights and lengths of the whole femurs. Four types of vibrations were considered: flexure in the plane of minimum stiffness (i = 1) and maximum stiffness (i = 2), longitudinal extension (i = 3) and torsion (i = 4). Note G’ = kG. The resonance frequencies for bending were not utilized in all cases because of the extensive numerical efforts involved in determining the mechanical parameters
Age length (cm) weight (g) density (g/cm3) cr (x lo5 cm/s) E/k,‘G c2 (x lo5 cm/s) Ejk,‘G cg (x lo5 cm/s)
E, ( x 10” dyn/cmz) cq (x lo5 cm/s) G’ (x 10” dyn/cm’)
.
.
24
46
53
54
61
61
46.0 377 2.10 3.32 6.07 3.23 5.17 3.29 2.21 1.80 6.80
37.5 319 2.11 3.22 4.06 3.07 3.86 3.34 2.35 1.77 6.60
40.0 370 2.05
41.8 278 2.02 3.40 6.45 3.39 7.50 3.34 2.25 1.58 5.05
43.2 319 1.97 3.29 7.06 3.00 5.00 3.18 1.99 1.70 5.69
40.2 310 2.00
40.0
3.32 2.20 1.66 5.52
co
l
.’
2’0 X
.
’ 60
40
3.30 2.23 1.75 6.27
l
80’
10'0 .
.
.
.
. to . . 1
20
40
.
. 77 E e &
80
60
-0
30
0.
.
1,8 1.6
I
20 AGE
40
60
80
[YEARS]
Fig. 6. Age dependence of E, G’ and p. Data are given for 10 female femur shafts. Relative variations of the longitudinal Young’s modulus E and density p with age are similar, whereas the modified shear modulus G’ exhibits a more rapid decay than E and p.
65
72
74
85
296 1.98
38.0 236 1.97
40.0 258 1.94
3.26 2.10 1.66 5.46
3.42 2.29 1.76 6.11
42.5 340 1.94 3.06 5.47 3.11 5.43 3.22 2.01 1.57 4.71
3.25 2.05 1.65 5.28
have to be corrected for the increase in the fluid mass within the bone tissue if we wish to evaluate Young’s modulus of the bone mineral. When such a correction is applied the discrepancy between their results and ours is insignificant. Hence Young’s modulus of compact human bone exhibits a very weak age dependence. Moreover, the elastic properties of bone mineral do not seem to be affected by the maceration process carried out with our bones. As mean value of E one obtains 2.2 x 10” dyn/cm2 which agrees with the results of Evans and Lebow (1951) who found by way of tensile tests for the middle third of dried human femurs a mean Young’s modulus of 2 x 10” dyn/cm’. The same mean value is given by Bundy (1974), whose data, however, ranges from 1.4 x 10” to 2.3 x 10” dyn/cm’. The faster decrease of p at menopause is supported by similar findings of Dequeker (1972) regarding the thickness of compact bone and of Cameron and Sorensen (1963) with respect to the total mineral content. The modified shear modulus G’ = kG shows a significantly larger decrease with age than does E. Because G’ contains the torsional rigidity factor k, which is unknown, we can not distinguish whether this is due to a change in geometry or due to a faster decrease of G. However, we know from Bundy’s data which was obtained from machined compacta samples of uniform geometry that G does indeed decrease faster. It is interesting to note that our dynamically determined values for Young’s modulus agree well with those found from tensile tests (Evans and Lebow, 1951). In addition Brash and Skorecki (1970) found with flexural and longitudinal resonance measurements on small specimens of compact bone from the bovine tibia that Young’s modulus was between 2 and 2.5 x 10” dyn/cm*. This range is compatible with our findings on the human femur shaft. From the relatively small decreases of the speed of bending and axial waves in our femur shafts with age we infer that these wave speeds are not sensitive parameters of osteoporosis. Moreover, the compacta
709
Elastic properties of cortical bone
density is likely to diminish more than the square of these speeds. Hence, as c2 = E/p, we may conclude that any decrease in Young’s modulus can be expected to be somewhat larger than that of the density. However, the error of the density measurement is only of the order of lo/, whereas that of the E determination is cu. 3-5x. Besides this, a possible clinical assessment of osteoporosis in terms of changes in the Young’s modulus of compact bone will be associated with a larger error than that of our laboratory measurements on excised bones. In addition, as reported by Bundy (1974), we know from experiments on bone samples from human femurs that compact bone has anisotropic elastic properties at least in so far as the lateral and circumferential parameters are significantly different from those in the axial direction. Moreover, the elastic parameters measured by Bundy, especially the ratio of Young’s modulus to shear modulus, are non-uniform as they vary over the bone cross section and along the length of the bone. Hence. the development of a non-invasive method of evaluating E of compact bone may be of questionable merit since according to our data it does not seem to offer more incisive information than that of a density measurement for which clinical techniques already exist (Cameron and Sorensen, 1963. Riiegsegger et al., 1974). Our results for the modified shear modulus G’ exhibit a more pronounced decrease with age than does E, yet the development of an in uiuo technique to measure G or G’ requires extensive efforts in view of the difficulties involved in generating and detecting shear waves. Finally. it should be stressed that the results and conclusions based on our investigations pertain exclusively to compact bone from human femurs. Spongy bone may exhibit an entirely different pattern with regard to changes in its mechanical material properties.
Evans, F. G. and Lebow, M. (1951) Regional differences in some of the physical properties of the human femur. J. Appl. Physiol. 3, 563. Fliigge, W. (Ed.), Handbook of Engineering Mechanics, McGraw-Hill. New York (19621. Goodbread J. Anliker M. ad Riidgsegger P. (1975) Experimental analysis of stress waves in beam-like samples with phase stabilized interferometers. J. Appl. Math. Phys. (ZAMP) 26, 735-747. Jurist. J. M. and Kianian, K. (1973) Three models of the vibrating ulna, J. Biomechanics 6, 331-342. Lang, S. B. (1969) Elastic coefficients of animal bone. Science 165, 287. Riiegsegger. P.. Elsasser, U., Anliker, M., Gnehm, H.. Kind. H. and Prader. A. Quantification of bone mineralization with the aid of computerized tomography, accepted for publication in Radiology. Riiegsegger, P., Niederer, P. and Anliker, M. (1974) An extension of classical bone mineral measurements, Ann. Biomed. Engng 2, 194. Sizgoric, S. and Gundjian, A. (1969) An optical homodyne technique for measurement of amplitude and phase of subangstroem ultrasonic vibrations, Proc. IEEE 57, 1313.
Spiegl, P. and Jurist, J. M. (1975) Prediction of ulnar resonant frequency. J. Biomechanics 8, 213-217. Thompson. G. A.. Anliker, M. and Young, D. R. (1970) Determi,lation of’ Mechanical Properties oj Excised Dog Radiifrom Lateral Vibration Experiments, J. R. Cameron (Ed.) CONF-700515 (Chicago), Oak Ridge: U.S.E.C.
APPENDIX The matrix approach used in the theoretical analysis of the vibrations of femur shafts, which can be subdivided into segments with uniform properties, is illustrated here for the case of small lateral motions in the direction of minimum stiffness (i = 1). We first consider an arbitrary beam segment without making use of the index j. According to Timoshenko’s theory of beam vibrations the total lateral displacements y1 (x,t), due to bending and shear, and the slope. $1 (x.t), due to bending, are governed by:
Ackr~owledgement-The
work described in this paper has been supported in part by the Swiss National Science Foundation under Grant No. 4.0600.72. M,=EAr:s
REFERENCES
Abendschein, W. and Hyatt. G. W. (1970) Ultrasonics and selected physical properties of bone, C/in. Or&p. 69, 294-301. Brash, J. I. and Skorecki, J. (1970) Determination of the modulus of elasticity of bone by a vibration method, Med. Biol. Engng. 8, 389-393. Bundy. K. (1974) Experimental studies of the nonuniformity and anisotropy of human compact bone, Dissertation submitted to the dept. of Materials Science and Engineering of Stanford University, December 1974. Cameron, J. R. and Sorensen, J. A. (1963) Measurement of bone mineral in uiuo: An improved method, Science 142. 230. Craven, J. D. and Costantini, M. (1973) in Metabolic Bone Disease, Excerpta Medica Amsterdam. Dequeker, J. (1972) Bone Loss in Normal and Pathological
Conditions, Leuven University Press.
ax ’
S,=-k’,GA
where MI and St denote the bending moment and transverse shearing force respectively. A is the cross-sectional area, rl its radius of gyration in the direction of minimal stiffness and k;, the corresponding geometric shear stiffness parameter. For uniform beams, the above equations yield the following differential equations of motion:
a*y, dt2
E k;G d2y, + E k;G W, p
E
a2
p
E
= o,
ax
If we consider harmonic vibrations by taking: y1(x, t) = Yl(X)cos(ot - E) il(X. t) = Yu,(x)cos(ot - E),
710
D.
VIANO, U. HELFENSTEIN, M. ANLIKER and
+ a2 sinhr+)
+ c,cosr$)
+ adsin
Y’,(x) = aiy, sinhr+)
-ay,
ard
ax
+ a,y, coshty)
+ a$, sinr?)
R~EGSEGGER
At the interfaces between neighboring segments we require continuity for y,(x, t) and $i(x, t) and also for the bending and shearing stresses. As E and G are assumed to be yniform over the entire length of the femur shaft and as the values of k;, A and rl at the interfaces apply to both of the neighboring segments, the continuity conditions for the bending and shearing stresses reduce to those for
we find: Y,(x) = a, coshr+)
P.
a+, -.
ax
Therefore we can formulate these interface conditions simply in the form
- a,& cos($),
ul(xi; j) =
U,(Xj;
j + 1).
When bending oscillations are enforced by applying a bending moment which varies sinusoidally in time at the end x = 0 of the shaft, we have as boundary conditions: 6
Z(O;1) = m,
=
1
’
Bll
?(O;
1) - Y,(O; 1) = 0
ay, x (kj,.,)
and where 1 denotes the length of the beam (or beam segment) considered and a, to a4 are defined by the boundary conditions. For the formulation of the boundary conditions and the continuity requirements between adjacent beam segments we introduce a vector ui whose components are the mode shapes Y,(x), and Yi(x), and their derivatives
am4 ax
and
With these equations the resonance frequencies and corresponding mode shapes are defined. The matrix approach for the forced bending vibrations in the direction of maximum stiffness is identical and that for the forced axial and torsional oscillations can be devised in a manner similar to that outlined above.
aw.4 dx’
If we make use of the expressions obtained for Y,(x) and Yi(x) and add the running index j to characterize the .. . arbitrary beim segi :nt, we obtain:
Y,(x)
Y,(x) au4 u,(x;j) = I ax
1-J au4
ax
where
j
cash(y)
sinh(7)
cos(F)
sinry)
yi sinh(y)
y1 cash(y)
6, sinF>
-6, cos(yl
ysinh(‘;‘>
ycosh($)
-qsin(y)
$cosr$)
y,Tsinh(F)
6,$cos(F)
6,qsin(y:
Wx;j) =
g,Fcosh(F)
= 0
1976. Vol.
9. pp. 703-710.
Pergamon
Press
Printed
m Great
Bntain
ELASTIC PROPERTIES OF CORTICAL BONE IN FEMALE HUMAN FEMURS* D. VIANO?, U. HELFENSTEIN, M. ANLIKERand P. R~~EGSEGGER Institut ftir Biomedizinische Technik der Universitlt Ziirich und der Eidgeniissischen Technischen Hochschule, Moussonstrasse 18, 8044 Ziirich. Switzerland
Abstract-The evaluation of the physiological and pathophysiological changes in the status of human bones is incomplete without information on the elastic properties of the bone material. To assess the merits of a corresponding noninvasive technique for cortical bone the mechanical parameters of macerated human female femur shafts are evaluated by way of measuring the resonance frequencies and associated mode shapes for the bending, axial and torsional motions. The shafts are considered to behave as hollow cylinders of variable cross section and the bone material is assumed to be isotropic and linearly elastic. It is characterized by its density, Young’s modulus and shear modulus. The density and the geometry of the cross sections are determined by digital tomography. Young’s modulus and the shear modulus are considered constant along the shaft and calculated on the basis of classical theories by matching iterative mathematical predictions with experimental data. Flexural, longitudinal and torsional motions are measured by means of a stabilized Michelson interferometer. Up to 16 eigenfrequencies in the range of 1 and 20 kHz are examined. Results for the age range of 24-85 years show that Young’s modulus of cortical bone decreases with age by approximately 10% whereas the compacta density diminishes by ca. 8%. In view of the uncertainty involved in determining Young’s modulus of compact bone and the relative change it exhibits with age the efforts of developing a noninvasive technique for its quantification may not be warranted.
INTRODUCTION The mechanical integrity of bones is defined by the bone mineral mass, its geometric distribution and the elastic properties of the bone material. Whereas noninvasive clinical methods of determining the mineral mass of certain bones and its geometric distribution have become available (Cameron and Sorensen, 1963; Riiegsegger et al., 1974), we are still lacking the corresponding validated techniques of measuring the elastic properties of the bone material. An evaluation of the physiological and pathophysiological changes in the status of human bones should include information on their elastic properties because they may not only depend on the amount of bone mineral present but also on the bonding between the constituents and the lamellar structure of bone. Estimates for the strength of the human ulna on the basis of resonance measurements and bone mineral mass determinations have been reported in the literature (Jurist and Kianian, 1973; Spiegl and Jurist, 1975). However, the method applied does not permit a separation of the effects of a variation of the cross-section along the bone, of the boundary conditions at the two ends of the ulna and of the elastic properties of the bone material on changes in the resonance frequency. Thompson et a[. (1970) found in their experiments on dog radii that variations in geometry along the length of the bone may significantly influence the resonance frequency. * Received 6 May 1976. t Now at: General Motors Warren, MI, U.S.A.
Research
Laboratories,
With this investigation we aim to assess the merit of a noninvasive method of measuring changes in the elastic properties of bone material. To this end we evaluate the elastic parameters of excised human femurs from their resonance frequencies for flexural, longitudinal, and torsional vibrations. We thereby make use of classical linear elastic beam theory and take into account the geometric variations of the cross section along the femur. These variations are determined nondestructively by the method described by Riiegsegger et al. (1974). As the ends of the femur are composed of both, cortical and spongy bone, we restrict our study for the sake of simplicity to the femur shaft which is primarily made up of compact bone. EXPERIMENTALPROCEDURE A schematic drawing of the experimental arrangement used to detect the resonance frequencies and the associated mode shapes of human femur shafts is given in Fig. 1. The principal component of the measuring system is a stabilized linear Michelson interferometer. An optical detection technique has the advantage that no contact is required with the moving structure. The interferometer is based on a design reported by Sizgoric and Gundjian (1969). With a refined stabilization against drift and low frequency vibrations described by Goodbread et al. (1975) it is ossible to resolve surface displacements as small as 0.01 1 Sinusoidal stress waves are generated by means of a piezoelectric transducer glued to one of the end sections of the femur shaft. The transducer is driven by a high frequency sweep generator (Wavetek, Model 144) with which the frequency is increased from 1 to 20 kHz in approx. 5 min. To measure the surface displacements, twenty equally spaced mirrors attached along the length of the bone (Fig. 2) are successively used as reflectors for one leg of the interferometer. A He-Ne Laser (Spectra
703
704
D. VIANO,U. HELFJZNSTEIN, M. ANLIKER and P. R~EGSEGGER
SHAFT DRIVER
0 STABkIZATlON
Fig. 1. Schematic drawing of the experimentul arrangement. A sweep generator excites stress waves in the femur shaft by means of a piezoelectric transducer. The lateral displacement of the bone surface is detected optically by a stabilized laser interferometer. The amplified detector voltage is fed in two lock-in amplifiers. The resulting phase-locked displacement signal is recorded together with the sweep voltage which is proportional to the frequency of the excitation signal.
longitudinal
Physics, Mod. 120; I = 6328 A) serves as a light source. A beam splitter sends 50% of the light to the mirror sensing the shaft displacement, the other 50% to the reference mirror. The reflected beams are recombined on a silicon pin photodiode (Monsanto MDl). While the low frequency component of its output signal is fed back to the stabilization circuit, the high frequency component is analyzed by two lock-in amplifiers (Brookdeal, Model 401) with a phase difference of 0” and 90” in relation to the driving signal. With these amplifiers the femur shaft movement is detected by multiplying the amplified photodiode voltage with the driver voltage and then filtering the product with a lowpass filter. This arrangement allows for the improvement of the signal to noise ratio up to 100 dB. Therefore surface movements on the order of 0.01 A are still measurable. A typical recording from one measuring site is given in Fig. 3. The surface displacements at resonance are on the order of several Angstroms. The accuracy of the resonance frequency determination depends on the peak separation. Good resolution is obtained especially for the lower modes for which the error is less than 1%. To minimize alterations of the bone resonances due to the effects of mirror and transducer masses, their total weight was kept below 1% of that of the femur shaft. THEORETICAL
ASPECTS
ELASTIC
Fig. 2. Mirror orientation for resonance and mode shape determination. Twenty mirrors are equally spaced at intervals of approx. 1.25 cm.
AND
EVALUATION
OF
PARAMETERS
For the purpose of mathematically analyzing the mechanical behavior of the femur shaft, we model it as a hollow cylinder composed of uniform compact
705
Elastic properties of cortical bone steady displacement (90)
phase.deg.
camp (0)
frequency (khz)
order.type
motion we can introduce a matrix w (x; j) such that II~ (x; j) = W$(X;j)bi (i).
16.43
L-th
torsion
15.12
2-nd
long
14 53
5-th
flexw
13 79
5-th
flexw
12 ?I
3-rd
torsion
As described in the appendix for the case of flexural vibrations, the matrix K (x; j) contains the eigenfunctions for the harmonic vibrations (Fltigge, 1962) and the components of the vector bi (j) are the unknown coefficients in the expression for the general mode shape for segment j. The vectors bi(~1are determined for each frequency of the enforced oscillatory motion by satisfying the boundary conditions and by requiring continuity between the segments: ui (xj; j) = ui (xj; ,j + I).
The vector bi (i) for a general segment is related to the vector bi (j + 1) of the adjacent segment by hi 0’ + 1) = fJi(.Hk where 2-nd
torsion
3-rd
flexw
1-st
torsion
U,(j) = Wi-‘(Xj; j + l)w(Xj, j = 1,2. . . . . (j,,, - 1).
Therefore the vectors b,(j) can all be expressed in
i. Fig. 3. Typktrl wcordiq of’ a displacrr~~olf~f~ryuolc! SMWI). Resonance frequencies and corresponding modes are indicated.
bone. The latter is taken as a linearly elastic, isotropic and homogeneous material, characterized by its density. p, Young’s modulus, E, and shear modulus, G. Variations in the geometric cross sectional properties along the length of the shaft are taken into account by subdividing the shaft into segments of uniform cross sectional properties. These are determined by a linear interpolation of the measured data as shown in Fig. 4. By requiring continuity of the displacements and stress resultants at the segment boundaries, a continuous femur shaft model is established. Four types of steady state forced harmonic vibrational motions of the femur shaft are considered: flexure in the direction of minimum stiffness (i = 1) and maximum stiffness (i = 2), longitudinal extension (i = 3) and torsion (i = 4). The corresponding mode shapes and stresses for each femur shaft segment j are defined by a vector ui (x; j) where x is the axial coordinate as shown in Fig. 2. For each type of
j)
:.
compacta
,\
area
(cm21
3. I.
moment
of inertia (cm‘)
1. ‘2 !.
/--‘I
1.0
radius
of gyration
(cm)
0.9
//./
0.6
,0
,5
,I0
position 115
km) ,20
Fig. 4. Sample
cross-sections and related geometrical parameters of a human femur shaft. Locations of the illustrated
cross-sections are indicated on the femur drawn in a reduced scale. Subscripts 1 and 2 refer to the principal directions of the moment of inertia.
106
D. VIANO,U.
HELFENSTEIN, M. ANLIKER
terms of the vector bi (1) of the first segment. Accordingly we have ui(O,l) = Il$(O;l)bi(l)
and P.
R~~EGSEGGER
mechanical parameters may be compared against additional experimental data and thereby serve as a check of the consistency and accuracy of the theory.
and udkjrnax) = WW,,,)~(j,,,,, - l)bi(l) whereF(.L, - 1)= ~i(jmax- l)K(j,,, - 2) and G(l) = U,(l). Finally, the boundary conditions at x = 0,l yield the frequency equation for resonance. To describe the flexural motions we make use of Timoshenko’s beam theory, whereby the geometric shear stiffness parameters k; and k; for the two principal directions are assumed constant over the entire length of the femur shaft (Fltigge, 1962). For longitudinal vibration Love’s theory is applied and for the torsional vibration the classical bar theory (Fltigge (1962)). In the latter case the geometric torsional stiffness parameter k is considered to be the same for all segments. With the aid of standard matrix techniques a computer algorithm is developed to determine the root of the frequency equations and the corresponding mode shapes. To evaluate the elastic parameters from the four types of motion, an iterative procedure is applied. As an initial step we assume a reasonable set of values for the parameters and determine the corresponding resonance frequencies as the roots of the appropriate frequency equations. The lowest theoretical and experimental resonance frequencies for each type of motion are compared. On the basis of the discrepancies we choose a new value for the wave speed ci = JE/p (i = 1,2,3) in each of the lowest bending and axial vibrations and also a new value for the torsional wave speed cq = ,/kGlp in case of the lowest torsional mode. The procedure is continued iteratively until the absolute difference between the theoretical and experimental resonance frequency is less than 1%. For the two bending directions the shear correction terms E/k’G are then adjusted by a similar iterative procedure which reduces the difference between the 4th theoretical and experimental resonance frequencies to 1% retaining the previously determined value for E/p. These values of E/p and E/k’G are used as new guesses, and the previous procedure involving the lowest and fourth modes is repeated until both E/p and E/k’G change less than 1%. In the case of longitudinal vibrations, Poisson’s ratio is given a fixed value of 0.32 (Lang, 1969). This means that cg is the only parameter determined iteratively from the lowest resonance frequency for longitudinal vibrations. Likewise, for torsional vibrations it is cq which is evaluated iteratively in a similar manner. After the wave speeds ci to cq and the two values for EIk’G have been determined as outlined, all other resonances and mode shapes in the frequency range 1-17 kHz are determined. Those theoretical predictions which have not been used to evaluate the
RESULTS
To determine the feasibility of our procedure, a macerated femur shaft of a 24 year old female was analyzed in detail. The cross-sectional data at ten stations equally spaced by 2.5 cm were obtained by densitometry (Rtiegsegger et al., 1974). The cross sectional image provided by this method allows for the determination of the compacta area, moments of inertia and radii of gyration (Fig. 4). These geometrical features are used for the calculation of the resonance frequencies, mode shapes and material properties (cl to cd, EIk;G and EIk;G) as described in the previous section. The 16 computed resonance frequencies are listed in Table 1 together with those obtained from measurements. The 10 resonance frequencies which are not used to determine the mechanical parameters of the models, as well as the entire set of mode shapes constitute a basis for assessing the validity of the theoretical models employed in describing the various motions of the femur shaft. The maximum difference between the theoretical and experimental resonance frequencies is 3%. From the corresponding mode shapes given in Fig. 5 it also follows that the measured and calculated amplitude patterns are in good agreement. The discrepancy observed for the fifth bending mode may be explained by the difficulty in separating closely spaced resonances at higher frequencies. Also included in Table 1 are the material properties deduced by the iterative procedure. The uncertainty of the wave speeds is between 2 and 3% and can be estimated from the uncertainties of the geometrical parameters and the precision of the resonance frequency determination. The mechanical properties for a set of female femurs are compiled in Table 2. Of these properties the density of the compacta, the longitudinal Young’s modulus E, = pci and the modified shear modulus G’ = kG are plotted against age in Fig. 6. Included in the table are also weight and length of the whole femur. DISCUSSION
AND CONCLUSIONS
The wave speeds ci, c1 and cj derived from the resonances for bending and longitudinal vibrations do not exhibit a significant age dependence. The values for c1 and c2 are between 3000 and 3400 m/s whereas cg ranges from 3200 to 34OOmls. These results are in general agreement with the average wave speed of 32OOm/s found by Craven and Costantini (1973) for the proximal radius. As evident from Fig. 6 the compacta density decreases by ca. 8% between the ages 24 and 85 whereas the Young’s modulus El = peg determined from the speed of longitudinal waves and the density decreases by approx. 10% over
.
707
Elastic properties of cortical bone Table 1. Resonance frequencies and material parameters of a 24 year old female femur shaft. The theoretical and experimental resonance frequencies are given for all four types of vibrations together with the corresponding material properties Vibration Order Type Flexure min.
Flexure max.
Longitudinal Torsion
Resonance frequencies Experimental Theoretical &Hz) NW
Material properties
I .67
1 2 3 4
1.67 4.26 1.29 10.44
4.17 1.22 10.45
5
13.79
13.80
I 2 3 4 5 1 2 1 2 3 4
1.73 4.32 7.51 10.88 14.53 7.78 15.12 3.91 8.14 12.37 16.43
1.73 4.3 1 1.47 10.89 14.39 7.78 15.15 4.03 8.14 12.30 16.45
c1 = 3.32 x IO’ cmjsec E/k,‘G = 6.07
c2 = 3.23 x 10’ cm/set E/k2’G = 5.17 cj = 3.29 x 10’ cmjsec
cq = 1.8 x 10’ cmjsec
this age range and thus diminishes slightly more than measurements diminish with decreasing density. does the density p. This observation appears to be However, the primary cause for a decrease in the in contradiction with the results obtained by Abendeffective density of their bone samples is an increase schein and Hyatt (1970) for fresh human tibia speci- in the volume fraction of pores filled with fluid, which mens. According to their data, both the static Young’s in effect means a loss in stress carrying material. Conmodulus and that derived from ultrasonic wave speed sequently the data given by Abendschein and Hyatt
flrxural
-----
mode shapes
tor5~onal
mode
shapes
throretical
I
longlfudinal
mode shapes
Fig. 5. Comparison of experimental and theoretical mode shapes. Solid lines represent experimentally determined modes, whereas the dashed lines are those predicted by theory.
I
708
D. VIANO,U. HEDENSTEIN,M. ANLIKERand P. RCJEGSEGGER
Table 2. Material properties of 10 macerated female femur shafts. The density p is a mean value for the compact bone. Also listed are weights and lengths of the whole femurs. Four types of vibrations were considered: flexure in the plane of minimum stiffness (i = 1) and maximum stiffness (i = 2), longitudinal extension (i = 3) and torsion (i = 4). Note G’ = kG. The resonance frequencies for bending were not utilized in all cases because of the extensive numerical efforts involved in determining the mechanical parameters
Age length (cm) weight (g) density (g/cm3) cr (x lo5 cm/s) E/k,‘G c2 (x lo5 cm/s) Ejk,‘G cg (x lo5 cm/s)
E, ( x 10” dyn/cmz) cq (x lo5 cm/s) G’ (x 10” dyn/cm’)
.
.
24
46
53
54
61
61
46.0 377 2.10 3.32 6.07 3.23 5.17 3.29 2.21 1.80 6.80
37.5 319 2.11 3.22 4.06 3.07 3.86 3.34 2.35 1.77 6.60
40.0 370 2.05
41.8 278 2.02 3.40 6.45 3.39 7.50 3.34 2.25 1.58 5.05
43.2 319 1.97 3.29 7.06 3.00 5.00 3.18 1.99 1.70 5.69
40.2 310 2.00
40.0
3.32 2.20 1.66 5.52
co
l
.’
2’0 X
.
’ 60
40
3.30 2.23 1.75 6.27
l
80’
10'0 .
.
.
.
. to . . 1
20
40
.
. 77 E e &
80
60
-0
30
0.
.
1,8 1.6
I
20 AGE
40
60
80
[YEARS]
Fig. 6. Age dependence of E, G’ and p. Data are given for 10 female femur shafts. Relative variations of the longitudinal Young’s modulus E and density p with age are similar, whereas the modified shear modulus G’ exhibits a more rapid decay than E and p.
65
72
74
85
296 1.98
38.0 236 1.97
40.0 258 1.94
3.26 2.10 1.66 5.46
3.42 2.29 1.76 6.11
42.5 340 1.94 3.06 5.47 3.11 5.43 3.22 2.01 1.57 4.71
3.25 2.05 1.65 5.28
have to be corrected for the increase in the fluid mass within the bone tissue if we wish to evaluate Young’s modulus of the bone mineral. When such a correction is applied the discrepancy between their results and ours is insignificant. Hence Young’s modulus of compact human bone exhibits a very weak age dependence. Moreover, the elastic properties of bone mineral do not seem to be affected by the maceration process carried out with our bones. As mean value of E one obtains 2.2 x 10” dyn/cm2 which agrees with the results of Evans and Lebow (1951) who found by way of tensile tests for the middle third of dried human femurs a mean Young’s modulus of 2 x 10” dyn/cm’. The same mean value is given by Bundy (1974), whose data, however, ranges from 1.4 x 10” to 2.3 x 10” dyn/cm’. The faster decrease of p at menopause is supported by similar findings of Dequeker (1972) regarding the thickness of compact bone and of Cameron and Sorensen (1963) with respect to the total mineral content. The modified shear modulus G’ = kG shows a significantly larger decrease with age than does E. Because G’ contains the torsional rigidity factor k, which is unknown, we can not distinguish whether this is due to a change in geometry or due to a faster decrease of G. However, we know from Bundy’s data which was obtained from machined compacta samples of uniform geometry that G does indeed decrease faster. It is interesting to note that our dynamically determined values for Young’s modulus agree well with those found from tensile tests (Evans and Lebow, 1951). In addition Brash and Skorecki (1970) found with flexural and longitudinal resonance measurements on small specimens of compact bone from the bovine tibia that Young’s modulus was between 2 and 2.5 x 10” dyn/cm*. This range is compatible with our findings on the human femur shaft. From the relatively small decreases of the speed of bending and axial waves in our femur shafts with age we infer that these wave speeds are not sensitive parameters of osteoporosis. Moreover, the compacta
709
Elastic properties of cortical bone
density is likely to diminish more than the square of these speeds. Hence, as c2 = E/p, we may conclude that any decrease in Young’s modulus can be expected to be somewhat larger than that of the density. However, the error of the density measurement is only of the order of lo/, whereas that of the E determination is cu. 3-5x. Besides this, a possible clinical assessment of osteoporosis in terms of changes in the Young’s modulus of compact bone will be associated with a larger error than that of our laboratory measurements on excised bones. In addition, as reported by Bundy (1974), we know from experiments on bone samples from human femurs that compact bone has anisotropic elastic properties at least in so far as the lateral and circumferential parameters are significantly different from those in the axial direction. Moreover, the elastic parameters measured by Bundy, especially the ratio of Young’s modulus to shear modulus, are non-uniform as they vary over the bone cross section and along the length of the bone. Hence. the development of a non-invasive method of evaluating E of compact bone may be of questionable merit since according to our data it does not seem to offer more incisive information than that of a density measurement for which clinical techniques already exist (Cameron and Sorensen, 1963. Riiegsegger et al., 1974). Our results for the modified shear modulus G’ exhibit a more pronounced decrease with age than does E, yet the development of an in uiuo technique to measure G or G’ requires extensive efforts in view of the difficulties involved in generating and detecting shear waves. Finally. it should be stressed that the results and conclusions based on our investigations pertain exclusively to compact bone from human femurs. Spongy bone may exhibit an entirely different pattern with regard to changes in its mechanical material properties.
Evans, F. G. and Lebow, M. (1951) Regional differences in some of the physical properties of the human femur. J. Appl. Physiol. 3, 563. Fliigge, W. (Ed.), Handbook of Engineering Mechanics, McGraw-Hill. New York (19621. Goodbread J. Anliker M. ad Riidgsegger P. (1975) Experimental analysis of stress waves in beam-like samples with phase stabilized interferometers. J. Appl. Math. Phys. (ZAMP) 26, 735-747. Jurist. J. M. and Kianian, K. (1973) Three models of the vibrating ulna, J. Biomechanics 6, 331-342. Lang, S. B. (1969) Elastic coefficients of animal bone. Science 165, 287. Riiegsegger. P.. Elsasser, U., Anliker, M., Gnehm, H.. Kind. H. and Prader. A. Quantification of bone mineralization with the aid of computerized tomography, accepted for publication in Radiology. Riiegsegger, P., Niederer, P. and Anliker, M. (1974) An extension of classical bone mineral measurements, Ann. Biomed. Engng 2, 194. Sizgoric, S. and Gundjian, A. (1969) An optical homodyne technique for measurement of amplitude and phase of subangstroem ultrasonic vibrations, Proc. IEEE 57, 1313.
Spiegl, P. and Jurist, J. M. (1975) Prediction of ulnar resonant frequency. J. Biomechanics 8, 213-217. Thompson. G. A.. Anliker, M. and Young, D. R. (1970) Determi,lation of’ Mechanical Properties oj Excised Dog Radiifrom Lateral Vibration Experiments, J. R. Cameron (Ed.) CONF-700515 (Chicago), Oak Ridge: U.S.E.C.
APPENDIX The matrix approach used in the theoretical analysis of the vibrations of femur shafts, which can be subdivided into segments with uniform properties, is illustrated here for the case of small lateral motions in the direction of minimum stiffness (i = 1). We first consider an arbitrary beam segment without making use of the index j. According to Timoshenko’s theory of beam vibrations the total lateral displacements y1 (x,t), due to bending and shear, and the slope. $1 (x.t), due to bending, are governed by:
Ackr~owledgement-The
work described in this paper has been supported in part by the Swiss National Science Foundation under Grant No. 4.0600.72. M,=EAr:s
REFERENCES
Abendschein, W. and Hyatt. G. W. (1970) Ultrasonics and selected physical properties of bone, C/in. Or&p. 69, 294-301. Brash, J. I. and Skorecki, J. (1970) Determination of the modulus of elasticity of bone by a vibration method, Med. Biol. Engng. 8, 389-393. Bundy. K. (1974) Experimental studies of the nonuniformity and anisotropy of human compact bone, Dissertation submitted to the dept. of Materials Science and Engineering of Stanford University, December 1974. Cameron, J. R. and Sorensen, J. A. (1963) Measurement of bone mineral in uiuo: An improved method, Science 142. 230. Craven, J. D. and Costantini, M. (1973) in Metabolic Bone Disease, Excerpta Medica Amsterdam. Dequeker, J. (1972) Bone Loss in Normal and Pathological
Conditions, Leuven University Press.
ax ’
S,=-k’,GA
where MI and St denote the bending moment and transverse shearing force respectively. A is the cross-sectional area, rl its radius of gyration in the direction of minimal stiffness and k;, the corresponding geometric shear stiffness parameter. For uniform beams, the above equations yield the following differential equations of motion:
a*y, dt2
E k;G d2y, + E k;G W, p
E
a2
p
E
= o,
ax
If we consider harmonic vibrations by taking: y1(x, t) = Yl(X)cos(ot - E) il(X. t) = Yu,(x)cos(ot - E),
710
D.
VIANO, U. HELFENSTEIN, M. ANLIKER and
+ a2 sinhr+)
+ c,cosr$)
+ adsin
Y’,(x) = aiy, sinhr+)
-ay,
ard
ax
+ a,y, coshty)
+ a$, sinr?)
R~EGSEGGER
At the interfaces between neighboring segments we require continuity for y,(x, t) and $i(x, t) and also for the bending and shearing stresses. As E and G are assumed to be yniform over the entire length of the femur shaft and as the values of k;, A and rl at the interfaces apply to both of the neighboring segments, the continuity conditions for the bending and shearing stresses reduce to those for
we find: Y,(x) = a, coshr+)
P.
a+, -.
ax
Therefore we can formulate these interface conditions simply in the form
- a,& cos($),
ul(xi; j) =
U,(Xj;
j + 1).
When bending oscillations are enforced by applying a bending moment which varies sinusoidally in time at the end x = 0 of the shaft, we have as boundary conditions: 6
Z(O;1) = m,
=
1
’
Bll
?(O;
1) - Y,(O; 1) = 0
ay, x (kj,.,)
and where 1 denotes the length of the beam (or beam segment) considered and a, to a4 are defined by the boundary conditions. For the formulation of the boundary conditions and the continuity requirements between adjacent beam segments we introduce a vector ui whose components are the mode shapes Y,(x), and Yi(x), and their derivatives
am4 ax
and
With these equations the resonance frequencies and corresponding mode shapes are defined. The matrix approach for the forced bending vibrations in the direction of maximum stiffness is identical and that for the forced axial and torsional oscillations can be devised in a manner similar to that outlined above.
aw.4 dx’
If we make use of the expressions obtained for Y,(x) and Yi(x) and add the running index j to characterize the .. . arbitrary beim segi :nt, we obtain:
Y,(x)
Y,(x) au4 u,(x;j) = I ax
1-J au4
ax
where
j
cash(y)
sinh(7)
cos(F)
sinry)
yi sinh(y)
y1 cash(y)
6, sinF>
-6, cos(yl
ysinh(‘;‘>
ycosh($)
-qsin(y)
$cosr$)
y,Tsinh(F)
6,$cos(F)
6,qsin(y:
Wx;j) =
g,Fcosh(F)
= 0
