.I Bmmechanics.
1975.
Vol.8.pp.363-367.
Pergamon Press. Printed in Great Bntain
THE DISTRIBUTION AND ANISOTROPY OF THE STIFFNESS OF CANCELLOUS BONE IN THE HUMAN PATELLA* P. R. TOWNSEND,P. RAUX?, R. M. ROSE, R. E. MIEGEL and E. L. Department of Metallurgy and Materials Science, Massachusetts Cambridge. Massachusetts, U.S.A.
RADIN~
Institute of Technology,
Abstract-The distribution and anisotropy of the stiffness of cancellous bone in the human patella was studied by compression testing of small cubes cut from autopsy specimens. The observed variations bear a definite relationship to the internal architecture which can be accounted for by a simple sheetand-strut model. The observed stiffness can be calculated with reasonable accuracy for loads approximately parallel to the sheets, with the proposed model. The spatial distribution of the stiffness was plotted and a close relationship to mechanical function is suggested.
INTRODUCHON Changes in the density of cancellous bone are associated with a number of diseases, the most obvious of which is osteoporosis; the accompanying reduction in strength has been well-documented (Bell et al., 1967). In the early stages of osteoarthritis there is a distinct stiffening of the subchondral cancellous bone
(Radin et al., 1970) which is due primarily to a change in trabecular connectivity rather than den&cation (Pugh et al., 1973a). By way of contrast, in chondromalacia patellae, which has at times been assumed to be somehow related to osteoarthritis due to certain similarities in the cartilaginous changes, the bony changes are essentially osteoporotic (Darracott and Vernon-Roberts, 1971). Since trabecular architecture depends on location and is generally anisotropic, so should be the physical properties, and in fact this is the case for the mechanical properties (Evans and King, 1961). It is likely that the principal determinant of the mechanical properties is the trabecular architecture, and that general structure-properties relations do exist, in this sense, for trabecular bone. Structural models to explain mechanical properties have been proposed based on an assumed porous block geometry (McElhaney et al., 1970) and on finite-element analysis of a small region under the subchondral plate (Pugh et al.. 1973b). Recently, on the basis of stereological analysis, the basic structural element for cancellous bone in the human patella was deduced by our group (Raux et al.. 1974). The geometry of this element is illustrated in Fig. 1. The orientation of the sheets was found to vary with position in the patella as shown in Fig. 2. Besides the variation in orientation
in the frontal
plane
as shown
figure, the sheets generally lie perpendicular to the articular surface until the anterior cortex is approached, where the sheets lie perpendicular to the cortical surface. It is to be expected that a structure such as that shown in Fig. 1 will exhibit considerable mechanical anisotropy, and consequently Fig. 2 suggests that a property such as elastic compliance or stiffness should vary with position both in magnitude and anisotropy, in a manner consistent with the structural data. It is this suggestion we set out to verify. EXPERIMENTALPROCEDURE 1. Material and preparation Seven grossly normal knee caps were obtained at autopsy from cadavers whose range of age was 2675 yr old with a mean of 55. There were four males. three females, four right and three left sides. In each case the cause of death was known to have no effect on skeletal joints. The following procedures were performed:
in this
* Received 7 December 1914.
t Present address: Department of Anatomy, Faculte Necker-Enfants Malades. Paris. France. $ Present address: Department of Orthopedic Surgery, Harvard Medical School, Boston, Massachusetts 02115, U.S.A.
Fig. I. Schematic of trabecular arrangement and resulting idealized model.
364
P. R.
TOWNSEND.
P. RAUX, R. M. ROSE,R. E. MIEGELand E. L. RADIN
PROXIMAL
differential
DISTAL
n\\\\\\\\\\u
Fig. 2. Arrangement of trabecular sheets under articular surface (after Raux et al., 1974).
1. While fresh, the patellae were cast into a block of polymethyl methacrylate (PMMA) made from a rapidly polymerizing mixture of PMMA powder and liquid methyl methacrylate monomer. The patellae were mounted to facilitate cutting of two types of cancellous bone cubes. One type, the ‘articular’ cubes, were oriented relative to the artitular surface of the patella; while the ‘non-articular’ cubes were each cut along sag&al, frontal and horizontal planes. The cancellous bone cubes were carefully cut on a diamond wheel ceramic saw 800pm in thickness with water lubrication and cooling to prevent excessive heat production and were oriented as follows: 2. Non-articular cubes were cut along purely sagittal, frontal and horizontal planes. The nine volumes (see Fig. 3), la-3c, were thus obtained. 3. The articular cubes, oriented relative to the articular surfaces, were cut with one face parallel to the lateral or medial articular surfaces or tangent to the articular surface of the crest. Horizontal and the mutually perpendicular third plane cuts were then made. 4. The cubes obtained were rejected if the final dimensions were not sufficient to assure accurate mechanical testing or if cortical or very dense cancellous bone (as de-
ANTERIOR
(a) LATER-D,
load cell LVDT
Fig. 5. Schematic of compression tester.
scribed by Raux et al., 1974) was apparent in the volume. The seven patellae cut produced 41 cubes for testing. The specimens were roughly cubical with edges ranging from 0.22 to 0.39 in.
2.
Compression
testing
The specimens were then tested on a specially constructed compression machine designed for C45lb load and total deflection of 0.05 in. The load was measured using a beam deflection load cell utilizing an LVDT in line with the specimen but with total deflection less than 1 per cent of the usual deflection of a cube specimen. A Keithley Model 140 Nanovolt amplifier was used for the output of the load cell. The deflection was measured using a second LVDT. The compression tester is shown in Fig. 4 and schematically in Fig. 5. Each specimen was tested in three directions and at a constant strain rate of 5.3 x 10S4cm/sec. Care was taken to obtain the modulus in all three directions before a particular direction was tested to failure. 6007
0-
/
AL
Fig. 3. (a) Specimen locations as seen in horizontal plane for articular oriented cubes and strictly coronal oriented cubes. (b) Volume code.
Fig. 6. Typical stress-strain curves. Top curve for loading along sheets, bottom curve for loading along struts.
Fig. 4. Specially constructed
(Facing p. 364)
compression
tester.
Stiffness of cancellous Table Articular NO. samples
Area la
3
lb
3
,. 1c
^ i
3a
3
3h
3
3c
3
All E, E2 E,
cubes
Non-articular
6, 6-78 + I.30 677 i 0.53 504 kO.13 4.87 k O-63 5.82 * 0.04 5.45 f I.30
1. Stiffness
E,
E,
3.9 1 *o. 40 5-5 2 i0, 52 5-c‘4 *o-39 3.55 kO.24 4-87 i l.‘fl 8.4 s 1,60
3-28 io-52 2-96 i o-41 l-88 to.19 3-94 2 I.16 3.38 i 0.20 1.76 fO.55
NO. samples
Area la
2
lb
2
Ic
2
3a
2
3b
2
3c
2
365
data
cubes
E, O-04 fO.38 5 92 +@42 2-41 kO.34 605 + 0.04 5-76 iO.67 3-76 ?r O-20
bone
Crest
E>
ES
7.18 f3.10 7.47 fO42 4.48 * I.50 5-75 *I-51 5.97 iO.21 3.18 f0.15
5.51 i O-52 449 * 0.44 3.36 *l-l5 448 iPI4 4.47 to-31 3.78 i I.05
NO. samples
Area
2a
E>
6.61 iO10 5.96 *OS1 5.1 ?_042
2h
I
4
E3
5.03
514
* 0.22
k 0.0s
5.12 + O-60 3.52 kO.32
3.59 + O-36 2.29 to20
stiffness values x 10“ psi. measured anterior-posterior. measured proximal- distal. measured lateral-medial. I%18
RESULTS
Typical stressstrain curves obtained are shown in Fig. 6. The maximum stiffness was measured from each curve. The initial portion of the curve is due to a seating of the loading faces of the highly porous samples under low loads. The maximum stiffness area was usually linear over a substantial range of stress and strain. Cubes cut from the crest area are of the same face orientation for both the articular and nonarticular cubes. The results of measurements on these cubes were, therefore, combined. The average modulus measured on all the cubes at the given strain rate are given in Table 1 and Figs. 7 and 8. The scheme used to display the data was deemed necessary for clarity in the dicussion of the results. Isobars of constant stiffness directly beneath and normal to the articular surface are shown in Fig. 9. DISCUSSION
1. StifSness
variation
with
6.78
551
6.04
6.6 1
13.55
4.87
6.0 5
5.9 6
3.76
LAT E RAL 8. Stiffness
CREST results,
596
fiB8
5.11
LATERAL 7. Stiffness
CREST results,
articular _” x IU’ PSI.
cubes.
All
values
surface are shown in Fig. 7. The articular surface on the lateral side is usually slightly concave. The surface on the medial side is quite convex. As we have said, the ‘sheets’ in the vicinity of the articular and cortical surfaces tend to lie perpendicular to those surfaces; the general directions of the sheets are shown in Fig. 2. From Figs. 2 and 7, the stiffness normal to the articular direction on the lateral side from proximal MAL
5.82
&2g
504
MEDIAL
non-articular x lo4 psi.
PROXI
6.77
5,7 6
241
location
15.03
(3.97
B-
Fig.
The maximum moduli, shown as stiffness vectors, for cubes with one set of faces parallel to the articular
Fig.
E,
4
2c
cubes
&76 545 MEDIAL cubes.
All
DISTAL values
Fig. 9. Constant
stiffness isobars under All values x 10’ psi
articular
surface.
366
P.
R. TOWNSI.NI>,
P.
RAUX.
R. M. ROSE.R. E.
to distal is the stiffness parallel to the sheets. The articular stiffness from volumes la and h are nearly the same and result from the same structure. The stiffness in area Ic is really associated with a gradual separation of the sheets (i.e. a decrease in density) from area h to c (Roux rl ~rl., 1974) The vertical stiffness is nearly constami. The lower value in volume la is probably due to the concavity of the surface forcing the sheets slightly out of alignment vertically. (Otherwise the structure model of sheets and struts would dictate identical I-esults in the articular and vertical direction.) The asticular blocks were cut with regard to the articular direction in particular and the vertical orientation of the sheets are not expected to be perpendicular to the horizontal surface. Note the general weakness of the lateral--medial (horizontal) direction. li’ubes from the crest area show a high stiffness in the articular direction with a more gradual weakening from proximal to distal. This matches quite well variation described previously with the structure (Raux et aJ.. 1974). The sheet structure in the crest area is much more constant than in the lateral side. The crest provides a vertical volume of nearly constant articular surface configuration. It is not surprising. then: to find the vertical stiffness of volumes 2u and b to be closer to the articular values. The marked decrease in 2c is probably due to the curving sheets shown ia Fig. 2. compression in the vertical direction would be met by fewer truly vertical sheets. The horizontal stiffness is, a.gain. much lower in the b and c Ievels. The high value in volume 20 may be due to the influence of the medial side’s horizontal sheets impinging on the crest area. but. more likely, this volume. the densest of ail” may simply be more isotropic than any other. On the medial side, the influence of nearly horjzontal sheets in 3a is confirmed. The vertical stiffness is now the lowest and in fact our previous work suggests that the sheets are not truly perpendicular in this case to the articular surface (Raux et al., 1974). Noduli of non-articular blocks are shown in Fig. 8. The magnitude differences between Fig. 7 and Fig. 8 cannot be compared due to patella to patella variations, only trends with the cube type can be compared. The non-articular cubes have strict sagittal, coronal and horizontal faces without regard for the sheet tendencies 10 curve from the anterior cortex to the irregular articular surface. For this reason there is much more variability but many of the general trends are the same. The maximum ratio of stiffnesses of the three directions is directly related to the anisotropy of the cubes. They are. however, related indirectly to the anisotropy of the structure since the sheet and strut orientations can be such as to produce quasi-isotropy. A comparison of these magnitudes for any one cube, therefore. demonstrates the tendency of sheets to be coordinated with the articular surface. The variation in stiffness thus complies with variations in structure. The scatter listed in Table 1 is explained by structure considerations given above and in any event is much
MIFF~L
and E. L. RAU~Z
less than the scatter in stiffness throughout the patella. The constant stiffness isobars (Fig. 9) divide the artitular surface into uniq~~e arcas. There are too areas of high stiffness gradients. in the vicinity between Lb and c and between, 2% and 3~1extending slightly distal. There are diverging stiffness gradients in t!-re vicinity of 3c and near 2h and 36. ‘We wc3itId expect drese features to be related to the mechanical function oT the patcIla.
The inherent function. or stiffness, can best be related to structure by considering the structure-stiffness data in the articular direction from the articular oriented cubes. As discussed above, the sheets run nearly perpendicular to the articular surface. Here the sheet structure runs generally parallel to ?he loading direction. Stereological data are better defined when the observation plane cuts the sheets on a nearly perpendicular orientation. Also. the stiffness measurements were done so that the loading directions were perpendicular to the structure observation planes. It should be stressed that average structural parameters taken from a number of patella specimens were chosen and average stiffnesses from different but norStructure parameters from mal pdtellas are correlated. the actual cubes tested fob; stiffness are not used. Although this procedure was dictated by experimental technique, it should be noted that any relationship derived thereby should be reasonably general. For loading parallel to the sheets, we have chosen the simplest set of assumptions: the struts carry no load, and bending and buckling are neglected. Thus the effective stiffness or modulus E, of the cancellous bone cube is given by
where E, is the modulus of the trabecular material (assumed to be that of cortical bone; we have taken 165 x IOh psi for this value). AT is ‘the area1 fraction of trabecular material as determined stereologically and the empirical constant C, is used to account for the distribution of orientations, i.e. the fact that the sheets in reality all deviate to some degree from parallelism, all curve. contain randomly located holes and uneven strut locations. so that the mode of deformation must include some degree of bending andjar buckling at some locations. in fact C, was approx. 0.2 and if this value is used together with the stereological values of AT determined previously by Raux et al. (1974) to calculate E,,r, the agreement with the measured value is quite good, as shown in Fig. 10.
1. The anisotropy bone in the patella definitely related to basic structural unit.
of the stiffness of the cancellous is a function of position, and is the changing orientation of the the sheet-and-strut model which
Stiffness of cancellous bone 8.0x104
361
Acknowledgements-This work was supported by the U.S. Office of Naval Research. In addition the authors also gratefully acknowledge the help of 1. M. Puffer of M.I.T. and J. Stukas of Massachusetts General Hospital. Dr Radin gratefully acknowledges the support of the Hartford Foundation and the National Institute of Arthritis, Metabolic and Digestive Diseases. REFERENCES
Bell, G. H., Dunbar, O., Beck, J. S. and Gibb, A. (1967) Variations in strength of vertebrae with age and their relation to osteoporosis. Calc. Tiss. Res. 1. 75-86. Darracott, J. and Vernon-Roberts, B. (1971) The bony changes in “chondromalacia patellae”. Rheum. Phys. Med. 11, 175-179. DaSilva, 0. L. and Bratt, J. F. (1970) Stress trajectories in the patella. Acta Orthop. Stand. 41. 608-618. Evans, F. G. and King, A. I. (1961) Regional differences in some physical properties of human spongy bone. In Biomechanical
Fig. 10. Predicted stiffness values vs those experimentally measured.
was deduced from our previous architectural study (Raux et al., 1974). 2. The variation of stiffness with position has been plotted for loads normal to the articular surface. Although it is likely that the observed variation is closely connected with the mechanical function of the patella, data on the latter is rather sparse, so that conclusions on such connections must wait. (Although there are, e.g. photoelastic studies (DaSilva and Brat< 1970), these do not consider intra-articular forces, which are quite important.) 3. For directions of stress approximately parallel to the sheets (in the model) the stiffness may be calculated with good accuracy from volume or area1 fractions of hard tissue. Thus the distributed stiffness can be calculated directly from architectural studies .in many cases.
Studies
of the Musculo-Skeletal
System
(Edited by Evans, F. G.), pp. 49-67. Thomas, Springfield, IL. McElhaney, J., Alem, N. and Roberts, V. (1970) A porous block model for cancellous bone. ASME Publication 70WA/BHF-2. Pugh, J. W., Rose, R. M. and Radin, E. L. (1973a) Elastic and viscoelastic properties of trabecular bone: dependence on structuie. j. Biomechanics 6, 475-485. Pugh. _; J. W.. Rose. R. M. and Radin. E. L. (1973b) \ , A structural model for the mechanical behavior of trabecular bone. J. Biomechanics 6, 657-610. Radin, E. L., Paul, I. L. and Tolkoff, M. (1970) Subchondral bone changes in patients with early degenerative joint disease. Arth. Rheum. 13, 4WO5. Raux, P., Townsend, P. R., Miegel, R., Rose, R. M. and Radin, E. L. (1975) Trabecular architecture of the human patella. J. Biomechanics 8, l-7. NOMENCLATURE
6
area fraction of bone modulus of a single trabeculae, psi maximum modulus or stiffness, psi stiffness measured in anterior-posterior direction stiffness measured in proximal-distal direction stiffness measured in lateral-medical direction distance between sheets trabecula thickness distance between struts stress, psi strain.
1975.
Vol.8.pp.363-367.
Pergamon Press. Printed in Great Bntain
THE DISTRIBUTION AND ANISOTROPY OF THE STIFFNESS OF CANCELLOUS BONE IN THE HUMAN PATELLA* P. R. TOWNSEND,P. RAUX?, R. M. ROSE, R. E. MIEGEL and E. L. Department of Metallurgy and Materials Science, Massachusetts Cambridge. Massachusetts, U.S.A.
RADIN~
Institute of Technology,
Abstract-The distribution and anisotropy of the stiffness of cancellous bone in the human patella was studied by compression testing of small cubes cut from autopsy specimens. The observed variations bear a definite relationship to the internal architecture which can be accounted for by a simple sheetand-strut model. The observed stiffness can be calculated with reasonable accuracy for loads approximately parallel to the sheets, with the proposed model. The spatial distribution of the stiffness was plotted and a close relationship to mechanical function is suggested.
INTRODUCHON Changes in the density of cancellous bone are associated with a number of diseases, the most obvious of which is osteoporosis; the accompanying reduction in strength has been well-documented (Bell et al., 1967). In the early stages of osteoarthritis there is a distinct stiffening of the subchondral cancellous bone
(Radin et al., 1970) which is due primarily to a change in trabecular connectivity rather than den&cation (Pugh et al., 1973a). By way of contrast, in chondromalacia patellae, which has at times been assumed to be somehow related to osteoarthritis due to certain similarities in the cartilaginous changes, the bony changes are essentially osteoporotic (Darracott and Vernon-Roberts, 1971). Since trabecular architecture depends on location and is generally anisotropic, so should be the physical properties, and in fact this is the case for the mechanical properties (Evans and King, 1961). It is likely that the principal determinant of the mechanical properties is the trabecular architecture, and that general structure-properties relations do exist, in this sense, for trabecular bone. Structural models to explain mechanical properties have been proposed based on an assumed porous block geometry (McElhaney et al., 1970) and on finite-element analysis of a small region under the subchondral plate (Pugh et al.. 1973b). Recently, on the basis of stereological analysis, the basic structural element for cancellous bone in the human patella was deduced by our group (Raux et al.. 1974). The geometry of this element is illustrated in Fig. 1. The orientation of the sheets was found to vary with position in the patella as shown in Fig. 2. Besides the variation in orientation
in the frontal
plane
as shown
figure, the sheets generally lie perpendicular to the articular surface until the anterior cortex is approached, where the sheets lie perpendicular to the cortical surface. It is to be expected that a structure such as that shown in Fig. 1 will exhibit considerable mechanical anisotropy, and consequently Fig. 2 suggests that a property such as elastic compliance or stiffness should vary with position both in magnitude and anisotropy, in a manner consistent with the structural data. It is this suggestion we set out to verify. EXPERIMENTALPROCEDURE 1. Material and preparation Seven grossly normal knee caps were obtained at autopsy from cadavers whose range of age was 2675 yr old with a mean of 55. There were four males. three females, four right and three left sides. In each case the cause of death was known to have no effect on skeletal joints. The following procedures were performed:
in this
* Received 7 December 1914.
t Present address: Department of Anatomy, Faculte Necker-Enfants Malades. Paris. France. $ Present address: Department of Orthopedic Surgery, Harvard Medical School, Boston, Massachusetts 02115, U.S.A.
Fig. I. Schematic of trabecular arrangement and resulting idealized model.
364
P. R.
TOWNSEND.
P. RAUX, R. M. ROSE,R. E. MIEGELand E. L. RADIN
PROXIMAL
differential
DISTAL
n\\\\\\\\\\u
Fig. 2. Arrangement of trabecular sheets under articular surface (after Raux et al., 1974).
1. While fresh, the patellae were cast into a block of polymethyl methacrylate (PMMA) made from a rapidly polymerizing mixture of PMMA powder and liquid methyl methacrylate monomer. The patellae were mounted to facilitate cutting of two types of cancellous bone cubes. One type, the ‘articular’ cubes, were oriented relative to the artitular surface of the patella; while the ‘non-articular’ cubes were each cut along sag&al, frontal and horizontal planes. The cancellous bone cubes were carefully cut on a diamond wheel ceramic saw 800pm in thickness with water lubrication and cooling to prevent excessive heat production and were oriented as follows: 2. Non-articular cubes were cut along purely sagittal, frontal and horizontal planes. The nine volumes (see Fig. 3), la-3c, were thus obtained. 3. The articular cubes, oriented relative to the articular surfaces, were cut with one face parallel to the lateral or medial articular surfaces or tangent to the articular surface of the crest. Horizontal and the mutually perpendicular third plane cuts were then made. 4. The cubes obtained were rejected if the final dimensions were not sufficient to assure accurate mechanical testing or if cortical or very dense cancellous bone (as de-
ANTERIOR
(a) LATER-D,
load cell LVDT
Fig. 5. Schematic of compression tester.
scribed by Raux et al., 1974) was apparent in the volume. The seven patellae cut produced 41 cubes for testing. The specimens were roughly cubical with edges ranging from 0.22 to 0.39 in.
2.
Compression
testing
The specimens were then tested on a specially constructed compression machine designed for C45lb load and total deflection of 0.05 in. The load was measured using a beam deflection load cell utilizing an LVDT in line with the specimen but with total deflection less than 1 per cent of the usual deflection of a cube specimen. A Keithley Model 140 Nanovolt amplifier was used for the output of the load cell. The deflection was measured using a second LVDT. The compression tester is shown in Fig. 4 and schematically in Fig. 5. Each specimen was tested in three directions and at a constant strain rate of 5.3 x 10S4cm/sec. Care was taken to obtain the modulus in all three directions before a particular direction was tested to failure. 6007
0-
/
AL
Fig. 3. (a) Specimen locations as seen in horizontal plane for articular oriented cubes and strictly coronal oriented cubes. (b) Volume code.
Fig. 6. Typical stress-strain curves. Top curve for loading along sheets, bottom curve for loading along struts.
Fig. 4. Specially constructed
(Facing p. 364)
compression
tester.
Stiffness of cancellous Table Articular NO. samples
Area la
3
lb
3
,. 1c
^ i
3a
3
3h
3
3c
3
All E, E2 E,
cubes
Non-articular
6, 6-78 + I.30 677 i 0.53 504 kO.13 4.87 k O-63 5.82 * 0.04 5.45 f I.30
1. Stiffness
E,
E,
3.9 1 *o. 40 5-5 2 i0, 52 5-c‘4 *o-39 3.55 kO.24 4-87 i l.‘fl 8.4 s 1,60
3-28 io-52 2-96 i o-41 l-88 to.19 3-94 2 I.16 3.38 i 0.20 1.76 fO.55
NO. samples
Area la
2
lb
2
Ic
2
3a
2
3b
2
3c
2
365
data
cubes
E, O-04 fO.38 5 92 +@42 2-41 kO.34 605 + 0.04 5-76 iO.67 3-76 ?r O-20
bone
Crest
E>
ES
7.18 f3.10 7.47 fO42 4.48 * I.50 5-75 *I-51 5.97 iO.21 3.18 f0.15
5.51 i O-52 449 * 0.44 3.36 *l-l5 448 iPI4 4.47 to-31 3.78 i I.05
NO. samples
Area
2a
E>
6.61 iO10 5.96 *OS1 5.1 ?_042
2h
I
4
E3
5.03
514
* 0.22
k 0.0s
5.12 + O-60 3.52 kO.32
3.59 + O-36 2.29 to20
stiffness values x 10“ psi. measured anterior-posterior. measured proximal- distal. measured lateral-medial. I%18
RESULTS
Typical stressstrain curves obtained are shown in Fig. 6. The maximum stiffness was measured from each curve. The initial portion of the curve is due to a seating of the loading faces of the highly porous samples under low loads. The maximum stiffness area was usually linear over a substantial range of stress and strain. Cubes cut from the crest area are of the same face orientation for both the articular and nonarticular cubes. The results of measurements on these cubes were, therefore, combined. The average modulus measured on all the cubes at the given strain rate are given in Table 1 and Figs. 7 and 8. The scheme used to display the data was deemed necessary for clarity in the dicussion of the results. Isobars of constant stiffness directly beneath and normal to the articular surface are shown in Fig. 9. DISCUSSION
1. StifSness
variation
with
6.78
551
6.04
6.6 1
13.55
4.87
6.0 5
5.9 6
3.76
LAT E RAL 8. Stiffness
CREST results,
596
fiB8
5.11
LATERAL 7. Stiffness
CREST results,
articular _” x IU’ PSI.
cubes.
All
values
surface are shown in Fig. 7. The articular surface on the lateral side is usually slightly concave. The surface on the medial side is quite convex. As we have said, the ‘sheets’ in the vicinity of the articular and cortical surfaces tend to lie perpendicular to those surfaces; the general directions of the sheets are shown in Fig. 2. From Figs. 2 and 7, the stiffness normal to the articular direction on the lateral side from proximal MAL
5.82
&2g
504
MEDIAL
non-articular x lo4 psi.
PROXI
6.77
5,7 6
241
location
15.03
(3.97
B-
Fig.
The maximum moduli, shown as stiffness vectors, for cubes with one set of faces parallel to the articular
Fig.
E,
4
2c
cubes
&76 545 MEDIAL cubes.
All
DISTAL values
Fig. 9. Constant
stiffness isobars under All values x 10’ psi
articular
surface.
366
P.
R. TOWNSI.NI>,
P.
RAUX.
R. M. ROSE.R. E.
to distal is the stiffness parallel to the sheets. The articular stiffness from volumes la and h are nearly the same and result from the same structure. The stiffness in area Ic is really associated with a gradual separation of the sheets (i.e. a decrease in density) from area h to c (Roux rl ~rl., 1974) The vertical stiffness is nearly constami. The lower value in volume la is probably due to the concavity of the surface forcing the sheets slightly out of alignment vertically. (Otherwise the structure model of sheets and struts would dictate identical I-esults in the articular and vertical direction.) The asticular blocks were cut with regard to the articular direction in particular and the vertical orientation of the sheets are not expected to be perpendicular to the horizontal surface. Note the general weakness of the lateral--medial (horizontal) direction. li’ubes from the crest area show a high stiffness in the articular direction with a more gradual weakening from proximal to distal. This matches quite well variation described previously with the structure (Raux et aJ.. 1974). The sheet structure in the crest area is much more constant than in the lateral side. The crest provides a vertical volume of nearly constant articular surface configuration. It is not surprising. then: to find the vertical stiffness of volumes 2u and b to be closer to the articular values. The marked decrease in 2c is probably due to the curving sheets shown ia Fig. 2. compression in the vertical direction would be met by fewer truly vertical sheets. The horizontal stiffness is, a.gain. much lower in the b and c Ievels. The high value in volume 20 may be due to the influence of the medial side’s horizontal sheets impinging on the crest area. but. more likely, this volume. the densest of ail” may simply be more isotropic than any other. On the medial side, the influence of nearly horjzontal sheets in 3a is confirmed. The vertical stiffness is now the lowest and in fact our previous work suggests that the sheets are not truly perpendicular in this case to the articular surface (Raux et al., 1974). Noduli of non-articular blocks are shown in Fig. 8. The magnitude differences between Fig. 7 and Fig. 8 cannot be compared due to patella to patella variations, only trends with the cube type can be compared. The non-articular cubes have strict sagittal, coronal and horizontal faces without regard for the sheet tendencies 10 curve from the anterior cortex to the irregular articular surface. For this reason there is much more variability but many of the general trends are the same. The maximum ratio of stiffnesses of the three directions is directly related to the anisotropy of the cubes. They are. however, related indirectly to the anisotropy of the structure since the sheet and strut orientations can be such as to produce quasi-isotropy. A comparison of these magnitudes for any one cube, therefore. demonstrates the tendency of sheets to be coordinated with the articular surface. The variation in stiffness thus complies with variations in structure. The scatter listed in Table 1 is explained by structure considerations given above and in any event is much
MIFF~L
and E. L. RAU~Z
less than the scatter in stiffness throughout the patella. The constant stiffness isobars (Fig. 9) divide the artitular surface into uniq~~e arcas. There are too areas of high stiffness gradients. in the vicinity between Lb and c and between, 2% and 3~1extending slightly distal. There are diverging stiffness gradients in t!-re vicinity of 3c and near 2h and 36. ‘We wc3itId expect drese features to be related to the mechanical function oT the patcIla.
The inherent function. or stiffness, can best be related to structure by considering the structure-stiffness data in the articular direction from the articular oriented cubes. As discussed above, the sheets run nearly perpendicular to the articular surface. Here the sheet structure runs generally parallel to ?he loading direction. Stereological data are better defined when the observation plane cuts the sheets on a nearly perpendicular orientation. Also. the stiffness measurements were done so that the loading directions were perpendicular to the structure observation planes. It should be stressed that average structural parameters taken from a number of patella specimens were chosen and average stiffnesses from different but norStructure parameters from mal pdtellas are correlated. the actual cubes tested fob; stiffness are not used. Although this procedure was dictated by experimental technique, it should be noted that any relationship derived thereby should be reasonably general. For loading parallel to the sheets, we have chosen the simplest set of assumptions: the struts carry no load, and bending and buckling are neglected. Thus the effective stiffness or modulus E, of the cancellous bone cube is given by
where E, is the modulus of the trabecular material (assumed to be that of cortical bone; we have taken 165 x IOh psi for this value). AT is ‘the area1 fraction of trabecular material as determined stereologically and the empirical constant C, is used to account for the distribution of orientations, i.e. the fact that the sheets in reality all deviate to some degree from parallelism, all curve. contain randomly located holes and uneven strut locations. so that the mode of deformation must include some degree of bending andjar buckling at some locations. in fact C, was approx. 0.2 and if this value is used together with the stereological values of AT determined previously by Raux et al. (1974) to calculate E,,r, the agreement with the measured value is quite good, as shown in Fig. 10.
1. The anisotropy bone in the patella definitely related to basic structural unit.
of the stiffness of the cancellous is a function of position, and is the changing orientation of the the sheet-and-strut model which
Stiffness of cancellous bone 8.0x104
361
Acknowledgements-This work was supported by the U.S. Office of Naval Research. In addition the authors also gratefully acknowledge the help of 1. M. Puffer of M.I.T. and J. Stukas of Massachusetts General Hospital. Dr Radin gratefully acknowledges the support of the Hartford Foundation and the National Institute of Arthritis, Metabolic and Digestive Diseases. REFERENCES
Bell, G. H., Dunbar, O., Beck, J. S. and Gibb, A. (1967) Variations in strength of vertebrae with age and their relation to osteoporosis. Calc. Tiss. Res. 1. 75-86. Darracott, J. and Vernon-Roberts, B. (1971) The bony changes in “chondromalacia patellae”. Rheum. Phys. Med. 11, 175-179. DaSilva, 0. L. and Bratt, J. F. (1970) Stress trajectories in the patella. Acta Orthop. Stand. 41. 608-618. Evans, F. G. and King, A. I. (1961) Regional differences in some physical properties of human spongy bone. In Biomechanical
Fig. 10. Predicted stiffness values vs those experimentally measured.
was deduced from our previous architectural study (Raux et al., 1974). 2. The variation of stiffness with position has been plotted for loads normal to the articular surface. Although it is likely that the observed variation is closely connected with the mechanical function of the patella, data on the latter is rather sparse, so that conclusions on such connections must wait. (Although there are, e.g. photoelastic studies (DaSilva and Brat< 1970), these do not consider intra-articular forces, which are quite important.) 3. For directions of stress approximately parallel to the sheets (in the model) the stiffness may be calculated with good accuracy from volume or area1 fractions of hard tissue. Thus the distributed stiffness can be calculated directly from architectural studies .in many cases.
Studies
of the Musculo-Skeletal
System
(Edited by Evans, F. G.), pp. 49-67. Thomas, Springfield, IL. McElhaney, J., Alem, N. and Roberts, V. (1970) A porous block model for cancellous bone. ASME Publication 70WA/BHF-2. Pugh, J. W., Rose, R. M. and Radin, E. L. (1973a) Elastic and viscoelastic properties of trabecular bone: dependence on structuie. j. Biomechanics 6, 475-485. Pugh. _; J. W.. Rose. R. M. and Radin. E. L. (1973b) \ , A structural model for the mechanical behavior of trabecular bone. J. Biomechanics 6, 657-610. Radin, E. L., Paul, I. L. and Tolkoff, M. (1970) Subchondral bone changes in patients with early degenerative joint disease. Arth. Rheum. 13, 4WO5. Raux, P., Townsend, P. R., Miegel, R., Rose, R. M. and Radin, E. L. (1975) Trabecular architecture of the human patella. J. Biomechanics 8, l-7. NOMENCLATURE
6
area fraction of bone modulus of a single trabeculae, psi maximum modulus or stiffness, psi stiffness measured in anterior-posterior direction stiffness measured in proximal-distal direction stiffness measured in lateral-medical direction distance between sheets trabecula thickness distance between struts stress, psi strain.
