Bone, 11, pp . 417-423 (1990)
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A Model of Vertebral Trabecular Bone Architecture and its Mechanical Properties K . S . JENSEN,' LIS MOSEKILDE2 AND LEIF MOSEKILDE3 t Physics Laboratory III, The Technical University of Denmark, DK-2800 Lyngby, Denmark `Department of Connective Tissue Biology, Institute of Anatomy, University of Aarhus, DK-8000 Aarhus, Denmark 'University Department of Endocrinology and Metabolism, Aarhus County Hospital, DK-S000 Aarhus, Denmark Address Jor correspondence and reprints : Klaus S . Jensen, Physics Laboratory 111, The Technical University of Denmark, DK-2800 Lyngby,
Denmark . closed-cell architecture as described by Whitehouse et al . (1971), and by Arnold (1980), is seen . Furthermore, the architecture varies strongly with the position in the vertebrae (Whitehouse et al . 1971) . Our analyses apply to trabecular bone from the central part of the vertebra (approx . 1 cm under the endplates), at which site vertical columns and horizontal supporting struts are dominating, even in younger individuals (Mosekilde 1989) . In the center of the vertebral body, vertical plates and horizontal struts dominate the architecture in young individuals, but at this site, too, the lattice changes with age to a rod-strut structure . The purpose of this paper is to put into perspective recent experimental data on trabecular architecture, bone mass, and mechanical behaviour, and to test current hypotheses on the relationship between bone mass and trabecular bone biomechanics . For our analyses we used an idealized, semi-analytic mathematical model . By assigning different values to the input parameters the model can be made to represent trabecular bone over a span of 40 years . In the study we placed emphasis on the sensitivity of mechanical behaviour to small changes in architecture . Bone research has for several years focused primarily on noninvasive measurements of bone density in relation to age and the menopause, and the techniques for measuring bone density have become increasingly accurate . On the other hand, it has become obvious that the biomechanical competence of boneand thereby the fracture risk-can only partly be derived from bone mass (Kleerekoper et al . 1985 ; Mosekilde et al . 1987) . The importance of bone architecture has been demonstrated in relation to aging in normal individuals (Mosekilde et al . 1987), but it is highly desirable to have a better understanding of the relationship between bone density, bone architecture, and bone strength for the evaluation of prophylactic and therapeutic modalities in osteoporotic states .
Abstract An idealized, structural model of vertebral trabecular bone is presented . The architecture of the model (thick vertical columns and thinner horizontal struts) is based on studies of samples taken from the central part of vertebral bodies from normal Individuals aged 30 to 90 years . With trabecular diameters and spacings typical for persons aged 40, 60, and 80 years respectively, the model accounts reasonably well for age-related changes in vertical and horizontal stiffness and trabecular bone volume, as seen in experimental data . By introducing a measure for the randomness of lattice joint positions in the modeled trabecular network, it is demonstrated that the apparent stiffness varies by a factor of between 5 and 10 from a perfect cubic lattice to a network of maximal Irregularity, even though trabecular bone volume remains almost constant . A considerable change in mechanical behaviour Is also seen, without changing the overall trabecular bone volume, when the bone material is slightly redistributed among vertical and horizontal trabeculae . It is concluded that measured bone mass should not be the sole indicator of trabecular bone biomechanical competence (stiffness and stress) . It is crucial that measurements of bone density are considered in combination with a detailed description of the architecture . Key Words : Vertebral trabecular bone-Architecture-Apparent bone density-Stiffness-Anisotropy-Mathematical model .
Introduction The age-related decline of vertebral trabecular bone strength and stiffness, and their connection to trabecular bone volume (TBV) have previously been studied (Mosekilde et al . 1987 ; Mosekilde and Mosekilde 1988), and a decline in bone strength that much exceeded the decline in apparent bone density was demonstrated . From histomorphometric studies on a large number of vertebral bodies (Mosekilde 1988, 1989), we have come to the conclusion that the trabecular architecture typically resembles a cubic lattice of thick vertical columns with thinner horizontal struts (Fig . 1) . This premise holds for persons aged approximately 40 years and older . In younger persons, a denser,
Materials and Methods The experimental data were from 79 normal autopsy cases, 39 females aged 30 to 91 years (mean 69) and 40 males aged 30 to 90 years (mean 65) . The data originate from earlier studies (Mosekilde et al . 1987 ; Mosekilde and Mosekilde 1988 ; Mosekilde 1988, 1989) . From each individual, the third lumbar vertebral body (L3 ) was removed and frozen at -20°C . Then four 417
41 8
K . S . Jensen et al . : Model of vertebral trabecular bone mechanics L,
r
r7 }
r rr
Fig. 2 . Unit cell of an idealized trabecular structure . L, and L c denote the distances between vertical and horizontal trabeculae, respectively . D, and D, are the corresponding diameters (thicknesses) .
Fig . 1 . Typical trabecular architecture as seen in the vertebra. Magnification on photo : 15 x .
plate at random rotation and by measuring the orthogonal intercepts . The model
trabecular cylinders (I = 5 mm, d = 7 mm) were obtained from the central part of the vertebral body (ca . I cm under the endplates) . Two of the cylinders were taken in a vertical direction and two in a horizontal direction .
Biomechanical testing The compression test was performed on one vertical and one horizontal cylinder on a materials testing machine (Alwetron®) . From the load deformation curves, maximum load, stress, and stiffness were read .
Trabecula bone density After bomechanical testing, the two cylinders were ashed at 105 °C for 2 hours and then at 580°C for 24 hours . The average apparent ash-density was expressed per unit total sample volume . Additionally, the second horizontal bone cylinder was embedded in methylmetacrylate and cut in 8-t um-thick sections for conventional histomorphometric assessment of trabecular bone volume (TBV) by point counting (Melsen et al . 1978) . Structural analysis From the second vertical bone cylinder the marrow was carefully cleaned with a water jet . The specimens were then embedded in methylmetacrylate and sawed in 400-p.m-thick "vertical sections" (Vesterby et al . 1987) on a diamond saw (Exakt®) . Two sections from the center of each specimen were investigated in polarized light with a X-filter inserted . From projected color photographs, the thicknesses of the horizontal and vertical trabeculae and the intertrabecular distances in the two directions were determined by using a Zeiss-integration
We modeled the trabecular architecture as a three-dimensional lattice with a unit cell geometry, as shown in Fig. 2 . The chosen geometry is typical for the trabecular lattice of the vertebral body underneath the endplates-at any age . In the center of the vertebral body, plates and struts are-after the age of 40-50 years-also converted to the lattice used in this model (Arnold 1980 ; Mosekilde 1989) . Curved plates, such as seen in the iliac crest (Whitehouse 1977) . are not typical for the vertebral body . In all the calculations, we assumed the solid bone material to be linearly elastic . Consequently, our analyses are restricted to mechanical behaviour corresponding to the steep, linear part of experimentally obtained stress-strain curves . A typical stressstrain curve for vertebral trabecular bone has a short initial phase where the stiffness increases to a constant value that lasts until collapse of trabeculae occurs at high loads ; thereafter stiffness decreases ( Mosekilde et al . 1987) . For a perfectly rectangular geometry, the structure's apparent stiffness in vertical or horizontal directions, E,,,, and E,,,, respectively, is a result of simple axial compression of the trabeculae : E° , . = rrD,2-El(4'L,'), E,,,, = a-D,2-El(4-L,-L,), where E is Young's Modulus for solid bone material, L, and L, are the intertmbecular distances, and D, and D, are the trabecular diameters (see Fig . 2) . At a certain applied maximum stress, failure occurs by either clastic or plastic buckling ; which type actually takes place depends on the characteristic yield stress of the solid material, and the slenderness ratio of the trabeculae under load . The slenderness ratio (for vertical trabeculae) is given by s=4-L,ID, . Small values of s give rise to plastic type buckling . From our experimental observations, however, it is clear that vertebral trabecular bone is not a perfectly columnar structure . The individual trabeculae are not quite parallel to the vertical or
K . S . Jensen et al . : Model of vertebral trabecular bone mechanics
OJAVUldhow
wwraugg
S
∎ i
a = 0 .3
a = 0.6
a =
1 .0
Fig . 3. Vertical (upper row) and horizontal (lower row) sections of the three-dimensional cancellous bone model for various degrees of relative lattice disorder, a . Dimensions used in all three examples are : L, = 750 µm . L, = 980 pm, D, = 200 µm, and Dc = 120 gin . The trabecular thickness is not drawn in scale,
horizontal axes, which means that bending will also contribute to the overall deformation . In order to make the model more realistic without destroying its generality we therefore defined a measure of "relative lattice disorder", a, to quantify the degree of deviation from a perfectly rectangular lattice . Using a, we can generate an irregular lattice from a regular one by adding a term, aiLX, to each component of the three-dimensional lattice joint coordinates . X is a random number drawn from a uniform distribution on the interval from - 1/2 to + 1/2 . L = L, for the two horizontal axes, while L = Lz in the vertical direction . Fig . 3 shows the appearance of three lattices created with different values of a, but all with the same mean trabecular dimensions . At maximum irregularity (a = 1), the possibility exists that some of the rectangles are transformed into triangles . It should be noticed that a does not measure the contiguity of the lattice . In our model contiguity, in the sense used by, for example, Pugh et al . 1974, is a function of L, and L 2 . The definition of a is, of course, only one of many ways to quantify deviations from rectangularity . We would like to emphasize that our definition is tailored to "generate" irregularity (i .e ., it is a model-based definition) . Trying to make an exact estimate of a for a given sample of experimental material could very well be a meaningless task, because the definition contains the assumption of a rectangular lattice, the position of which may be impossible to determine uniquely for an actual trabecular bone specimen . It is crucial for the interpretation of the calculated results that we use a solely to get a more realistic description of the lattice geometry (which is important in order to find the distribution of compression and bending forces) . Any attempt to model other factors, such as discontinuities in the trabecular lattice, by adjusting a should be avoided . The proper way to treat a is thus to make an independent estimate of its value before the model is used to calculate the mechanical behaviour of the lattice . For our purpose we do not need best-fit values of a . By simply comparing photographs of sections of bone specimens (Mosekilde 1988) with computer generated lattices, we found that values between 0 .4 and 0 .8 gave a satisfactory representation of the typical trabecular geometry . The absolute size of the model (measured in number of unit cells) is of no importance for the results when a = 0 . For a > 0, however, our semi-random description of the irregularity can give rise to some variability of the calculated mechanical
419
properties when the lattice is not infinite . Therefore we chose a lattice of approximately the same overall dimensions as the tested bone specimens . In this way we could study whether or not the actual size of the tested structure significantly influenced the variability of the calculated or measured mechanical behaviour . For reasons of convenience, however, we used a box geometry instead of a cylinder (Fig . 3) . The model, put into a set of lattice equations (BernoulliEuler beam theory), was analyzed on a computer using a structural design program that enabled us to calculate the displacement, rotation, and reaction of any joint in the lattice . During the calculations, the lattice joints corresponding to the top and the bottom of the bone sample were displaced 1% of the total modeled sample height . The displacement was only specified on the axis of compression (i .e ., the top-bottom surface joints were allowed to move (and rotate) in the plane perpendicular to the compression axis) . We thus simulated a test machine where the friction between piston and bone sample is small compared with the internal forces generated by deformation of the trabeculae . The program also allowed us to find the forces, moments, and stresses for every beam element (trabecula) in the construction . A detailed analysis of trabecular collapse, however, requires knowledge about several material parameters, such as tensile, compressive, and flexural strength, and is beyond the scope of the present work . Such parameters have been experimentally determined for compact bone, but there is a wide variation in the measured values (Carter and Spengler 1978 ; Gibson 1985) .
Results Ignoring any age-related changes in solid bone material properties, we used the same size of Young's Modulus whether modeling a "young" or an "old" trabecular lattice . The main input parameters were therefore the trabecular diameters and spacings, to which we assigned measured mean values . In Fig . 4, two sets of trabecular dimensions were used, corresponding to a typical lattice for a 60-year-old person (Fig . 4a) and that for a person aged 80 years (Fig . 4b) . The values for L„ L2 , D„ and D, were derived from the experimental material previously described (Mosekilde 1989) . For Young's Modulus we used the value E = 11 .4 GPa as found by Townsend et al . 1975b, because this is the only determination we know of that has been done directly on solid trabecular bone material . The model, however, has one more degree of freedom : the degree of irregularity, a . As illustrated in Fig . 4, the mechanical behaviour is dramatically affected by a . It should be noted that the apparent stiffness in both vertical and horizontal directions decreases by a factor of 5 for the 60-year-old structure when a is increased from 0 to I (i .e ., when a less rectangular architecture is specified) . This is due to a change in mode of deformation from simple axial compression of the trabeculae to a mixture of compression and bending . As shown in Fig . 4b, the effect is even more pronounced for a less dense, but more anisotropic lattice . Here, the stiffness varies by a factor of 10 between a perfectly rectangular architecture and a lattice of maximum irregularity . The apparent, relative density of the modeled bone structure (equivalent to TBV) is also a function of a, but the dependency is much weaker than that seen for the stiffness, The density increases with a simply because the sum of trabecular lengths is slightly larger in an irregular lattice than in a rectangular one . The error bars in Fig . 4 indicate extreme values (not standard deviation) obtained from calculations on five separate lattices
420
K . S . Jensen et al . : Model of vertebral trabecular hone mechanics
(a) 5000 a 400-300-200-100-0-200-M
cu
150.09-0 100-50-0-0 .08-M
0 .07-
0,0
1 .0
0 .5
Rel . latt. disorder, a
I 0 .0
I 0 .5
1 .0
Rel . latt. disorder, a
Fig . 4 . Vertical and horizontal stiffness, and relative density (TBV), as calculated from the model with Young's Modulus E = 11 .4 GPa (a) : With trabecular dimensions as in Fig . 3, typical for a person of age 60 years . (b) : With L, = 1200 µm . L, = 1900 µm, D, = 200 µm, Dc = 80 µm (80-year-old person (Fig . 5b)) .
with the same a-values, but generated from different random sequences, X, as explained earlier . For small values of a, this scatter is less than the height of the solid square markers used in Fig . 4, The moderate variability suggests that the actual, applied size of our bone test specimens is unlikely to cause statistical fluctuations of a magnitude that would veil the age-related, qualitative differences in mechanical behaviour . The calculated stiffnesses displayed in Fig . 4 are considerably higher than found experimentally, except for a = 1 . From visual inspection of the experimental bone specimens, however . we found that the trabecular architecture in general is more regular than a lattice with a = 1 . This holds for the entire range 30 to 90 years . As previously mentioned, an a-value between 0 .4 and 0 .8 seems appropriate . We have not been able to observe any obvious difference in lattice disorder between bone samples of different ages that would justify a differentiation in the sense of a-values . In the
1 (a)
(b)
Fig. 5. Vertical section of model (a = 0 .6) showing a typical lattice for persons of age (a) 40 years, and (h) 80 years .
rest of the modeling studies we therefore used a = 0 .6 as the standard value (typical lattices are displayed in Fig . 5) . This gives rise to calculated stiffnesses about three times higher than normally seen, which may be due to an over-idealized architecture, wrong trabecular dimensions and/or too high a value for Young's Modulus . We believe, though, that the model reflects the actual trabecular structure to an extent that allows us to study the relative dependence of stiffness and anisotropy on trabeeular dimensions and lattice irregularity . In order not to cause confusion by showing unnaturally high stiffnesses . we chose to reduce Young's Modulus in all other calculations to E = 3 .8 GPa-one third the value used in Fig . 4 . Figure 6 displays values for stiffness and density for 40, 60, and 80-year-old individuals when modeled with experimental mean values for trabecular diameters and spacings, as shown in Table I . Also shown in Fig . 6 are the corresponding, expcrimentally found values for stiffness and density (Mosekilde et al . 1987 ; Mosekilde and Mosekilde 1988) . In general, the modeled TB V is smaller than the corresponding experimental value . This mass difference may originate from the idealized architecture, in which only whole trabecular structures without perforations have been taken into account . From the stiffness data we found the degree of mechanical anisotropy (vertical/horizontal stiffness ratio) as listed in Table II . It can be seen that both model and experimental data exhibit an increasing anisotropy as a function of age, although the increase is more pronounced for the model . Table III shows the results of fitting a power function to stiffness and TBV . As seen from the powers, the model gives a linear relation between density and stiffness in the vertical direction . In the horizontal direction, the relationship lies some-
K . S . Jensen et al . : Model of vertebral trabecular bone mechanics
421
Table I . Mean trabecular dimensions measured in "typical" persons aged approximately 40, 60, and 80 years . These data correspond to the results shown in Fig . 6 . (The data are derived from Mosekilde 1988) . Trabecular dimension (Am)
Lt
1-2
Dl
D2
Age (years) 40
720
770
210
150
60
870
1110
210
125
80
990
1450
210
95
where between quadratic and cubic . The higher power for the horizontal direction is also a typical feature of the experimental data, but here we find a quadratic relationship (instead of a linear) for the vertical direction . In this case, though, using linear regression gives just as good a fit as the power function . Including experimental data for 20-year-old persons (not shown here) does not alter the calculated powers significantly . The horizontal and vertical stiffness/density ratios . which determine the powers listed in Table III, are, however, very sensitive to changes in trabecular dimensions that cause a redistribution of the mass, as illustrated by Fig . 7 . Here we have modeled the stiffness in the two directions for three sets of trabecular dimensions, all of which give the same density . The three sets of diameters and spacings are all within the range of values that are representative of 60-year-old persons (the middle set was used for the results pictured in Fig . 3 and Fig . 4a) . It can be seen that the vertical/horizontal stiffness ratio changes from 1 .5 to 7 .8 when D, is increased from 180 ism to 220 Am for the chosen combinations of diameters and spacings .
the concomitant changes in trabecular architecture without the need for assuming age-related shifts in material properties (i .e ., Young's Modulus) . In the following paragraphs we discuss some of the quantitative discrepancies between model and experimental data . In order to obtain reasonable numbers for apparent stiffness, we had to reduce Young's Modulus to about one third the value determined by Townsend et al . 1975b . We have no obvious explanation for this,' but it should be noted that the authors themselves needed to multiply their values by one fifth when modeling the stiffness of the patella (Townsend et al . 1975a) . According to our findings from using various degrees of trabec-
Discussion Unlike some other published models (e .g ., Williams and Lewis 1982), the model architecture described here is not a copy of the exact trabecular geometry of a single bone test specimen, but an idealized archetype representing the vertebral trabecular bone structure over a span of years (from 40 to 80 years of age) . By scaling the trabecular dimensions for vertical and horizontal directions separately in accordance with experimental data . we have used the model to study the theoretical relationship between trabecular dimensions, relative bone density, and apparent stiffness-with special emphasis on anisotropy . In another theoretical study (Gibson 1985), the anisotropy of trabecular scaling was ignored, leading to different results for stiffness as a function of density . Taking into consideration the simplifications and idealizations made, it is only natural that our model cannot exactly minor the experimental data in every respect . However, qualitatively and, to some extent, quantitatively, it accounts for the observed relationships between changes in trabecular spacings and diameters, TBV, and stiffness in the vertical and horizontal directions, respectively . The model thus explains the age-related changes in mechanical behaviour as a natural consequence of
1 40
1
60
I 80 Age (year)
'After submission of the manuscript we became aware of a recent study by Kuhn et al . (1989) in which Young's Modulus for solid, trabecular bone from the iliac crest is found to be 3 .81 GPa . This is exactly the value we had to use in our model to get reasonable results .
Stiffness and relative density as functions of age . Dotted curves are experimental data . Solid and dashed curves are calculated from the model (E = 3 .8 GPa) with a = 0 .6 and a = 1 .0, respectively . The trabecular dimensions are listed in Table 1 . Fig. 6.
42 2
K . S . Jensen et al . : Model of vertebral trabecular bone mechanics
Table II . Vertical/horizontal stiffness ratio calculated from data shown in Fig . 6 . (Experimental data are derived from Mosekilde et al . 1987 .)
Experimental data
Model a=0.6
Model a=1 .0
Age (years) 40
2 .8
1 .7
1 .5
60
4 .0
3 .0
2 .9
80
4 .5
6 .2
5 .7
ular lattice irregularity, this can to some extent, but not fully, be related to their assumption of pure axial compression of the trabecular plates . Similar comments apply to the modeling work by Williams and Lewis 1982 . They needed a value as low as E = 1 .3 GPa which, in their case, is most unlikely to be caused solely by ignoring the deformations by bending that will inevitably exist in an "almost perfect" columnar architecture . The fact that the calculated stiffness anisotropy increases more strongly with age than seen experimentally is possibly due to the way we modeled the age-related thinning of the trabecular network . Instead of removing entire trabeculae (at random) from the lattice, as done in the bone remodeling processes (Parfitt 1984, 1987), we simply scaled the grid size of the lattice while preserving the initially chosen architecture (six beams radiating from every lattice joint) . It is important to realize that the growing dispersion of trabecular spacings does not change the angle between trabeculae at the joints and therefore can not be described by increasing the a-value in our model . In general, the modeled TBV is smaller than found experimentally for the same trabecular dimensions . Part of the discrepancy could arise from the design of the model . In the experimental data, the extreme age-related increase in distance between trabeculae is partly caused by perforations of trabeculae and partly by the subsequent removal of whole trabeculae . In the model, however, all increases in inter-trabecular distances are deemed to be caused solely by increasing the space between existing trabeculae . In consequence, the trabecular bone mass is
relatively underestimated in the model . At this point we would like to mention that in all our model calculations on lattice compression, the highest absolute values of normal stress on the trabecular surfaces always appeared at the joints . This is in close agreement with the experimental observation of trabecular thickening near the joints : the remodeling processes will cause a sufficient amount of bone material to be placed here to equate stress and strain (Frost 1989) to the values found elsewhere on the trabecula . As argued earlier, a numerical analysis of the strength of the modeled lattice would force us into a number of assumptions about the size of various material properties . However, from the theoretical arguments given for open cell structures by Gibson (1985) . it is clear that strength depends on architecture and trabecular dimensions in much the same way as stiffness does . Osteoclastic and oseoblastic activity, locally governed by mechanical strain and stress (Frost 1983, 1987, 1989 ; Parfitt 1982; Lanyon 1981, 1984), determine the trabecular dimensions in both vertical and horizontal directions . This, of course, imposes some limitations on the variation in anisotropy of vertebral trabecular bone with the same TBV . There is still room, though, for a substantial spreading in stiffness and strength, as illustrated by some of the experimental data displayed by Carter and Hayes (1977) and Gibson (1985) . Taking into account the calculated high sensitivity of mechanical behaviour to changes in architectural anisotropy (Fig . 7), we therefore conclude that bone density measurements alone do not reveal
Table III . Best-fit-power, x, obtained by fitting corresponding values of stiffness (E.) and trahecular bone density (TBV) to the function Eo = C - (TBV)x .
Experimental data
Model a=0 .6
Model a=1 .0
Vertical direction
1 .8a
1 .0
1 .0
Horizontal direction
2 .8b
25
2 .7
'No significant difference between power function (x = 1 .8) and linear regression analysis . b Power function (x = 2.8) was significantly better than linear regression analysis .
K . S . Jensen et aL : Model of vertebral trabecular bone mechanics
80 -
vertical
60 -t
40 horizontal 20 -
0180
1 200
1 220
D, (Jim)
850 145 900
750 120 980
700 100 1200
L, (Jim) D, ()im) L, (pin)
Fig. 7. Calculated stiffness (using E = 3 .8 GPa) for three combinations of trabecular distances and diameters giving the same relative density of 0 .076 .
very much about biomechanical competence (i .e ., stiffness and strength) . This conclusion is also supported by our findings shown in Fig . 4 : When the architecture becomes more columnar (for unchanged TBV as well as mean trabecular dimensions) a much greater stiffness is obtained . This suggests that bone density measurements need to be combined with information about the architecture in order to give optimum indication of the mechanical condition of trabecular bone . The concept is important not only in relation to normal age-related changes but to an even greater extent in relation to osteoporosis and the evaluation of prophylactic and therapeutic modalities . Acknowledgments : The technical assistance of Inger Vang Magnussen, Lotte P . Kristensen, Aase Young, and Lotte Brohm is gratefully acknowledged, as is the help of the University Institute of Pathology, Aarhus County Hospital, and the Department of Forensic Medicine, Aarhus Kommunehospital, in providing the specimens . We are also thankful to Rasmus Feldberg and Caraten Knudsen for making the graphics software to generate Figs . 3 and 5 . The investigation was supported by grants from the Danish Medical Research Council (J .no . 12-6164 and J-no . 12-6836), Sygekassemes Helsefond (J .no H I1/ 218-88), L .F . Foght's Fund, NOVO's Fund, NOVO and Aarhus University Research Foundation (] .no . 1987-7131/01-6-LF 1-29) .
42 3
porous structure . J. Bone Joint Surg . 59-A, 7 :954-962; 1977 . Caner, D .R . ; Spengler, D .M . Mechanical properties and composition of cortical bone. Clin . Onhop . Rel . Res. 135:192-197 ; 1978 . Frost, H .M . The skeletal intermediary organization . Merab . Bone Dis . Relar . Res. 4 :281-290 ; 1983 . Frost, H .M . The mechanostat : a proposed pathogenic mechanism of osteoporosis and the bone mass effects of mechanical and nonmechanica] agents . Bone and Mineral 2 :73-85 ; 1987 . Frost, H .M . Some ABCs of skeletal pathophysiology III : Bone balance and the B .BMU . Calcif. Tissue Int . 45 :131-133 ; 1989 . Gibson, L.I . The mechanical behaviour of cancellous hone . J. Biomech . 18, 5 :317-328 ; 1985 . Kleerekoper, M . ; Villanueva, A .R. : Stands . J . ; Rao, D .S . ; Parfait, A .M . The role of three-dimensional trabecular microswcture in the pathogenesis of vertebral compression fractures . Calcif. Tissue lot. 37:594-597 ; 1985 . Kuhn, LL . : Goldstein, S .A . ; Choi, K . : London, M . ; Feldkamp, L .A . ; Matthews, L .S . Comparison of the tmbecular and conical tissue moduli from human iliac crest . J . Orrhop . Res . 7 :876-884 ; 1989 . Lanyon, L .E . Bone remodelling, mechanical stress, and osteoporosis . DeLuca, H .F . : Frost, H .M . : Jee, W.S .S . : Johnston, C .C . Jr . ; Partiitr . A .M . . eds. Osteoporosis recent advances in pathogenesis and treatment . Baltimore : University Park Press ; 1981 :129, 138 . Lanyon, L .E . Functional strain as a determinant for bone remodeling . Calcif. Tissue lm . 36 :556-S61 ; 1984 . Melsen, F . ; Melsen, B . ; Mosekilde, Le . ; Bergmann, S . Histomorphometric analysis of normal bone from the iliac crest . Acra Parhol. Microbial . Scand. 86 :70-81 ; 1978. Mosekilde, Li . Age-related changes in vertebral trabecular bone architectureassessed by a new method . Bone 9:247-250; 1988 . Mosekilde, Lt. Sex differences in age-related loss of vertebral trabecular bone mass and structure-Biomechanical consequences . Bone 10 :425-432 ; 1989 . Mosekilde, Li . ; Mosekilde, Le . Iliac crest trabecular bone volume as predictor for vertebral compressive strength, ash density and trabecular bone volume in normal individuals . Bone 9 :195-199 ; 1988 . Mosekilde, U . ; Mosekilde . Le . ; Danielsen, C .C . Biomechanica] competence of vertebral trabecular bone in relation to ash density and age in normal individuals . Bone 8 :79-85 ; 1987 . Parfin, A .M . The coupling of bone formation to bone resorption : A critical analysis of the concept and of its relevance to the pathogenesis of osteoporosis . Memb . Bone Dis . Relor. Res . 4 :1-6 ; 1982 . Parfitt, A .M . Age-related structural changes in trabecular and conical bone : Cellular mechanisms and biomechanical consequences . Calcif. Tissue Inc. 36 :5123-5128 ; 1984. Partiu, A.M . Trabecular bone architecture in the pathogenesis and prevention of fracture . Am . J . Med. 82 :68-72 ; 1987 . Pugh, 1 . W. ; Radin, E.L . ; Rose, R .M . Quantitative studies of human subchondml cancellous bone . J. Bone Joint Surg . 56-A, 2 :313-321 ; 1974. Townsend, P .R . ; Raux P . ; Rose, R.M . ; Miegel, R.E . ; Radin, E .L . The distribution and anisotopy of the stiffness of cancellous bone in the human patella . J . Biome cit . 8:363-367 : 1975a. Townsend, P .R . : Rose, R .M . ; Radin . E .L . Buckling studies of single human trabeculae . J . Biomech . 8 :199-201 ; 1975b . Vesterby, A . : Kragstmp, J . ; Gundersen, H .J .G . ; Melsen, F . Unbiased stereologic estimation of surface density in bone using vertical sections . Bone 8 :13-17 : 1987 . Whitehouse, W.J . Cancellous bone in the anterior pan of the iliac crest. Colt . Tiss. Res. 23 :67-76 ; 1977 . Whitehouse, W .1 . ; Dyson, E .D. ; Jackson, C .K . The scanning electron microscope in studies of trabecular bone from a human vertebral body . J . Anat. 108 : 481-496; 1971 . Williams, 1 .L. ; Lewis, J .L . Properties and an anisotropic model of cancellous bone from the proximal tibial epiphysis . J. Biomech . Eng . 104 :50-56 ; 1982 .
References Arnold. J .5 . Trabecular pattern and shapes in aging and osteoporosis . Metab . Bone Div. et Rel . Res . 2:5297-5308 ; 1980. Carter, D .R . ; Hayes, W .C . The compressive behaviour of bone as a two-phase
Received: January 18, 1990 Revised : May 17, 1990 Accepted : May 29, 1990
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Copyright © 1990 Pergamon Press plc
A Model of Vertebral Trabecular Bone Architecture and its Mechanical Properties K . S . JENSEN,' LIS MOSEKILDE2 AND LEIF MOSEKILDE3 t Physics Laboratory III, The Technical University of Denmark, DK-2800 Lyngby, Denmark `Department of Connective Tissue Biology, Institute of Anatomy, University of Aarhus, DK-8000 Aarhus, Denmark 'University Department of Endocrinology and Metabolism, Aarhus County Hospital, DK-S000 Aarhus, Denmark Address Jor correspondence and reprints : Klaus S . Jensen, Physics Laboratory 111, The Technical University of Denmark, DK-2800 Lyngby,
Denmark . closed-cell architecture as described by Whitehouse et al . (1971), and by Arnold (1980), is seen . Furthermore, the architecture varies strongly with the position in the vertebrae (Whitehouse et al . 1971) . Our analyses apply to trabecular bone from the central part of the vertebra (approx . 1 cm under the endplates), at which site vertical columns and horizontal supporting struts are dominating, even in younger individuals (Mosekilde 1989) . In the center of the vertebral body, vertical plates and horizontal struts dominate the architecture in young individuals, but at this site, too, the lattice changes with age to a rod-strut structure . The purpose of this paper is to put into perspective recent experimental data on trabecular architecture, bone mass, and mechanical behaviour, and to test current hypotheses on the relationship between bone mass and trabecular bone biomechanics . For our analyses we used an idealized, semi-analytic mathematical model . By assigning different values to the input parameters the model can be made to represent trabecular bone over a span of 40 years . In the study we placed emphasis on the sensitivity of mechanical behaviour to small changes in architecture . Bone research has for several years focused primarily on noninvasive measurements of bone density in relation to age and the menopause, and the techniques for measuring bone density have become increasingly accurate . On the other hand, it has become obvious that the biomechanical competence of boneand thereby the fracture risk-can only partly be derived from bone mass (Kleerekoper et al . 1985 ; Mosekilde et al . 1987) . The importance of bone architecture has been demonstrated in relation to aging in normal individuals (Mosekilde et al . 1987), but it is highly desirable to have a better understanding of the relationship between bone density, bone architecture, and bone strength for the evaluation of prophylactic and therapeutic modalities in osteoporotic states .
Abstract An idealized, structural model of vertebral trabecular bone is presented . The architecture of the model (thick vertical columns and thinner horizontal struts) is based on studies of samples taken from the central part of vertebral bodies from normal Individuals aged 30 to 90 years . With trabecular diameters and spacings typical for persons aged 40, 60, and 80 years respectively, the model accounts reasonably well for age-related changes in vertical and horizontal stiffness and trabecular bone volume, as seen in experimental data . By introducing a measure for the randomness of lattice joint positions in the modeled trabecular network, it is demonstrated that the apparent stiffness varies by a factor of between 5 and 10 from a perfect cubic lattice to a network of maximal Irregularity, even though trabecular bone volume remains almost constant . A considerable change in mechanical behaviour Is also seen, without changing the overall trabecular bone volume, when the bone material is slightly redistributed among vertical and horizontal trabeculae . It is concluded that measured bone mass should not be the sole indicator of trabecular bone biomechanical competence (stiffness and stress) . It is crucial that measurements of bone density are considered in combination with a detailed description of the architecture . Key Words : Vertebral trabecular bone-Architecture-Apparent bone density-Stiffness-Anisotropy-Mathematical model .
Introduction The age-related decline of vertebral trabecular bone strength and stiffness, and their connection to trabecular bone volume (TBV) have previously been studied (Mosekilde et al . 1987 ; Mosekilde and Mosekilde 1988), and a decline in bone strength that much exceeded the decline in apparent bone density was demonstrated . From histomorphometric studies on a large number of vertebral bodies (Mosekilde 1988, 1989), we have come to the conclusion that the trabecular architecture typically resembles a cubic lattice of thick vertical columns with thinner horizontal struts (Fig . 1) . This premise holds for persons aged approximately 40 years and older . In younger persons, a denser,
Materials and Methods The experimental data were from 79 normal autopsy cases, 39 females aged 30 to 91 years (mean 69) and 40 males aged 30 to 90 years (mean 65) . The data originate from earlier studies (Mosekilde et al . 1987 ; Mosekilde and Mosekilde 1988 ; Mosekilde 1988, 1989) . From each individual, the third lumbar vertebral body (L3 ) was removed and frozen at -20°C . Then four 417
41 8
K . S . Jensen et al . : Model of vertebral trabecular bone mechanics L,
r
r7 }
r rr
Fig. 2 . Unit cell of an idealized trabecular structure . L, and L c denote the distances between vertical and horizontal trabeculae, respectively . D, and D, are the corresponding diameters (thicknesses) .
Fig . 1 . Typical trabecular architecture as seen in the vertebra. Magnification on photo : 15 x .
plate at random rotation and by measuring the orthogonal intercepts . The model
trabecular cylinders (I = 5 mm, d = 7 mm) were obtained from the central part of the vertebral body (ca . I cm under the endplates) . Two of the cylinders were taken in a vertical direction and two in a horizontal direction .
Biomechanical testing The compression test was performed on one vertical and one horizontal cylinder on a materials testing machine (Alwetron®) . From the load deformation curves, maximum load, stress, and stiffness were read .
Trabecula bone density After bomechanical testing, the two cylinders were ashed at 105 °C for 2 hours and then at 580°C for 24 hours . The average apparent ash-density was expressed per unit total sample volume . Additionally, the second horizontal bone cylinder was embedded in methylmetacrylate and cut in 8-t um-thick sections for conventional histomorphometric assessment of trabecular bone volume (TBV) by point counting (Melsen et al . 1978) . Structural analysis From the second vertical bone cylinder the marrow was carefully cleaned with a water jet . The specimens were then embedded in methylmetacrylate and sawed in 400-p.m-thick "vertical sections" (Vesterby et al . 1987) on a diamond saw (Exakt®) . Two sections from the center of each specimen were investigated in polarized light with a X-filter inserted . From projected color photographs, the thicknesses of the horizontal and vertical trabeculae and the intertrabecular distances in the two directions were determined by using a Zeiss-integration
We modeled the trabecular architecture as a three-dimensional lattice with a unit cell geometry, as shown in Fig. 2 . The chosen geometry is typical for the trabecular lattice of the vertebral body underneath the endplates-at any age . In the center of the vertebral body, plates and struts are-after the age of 40-50 years-also converted to the lattice used in this model (Arnold 1980 ; Mosekilde 1989) . Curved plates, such as seen in the iliac crest (Whitehouse 1977) . are not typical for the vertebral body . In all the calculations, we assumed the solid bone material to be linearly elastic . Consequently, our analyses are restricted to mechanical behaviour corresponding to the steep, linear part of experimentally obtained stress-strain curves . A typical stressstrain curve for vertebral trabecular bone has a short initial phase where the stiffness increases to a constant value that lasts until collapse of trabeculae occurs at high loads ; thereafter stiffness decreases ( Mosekilde et al . 1987) . For a perfectly rectangular geometry, the structure's apparent stiffness in vertical or horizontal directions, E,,,, and E,,,, respectively, is a result of simple axial compression of the trabeculae : E° , . = rrD,2-El(4'L,'), E,,,, = a-D,2-El(4-L,-L,), where E is Young's Modulus for solid bone material, L, and L, are the intertmbecular distances, and D, and D, are the trabecular diameters (see Fig . 2) . At a certain applied maximum stress, failure occurs by either clastic or plastic buckling ; which type actually takes place depends on the characteristic yield stress of the solid material, and the slenderness ratio of the trabeculae under load . The slenderness ratio (for vertical trabeculae) is given by s=4-L,ID, . Small values of s give rise to plastic type buckling . From our experimental observations, however, it is clear that vertebral trabecular bone is not a perfectly columnar structure . The individual trabeculae are not quite parallel to the vertical or
K . S . Jensen et al . : Model of vertebral trabecular bone mechanics
OJAVUldhow
wwraugg
S
∎ i
a = 0 .3
a = 0.6
a =
1 .0
Fig . 3. Vertical (upper row) and horizontal (lower row) sections of the three-dimensional cancellous bone model for various degrees of relative lattice disorder, a . Dimensions used in all three examples are : L, = 750 µm . L, = 980 pm, D, = 200 µm, and Dc = 120 gin . The trabecular thickness is not drawn in scale,
horizontal axes, which means that bending will also contribute to the overall deformation . In order to make the model more realistic without destroying its generality we therefore defined a measure of "relative lattice disorder", a, to quantify the degree of deviation from a perfectly rectangular lattice . Using a, we can generate an irregular lattice from a regular one by adding a term, aiLX, to each component of the three-dimensional lattice joint coordinates . X is a random number drawn from a uniform distribution on the interval from - 1/2 to + 1/2 . L = L, for the two horizontal axes, while L = Lz in the vertical direction . Fig . 3 shows the appearance of three lattices created with different values of a, but all with the same mean trabecular dimensions . At maximum irregularity (a = 1), the possibility exists that some of the rectangles are transformed into triangles . It should be noticed that a does not measure the contiguity of the lattice . In our model contiguity, in the sense used by, for example, Pugh et al . 1974, is a function of L, and L 2 . The definition of a is, of course, only one of many ways to quantify deviations from rectangularity . We would like to emphasize that our definition is tailored to "generate" irregularity (i .e ., it is a model-based definition) . Trying to make an exact estimate of a for a given sample of experimental material could very well be a meaningless task, because the definition contains the assumption of a rectangular lattice, the position of which may be impossible to determine uniquely for an actual trabecular bone specimen . It is crucial for the interpretation of the calculated results that we use a solely to get a more realistic description of the lattice geometry (which is important in order to find the distribution of compression and bending forces) . Any attempt to model other factors, such as discontinuities in the trabecular lattice, by adjusting a should be avoided . The proper way to treat a is thus to make an independent estimate of its value before the model is used to calculate the mechanical behaviour of the lattice . For our purpose we do not need best-fit values of a . By simply comparing photographs of sections of bone specimens (Mosekilde 1988) with computer generated lattices, we found that values between 0 .4 and 0 .8 gave a satisfactory representation of the typical trabecular geometry . The absolute size of the model (measured in number of unit cells) is of no importance for the results when a = 0 . For a > 0, however, our semi-random description of the irregularity can give rise to some variability of the calculated mechanical
419
properties when the lattice is not infinite . Therefore we chose a lattice of approximately the same overall dimensions as the tested bone specimens . In this way we could study whether or not the actual size of the tested structure significantly influenced the variability of the calculated or measured mechanical behaviour . For reasons of convenience, however, we used a box geometry instead of a cylinder (Fig . 3) . The model, put into a set of lattice equations (BernoulliEuler beam theory), was analyzed on a computer using a structural design program that enabled us to calculate the displacement, rotation, and reaction of any joint in the lattice . During the calculations, the lattice joints corresponding to the top and the bottom of the bone sample were displaced 1% of the total modeled sample height . The displacement was only specified on the axis of compression (i .e ., the top-bottom surface joints were allowed to move (and rotate) in the plane perpendicular to the compression axis) . We thus simulated a test machine where the friction between piston and bone sample is small compared with the internal forces generated by deformation of the trabeculae . The program also allowed us to find the forces, moments, and stresses for every beam element (trabecula) in the construction . A detailed analysis of trabecular collapse, however, requires knowledge about several material parameters, such as tensile, compressive, and flexural strength, and is beyond the scope of the present work . Such parameters have been experimentally determined for compact bone, but there is a wide variation in the measured values (Carter and Spengler 1978 ; Gibson 1985) .
Results Ignoring any age-related changes in solid bone material properties, we used the same size of Young's Modulus whether modeling a "young" or an "old" trabecular lattice . The main input parameters were therefore the trabecular diameters and spacings, to which we assigned measured mean values . In Fig . 4, two sets of trabecular dimensions were used, corresponding to a typical lattice for a 60-year-old person (Fig . 4a) and that for a person aged 80 years (Fig . 4b) . The values for L„ L2 , D„ and D, were derived from the experimental material previously described (Mosekilde 1989) . For Young's Modulus we used the value E = 11 .4 GPa as found by Townsend et al . 1975b, because this is the only determination we know of that has been done directly on solid trabecular bone material . The model, however, has one more degree of freedom : the degree of irregularity, a . As illustrated in Fig . 4, the mechanical behaviour is dramatically affected by a . It should be noted that the apparent stiffness in both vertical and horizontal directions decreases by a factor of 5 for the 60-year-old structure when a is increased from 0 to I (i .e ., when a less rectangular architecture is specified) . This is due to a change in mode of deformation from simple axial compression of the trabeculae to a mixture of compression and bending . As shown in Fig . 4b, the effect is even more pronounced for a less dense, but more anisotropic lattice . Here, the stiffness varies by a factor of 10 between a perfectly rectangular architecture and a lattice of maximum irregularity . The apparent, relative density of the modeled bone structure (equivalent to TBV) is also a function of a, but the dependency is much weaker than that seen for the stiffness, The density increases with a simply because the sum of trabecular lengths is slightly larger in an irregular lattice than in a rectangular one . The error bars in Fig . 4 indicate extreme values (not standard deviation) obtained from calculations on five separate lattices
420
K . S . Jensen et al . : Model of vertebral trabecular hone mechanics
(a) 5000 a 400-300-200-100-0-200-M
cu
150.09-0 100-50-0-0 .08-M
0 .07-
0,0
1 .0
0 .5
Rel . latt. disorder, a
I 0 .0
I 0 .5
1 .0
Rel . latt. disorder, a
Fig . 4 . Vertical and horizontal stiffness, and relative density (TBV), as calculated from the model with Young's Modulus E = 11 .4 GPa (a) : With trabecular dimensions as in Fig . 3, typical for a person of age 60 years . (b) : With L, = 1200 µm . L, = 1900 µm, D, = 200 µm, Dc = 80 µm (80-year-old person (Fig . 5b)) .
with the same a-values, but generated from different random sequences, X, as explained earlier . For small values of a, this scatter is less than the height of the solid square markers used in Fig . 4, The moderate variability suggests that the actual, applied size of our bone test specimens is unlikely to cause statistical fluctuations of a magnitude that would veil the age-related, qualitative differences in mechanical behaviour . The calculated stiffnesses displayed in Fig . 4 are considerably higher than found experimentally, except for a = 1 . From visual inspection of the experimental bone specimens, however . we found that the trabecular architecture in general is more regular than a lattice with a = 1 . This holds for the entire range 30 to 90 years . As previously mentioned, an a-value between 0 .4 and 0 .8 seems appropriate . We have not been able to observe any obvious difference in lattice disorder between bone samples of different ages that would justify a differentiation in the sense of a-values . In the
1 (a)
(b)
Fig. 5. Vertical section of model (a = 0 .6) showing a typical lattice for persons of age (a) 40 years, and (h) 80 years .
rest of the modeling studies we therefore used a = 0 .6 as the standard value (typical lattices are displayed in Fig . 5) . This gives rise to calculated stiffnesses about three times higher than normally seen, which may be due to an over-idealized architecture, wrong trabecular dimensions and/or too high a value for Young's Modulus . We believe, though, that the model reflects the actual trabecular structure to an extent that allows us to study the relative dependence of stiffness and anisotropy on trabeeular dimensions and lattice irregularity . In order not to cause confusion by showing unnaturally high stiffnesses . we chose to reduce Young's Modulus in all other calculations to E = 3 .8 GPa-one third the value used in Fig . 4 . Figure 6 displays values for stiffness and density for 40, 60, and 80-year-old individuals when modeled with experimental mean values for trabecular diameters and spacings, as shown in Table I . Also shown in Fig . 6 are the corresponding, expcrimentally found values for stiffness and density (Mosekilde et al . 1987 ; Mosekilde and Mosekilde 1988) . In general, the modeled TB V is smaller than the corresponding experimental value . This mass difference may originate from the idealized architecture, in which only whole trabecular structures without perforations have been taken into account . From the stiffness data we found the degree of mechanical anisotropy (vertical/horizontal stiffness ratio) as listed in Table II . It can be seen that both model and experimental data exhibit an increasing anisotropy as a function of age, although the increase is more pronounced for the model . Table III shows the results of fitting a power function to stiffness and TBV . As seen from the powers, the model gives a linear relation between density and stiffness in the vertical direction . In the horizontal direction, the relationship lies some-
K . S . Jensen et al . : Model of vertebral trabecular bone mechanics
421
Table I . Mean trabecular dimensions measured in "typical" persons aged approximately 40, 60, and 80 years . These data correspond to the results shown in Fig . 6 . (The data are derived from Mosekilde 1988) . Trabecular dimension (Am)
Lt
1-2
Dl
D2
Age (years) 40
720
770
210
150
60
870
1110
210
125
80
990
1450
210
95
where between quadratic and cubic . The higher power for the horizontal direction is also a typical feature of the experimental data, but here we find a quadratic relationship (instead of a linear) for the vertical direction . In this case, though, using linear regression gives just as good a fit as the power function . Including experimental data for 20-year-old persons (not shown here) does not alter the calculated powers significantly . The horizontal and vertical stiffness/density ratios . which determine the powers listed in Table III, are, however, very sensitive to changes in trabecular dimensions that cause a redistribution of the mass, as illustrated by Fig . 7 . Here we have modeled the stiffness in the two directions for three sets of trabecular dimensions, all of which give the same density . The three sets of diameters and spacings are all within the range of values that are representative of 60-year-old persons (the middle set was used for the results pictured in Fig . 3 and Fig . 4a) . It can be seen that the vertical/horizontal stiffness ratio changes from 1 .5 to 7 .8 when D, is increased from 180 ism to 220 Am for the chosen combinations of diameters and spacings .
the concomitant changes in trabecular architecture without the need for assuming age-related shifts in material properties (i .e ., Young's Modulus) . In the following paragraphs we discuss some of the quantitative discrepancies between model and experimental data . In order to obtain reasonable numbers for apparent stiffness, we had to reduce Young's Modulus to about one third the value determined by Townsend et al . 1975b . We have no obvious explanation for this,' but it should be noted that the authors themselves needed to multiply their values by one fifth when modeling the stiffness of the patella (Townsend et al . 1975a) . According to our findings from using various degrees of trabec-
Discussion Unlike some other published models (e .g ., Williams and Lewis 1982), the model architecture described here is not a copy of the exact trabecular geometry of a single bone test specimen, but an idealized archetype representing the vertebral trabecular bone structure over a span of years (from 40 to 80 years of age) . By scaling the trabecular dimensions for vertical and horizontal directions separately in accordance with experimental data . we have used the model to study the theoretical relationship between trabecular dimensions, relative bone density, and apparent stiffness-with special emphasis on anisotropy . In another theoretical study (Gibson 1985), the anisotropy of trabecular scaling was ignored, leading to different results for stiffness as a function of density . Taking into consideration the simplifications and idealizations made, it is only natural that our model cannot exactly minor the experimental data in every respect . However, qualitatively and, to some extent, quantitatively, it accounts for the observed relationships between changes in trabecular spacings and diameters, TBV, and stiffness in the vertical and horizontal directions, respectively . The model thus explains the age-related changes in mechanical behaviour as a natural consequence of
1 40
1
60
I 80 Age (year)
'After submission of the manuscript we became aware of a recent study by Kuhn et al . (1989) in which Young's Modulus for solid, trabecular bone from the iliac crest is found to be 3 .81 GPa . This is exactly the value we had to use in our model to get reasonable results .
Stiffness and relative density as functions of age . Dotted curves are experimental data . Solid and dashed curves are calculated from the model (E = 3 .8 GPa) with a = 0 .6 and a = 1 .0, respectively . The trabecular dimensions are listed in Table 1 . Fig. 6.
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K . S . Jensen et al . : Model of vertebral trabecular bone mechanics
Table II . Vertical/horizontal stiffness ratio calculated from data shown in Fig . 6 . (Experimental data are derived from Mosekilde et al . 1987 .)
Experimental data
Model a=0.6
Model a=1 .0
Age (years) 40
2 .8
1 .7
1 .5
60
4 .0
3 .0
2 .9
80
4 .5
6 .2
5 .7
ular lattice irregularity, this can to some extent, but not fully, be related to their assumption of pure axial compression of the trabecular plates . Similar comments apply to the modeling work by Williams and Lewis 1982 . They needed a value as low as E = 1 .3 GPa which, in their case, is most unlikely to be caused solely by ignoring the deformations by bending that will inevitably exist in an "almost perfect" columnar architecture . The fact that the calculated stiffness anisotropy increases more strongly with age than seen experimentally is possibly due to the way we modeled the age-related thinning of the trabecular network . Instead of removing entire trabeculae (at random) from the lattice, as done in the bone remodeling processes (Parfitt 1984, 1987), we simply scaled the grid size of the lattice while preserving the initially chosen architecture (six beams radiating from every lattice joint) . It is important to realize that the growing dispersion of trabecular spacings does not change the angle between trabeculae at the joints and therefore can not be described by increasing the a-value in our model . In general, the modeled TBV is smaller than found experimentally for the same trabecular dimensions . Part of the discrepancy could arise from the design of the model . In the experimental data, the extreme age-related increase in distance between trabeculae is partly caused by perforations of trabeculae and partly by the subsequent removal of whole trabeculae . In the model, however, all increases in inter-trabecular distances are deemed to be caused solely by increasing the space between existing trabeculae . In consequence, the trabecular bone mass is
relatively underestimated in the model . At this point we would like to mention that in all our model calculations on lattice compression, the highest absolute values of normal stress on the trabecular surfaces always appeared at the joints . This is in close agreement with the experimental observation of trabecular thickening near the joints : the remodeling processes will cause a sufficient amount of bone material to be placed here to equate stress and strain (Frost 1989) to the values found elsewhere on the trabecula . As argued earlier, a numerical analysis of the strength of the modeled lattice would force us into a number of assumptions about the size of various material properties . However, from the theoretical arguments given for open cell structures by Gibson (1985) . it is clear that strength depends on architecture and trabecular dimensions in much the same way as stiffness does . Osteoclastic and oseoblastic activity, locally governed by mechanical strain and stress (Frost 1983, 1987, 1989 ; Parfitt 1982; Lanyon 1981, 1984), determine the trabecular dimensions in both vertical and horizontal directions . This, of course, imposes some limitations on the variation in anisotropy of vertebral trabecular bone with the same TBV . There is still room, though, for a substantial spreading in stiffness and strength, as illustrated by some of the experimental data displayed by Carter and Hayes (1977) and Gibson (1985) . Taking into account the calculated high sensitivity of mechanical behaviour to changes in architectural anisotropy (Fig . 7), we therefore conclude that bone density measurements alone do not reveal
Table III . Best-fit-power, x, obtained by fitting corresponding values of stiffness (E.) and trahecular bone density (TBV) to the function Eo = C - (TBV)x .
Experimental data
Model a=0 .6
Model a=1 .0
Vertical direction
1 .8a
1 .0
1 .0
Horizontal direction
2 .8b
25
2 .7
'No significant difference between power function (x = 1 .8) and linear regression analysis . b Power function (x = 2.8) was significantly better than linear regression analysis .
K . S . Jensen et aL : Model of vertebral trabecular bone mechanics
80 -
vertical
60 -t
40 horizontal 20 -
0180
1 200
1 220
D, (Jim)
850 145 900
750 120 980
700 100 1200
L, (Jim) D, ()im) L, (pin)
Fig. 7. Calculated stiffness (using E = 3 .8 GPa) for three combinations of trabecular distances and diameters giving the same relative density of 0 .076 .
very much about biomechanical competence (i .e ., stiffness and strength) . This conclusion is also supported by our findings shown in Fig . 4 : When the architecture becomes more columnar (for unchanged TBV as well as mean trabecular dimensions) a much greater stiffness is obtained . This suggests that bone density measurements need to be combined with information about the architecture in order to give optimum indication of the mechanical condition of trabecular bone . The concept is important not only in relation to normal age-related changes but to an even greater extent in relation to osteoporosis and the evaluation of prophylactic and therapeutic modalities . Acknowledgments : The technical assistance of Inger Vang Magnussen, Lotte P . Kristensen, Aase Young, and Lotte Brohm is gratefully acknowledged, as is the help of the University Institute of Pathology, Aarhus County Hospital, and the Department of Forensic Medicine, Aarhus Kommunehospital, in providing the specimens . We are also thankful to Rasmus Feldberg and Caraten Knudsen for making the graphics software to generate Figs . 3 and 5 . The investigation was supported by grants from the Danish Medical Research Council (J .no . 12-6164 and J-no . 12-6836), Sygekassemes Helsefond (J .no H I1/ 218-88), L .F . Foght's Fund, NOVO's Fund, NOVO and Aarhus University Research Foundation (] .no . 1987-7131/01-6-LF 1-29) .
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porous structure . J. Bone Joint Surg . 59-A, 7 :954-962; 1977 . Caner, D .R . ; Spengler, D .M . Mechanical properties and composition of cortical bone. Clin . Onhop . Rel . Res. 135:192-197 ; 1978 . Frost, H .M . The skeletal intermediary organization . Merab . Bone Dis . Relar . Res. 4 :281-290 ; 1983 . Frost, H .M . The mechanostat : a proposed pathogenic mechanism of osteoporosis and the bone mass effects of mechanical and nonmechanica] agents . Bone and Mineral 2 :73-85 ; 1987 . Frost, H .M . Some ABCs of skeletal pathophysiology III : Bone balance and the B .BMU . Calcif. Tissue Int . 45 :131-133 ; 1989 . Gibson, L.I . The mechanical behaviour of cancellous hone . J. Biomech . 18, 5 :317-328 ; 1985 . Kleerekoper, M . ; Villanueva, A .R. : Stands . J . ; Rao, D .S . ; Parfait, A .M . The role of three-dimensional trabecular microswcture in the pathogenesis of vertebral compression fractures . Calcif. Tissue lot. 37:594-597 ; 1985 . Kuhn, LL . : Goldstein, S .A . ; Choi, K . : London, M . ; Feldkamp, L .A . ; Matthews, L .S . Comparison of the tmbecular and conical tissue moduli from human iliac crest . J . Orrhop . Res . 7 :876-884 ; 1989 . Lanyon, L .E . Bone remodelling, mechanical stress, and osteoporosis . DeLuca, H .F . : Frost, H .M . : Jee, W.S .S . : Johnston, C .C . Jr . ; Partiitr . A .M . . eds. Osteoporosis recent advances in pathogenesis and treatment . Baltimore : University Park Press ; 1981 :129, 138 . Lanyon, L .E . Functional strain as a determinant for bone remodeling . Calcif. Tissue lm . 36 :556-S61 ; 1984 . Melsen, F . ; Melsen, B . ; Mosekilde, Le . ; Bergmann, S . Histomorphometric analysis of normal bone from the iliac crest . Acra Parhol. Microbial . Scand. 86 :70-81 ; 1978. Mosekilde, Li . Age-related changes in vertebral trabecular bone architectureassessed by a new method . Bone 9:247-250; 1988 . Mosekilde, Lt. Sex differences in age-related loss of vertebral trabecular bone mass and structure-Biomechanical consequences . Bone 10 :425-432 ; 1989 . Mosekilde, Li . ; Mosekilde, Le . Iliac crest trabecular bone volume as predictor for vertebral compressive strength, ash density and trabecular bone volume in normal individuals . Bone 9 :195-199 ; 1988 . Mosekilde, U . ; Mosekilde . Le . ; Danielsen, C .C . Biomechanica] competence of vertebral trabecular bone in relation to ash density and age in normal individuals . Bone 8 :79-85 ; 1987 . Parfin, A .M . The coupling of bone formation to bone resorption : A critical analysis of the concept and of its relevance to the pathogenesis of osteoporosis . Memb . Bone Dis . Relor. Res . 4 :1-6 ; 1982 . Parfitt, A .M . Age-related structural changes in trabecular and conical bone : Cellular mechanisms and biomechanical consequences . Calcif. Tissue Inc. 36 :5123-5128 ; 1984. Partiu, A.M . Trabecular bone architecture in the pathogenesis and prevention of fracture . Am . J . Med. 82 :68-72 ; 1987 . Pugh, 1 . W. ; Radin, E.L . ; Rose, R .M . Quantitative studies of human subchondml cancellous bone . J. Bone Joint Surg . 56-A, 2 :313-321 ; 1974. Townsend, P .R . ; Raux P . ; Rose, R.M . ; Miegel, R.E . ; Radin, E .L . The distribution and anisotopy of the stiffness of cancellous bone in the human patella . J . Biome cit . 8:363-367 : 1975a. Townsend, P .R . : Rose, R .M . ; Radin . E .L . Buckling studies of single human trabeculae . J . Biomech . 8 :199-201 ; 1975b . Vesterby, A . : Kragstmp, J . ; Gundersen, H .J .G . ; Melsen, F . Unbiased stereologic estimation of surface density in bone using vertical sections . Bone 8 :13-17 : 1987 . Whitehouse, W.J . Cancellous bone in the anterior pan of the iliac crest. Colt . Tiss. Res. 23 :67-76 ; 1977 . Whitehouse, W .1 . ; Dyson, E .D. ; Jackson, C .K . The scanning electron microscope in studies of trabecular bone from a human vertebral body . J . Anat. 108 : 481-496; 1971 . Williams, 1 .L. ; Lewis, J .L . Properties and an anisotropic model of cancellous bone from the proximal tibial epiphysis . J. Biomech . Eng . 104 :50-56 ; 1982 .
References Arnold. J .5 . Trabecular pattern and shapes in aging and osteoporosis . Metab . Bone Div. et Rel . Res . 2:5297-5308 ; 1980. Carter, D .R . ; Hayes, W .C . The compressive behaviour of bone as a two-phase
Received: January 18, 1990 Revised : May 17, 1990 Accepted : May 29, 1990
