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Abstract

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MATERIALS AND METHODS
  5. RESULTS
  6. DISCUSSION
  7. Acknowledgements
  8. References

A finite-element model was used to explore the relationship between connectivity density and the elastic modulus of trabecular bone. Six cubic specimens of trabecular bone, three prepared from human distal radii and three from L1 vertebrae, were imaged with synchrotron microtomography. The three-dimensional images were reconstructed into binary volumes of mineralized bone and soft tissue, and incorporated into the finite-element model. The relationship between three-dimensional connectivity and elastic modulus was explored by uniform thinning (atrophy) and thickening (recovery) of the trabecular bone. Though no functional relationship was found between connectivity and elastic modulus, there was a linear relationship, after a full cycle of atrophy and recovery, between the loss of elastic modulus and the overall loss of connectivity. The results indicate that recovery of mechanical function depends on preserving or restoring trabecular connectivity.


INTRODUCTION

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MATERIALS AND METHODS
  5. RESULTS
  6. DISCUSSION
  7. Acknowledgements
  8. References

OSTEOPOROSIS IS A SIGNIFICANT public health problem. The World Health Organization defines osteoporosis as a bone mass 2.5 SD less than the mean established in young adults.1 It is known that fracture risk is strongly correlated with the loss of bone mass, but the majority of patients with low bone mass do not suffer from osteoporotic fractures. Even for a given bone mass, there is an age-related increase in fracture risk.2 This has led investigators to search for factors other than low bone mass, such as changes in geometric structure, accumulated microdamage, and differences in tissue properties, to help explain the occurrence of fracture in some osteoporotics and not in others.3,4 Much of this research has focused on trabecular bone.

Connectivity, the number of closed loops in the trabecular bone network, is a topological measure that attempts to relate biomechanical behavior with architecture. Although much has been written about the many two5 and three-dimensional6 methods for estimating the connectivity of trabecular bone, relatively few studies have attempted to relate connectivity to mechanical properties. One such attempt was reported by McCalden et al. where connectivity was observed to have little independent effect on the strength of trabecular bone.7 Despite this, connectivity is often reported in studies of bone loss. The purported ability to recover lost connectivity with anabolic agents is considered an important indicator of therapeutic efficacy.8

The purpose of the present study was to determine whether or not there is a functional relationship between connectivity and elastic modulus in trabecular bone. The study used finite-element simulations based on high-resolution three-dimensional images of trabecular bone from human distal radii and vertebrae. The connectivity scaling was explored under the most ideal conditions—uniform thinning (atrophy) and thickening (recovery) of the trabecular bone. The atrophy and recovery were performed by two different methods. In the first method, trabecular bone was added and removed uniformly from all surfaces without regard to connectivity. In the second method, bone was added or removed subject to the constraint that the connectivity of the network was maintained. In this manner, we were able to separate the usually coupled effects of mass and connectivity on the elastic properties of trabecular bone.

MATERIALS AND METHODS

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MATERIALS AND METHODS
  5. RESULTS
  6. DISCUSSION
  7. Acknowledgements
  8. References

Cubic specimens of trabecular bone were prepared from human distal radii (specimens DR1–DR3) and vertebrae (specimens V1–V3). The specimens were cut under irrigation with a precision diamond saw. The distal radii were cut to a dimension of 12 mm and the L1 vertebrae were cut from the superior–anterior region to a dimension of 8 mm. The orientations of the cubes with respect to the principal anatomic axes were noted. The radii specimens were obtained from adults of unknown age and gender; the vertebrae were from males aged 63–80 years.

After preparation, the specimens were placed in distilled water and imaged with synchrotron radiation. The radiation was made monochromatic at 25 keV with two silicon single crystals whose faces were cut parallel to the (220) diffraction planes. Tomography was performed with the X-ray tomographic microscope according to a protocol described in detail elsewhere.9 Two-dimensional radiographs of the specimens were obtained at half-degree rotational increments, and the data were reconstructed into three-dimensional images by a Fourier-filtered back-projection technique. Although the maximum spatial resolution of the instrument is 2 μm, for the purposes of finite-element modeling the data were reconstructed into cubic volume elements (voxels) about 20 μm on edge. The exact voxel dimensions were 17.7 and 23.4 μm for the radii and vertebrae, respectively.

Once reconstructed, ∼3.5 and 4.5 mm cubic subvolumes (8 million voxels) were extracted from the complete volumetric images of the radii and vertebrae, respectively. These subvolumes, which were used for all subsequent calculations, were segmented into voxels containing bone and voxels containing marrow. A single threshold value was used in this segmentation: a voxel was defined as bone if half or more of the voxel was occupied by bone and defined as marrow if half or more of the voxel was occupied by marrow. The segmented volumes were then cluster analyzed to remove any bone that was not continuously connected between the loading surfaces. Details of the cluster analysis method can be found elsewhere.10

Connectivity scaling was explored by thinning and thickening the trabecular bone in each volume set. Surface elements were removed one voxel at a time (∼20 μm). In the first method, all surface elements were identified and then either removed (atrophy) or added to (recovery) without any regard to the connectivity of the trabecular network. In this manner, trabecular connections could be formed or removed, and plates could be fenestrated or filled. In the second method, the surface elements were removed or added subject to the constraint that the connectivity of the trabecular lattice did not change. Hence, with this method mass was added or removed without breaking or adding connections or fenestrating trabecular plates. Each specimen was thinned three times (with the exception of DR3, which, because of its much higher initial trabecular bone volume [TBV], could be thinned four times) with each method, providing of total of 50 data vol. An additional 72 data vol were prepared by thickening each specimen five times from its fully atrophied condition. Finally, the original unthinned volumes were thickened twice in order to establish connectivity and modulus values that could be used in interpolating the results. In all, a total of 146 data vol were prepared for finite-element analysis.

The connectivity, β1, was calculated from the relationship

  • equation image(1)

β0, β1, and β2, are the Betti numbers, and X is the Euler–Poincare characteristic of the network. The Betti numbers are topological parameters that are invariant with respect to a homeomorphic transformation of the structure. β0 quantifies the number of separate (not connected) trabeculae in the image, β1 quantifies the number of handles or closed loops in the trabecular network (connectivity), and β2 quantifies the number of encapsulated voids within the trabecular tissue. β0 and β2 were found with a cluster labeling algorithm. Because the calculations were performed on a single, continuously connected trabecular network, β0 was always one. In the bone that we examined, there was never evidence of encapsulated voids within the trabecular tissue, and β2 was always zero. Hence, the connectivity depended only on the Euler characteristic, which was determined by applying Feldkamp's algorithm.11

Recovery of connectivity was estimated from the difference in connectivity at the beginning and end of a cycle of atrophy and recovery. The initial connectivity density was determined, by interpolation, at a reference density of 20% TBV by thickening from the original bone volume. Upon recovery, the connectivity density was redetermined at the same value of TBV, and the connectivity loss was defined as the percentage difference between these two values. The value of 20% TBV was selected as a reference because it represented a plateau region in the connectivity density curves and appeared less sensitive to interpolation errors.

Two-dimensional histomorphometry parameters were obtained from a single section of each specimen, cut parallel with the anterior face of the cubes. Trabecular bone area and perimeter were measured, and trabecular separation, width, and number were calculated with the plate model.12 Marrow star volumes were obtained from three unbiased sections for comparison with three-dimensional connectivity values.13 One hundred randomly selected points from each slice were used for the star volume calculations.

The prepared volumes were directly incorporated into a finite-element model to determine the elastic properties of the trabecular networks using a method described by Hollister and Riemer.14 We used a finite element derived from a spring network model that we have described elsewhere.15,16 The advantage of this element over a standard cubic element, which uses an exact integration of the strain energy,17 is an order of magnitude increase in computational efficiency. For the present calculations, the boundary conditions of the load surfaces assumed a fixed grip (no transverse displacements on the load surfaces) and surface displacements of order 10−6. The local elastic stiffness was assumed to be constant in these calculations.

An apparent Young's modulus that was normalized to the modulus of the fully dense tissue, Ei/Eo, was calculated for the inferior–superior direction. Because of the increased efficiency afforded by the 18-spring element, the calculations could be performed on conventional work stations. The number of elements in each data set ranged from 150,000 elements for the thinned volumes to more than 2 million elements for the data sets with thickened trabeculae. The calculations were distributed on available processors that included DEC Alpha (Digital Equipment Corp.) and Sun UltraSparc workstations (SUN Microsystems, Mountain View, CA, U.S.A.).

RESULTS

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MATERIALS AND METHODS
  5. RESULTS
  6. DISCUSSION
  7. Acknowledgements
  8. References

Three-dimensional tomographic reconstructions of a subvolume from each specimens are shown in Fig. 1. All trabecular networks gave the appearance of having fenestrated plate-like structures interconnected by rod-like struts, although there was considerable variation in geometric structure among specimens. Figures 2A and 2B are reconstructed images of a single vertebral specimen that has been thinned in a single pass with the connectivity preserving algorithm (Fig. 2A) and with the algorithm that does not conserve connectivity (Fig. 2B). Close inspection of Fig. 2B shows that even with a single thinning operation there are several microfenestrations of the trabecular plates. These microfenestrations caused an increase in the connectivity density of the specimen with the first few thinning passes.

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Figure FIG. 1. Subsets of the three-dimensional X-ray microtomography images used in the finite element calculations. Specimens DR1–DR3 and V1–V3 were prepared from the distal radius and the L1 vertebra, respectively. The orientation is such that the inferior–superior axis is approximately top to bottom, and the lateral axis is upper left to lower right in the figures.

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Figure FIG. 2. Three-dimensional images of specimen V2 after thinning with the connectivity preserving algorithm (A) and the nonconserving algorithm (B). Fine scale fenestrations, such as shown in the circled region of (B), caused the increase in the connectivity density seen with thinning by the nonconserving method.

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Histomorphometric parameters of the original trabecular architectures are provided in Table 1 for each of the specimens used in this study. Trabecular bone volume ranged from ∼7–17% in the radii and from 6–10% in the vertebrae. Trabecular width was similar in all specimens, ranging from 100–120 μm. Trabecular number was much higher in the radii, corresponding to a closer spacing of the trabeculae. Trabecular connectivity varied by as much as a factor of two within each anatomic site, with the distal radii having a higher average connectivity density than the vertebrae.

Table Table 1. Listing of Specimens with Initial Values of Histomorphometry Parameters
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Without connectivity preservation, the atrophy model led to an increase in connectivity with decreasing trabecular density, as shown in Fig. 3. The increase in connectivity was caused by fenestrations of the trabecular plates; however, trabecular rods were being severed at the same time, but in smaller numbers. Upon recovery, the fenestrations were removed, but the severed rods were not reconnected. This led to a lower connectivity in the recovered structure (see Fig. 4). Because of inherent differences in specimen morphology, there was a wide variation in the amount of connectivity loss, from a few percentage points to as much as 60% loss in connectivity was observed.

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Figure FIG. 3. Connectivity density as a function of trabecular bone volume for all six specimens. The starting density is indicated by the open circle with the exception of DR3, where all of the points are labeled. The increase in connectivity density with thinning is a result of plate fenestration. One specimen, DR3, could be thinned enough such that the connectivity began to decrease.

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Figure FIG. 4. The loss of connectivity upon regrowth of trabecular bone. The connectivity loss was calculated as the difference in connectivity between the original network and the regrown network in the plateau region, taken at 20% trabecular bone volume.

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Both thinning methods led to a decrease in elastic modulus with trabecular bone density. There was no correlation between connectivity number and the scaling of the modulus with density. However, in the samples that did not recover lost connectivity, the original modulus for the equivalent trabecular bone density was also not recovered. There was a linear relationship between the amount of connectivity loss, in a full cycle of atrophy and recovery, and the reduction in elastic modulus (R2 = 0.96; p < 0.001), which is illustrated in Fig. 5, where the percentage loss of elastic modulus is graphed as a function of connectivity loss. Also graphed, but not used in the regression, are recovery values for the case where connectivity was preserved with atrophy. When there was no loss in connectivity, the elastic modulus recovered from 92–100% of its original value.

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Figure FIG. 5. The loss of elastic modulus as a function of the loss of connectivity upon recovery to the original trabecular bone volume. The x symbols represent the case of atrophy without breaking connections, hence, connectivity was never lost. When connectivity was preserved, between 92 and 100% of the elastic modulus was recovered.

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DISCUSSION

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MATERIALS AND METHODS
  5. RESULTS
  6. DISCUSSION
  7. Acknowledgements
  8. References

In this study, a finite-element model was used to calculate variations in the elastic modulus of trabecular bone during a simulated atrophy and regrowth process. From previous investigations, we have demonstrated that the finite element model can accurately calculate the elastic modulus of a given trabecular bone architecture, at least within the experimental errors of mechanical testing.15,16,18 The finite-element method, coupled with our idealized model of bone atrophy and regrowth, allowed us to separate the effects of connectivity from those of variable trabecular mass and orientation. In addition, because the load surface boundary conditions were never changed, the often serious experimental errors associated with load platen variations were avoided.19,20

Our results showed that there is no functional relationship between connectivity and elastic modulus. This can be explained by the fact that connectivity is independent of the contact area, whereas mechanical load transfer depends on the contact area. However, a more important reason for the apparent lack of a relationship between modulus and connectivity is that a global measure of connectivity does not discriminate between rod-like connections and fenestrated plates. This ambiguity is illustrated in Fig. 6. At the top of the figure is a trabecular plate with no fenestrations. The connectivity, β1, is zero. Moving to the lower left, it can be seen that a fenestration has increased the connectivity but reduced the strength. With the further transformation to the right, the severing of the strut creates a decrease in connectivity and a further decrease in strength. There is no difference in connectivity between this structure and the original, even though the transformed plate is significantly weaker. Since fenestration and severing can occur simultaneously, it is possible that significant changes can occur to the geometric structure and mechanical properties of the trabecular bone lattice without any change in the connectivity number.

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Figure FIG. 6. The solid plate (upper structure) has the highest strength. Connectivity, as defined by the number of closed loops or handles, was increased by adding a fenestration (moving to lower left), resulting in lower strength. But decreasing the connectivity also decreased the strength (moving from left to right). Hence, there is no single valued function that relates connectivity to strength.

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Although absolute measures of connectivity are not directly related to mechanical strength, there is an important connection between changes in connectivity and changes in elastic stiffness: specimens that do not recover connectivity do not recover the full elastic modulus for the equivalent trabecular bone volume. We observed a linear proportionality between connectivity loss and modulus loss that was independent of trabecular bone volume. Specimens thinned to the same degree of bone atrophy (loss of trabecular bone density), but preserving connectivity, recovered most if not all of their original modulus. This relationship between connectivity and modulus is not an obvious one and has perhaps escaped attention because of the impossibility of determining the unrecovered connectivity from histologic examinations at a single time point, coupled with the fact that histology can never be performed on the same specimen used in mechanical testing.

A complete cycle of bone atrophy and regrowth (see Fig. 4) led to an overall loss of connectivity as a result of failure to re-establish severed trabecular connections. A two-dimensional example of this irreversibility is provided in Fig. 7. The original structure, which consisted of two plates connected by a single rod, was thinned without conserving connectivity. After six thinning passes, the connecting strut disappeared, leaving behind two thinned plates. This thinned structure was then thickened in the same manner. The plates recovered to their original width, but the connection was never re-established. This operation, erosion followed by dilation, is a morphological operation known as opening.21 In more realistic architectures, this observation implies that the connectivity increase due to plate fenestration is reversed, but trabecular breaking is not. Thus, the measure of connectivity loss in an atrophy–regrowth cycle reflects mostly trabecular breakage.

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Figure FIG. 7. A two-dimensional example of how thinning can lead to irreversible loss in connectivity in a simple atrophy model. Once the center span is severed, it cannot be regrown by reversing the process.

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An inability to reform lost trabecular connections has also been observed in two-dimensional simulations of bone remodeling, indicating that bone resorption and formation might lead to irreversible connectivity loss.22 However, this observation has not been convincingly demonstrated in vivo, where contradictory observations have been reported. In the work of Shen et al.8 connectivity was restored by combined parathyroid hormone (PTH) and estradiol therapy, whereas in the work of Lane et al.,23 connectivity was not restored with the application of PTH therapy alone. In the latter study, PTH therapy was withheld for 8 weeks post-ovariectomy; whereas, in the work of Shen et al., treatment was started at 3 and 5 weeks post-ovariectomy. It is possible that less irreversible bone loss, such as smaller plate fenestrations and less rod severing, occurred in Shen's experiments.

Initially, the trabecular connectivity in all six specimens increased greatly with thinning because, for uniform surface removal of trabecular tissue, plate fenestration overwhelmed rod severing as the mechanism of architectural change. Others have speculated that connectivity should also increase in the early stages of hypoestrogenemic bone loss and that the failure to observe this increase in small animal models was because bone morphometry had not been performed at early enough time points.24 Recently, Dempster et al. detected a net connectivity decrease in rat tibias within 5 days post-ovariectomy.25 A similar result was found by Lane et al. in a three-dimensional in vivo study of the early stages of bone architectural changes in the rat.26 These studies, performed by entirely different methods, established that trabecular bone begins to loose connectivity immediately following ovariectomy, indicating that rod severing may be the primary cause of early trabecular deterioration in hypoestrogenemia.

Our study has highlighted a major weakness in using topological connectivity as a measure of trabecular bone architecture, namely, even when measured in three dimensions, connectivity cannot distinguish plate fenestrations from rod severing. Because of the possible seriousness of connectivity loss, we believe that efforts should be made to develop new morphometry procedures that can better distinguish between irreversible rod severing and reversible plate fenestration. In addition, evaluations of the effects of anabolic agents on the mechanical properties of trabecular bone should attempt to correlate the recovery of connectivity density with the recovery of mechanical strength at the same anatomic location.

Meng et al. recently published results of an experiment in which ovariectomized rats were treated with PTH(1–34).27 Histology of the proximal tibia indicated that trabecular number, an indicator of trabecular connectivity, was not recovered in the ovariectomized animals. Trabecular bone volume, however, was recovered due largely to thickening of the remaining trabecular plates. Mechanical tests on the distal femur indicated that strength increased greatly in the PTH treated animals, even though connectivity in the tibia remained unrecovered. Based on this single experimental observation, Dempster speculated that recovery of connectivity may not be necessary to restore bone strength.28

There are several difficulties with Dempster's argument. First, histology was performed on a different site from that used to measure mechanical strength. It was not demonstrated that changes in connectivity within the proximal tibia reflect those in the distal femur. Second, measurement of strength on small specimens of trabeculae are difficult at best, and differences in load platen boundary conditions between thick trabeculae and thin trabeculae could lead to sizeable errors. Nevertheless, Dempster's argument that trabecular plate thickening increases bone strength irrespective of connectivity may be valid. In our study, trabecular plates were not regrown to a thickness greater than the original, and the absence of interconnecting rods may have lowered the ability of the trabecular bone to resist compression. In the absence of interconnecting rods, thickening of the trabeculae might help increase the resistance to compression out of proportion to the gain in mass.

It must be emphasized that our computer model of atrophy and recovery does not simulate adaptive bone remodeling, where loss and recovery from trabecular surfaces is known to be nonuniform. Nevertheless, we believe that our observations may have important implications for the management of osteoporosis. The key point is that connectivity loss during a bone atrophy–bone regrowth cycle primarily reflects trabecular breakage and is, therefore, directly related to loss of mechanical stiffness and strength. It remains to be seen if two-dimensional measures of changes in connectivity can distinguish between trabecular breakage and plate fenestration. Since irreversible connectivity reduction is one of the earliest manifestations of estrogen loss, our observations support early intervention to prevent possibly irreversible deterioration of trabecular architecture after menopause. Further study is warranted.

Acknowledgements

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MATERIALS AND METHODS
  5. RESULTS
  6. DISCUSSION
  7. Acknowledgements
  8. References

The authors acknowledge the help of A. Geiss for preparation of the cubes from the distal radii and S.A. Goldstein for providing the specimens from the vertebrae. We also thank A. Odgaard for suggesting a simple version of a connectivity preserving atrophy algorithm and for encouraging us to compare the two methods of thinning and thickening. The authors would also like to thank J.J. Kaufman for technical discussions, T.M. Breunig for help with specimen imaging, and D.L. Haupt with reconstructions. This work was performed under the auspices of the U.S. Department of Energy through contract W–7405–ENG48 with the University of California, Lawrence Livermore National Laboratory.

References

  1. Top of page
  2. Abstract
  3. INTRODUCTION
  4. MATERIALS AND METHODS
  5. RESULTS
  6. DISCUSSION
  7. Acknowledgements
  8. References
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