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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

Quantitative information about bone tissue-level loading is essential for understanding bone mechanical behavior. We made microfinite element models of a healthy and osteoporotic human femur and found that tissue-level strains in the osteoporotic femoral head were 70% higher on average and less uniformly distributed than those in the healthy one.

Introduction: Bone tissue stresses and strains in healthy load-adapted trabecular architectures should be distributed rather evenly, because no bone tissue is expected to be overloaded or unused. In this study, we evaluate this paradigm with the use of microfinite element (μFE) analyses to calculate tissue-level stresses and strains for the human femur. Our objectives were to quantify the strain distribution in the healthy femur, to investigate to what extent this distribution is affected by osteoporosis, to determine if osteoporotic bone is simply bone adapted to lower load levels, and to determine the “safety factor” for trabecular bone.

Materials and Methods: μFE models of a healthy and osteoporotic proximal femur were made from microcomputed tomography images. The models consisted of over 96 and 71 million elements for the healthy and osteoporotic femur, respectively, and represented their internal and external morphology in detail. Stresses and strains were calculated for each element and their distributions were calculated for a volume of interest (VOI) of trabecular bone in the femoral head.

Results: The average tissue-level principal strain magnitude in the healthy VOI was 304 ± 185 microstrains and that in the osteoporotic VOI was 520 ± 355 microstrains. Calculated safety factors were 8.6 for the healthy and 4.9 for the osteoporotic femurs. After reducing the force applied to the osteoporotic model to 59%, the average strain compared with that of the healthy femur, but the SD was larger (208 microstrains).

Conclusions: Strain magnitudes in the osteoporotic bone were much higher and less uniformly distributed than those in the healthy one. After simulated joint-load reduction, strain magnitudes in the osteoporotic femur were very similar to those in the healthy one, but their distribution is still wider and thus less favorable.


INTRODUCTION

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

Already over a century ago, the anatomist Meyer(1) and the engineer Culmann(2) discovered a remarkable similarity between the trabecular architecture of the proximal femur and the patterns of stress trajectories, calculated with the new theory of “Graphical Statics,” developed by Culmann.(2) On these results, Wolff(3) based his “trajectorial theory,” the match of trabecular morphology and stress trajectories and the essence of Wolff's law. Accordingly, trabecular architecture is assumed to minimize both bone stress and weight. This paradigm at least suggests that, normally, stresses and strains should be distributed rather evenly over the trabecular architecture.

So far, however, there have been no possibilities for a quantitative evaluation of this paradigm. It has been argued that the premise of Wolff's law is a false one and that the correspondence between trabecular architecture and stress trajectories is just an optical illusion.(4) The “Graphical Statics” technique for analysis of stress transfer in solids is based on requirements for continuity of the material. The results of these analyses only tell us about the courses of the stress trajectories if the bone would be made out of a single homogeneous and isotropic material with no trabecular architecture. It has been argued as well that the similarity between trabecular orientation and stress trajectories is circumstantial rather than causal and that there are no mathematical rules for bone architecture.(5) Over the last decades, far more advanced computational models, as based on finite element (FE) analysis, were developed to calculate stresses and strains in complex three-dimensional (3D) structures. Where analyses of bones are concerned, however, this approach still suffers from the same limitations: bone material can only be represented as a homogenized continuum. Although such methods can account for the local porosity and anisotropy of the material, they can only provide stresses and strains at the homogenized level and not those in individual trabeculae.

Because of the lack of stress and strain information at this small level, it is not known if, and to what extent, even distributions are possible for the actual tissue stresses and strains, or whether this is only possible for the average tissues stresses and strains over a loading cycle. Nor is it known to what extent a potential even distribution of tissue stresses and strains is affected by structural changes caused by osteoporosis and what the “safety factor” of the bone is for changes in loads.

In this study, we aim at finding answers to these questions by calculating bone tissue stresses and strain in the femoral head of a healthy and osteoporotic femur using high-resolution FE models that can represent individual trabeculae in the proximal end of the femur. With this technique, high-resolution sequential images obtained from, for example, microcomputed tomography (μCT) scanners are used as a basis for the geometry of a 3D μFE model that can represent the trabecular structure in detail.(6) Because models generated in this way will in generally consist of a very large number of elements (on the order of 105-106/cm3), special iterative FE solvers and supercomputers are generally required.(7,8) To date, this technique has only been used to analyze test samples of bone (loaded in a non-physiological way) and small animal bones because of limitations on CT scanning volume and computer resources. For the calculation of physiological tissue loading, natural boundary conditions must be applied, which is possible only when μFE models can represent whole bones. With recently developed μCT scanners for large pieces of bone and new parallel computers with many gigabytes of memory, this is now possible. The feasibility of this μFE approach to calculate tissue level stresses and strains in human bones with this new hardware is explored here.

Specific purposes of this study are to quantify and compare strain distributions in the healthy and osteoporotic femoral head, to investigate to what extent an even strain distribution can be obtained and to what extent this distribution is affected by osteoporosis, to determine if osteoporotic bone is simply bone adapted to lower load levels, and to determine the safety factor for 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

Bone mineral density (BMD) values of the femoral neck were measured (DPX-L, DXA scanner; Lunar) in situ in 80 elderly cadavers from an anatomic dissection course. Based on these measurements, a healthy femur (T-score: −0.5) and a severely osteoporotic (T-score: −4.0) femur were selected, with closely matched age (healthy, 82 years; osteoporotic, 89 years), body weight (healthy, 63 kg; osteoporotic, 57 kg), length (healthy, 1.60 m; osteoporotic, 1.61 m), and femoral head diameter (45 mm for both femurs) of the female donors.

Sequential cross-sectional images of the proximal 10 cm of these femurs were made using a μCT scanner (μCT80; Scanco, Bassersdorf, Switzerland) with an 80-mm field-of-view. The resolution of the images was chosen as 80 μm so that the entire volume could be captured with images of 1024 × 1024 pixels in size. A total of 1154 and 1152 images were made for the healthy and osteoporotic femur, respectively, covering a length of approximately 92 mm of each femur. The slice distance was chosen the same as the pixel size at 80 μm. The total time needed for imaging and reconstruction was about 200 s per cross-section, resulting in a total scan time of approximately 3 days for each femur. After reconstruction of the bone in a voxel grid, a modest Gauss filtering algorithm followed by a segmentation algorithm was applied to reduce the noise and to extract the bone tissue, respectively. The resulting 3D reconstructions of the healthy and osteoporotic proximal femur represented the internal and external architecture of the bone in great detail (Fig. 1). By comparing simulated radiographs made from the computer reconstruction data (Fig. 2) with real radiographs of the bone, it was concluded that all relevant features of the bone as seen on the real radiographs were clearly recognizable in the simulated radiographs.

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Figure FIG. 1.. μFE models of the (A) healthy and (B) osteoporotic femurs. In the larger images, only the half-models are shown to show the trabecular bone regions. Insets: rendered image of the full model.

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Figure FIG. 2.. Simulated radiographs of the FE models of the (A) healthy and (B) osteoporotic femurs. Grayscale represents the density in each pixel of the projection image. The same scale is used for both images. The circles drawn in the femoral heads indicate the outer dimension of the spherical VOI in each femoral head.

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Bone voxels were converted to equally sized brick elements, resulting in μFE models with over 96.8 and 71.6 million elements and 130 and 100 million nodes for the healthy and osteoporotic femurs, respectively. In each femur, cortical and cancellous bone regions were identified based on the number of bone voxels in the neighborhood of each voxel. Elements were assigned a stiffness of 22.5 GPa for cortical and 15 GPa for trabecular bone tissue.(9,10) A physiological distributed load was applied normal to the femoral head surfaces. The radius and the center of the femoral head sphere were found by calculating the radius and center of the circle that best fits the contour of the femoral head, as seen in the simulated radiographs in three spatial directions.

The direction and magnitude of the resultant external force acting on the femoral head were chosen to represent the stance phase of walking and were obtained from in vivo telemetry measurement.(11) The force magnitude was 234% of body weight (BW), corresponding to 1446N and 1308N for the healthy and osteoporotic femurs, respectively. The force was distributed normal to the surface of the femoral head with its magnitude following a cosine function of the polar angle as obtained from the telemetry data. The distal ends of both models were fully constrained.

The μFE analyses were performed with an iterative element-by-element solver that was optimized for parallel processing.(7,8) Thirty processors of an SG/Cray Origin2000 computer, with a total of 128 processors and 57 GB of memory, were used for the calculations. Memory requirements were 17 GB for the healthy femur and 13 GB for the osteoporotic one. About 30,000 iterations were needed to obtain sufficiently converged results.(12) Total cpu time for solving the healthy femur model was about 25.000 h, and total wall-clock time was approximately 6 weeks. For the osteoporotic model, these figures were approximately 25% less.

Principal stresses and strains were calculated for each element in the model, and the component of the tissue principal strain with the largest magnitude was represented in 3D contour plots. Further analyses of tissues stresses and strains in the trabecular bone tissue concentrated on a spherical volume of interest (VOI) located within the femoral head (Fig. 2). This sphere, with a radius of 16 mm, comprises most of the trabecular bone in the femoral head of both femurs. The number of elements in this sphere was 15.3 and 8.4 million for the healthy and osteoporotic model, respectively. From these numbers, it was calculated that the trabecular bone volume fraction was 0.46 for the VOI in the healthy femur model and 0.25 for the VOI in the osteoporotic model, which clearly shows the large difference in bone density in this region. Histogram plots were created for the component of the principal strain with the largest magnitude, further indicated as ±|ϵmath image|max, with the sign dependent on the sign of ϵmath image and i = 1, 2, or 3, representing the three components of the principal strain. The same indicators were used for the component of the principal stress with the largest magnitude, further indicated as ±|σmath image|max, for the strain energy density (SED), for the Von Mises equivalent stress, indicated as σVM, and for the maximal principal strain magnitude, indicated as |ϵmath image|max.

To check whether the osteoporotic architecture is adapted to reduced load, we investigated if the strain distribution in the osteoporotic bone would be similar to that in healthy bone for a reduced hip-joint load. For this purpose, the stress and strain calculations for the osteoporotic bone were redone after scaling the external load such that the average value of the principal strain magnitude in the VOIs was the same in the healthy and osteoporotic case. Because these are linear FE analyses, the scaling of external loading conditions can be done without performing additional FE analyses. The calculated displacements and forces could simply be scaled by the same factor, and the indices for the reduced hip-joint load were obtained by recalculation of the stress and strain values.

Finally, to determine a safety factor for bone in the femoral head, we calculated the magnitude of the external load for which local bone failure would occur. For this purpose, we used a failure criterion that was developed in an earlier study of experimentally failure prediction in the human radius.(13) A good prediction of bone failure could be obtained from linear FE analyses, assuming that bone failure would occur for an external force resulting in a tissue strain distribution with 2% of the bone tissue loaded beyond 7000 microstrains. In this study, we calculated the safety factor as the number by which the external load magnitude must be multiplied to reach this tissue-level strain criterion.

RESULTS

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

The contour plot for the component of the tissue principal strain with the largest magnitude ±|ϵmath image|max in the healthy femur (Fig. 3A) shows that compression is found for trabeculae running from the femoral head to the medial cortex and tension for trabeculae in the perpendicular direction. A similar loading pattern can be observed for the osteoporotic femur (Fig. 3B), although the strains in this bone are generally higher.

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Figure FIG. 3.. Calculated distribution of the principal strain component with the largest magnitude in the (A) healthy and the (B) osteoporotic femurs for a physiological joint force representing the stance-phase of walking. Yellow-to-red represent increasing compressive strains; blue-to-green, increasing tensile strains.

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The average tissue-level principal strain magnitude |ϵmath image|max in the healthy VOI was 304 microstrains and that for the osteoporotic VOI was 520 microstrains. Hence, to have the same average value in the healthy and osteoporotic VOIs, the magnitude of the external load working on the osteoporotic femur had to be reduced to 767N or 59% of its original value.

Histograms of the bone tissue level loading distribution as represented by the parameters ±|ϵmath image|max, ±|σmath image|max, SED, σVM, and |ϵmath image|max in the VOI (Fig. 4) clearly show the differences between the healthy and osteoporotic femoral heads. For the osteoporotic VOI, the distribution curves are much wider, indicating that more material is subjected to high stress than for the healthy VOI. After downscaling the load to 59% of its original value (such that the average of |ϵmath image|max is the same as in the healthy situation), however, the curves for the osteoporotic bone and those for the healthy bone closely match for all investigated parameters.

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Figure FIG. 4.. Histograms of the (A) maximum principal strain, (B) the maximum principal stress, (C) strain energy density, (D) von Mises equivalent stress, and (E) maximum principal strain magnitude for the VOI in the femoral heads. Thick white lines represent results for the healthy femur, thick black lines for the osteoporotic femur, and narrow black lines for the osteoporotic femur after reducing the joint force until the average strain was the same as in the healthy case.

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From the average values of the parameters investigated and their SDs (Table 1), it was calculated that, on average, the stress and strain components in the osteoporotic VOI are increased by 57–71% relative to the healthy one. The increase in the SED was much larger at 208%. After reducing the magnitude of the external load applied to the osteoporotic femur to 59%, differences between the average values in the healthy and osteoporotic VOI were less than 8% for any of the parameters investigated. Nevertheless, the SDs of the parameters calculated for the osteoporotic bone were larger in all cases (by 12–25%) than in the healthy VOI. Even for parameter |ϵmath image|max, which has exactly the same average value as in the healthy situation because its value was used as the reference for the load scaling, the SD was 13% higher than in the healthy case. The safety factor calculated for the healthy femur was 8.62. This means that the external load magnitude of 1446N should be multiplied with a factor of 8.62, corresponding to a force magnitude of 12.5 kN, to reach a state where the tissue level strains exceed 7000 microstrains for 2% of the bone tissue. For the osteoporotic femur, this factor was much less at 4.9, corresponding to a force magnitude of 6.4 kN. For the osteoporotic femur with reduced loading, this factor was only slightly less than that for the healthy case at 7.94. The corresponding magnitude of the load, however, was the same as for the osteoporotic bone; the scaling factor is increased only because the magnitude of the external load was reduced in this case.

Table Table 1. Statistical Description of the Distributions Found for the Investigated Parameters
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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

One of the purposes of this study was to investigate to what extent an even distribution of tissue-level strains can be expected for a single external load case. From the results obtained, it is clear that there is considerable variation in the tissue strains for the most common and largest external load (the stance phase of walking) working on the femoral head. Obviously, a perfectly even distribution cannot be expected for one single load case because the femur is subjected to varying loading magnitudes and directions during normal daily activities. The actual trabecular structure that is caused by load adaptive remodeling should reflect all these different loading configurations in a weighted manner. In this light, the distribution found in this study is surprisingly small. Less than 2% of the tissue is loaded below 50 microstrains, and only 0.0039% of the tissue is loaded beyond 3000 microstrains.

The tissue-level stress and strain distributions calculated in this study compare very well to those obtained in an earlier study of a beagle femur.(14) The average strain magnitude calculated in this study (304 microstrains) is only 8.9% higher than that calculated for the beagle (279 microstrains). The average Von Mises stress and strain energy density found in the present study were 4.66 MPa and 1011 Jm−3, respectively, which are almost the same as in the canine study (4.60 MPa and 1020 Jm−3, respectively). It is interesting to note that the SDs for the strain magnitude in the canine femur (212 microstrains) are higher than those found in the present study (139 microstrains). This could be explained by the fact that magnitudes and directions of joint loads in the dog vary much more than those in humans.(15) It should be noted at this point that the sample size for each of the investigated distributions is very large, and that, accordingly, any suitable statistical test would indicate that average values differ significantly in case that these are only a fraction apart. However, statistical analyses are not very useful here because we have only analyzed one healthy and one osteoporotic bone. Hence, our results are exemplary.

A second purpose was to investigate to what extent the stress and strain distribution would be affected by osteoporosis. The strain distribution in the osteoporotic femur was less favorable than that in the healthy femur, with strains 70% higher on average, and with a much higher (though still small) fraction of the tissue (0.045%) loaded beyond 3000 microstrains. Interestingly, the amount of low-loaded bone with a strain magnitude less than 50 microstrains (2.63%) was higher in the osteoporotic than in the healthy bone (1.98%), suggesting that osteoporosis does lead to the formation of loose trabecular ends.

A third purpose of this study was to determine if osteoporotic bone is simply bone adapted to reduced load levels. To obtain the same average tissue level strain magnitudes, the external load applied to the osteoporotic femoral head had to be reduced considerably to 59% of its original value. After this load reduction, the histograms of all investigated parameters were very similar to those of the healthy femoral head, although the distributions were still somewhat wider, as indicated by the larger SDs. It is worth noting, however, that the amount of low-loaded bone with a strain magnitude less than 50 microstrains (4.69%) is more than twice as high as in the healthy case (1.98%), indicating that after this load reduction an increased amount of bone would be subjected to very small strains. Hence, although the tissue stress and strain distributions after load reduction look very similar to those of the healthy femur, the larger SDs and the larger amount of low-loaded bone indicate that osteoporotic bone is not the same as healthy bone adapted to lower loads. Following the paradigm that, normally, stresses and strains should be distributed rather evenly over the trabecular architecture, the larger SD found for the osteoporotic bone seems unfavorable. The interpretation of these findings also depends on the physical activities of both donors, which unfortunately were not known. If both donors were equally active, the load magnitudes originally applied to the FE models, which were based on in vivo telemetry measurements in active subjects, should be realistic. In this case, the bone of the osteoporotic donor is clearly higher loaded and will be more susceptible to fractures. It is possible, however, that the osteoporotic donor was (much) less active than the healthy one, such that (part of) the osteoporotic bone loss is caused by adaptation to lower load levels. In this case, the FE analyses with the reduced load would be more appropriate, and the bone loading during normal activities would be very similar to the healthy femur. It should be noted, however, that also in the latter case, the bone of the osteoporotic donor would be much more susceptible to fractures caused by non-physiological forces, such as falls, where the forces are largely determined by body weight, which was similar for both subjects.

A final purpose of this study was to establish a safety factor for trabecular bone in the human femur. We calculated a safety factor of 8.62 for the healthy bone and a factor of 4.90 for the osteoporotic bone. These values correspond to 20 times BW and 11.5 times BW for the healthy and osteoporotic femur, respectively. In their study, Bergmann et al.(11) reported a maximum load magnitude measured during all measurements of 8.7 times BW (for stumbling). This indicates that the bone in the femoral head is unlikely to fracture for any of the activities that have been measured in these studies, including stumbling. It should be noted, however, that femoral fractures usually do not occur in he femoral head but rather in the neck or trochanteric region and usually are not caused by physiological loading, but to an unusual load, such as in a fall. As such, the numbers that we mention here are not directly representative for the bone fracture risk, but they are an indication for the safety factor of bone in the femoral head for physiological loading conditions.

Some limitations of this study should be mentioned. First, muscle forces were not included in the model. However, their net effect is included in the joint force because these were based on in vivo measurements. Hence, stresses and strains in the femoral head and neck, which are determined only by the joint force, are correct. In the trochanteric and more distal regions, however, the muscle forces will contribute to the stresses and strains as well. Hence, the results as represented in Fig. 3 represent physiological values in the femoral head and neck that are determined by the joint force alone, but do not represent physiological values in other regions that are determined by the muscle forces as well. Second, in this study, we have assumed that the tissue level material properties in the healthy and osteoporotic situation are the same. This assumption is supported by recent studies that reported no significant changes in bone tissue material properties with osteoporosis.(16–18) It is possible to simply scale the results of the osteoporotic analyses to accommodate for changes in material behavior as long as these take place uniformly. For example, a 10% higher modulus in the osteoporotic case would result in 10% lower strains in the tissue. Third, numerical errors and the relatively large 80-μm voxel size can limit the accuracy of the μFE models.(19) However, earlier convergence studies have indicated that high-resolution FE models with an element size of 80 μm can produce accurate calculations in the femoral head.(20,21) Although local errors in the strains might exist, in particular near the surface of the trabeculae, their effects on histograms and plots are small.(7,14) Fourth, we used linear elastic FE analyses to predict bone failure. In an earlier study, using similar FE analyses in combination with experimental data provided good results for the prediction of the bone failure load measured in experiments.(13) The concept that bone failure would occur as soon as 2% of the tissue is overloaded also agrees with findings by others, who used nonlinear μFE to predict bone yield behavior.(22,23) Nevertheless, it is not known if the same values would provide similar results for bone at other sites. The values reported here thus can only be used as a first estimate of the actual failure load. Finally, computational requirements for these analyses are still excessive. The equivalent of 2.9 years of cpu-time was used for each of the analyses. Consequently, large parallel computer systems are required to perform these analyses in a reasonable amount of wall-clock time.

Acknowledgements

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

This study was supported by an ETH research grant. We thank Dr Bruno Koller from Scanco Medical (Bassersdorf, Switzerland) for making the μCT scans of the bones.

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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