W. PISTOIA, 1 B. VAN RIETBERGEN, 2 E.-M. LOCHMÜLLER, 3 C. A. LILL, 4,5 F. ECKSTEIN, 6 and P. RÜEGSEGGER 1

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1 Estimation of Distal Radius Failure Load With Micro-Finite Element Analysis Models Based on Three-dimensional Peripheral Quantitative Computed Tomography Images W. PISTOIA, 1 B. VAN RIETBERGEN, 2 E.-M. LOCHMÜLLER, 3 C. A. LILL, 4,5 F. ECKSTEIN, 6 and P. RÜEGSEGGER 1 1 Institute for Biomedical Engineering, University of Zürich and Swiss Federal Institute of Technology (ETH), Zürich, Switzerland 2 Department of Biomedical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands 3 Universitätsfrauenklinik der LMU, München, Germany 4 AO Research Institute, Davos, Switzerland 5 Department of Orthopedic Surgery, University of Heidelberg, Heidelberg, Germany 6 Institute of Anatomy der LMU, München, Germany Address for correspondence and reprints: Dr. Walter Pistoia, Institute for Biomedical Engineering, University of Zürich and Swiss Federal Institute of Technology (ETH), Moussonstrasse 18,8044 Zürich, Switzerland. pistoia@biomed.ee.ethz.ch There is increasing evidence that, in addition to bone mass, bone microarchitecture and its mechanical load distribution are important factors for the determination of bone strength. Recently, it has been shown that new high-resolution imaging techniques in combination with new modeling algorithms based on the finite element (FE) method can account for these additional factors. Such models thus could provide more relevant information for the estimation of bone failure load. The purpose of the present study was to determine whether results of whole-bone micro-fe ( FE) analyses with models based on three-dimensional peripheral quantitative computer tomography (3D-pQCT) images (isotropic voxel resolution of 165 m) could predict the failure load of the human radius more accurately than results with dual-energy X-ray absorptiometry (DXA) or bone morphology measurements. For this purpose, FE models were created using 54 embalmed cadaver arms. It was assumed that bone failure would be initiated if a certain percentage of the bone tissue (varied from 1% to 7%) would be strained beyond the tissue yield strain. The external force that produced this tissue strain was calculated from the FE analyses. These predictions were correlated with results of real compression testing on the same cadaver arms. The results of these compression tests were also correlated with results of DXA and structural measurements of these arms. The compression tests produced Colles-type fractures in the distal 4 cm of the radius. The predicted failure loads calculated from the FE analysis agreed well with those measured in the experiments (R p < 0.001). Lower correlations were found with bone mass (R , p < 0.001) and bone structural parameters (R p < 0.001). We conclude that application of the techniques investigated here can lead to a better prediction of the bone failure load for bone in vivo than is possible from DXA measurements, structural parameters, or a combination thereof. (Bone 30: ; 2002) Science Inc. All rights reserved by Elsevier Key Words: Colles fracture; Dual-energy X-ray absorptiometry (DXA); Peripheral quantitative computed tomography (pqct); Finite element (FE) method. Introduction Osteoporosis reduces bone strength through a loss of bone mass and a diminution of structural integrity. Bone loss can be assessed easily with dual-energy X-ray absorptiometry (DXA). For this reason prediction of bone failure load is nowadays merely based on bone mineral density (BMD) measurements. BMD alone, however, is not sufficient to predict bone failure load for individual patients. 1,18,26 An improved bone strength prediction should be based on the mechanical properties of bone that can account for the bone trabecular structure. 6,11,24,29,30 Recently developed micro-finite element ( FE) techniques 9,14,37 enable a precise determination of the mechanical properties of bone. At the basis of this approach are highresolution digital images that represent the trabecular architecture in detail. Such images (pixel size on the order of 10 m) can be obtained from micro-computed tomography scanning, 4,8,27 serial sectioning, 23 or serial grinding. 3 By stacking a large number of sequential images, a three-dimensional (3D) reconstruction of the bone structure can be obtained. With the FE approach, such reconstructions are converted to FE models by converting voxels representing bone tissue to an equally sized brick element in an FE model. Mechanical properties of the reconstructed bone can then be calculated by simulating compression tests with the FE models. 35 So far, the determination of bone mechanical properties with this technique has been limited to excised bones and bone specimens. Recent developments in high-resolution imaging, however, allow the application of FE techniques to patient examinations. Two in vivo imaging techniques are presently available: a threedimensional peripheral quantitative computed tomography (3DpQCT) technique, providing images with an isotropic resolution of 165 m 15 ; and a magnetic resonance (MR) technique, pro by Elsevier Science Inc /02/$22.00 All rights reserved. PII S (02)

2 W. Pistoia et al. Estimation of radial loading with micro-fe 843 viding a resolution of approximately m. 16,38 Although the resolution of these in vivo techniques is not as good as that of the in vitro imaging techniques mentioned earlier, it suffices for visualization of the trabecular network. 16,19,31 Furthermore, it has been shown that this level of resolution also suffices for ensuring accurate results of FE analysis, 21,25 albeit correction factors may also be needed to predict correct values in an absolute sense. In a recent study, Ulrich et al. 33 for the first time analyzed the load transfer in a human radius in vivo using a FE model of the distal radius generated from 3D-pQCT images that represented the actual trabecular structure. Using this model, the investigators were able to calculate the tissue-loading conditions for external loading conditions representing a fall on the outstretched hand, an event that can lead to a typical Colles-type fracture in osteoporotic subjects. In their study, however, only one (nonosteoporotic) radius was modeled, and it was not possible determine whether these analyses could lead to a better estimation of bone fracture load than the one based on bone mass alone. The purpose of the present study was to determine whether results of FE analyses of bone in vivo can lead to a better estimation of bone fracture load than estimations based on bone mass alone or estimations based on structural parameters. We used an approach introduced by Ulrich et al. and applied it to a large number of cadaver arms for which, after 3D-pQCT examination, the failure load to produce a Colles-type fracture was experimentally measured. This setup enabled us to determine the predictive value of the FE-calculated parameters for the bone failure load and to compare them with the predictive value of bone mass alone, as measured by DXA, and that of a combined statistical model based on structural parameters, which are obtainable from the 3D-pQCT images. Materials and Methods Materials, Imaging, and Measurement of Structural Indices Figure 1. (a) The radius and the two adjacent carpal bones, together with artificially modeled cartilage, colored for better visibility (surface-rendered image from a 3D-pQCT measurement). (b) Typical three-dimensional representation of the radius and the two adjacent cartilage layers of the carpal bones as input for the FE analysis. The arrows symbolize the applied load on the radius representing a near-neutral hand position: 400 N force applied on the lunate side and 600 N force on the scaphoid side. (c) Top view of the same radius. A total of 70 embalmed cadaver arms were collected at the Institute of Anatomy, LMU, Munich, Germany. BMD and bone mineral content (BMC) values of the forearms under in situ conditions with soft tissue were obtained using Lunar DXA equipment (DPX-L, Lunar Corp., Madison, WI; scanned region 33% proximal). All cadaver arms (mean SD age: years) were transported to the Institute for Biomedical Engineering, IBT, Zurich, where high-resolution 3D-pQCT images of the distal radii were made to measure their 3D microarchitecture. The 3D-pQCT scanner developed in our lab enables the simultaneous acquisition of a stack of parallel computed tomography (CT) slices (165 m side length) with a nominal in-plane resolution of 165 m, which is sufficient to represent the microarchitecture of cancellous bone and to determine certain structural indices. 15 The distal 4 cm region of each forearm was covered with a stack of 240 CT images and transformed to a 3D voxel grid with cubic voxels of 165 m side length. Structural indices, such as cortical thickness (C.Th), trabecular number (Tb.N), trabecular thickness (Tb.Th), trabecular separation (Tb.Sp), and relative bone volume (BV/TV), were extracted from these images. The gray-value images from the original 3D-pQCT measurements were segmented using a low-pass gaussian filter and a fixed threshold to extract the bone tissue. Image processing also included the removal of unconnected parts and structure thickening to restore the original bone density. 32 For 11 of the 70 arms, the image quality was inferior to that obtained in patients, 28 presumably because, depending on the morphology of the bone, the prolonged exposure to formalin changed the composition of bone marrow and hence diminished the bone/marrow contrast. These 11 samples were excluded from further analysis. Another 5 arms were excluded because the loading experiment did not produce a Colles-type fracture (see later). For the remaining 54 radii (21 men, 33 women; mean SD age: 82 9 years) 3D FE models were created. The images contained parts of the carpal bones. The reason for including these carpal bones in the FE models was to apply realistic load conditions to the radius. The 3D-pQCT images, however, did not show the cartilage, thus leaving a gap between carpal bones and radius. To restore contact, voxels representing a cartilage layer were added to the articular surface of the bones. Figure 1a shows a typical model of a radius and two carpal bones with the artificially generated layers of cartilage, which are shown in color for better visibility. After producing the cartilage layer and saving the anatomical features of the carpal bones in this way, the two bones were removed to reduce the number of elements involved in the FE analysis (Figure 1b,c). Experiment After 3D-pQCT imaging, the arms were transported to the AO Research Institute, Davos, Switzerland, where all arms were loaded in a testing machine to produce a Colles-type fracture in the radii. The forearms with intact soft tissues were subjected to a compression test using an Instron uniaxial-driven mechanical testing machine (Type 4302, Instron, Ltd., High Wycombe, UK). The proximal part of the radius of each arm was embedded in epoxy resin and clamped to the bottom anvil of the testing machine such that the arms were in a vertical position with the hand placed in hyperextension (Figure 2). A specially developed wooden wedge was attached to the top-indenter of the test machine, such that the bottom plane of the indenter conformed to the position of the hand. The indenter with attached wedge was then moved downward at a speed of 200 mm/min until complete fracture (Figure 3). The forces applied to the hand with this setup are similar in magnitude and direction to those that occur during a fall on the outstretched hand; the type of fall that usually leads to a Colles-type osteoporotic bone fracture of the radius. However, the strain rate in the experiment was much lower than that during a real fall. Nevertheless, the experiment resulted in realistic Colles-type fractures. For five arms, however, these compression tests produced a scaphoid fracture. The occurrence of other types of fractures could possibly indicate an error in the

3 844 W. Pistoia et al. Bone Vol. 30, No. 6 Estimation of radial loading with micro-fe Figure 4. Conceptual histogram of the effective strain distribution. The tissue yield strain was set to 0.7% (7000 microstrain). Bone failure was assumed if a significant part of the bone tissue volume (here set to 5%) was overloaded. Figure 2. Typical experimental setup. For this test, the forearm was positioned such that a load was applied on the ball of the outstretched hand while clamping the proximal end of radius, until complete fracture was obtained. specific experimental setup. For this reason, these arms were excluded from further analyses. Displacements were measured at the crosshead of the machine using the internal LVDT (accuracy 0.1 mm), and forces were measured using a 10 kn load cell (accuracy 0.5%). The bone failure load was obtained from the recorded force-displacement curves (Figure 3). For some of the arms, the force-displacement curves showed several peaks before the ultimate force was reached. In such cases, the bone failure load was taken as the load peak after which a decrease in load of least 5% was observed. FE analyses FE models of the radii were created by converting bone and cartilage voxels to brick elements. 37 Material properties chosen were isotropic and linear-elastic. Young s modulus (E) and Poisson s ratio ( ) were: E 10 GPa and 0.3 for bone tissue, and E 1 MPa and 0.49 for cartilage. 2,34 The model was loaded by a distributed force applied to the cartilage layers of the carpal bones while the proximal end was fully constrained. Loading conditions were chosen to simulate the test conditions. The magnitude of the total applied force was initially chosen to be 1000 N. For an even load distribution, 600 N acted on the scaphoid side and 400 N on the lunatum side. 33 The directions of the nodal forces were perpendicular to the articular surface of the radius (Figure 1b). A special-purpose FE solver was used to solve these very large FE problems with up to 1.7 million elements, 37 and the tissue-level stresses and strains were calculated using a Cray J90 machine for calculations. Approximately 50 h of user cpu time and 500 megabytes of memory were necessary for each analysis. To predict the failure load of the radius, we assumed that bone failure would be initiated as soon as a significant part of the bone tissue was strained beyond a critical limit, which was taken as the tissue yield strain. In these analyses, the tissue-level effective strain, eff, which was calculated from the strain-energy density, U, and the Young s modulus, E, of the bone tissue was used as a measure of tissue strain: eff 2U E. (1) A bone tissue yield strain of 7000 microstrain was chosen, based on literature values. 20,36 By linearly scaling the tissue strains, it was calculated for which external load more than 5% of bone tissue was strained beyond the yield strain. In other words, it was calculated for which external loads the gray area in the strain histograms (Figure 4) would exceed 5% of the total bone tissue. The external load thus determined was taken as the predicted failure load of the radius and was compared with the failure load measured in the experiment. The value of 5% was assumed somewhat arbitrarily because it is not known what percentage of bone tissue actually exceeds yield strain when bone fractures occur. To determine whether another value would yield a better estimation of the measured bone failure load, the fracture load was calculated as well, using the assumptions that fracture occurs when 1%, 2%, 3%, 4%, 6%, or 7% of the bone tissue exceeds the yield strain. Statistical Analyses Figure 3. Typical force-displacement curve. The bone failure load was defined as the load for which the recorded force-displacement curves showed the first clear signs of failure. The predictive value of the fracture load calculated from the FE analyses was determined by performing a regression analysis of the results with the measured failure loads. For each of the seven failure criteria, which differ in the amount of overloaded tissue volume that was assumed at the onset of fracture (ranging from 1% to 7% of the total bone tissue in 1% intervals), the coefficient of determination and the slope and intercept of the regression line were calculated. The predictive value of the statistical model based on bone mass or structural indices was investigated in a first step using

4 W. Pistoia et al. Estimation of radial loading with micro-fe 845 Figure 5. Contour plots for the calculated Von Mises stress in bone tissue of a typical model. single linear regression analyses. As predictor variables we used those obtained from DXA measurements (BMD and BMC) and those obtained from 3D structural measurements of the 3DpQCT images (C.Th, Tb.N, Tb.Th, Tb.Sp, and BV/TV). In a second step, multiple linear regression analysis was used to determine whether a better correlation could be obtained when including combined structural parameters. Results FE Analyses Figure 5 shows contour plots for the calculated Von Mises stresses at the bone tissue for one typical model. Red coloration indicates high stress levels, and white indicates low stresses. The plots demonstrate how the loads are transferred through the trabecular network in the distal part of the radius to the cortical bone in the more proximal region. The results shown in Figure 5 are for a total external force of 1000 N, but they can be linearly scaled for any other force because of the linearity of these FE analyses. Figure 6. (a) (d) Experimental failure load for the 54 forearms (horizontal axes) vs. those predicted from the FE analysis and different failure criterion (vertical axes). The percentage of the bone material strained beyond the yield strain characterizes the failure criteria and is reported in the corner of the graphs. Correlation coefficients and regression lines indicated. Prediction of Failure Load From FE-based Failure Criteria A good correlation was found between the failure load calculated from the FE models and that measured in the experiments (Figure 6). The coefficient of determination (R 2 ) ranged between 0.70 and 0.75, depending on the chosen failure criterion (p for all cases). The slope and intercept of the regression lines are reported in the graphs. The best agreement (R ) was found for the criterion that allowed 2% of the bone material to be strained beyond the yield strain at the onset of fracture (Table 1). As expected, the predicted failure load increased when a larger region of overloaded bone was assumed at the onset of fracture (Table 2). An average failure load of 1.40 kn was predicted for all 54 arms when the assumed overloaded volume was set to 1%, which increased to 2.04 kn for an assumed overloaded volume of 7% (Table 2). The average failure load for the 54 embalmed arms measured in the experiments was 1.24 kn, indicating that, on average, the predicted failure load overestimated the measured load by 13% 64%. Prediction of Failure Load From Stochastic Models Including Bone Mass and Structural Indices The prediction of the measured failure load from a regression model with bone mass or structural parameters as the predictor variable was less favorable (Table 3). The best prediction based on bone mass was obtained from a stochastic model with BMC as the predictor variable (R , p 0.001). In contrast, BMD was a poor predictor (R , p 0.001). The best agreement between measured failure load and single structural indices was found for C.Th (R , p 0.001). Using multivariate regression analysis, the best prediction of measured failure load was obtained when the combination of the following structural indices was chosen: Tb.N, Tb.Th, BV/TV, and C.Th (R , p 0.001). Discussion In this study we aimed to answer the question of whether results of FE analyses of the distal radius leads to a better estimation of bone fracture load than predictions based on bone mass or structural parameters. Based on the results we conclude that tissue-level strains calculated from FE analyses can be a considerably better predictor of bone fracture load (R ) than bone mass (R ) or structural indices (R ). A major advantage of the approach introduced herein over stochastic models for bone failure estimation, such as those based on DXA data, is that FE models directly account for the mechanical aspects of the full, patient-specific bone architecture, making it potentially a more precise and versatile approach. The somewhat poor correlation of bone fracture load from bone mass or structural indices may seem to disagree with the results of earlier studies that reported much higher coefficients of determination. 5,22 However, these earlier studies involved the testing of relatively homogeneous bone specimens rather than whole bones, such as those in this study. Furthermore, we suspect that, if an adequate number of younger subjects with higher bone mass would have been included, then a higher correlation would have been found. It must be noted, however, that the age group

5 846 W. Pistoia et al. Bone Vol. 30, No. 6 Estimation of radial loading with micro-fe Table 1. Results of regression analyses a Exp. load Predicted load: failure criteria 1% 2% 3% 4% 5% 6% 7% R 2 (p 0.001) x x 0.70 x x 0.75 x x 0.74 x x 0.73 x x 0.73 x x 0.71 x x 0.70 a The predicted parameter is the experimentally measured bone failure load (first column). The independent parameters are given in the subsequent columns and consist of the FE-predicted load calculated with different failure criteria. x indicates which criterion is included in the same regression analyses. The subsequent column lists the coefficients of determination. examined here ( years) is the one most heavily affected by fractures and thus does constitute a relevant study sample. In this study we assumed that bone failure would occur if a certain amount of bone tissue (varied from 1% to 7%) were to be strained beyond a level commonly reported as dangerous for bone tissue (7000 microstrain). We further assumed that the tissue-level strains at the onset of fracture could be calculated with sufficient accuracy from linear elastic FE analyses. Although we cannot presently verify the accuracy of these assumptions, we have demonstrated that a bone failure criterion based on these assumptions can produce good results for the estimation of bone failure load. We found that the that best prediction of bone failure was obtained when it was assumed that 2% of the bone tissue is loaded beyond a yield strain of 7000 microstrain at the onset of fracture. This value is in excellent agreement with the results of nonlinear FE analyses for the simulation of compression tests on bone specimens by Niebur et al., 20 who reported that 2.5% of the tissue exceeded the tissue yield strain when reaching the apparent compressive yield point. With our 2% criterion, however, the predicted fracture load overestimated the measured fracture load by 29%. This overestimation could be reduced when assuming that a smaller percentage of the tissue is overloaded at the onset of fracture, but this reduces the coefficients of determination of the correlation. In relation to this, we note that the values measured in the present experiment might not be fully representative of an actual fall. First, the strain rate in the experiment was lower than that during a fall. As a result, it is likely that the actual failure load during a fall would be higher than the load measured herein. Second, fixation of the arms in formalin could have affected the mechanical properties of the bone. It has been described that formalin can reduce the strength of bone by 12%, 17 although other studies reported an Table 2. Average values of experimental and predicted failure loads ( SD) (relative differences between different predicted failure loads and the experimental load also shown) Mean values (kn) Difference (%) Exp. load Predicted failure load 1% % % % % % % increase in strength. 7 Because the mechanical properties of embalmed specimens indeed differ from those of fresh ones, this would change the slope of the regression as presented in Figure 6; however, it would likely not affect the coefficients of determination. All our conclusions are based on these coefficients of determination and we thus believe that the use of embalmed specimens would not have affected our results. Due to these uncertainties, we do not know whether the criterion for which the best failure load estimation was found in the present study would produce the best results in general, but it is proposed as a good starting point. Some limitations of our study have to be mentioned. First, due to insufficient image quality we had to exclude 11 of the 70 arms. Visual inspection of in vivo measurements made with the same instruments showed better image quality than that obtained for our cadaveric arms. We suspect that fixation in formalin changed the bone/soft tissue contrast. What was inconvenient for our study might be positive for future FE-based failure load estimations of patients all the more as new generations of 3D-pQCT scanners 12 promise greatly improved spatial resolution and faster scanning time. Second, it is known that the results of FE models can be inaccurate near jagged boundaries. 10,13 With the resolution of the images used in this study, most elements representing trabecular bone tissue in the FE models would be at or near the size of the trabeculae they represent. It thus is possible that the tissue-level strains calculated at a specific point in the trabecular bone tissue would be inaccurate. However, it has been demonstrated that the histograms showing tissue load distribution are not much affected by these errors, 25,35,37 and that a failure criterion based on the histograms, as used in this study, should be accurate as well. Third, it was sometimes unclear at which load the fracture occurred, because, for some arms, the recorded force-displacement curves showed several peaks prior to the ultimate load. Due to the compressive nature of the experiment, it is possible that the recorded force continues to increase after the fracture, and thus ultimate load does not represent the initial fracture load but rather the load at which substantial damage has accumulated to such an extent that the bone collapses. In this study, we used the first drop in force as an indicator of fracture. It is possible, however, that such peaks were, in fact, due to movement or failure of hand bones, or due to fractures at other locations. Considering the uncertainties involved in this experiment, we expect that the coefficient of determination found for the FE-predicted failure load in the present study is close to the best value that can be reached at all. Finally, to simulate the failure process in more detail, nonlinear analyses will need to be performed, as introduced by Niebur et

6 W. Pistoia et al. Estimation of radial loading with micro-fe 847 Table 3. Results of the regression analysis a Exp. load Three-dimensional structural parameters DXA parameters C.Th Tb.N Tb.Th Tb.Sp BV/TV BMD BMC R 2 (p 0.001) x x 0.53 x x 0.47 x x 0.20 x x 0.39 x x 0.44 x x 0.31 x x 0.48 x x x x x 0.57 a The predicted parameter is the experimentally measured bone failure load (first column). The independent parameters are given in the subsequent columns and consist of the three-dimensional structural parameters supplemented by the dual-energy X-ray absorptiometry (DXA) parameters of bone mineral density (BMD) and bone mineral content (BMC). x indicates that the parameters are included in the same regression analyses. The last column lists the coefficients of determination. al. 20 The resolution of the models used in their study, however, was much higher than that of the models used herein and it is not known how realistic such simulations would be at this resolution. Furthermore, even if higher resolution models of bone were available, it is unlikely that these could be solved within a reasonable cpu time. For these reasons, we believe that such analyses are currently unfeasible. Presently, only a specialized 3D-pQCT or magnetic resonance procedure can provide the resolution required to resolve the trabecular network in vivo, and a high-speed computer is needed to solve the numerical problems. Although these requirements inhibit routine patient examination at this stage, we expect that this will be possible in the near future or when these imaging techniques and special-purpose analysis software become more widely available. Based on the results of this study, we expect that such analyses will considerably improve the estimation of bone failure load. Acknowledgments: This study was supported by the Swiss National Science Foundation, Grant No References 1. Ahmed A. I., Blake G. M., and Rymer J. M. 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