Incorporating uncertainty in mechanical properties for finite element-based evaluation of bone mechanics

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1 Journal of Biomechanics 40 (2007) Incorporating uncertainty in mechanical properties for finite element-based evaluation of bone mechanics Peter J. Laz a,, Joshua Q. Stowe a, Mark A. Baldwin a, Anthony J. Petrella b, Paul J. Rullkoetter a a Computational Biomechanics Laboratory, University of Denver, 2390 S. York Street, Denver, CO 80208, USA b DePuy, A Johnson & Johnson Company, Warsaw, IN, USA Accepted 14 March 2007 Abstract Finite element (FE) models of bone, developed from computed tomography (CT) scan data, are used to evaluate stresses and strains, load transfer and fixation of implants, and potential for fracture. The experimentally derived relationships used to transform CT scan data in Hounsfield unit to modulus and strength contain substantial scatter. The scatter in these relationships has potential to impact the results and conclusions of bone studies. The objectives of this study were to develop a computationally efficient probabilistic FE-based platform capable of incorporating uncertainty in bone property relationships, and to apply the model to a representative analysis; variability in stresses and fracture risk was predicted in five proximal femurs under stance loading conditions. Based on published variability in strength and modulus relationships derived in the proximal femur, the probabilistic analysis predicted the distributions of stress and risk. For the five femurs analyzed, the 1 and 99 percentile bounds varied by an average of 17.3 MPa for stress and by 0.28 for risk. In each femur, the predicted variability in risk was greater than 50% of the mean risk calculated, with obvious implications for clinical assessment. Results using the advanced mean value (AMV) method required only seven analysis trials (1 h) and differed by less than 2% when compared to a 1000-trial Monte-Carlo simulation (400 h). The probabilistic modeling platform developed has broad applicability to bone studies and can be similarly implemented to investigate other loading conditions, structures, sources of uncertainty, or output measures of interest. r 2007 Elsevier Ltd. All rights reserved. Keywords: Bone mechanics; CT scan; Probabilistic; Mechanical properties; Fracture 1. Introduction Finite element (FE) models developed from computed tomography (CT) scans have become an important tool to evaluate mechanical stresses and strains in bone (Taddei et al., 2004; Hernandez and Keaveny, 2006; Bevill et al., 2006), load transfer related to implant fixation and repair (Taylor, 2006; Haider et al., 2006), bone cement interface mechanics (Mann et al., 2001; Mann and Damron, 2002), and fracture risk (Perillo-Marcone et al., 2003; Keyak and Falkinstein, 2003; Keyak et al., 2001). Bone fracture continues to be an important issue affecting aging populations and patients with bone diseases, and an Corresponding author. Tel.: ; fax: address: plaz@du.edu (P.J. Laz). understanding of the local bone quality and properties is important in making assessments of fracture potential or implant performance. These FE models utilize CT intensity in Hounsfield unit (HU) to determine the material properties in a specific finite element or voxel. Numerous studies have sought to define relationships between HU and density, density and Young s modulus, and density and bone strength (e.g. Carter and Hayes, 1977; Bentzen et al., 1987; Hvid et al., 1989; Snyder and Schneider, 1991; Keller, 1994; Rho et al., 1995; Hernandez et al., 2001; Morgan et al., 2003) for various bones, with average relationships fit to the experimental data. In all of these references, large amounts of scatter are present in the experimental data. For example, Keller (1994) reported differences from the mean commonly around 100% (and up to 400%) for modulus /$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi: /j.jbiomech

2 2832 P.J. Laz et al. / Journal of Biomechanics 40 (2007) and strength. The scatter in these relationships has potential to impact the results and conclusions of bone studies. Mupparapu et al. (2006) examined the effects of using different HU modulus relationships on predicted stress in a proximal tibia with a unicondylar implant and found large differences in modulus assignment and resulting displacements and strains predicted. Using Monte-Carlo simulation, Taddei et al. (2006a) found that bone stresses and strains in the proximal femur were more sensitive to uncertainties in the geometric representation than material properties. The objectives of the current study were to develop a probabilistic FE-based platform to incorporate uncertainty in bone property estimation for prediction of bone mechanics and fracture risk, and to apply the model to evaluate resulting stresses and risk in the proximal femur under stance loading conditions. In addition, this study evaluated the use of efficient probabilistic methods, as an alternative to the computationally intensive Monte-Carlo method, in making accurate predictions of bone mechanics. The probabilistic model predicts the level of performance (stress or fracture risk) and its likelihood, which provides a more comprehensive evaluation of the physical system and may impact bone study findings as well as clinical assessment. 2. Methods An automated probabilistic platform was developed that linked probabilistic modeling software (Nessus, SWRI, San Antonio, TX), bone Nessus Material Relations CT Data Bonemat Abaqus Stress Risk Fig. 1. Automated probabilistic framework was developed by linking probabilistic modeling software (Nessus, SWRI, San Antonio, TX), bone material property assignment (Bonemat, Instituti Ortopedici Rizzoli, Italy), and finite element analysis (Abaqus, Inc., Providence, RI). material property assignment (Bonemat, Instituti Ortopedici Rizzoli, Bologna, Italy), and FE analysis (Abaqus, Inc., Providence, RI) to predict the distributions of stresses and risk (Fig. 1). Probabilistic analyses were performed on FE models of five human proximal femurs extracted from CT scans. The CT scans were from three cadavers with an average age of 50. Slice thickness was 3 mm and the field of view was pixels for all scans. Scan resolutions were mm for two patients and mm for one patient. Surface geometry was extracted with segmentation of the CT scans using a grayscale-based edge detection algorithm in Scan IP (Simpleware Ltd., Exeter, UK) and then meshed with tetrahedral elements using Hypermesh (Altair Engineering, Inc., Troy, MI). Once meshed, the Bonemat software was used to assign material properties to each individual element. Perillo-Marcone et al. (2003) recommended element edge lengths similar to the CT slice thickness and demonstrated converged stiffness, stress and risk results with an element size of 1.4 mm for CT data with a slice thickness of 1 mm. A mesh convergence study was performed on one femur with element edge lengths of 6, 4.5, and 3 mm (slice thickness), corresponding to 4613, 8776, and 33,150 elements, respectively. The results for the 4.5 and 3 mm element meshes exhibited convergence with details provided in Section 3. The relationships for Young s modulus and strength, as a function of the apparent ash density, were developed from experimental data for the femur (Keller, 1994). Power law relations between modulus and density and strength and density were utilized (Keller, 1994). The four probabilistic variables were the coefficients and exponents for the modulus and strength relationships (Table 1, Fig. 2). Each probabilistic variable was modeled as a Gaussian distribution with mean values based on the femoral data from 297 specimens with varying mineral contents from Keller (1994). Standard deviation levels (Table 1) for the exponents were assigned the values reported by Keller (1994); standard deviations for the coefficients were set at a level so that a Monte-Carlo analysis of 1000 trials performed with variability in all of the parameters closely bounded the observed scatter in both the experimental modulus and strength data. In each trial of the probabilistic analysis, values for each parameter (a, b, c, d) were generated according to their distributions. Based on Peng et al. (2006), a relationship between HU and apparent bone density of r a (g/cm 3 ) ¼ HU was used to assign the density to individual elements from the CT data. The material was modeled as isotropic and a Poisson ratio of 0.3 was assumed. The loading conditions were based on stance phase gait loading and have been shown experimentally to produce clinically relevant fractures (Keyak, 2000; Keyak et al., 2001; Keyak and Falkinstein, 2003). The distal end of each femur was fixed, while a load of 10 kn was applied to the femoral head at an angle of 201 to the shaft axis in the frontal plane (Fig. 3). The applied load of 10 kn was an experimentally measured fracture load under stance conditions (Keyak, 2000). The load was distributed evenly among nodes on the femoral head that were located within 1.5 cm from the center of load application (Keyak et al., 2001) Modulus [GPa] Strength [MPa] Density [g/cm 3 ] Density [g/cm 3 ] Fig. 2. Variability (1 99% bounds) in the modulus and strength versus density relationships (see Table 1 for corresponding data).

3 P.J. Laz et al. / Journal of Biomechanics 40 (2007) Table 1 Material property relationships and distribution parameters based on Keller (1994) mm 4.5 mm 3 mm 0.7 Relationship Coefficient Exponent E (GPa) ¼ ar b a (m ¼ 1.99, s ¼ 0.30) b (m ¼ 3.46, s ¼ 0.12) S (MPa) ¼ cr d c (m ¼ 26.9, s ¼ 2.69) d (m ¼ 3.05, s ¼ 0.09) Maximum Stress [MPa] Risk 75 Max. Stress Avg. Risk Degrees of Freedom Fig. 4. Mesh convergence results with maximum stress and risk versus degrees of freedom for femur 3L. Meshes had element edge lengths of 6, 4.5, and 3 mm. Fig. 3. Distribution of modulus in sectioned femur and loading conditions (a), stress contours (b), and locations of highest risk (c) for femur 1L. Maximum von Mises stress was computed for each trial, while excluding the distal end of the femur due to discontinuities associated with the fixed boundary condition. Risk ratio, defined as the ratio of von Mises stress computed in the FE model and strength determined based on the density, was evaluated for each element (Perillo-Marcone et al., 2003). Since failure of a single element may be due to an artifact of the CT scan and not constitute structural failure, the reported risk was averaged over 500 elements with the greatest risk ratios, representing a volume of 1600 mm 3. Analyses were performed using the Monte-Carlo and advanced mean value (AMV) methods (Haldar and Mahadevan, 2000; Wu et al., 1990; Easley et al., 2007) within Nessus. Briefly, the efficient AMV method combines optimization and reliability techniques to determine the most probable point (MPP), which represents the combination of parameter values that predict performance at a specific probability level (e.g. p ¼ 0.01). The implementation evaluates the FE model with the variables at the mean and then perturbs each variable individually. Based on the stress or risk values, the method then approximates the MPP for a specified probability level using an optimization algorithm. After evaluating the model at the MPP, the results are then corrected for higher order terms. Accordingly, the results for the AMV method are discrete and the number of trials required is 1+the number of variables+the number of probability levels desired. In the current study with four probabilistic variables, only seven trials are required to determine the bounds corresponding to the 1 and 99 percentiles. In addition, sensitivity factors can be computed from the unit vector to the MPP and represent the relative contributions of each variable to the variability in performance. The AMV method, while approximate, is able to achieve accurate results for monotonic systems within a small number of trials. After quantifying the excellent agreement with the Monte-Carlo results for one femur (presented in Section 3), probabilistic analyses were performed using the AMV method for the other femurs. 3. Results The model predicted the modulus, strength, stress, and risk for every element in each femur (Fig. 3). The results of Table 2 Mean density and mean and bounds (1 and 99 percentile) of average modulus for the five femurs analyzed Femur Mean density (g/cm 3 ) Average modulus (MPa) Mean (50%) Bounds (1 99%) 1L ,012 1R ,996 2L ,036 3L ,630 3R ,570 the mesh convergence study (Fig. 4) exhibited convergence for the 4.5 mm mesh with differences less than 1% for maximum stress and 1.2% for predicted risk compared to the 3 mm mesh. Results presented for all femurs were based on the 3 mm results. Modulus was higher in elements in cortical regions of the bone and stresses were highest in the femoral neck region (Fig. 3). Based on the uncertainty in the material relationships, the average modulus in each femur varied by as much as 5950 MPa between the 1 and 99 percentiles (Table 2). Results for the distributions of maximum stress and risk are presented as a cumulative distribution function (CDF) 1 for one femur (Fig. 5), and as mean (50 percentile) and bounds (1 and 99 percentiles) for all femurs (Table 3). Comparisons between results from the AMV and the Monte-Carlo (1000 trials) methods differed by less than 2% (Fig. 5), which is smaller than the errors associated with sampling in the Monte-Carlo method. While the Monte-Carlo analysis required over 400 h, the AMV method required less than 2 h for a full CDF (14 trials) 1 The CDF is obtained by ordering the values of stress or risk from smallest to largest and plotting them versus the value index divided by the number of trials+1. The F(x) value corresponds to the specific probability level or percentile.

4 2834 P.J. Laz et al. / Journal of Biomechanics 40 (2007) a 1.0 b F (x) AMV Monte Carlo F (x) AMV Monte Carlo Stress [MPa] Risk Fig. 5. CDF plots of maximum stress (a) and risk (b) for femur 1L with the Monte-Carlo (1000 trials) and AMV (14 trials) methods. Table 3 Mean and bounds (1 and 99 percentile) of maximum stress and risk for the five femurs analyzed Femur Maximum stress (MPa) Risk Mean (50%) Bounds (1 99%) Mean (50%) Bounds (1 99%) 1L R L L R and less than 1 h (seven trials) to predict the performance bounds for each femur. Because of the excellent agreement between the two methods, the results in Table 3 are reported with the more efficient AMV method. While the mean for maximum stress was expected to vary for the different bones, the range of maximum stress between the 1% and 99% bounds was reasonably consistent and averaged 17.3 MPa for the five femurs (Table 3). The location of the elements with the highest risk corresponded to the region of highest stress and was consistent between femurs. The risk ratio was less than 1 for each femur, indicating no failure for the applied load of 10 kn. However, the results could also be presented as predicted mean failure loads from 15.2 to 25.2 kn corresponding to a risk ratio of 1 in the five femurs; the predicted failure loads were at the upper bound of those experimentally measured (Keyak, 2000). The distribution of risk ratio, accounting for variability in modulus and strength at each element, varied in mean due to the individual bone geometry and properties for each femur (Table 3). The range of predicted risk between the 1% and 99% bounds averaged 0.28 (Table 3). For each of the femurs studied, the predicted variability in risk was greater than 50% of the mean risk calculated. The bounds of predicted risk also correspond to bounds of predicted failure loads (e.g. loads from 14.1 to 23.4 kn for a risk ratio of 1 in specimen 1L). The probabilistic sensitivity factors, a measure of the contribution of each input parameter to the output, indicated that the exponent (b) for the modulus relationship had the greatest impact (0.99) on predicted stress. The strength parameters (c and d) had a greater impact (0.92, 0.37) on risk, because they affected risk more directly than modulus. The parameters with high-sensitivity factors contributed most to the variability in predicted performance. 4. Discussion A probabilistic analysis platform was developed to quantitatively assess the effects of variability in relationships between CT and bone mechanical properties on outputs of stress and risk. While the model was applied to hip fracture risk incorporating variability in the conversion of CT data to bone material properties, other loading conditions, bones, sources of uncertainty, or output metrics could be similarly evaluated. The substantial differences in reported relationships for modulus and strength, as well as the observed scatter in experimental data, have the potential to impact the findings of bone studies. Based on the experimentally derived variability levels with coefficients of variation (standard deviation divided by the mean) less than 0.15 for all parameters, the predicted range between the 1% and 99% bounds averaged 17.3 MPa for stress and 0.28 for fracture risk for the five femurs analyzed. Consideration of this uncertainty may be especially important when applying laboratory-developed relationships to patient-specific models, where this level of uncertainty in stress or risk may impact clinical decisions and/or study conclusions.

5 P.J. Laz et al. / Journal of Biomechanics 40 (2007) In the current study, the hip fracture risk analysis performed was used largely to demonstrate the application of the probabilistic analysis framework. Future analyses could include greater complexity and realism in the bone model, including, for example, individual relationships for trabecular and cortical bone (Rho et al., 1995; Morgan et al., 2003; Perillo-Marcone et al., 2003) and more physiological loading (Taddei et al., 2006b; Duda et al., 2001). In addition, uncertainty was included in relationships of density to modulus and density to strength, but was not included in the relationship between HU and density. The inclusion of additional uncertainty would result in larger predicted bounds of stress and risk. Although both modulus and strength were functions of density, the risk ratio showed greater variability than if only the impact of modulus on stress were considered. Specifically, the risk ratios predicted with the probabilistic approach resulted in a wider envelope than merely predicting the bounds of stress and dividing them by a mean strength. This finding highlights the importance of using a probabilistic approach that includes variability in both modulus and strength. The predicted variability in stress from this study was comparable to the levels predicted by Taddei et al. (2006a). For comparison purposes, assuming six standard deviations (73s) approximated the 1 99 percentile range, the coefficient of variation for von Mises stress averaged 1.95% in this study, compared to 3.16% from Taddei et al. (2006a), which included both material and geometric variability. The current study utilized a greater level of variability in the material properties, based on using all femoral data instead of the higher mineral density data from Keller (1994). While the coefficient of variation levels reported seems small, it is important to recognize that only approximately 68% of data is captured within 71 standard level. Considering 1% and 99% bounds, which capture approximately 99.5% of the population, results in larger variability in stress (75.9% and a range of 17.3 MPa for this study, or 79.5% in Taddei et al., 2006a) and may be important when applying these techniques to patientspecific analyses. While stress-based analyses are most common for assessing load transfer, recent studies have implemented strain-based failure criteria (Taylor, 2006; Gupta et al., 2006; Wong et al., 2005). Experimental studies have shown yield strain to be isotropic (Chang et al., 1999) and uniform within an anatomic site (Morgan and Keaveny, 2001). The probabilistic platform could be similarly applied to modulus and yield strain, based on, for example, standard deviations associated with yield strain (Chang et al., 1999). In comparison to the Monte-Carlo method with 1000 trials, the AMV method required only seven trials to predict the bounds. This efficient approach makes probabilistic analysis quite feasible, even when considering computationally intensive bone models developed from micro-ct scans (e.g. Hernandez and Keaveny, 2006; Bevill et al., 2006; Morgan et al., 2004). In closing, this study developed a probabilistic platform to assess the impact of uncertainty in bone property relationships on performance and applied the platform to predict the variability in bone stresses and fracture risk for five proximal femurs. The computationally efficient approach can provide a more thorough evaluation, as the likelihood of a particular level of performance can be considered when making clinical assessments. The probabilistic approach may prove useful in modeling bone performance in aging patients and/or patients with disease where bone quality may be more variable. 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