Joseph E. Langenderfer Ph.D Peter J. Laz. Anthony J. Petrella DePuy, a Johnson and Johnson Company, Warsaw, IN 46581

Transcription

1 An Efficient Probabilistic Methodology for Incorporating Uncertainty in Body Segment Parameters and Anatomical Landmarks in Joint Loadings Estimated From Inverse Dynamics Joseph E. Langenderfer Ph.D Peter J. Laz Computational Biomechanics Laboratory, Department of Mechanical and Materials Engineering, University of Denver, Denver, CO Anthony J. Petrella DePuy, a Johnson and Johnson Company, Warsaw, IN Paul J. Rullkoetter Computational Biomechanics Laboratory, Department of Mechanical and Materials Engineering, University of Denver, Denver, CO Inverse dynamics is a standard approach for estimating joint loadings in the lower extremity from kinematic and ground reaction data for use in clinical and research gait studies. Variability in estimating body segment parameters and uncertainty in defining anatomical landmarks have the potential to impact predicted joint loading. This study demonstrates the application of efficient probabilistic methods to quantify the effect of uncertainty in these parameters and landmarks on joint loading in an inversedynamics model, and identifies the relative importance of the parameters and landmarks to the predicted joint loading. The inverse-dynamics analysis used a benchmark data set of lowerextremity kinematics and ground reaction data during the stance phase of gait to predict the three-dimensional intersegmental forces and moments. The probabilistic analysis predicted the 1 99 percentile ranges of intersegmental forces and moments at the hip, knee, and ankle. Variabilities, in forces and moments of up to 56% and 156% of the mean values were predicted based on coefficients of variation less than 0.20 for the body segment parameters and standard deviations of 2mmfor the anatomical landmarks. Sensitivity factors identified the important parameters for the specific joint and component directions. Anatomical landmarks affected moments to a larger extent than body segment parameters. Additionally, for forces, anatomical landmarks had a larger effect than body segment parameters, with the exception of segment masses, which were important to the proximal-distal joint forces. The Manuscript received November 2, 2006; final manuscript received May 11, 2007; published online February 5, Review conducted by Jeffrey A. Weiss. Paper presented at the 2006 Summer Bioengineering Conference BIO2006, Amelia Island, FL, Jun , probabilistic modeling approach predicted the range of possible joint loading, which has implications in gait studies, clinical assessments, and implant design evaluations. DOI: / Keywords: joint kinetics, inverse dynamics, body segment parameters, anatomical landmarks, joint loading 1 Introduction Lower-extremity joint loadings can be measured directly in vivo for those few patients with instrumented hip or knee total joint replacements 1,2. However, inverse dynamics remains the time-tested standard for determining intersegmental joint loading during functional activities and motions 3,4. Inverse-dynamics analyses apply Newton s and Euler s equations to determine the intersegmental forces and moments, respectively, from subject kinematic data, ground reaction forces, and moments, and body segment parameters. Inter-segmental forces and moments when combined with muscle forces can be used to determine the joint loading. Body segment parameters BSPs; masses, moments of inertia, and center-of-mass locations can be measured directly in subjects using gamma-ray scanning or dual-energy X-ray absorptiometry, but are more commonly estimated instead from regression models of in vivo or cadaveric data 5 7. The different techniques for determining BSPs can yield remarkably different values for a given subject 5,7. For example, the coefficient of variation across different techniques can be 0.12 for segment masses 6. Uncertainty in BSPs affects intersegmental forces and moments calculated in an inverse-dynamics analysis. The resulting uncertainty in intersegmental loads has the potential to affect muscle force predictions from models, which use inverse-dynamics results as inputs, such as forward-driven models 8. Other recent studies have investigated how different methods for estimating BSPs affect predicted joint loadings using kinematics collected in vivo 5,6. Impact of variation in BSPs on total work has also been evaluated with an inverse-dynamics sensitivity study 9. An additional source of uncertainty affecting inverse-dynamics results is the identification and digitization of anatomical landmarks ALs. Small variability in identifying, locating, and digitizing ALs can affect construction of the anatomical coordinate systems whose motion is used to describe kinematics 10. Digitization is affected by tester experience and expertise, subject morphology, soft tissue overlying the AL, and the specific landmark to be digitized. For example, many ALs are not discrete points, but large areas of varying topology resulting in location uncertainty upon palpation 11. Consequently, in vivo identification of landmarks is never perfect 12. Several studies have examined the accuracy of identifying ALs for constructing coordinate systems and have reported substantial inter- and intra-observer variabilities 11,13,14. Previous analyses have been conducted to understand how landmark uncertainty affects the description of lowerextremity kinematics and kinetics 15,16, but have not considered sensitivity factors. Probabilistic simulation is a holistic analytical technique that can efficiently assess the combined impact of multiple variables. Most earlier studies considered BSPs and AL locations independently. However, a recent study utilized Monte Carlo simulation to analyze the combined effects of uncertainties in segmental joint parameters, i.e., joint positions and orientations, and BSPs on inverse-dynamics results for patient specific models 17. The prior work did not consider the uncertainty in ALs, which are the most commonly used method of constructing coordinate systems, nor sensitivity factors in assessing the most important parameters. Accordingly, the objectives of the current study were twofold: to demonstrate the application of an efficient probabilistic method, the advanced mean value method, to quantify the effects of uncertainties in BSP estimation and in AL locations on inverse- Journal of Biomechanical Engineering Copyright 2008 by ASME FEBRUARY 2008, Vol. 130 /

2 Fig. 1 Uncertainties in BSPs and ALs were simulated with a probabilistic analysis of inverse dynamics. Deterministic inputs were a kinematics calculated from the motion of ALs attached to the lower-extremity and b ground reaction data 18. Probabilistic inputs were c BSPs: segment masses, moments of inertia, and proximal-distal locations of centers of mass, as well as d nine lower-extremity ALs. Inverse-dynamics analysis e was conducted using Newton s and Euler s equations. Outputs of the probabilistic analysis were f distributions of intersegmental forces and moments for the hip, knee, and ankle, and g sensitivities of forces and moments to inputs. dynamics results, and to determine the sensitivities of joint loading to the BSPs and ALs, thereby assigning relative importance to input parameters for calculations of ankle, knee, and hip intersegmental loads. The range of possible joint loadings for gait studies and an understanding of the important variables affecting them is useful in making accurate clinical assessments, as well as in developing muscle models and evaluating implant designs. 2 Methods Intersegmental forces and moments at the ankle, knee, and hip during the stance phase of gait were calculated with a threedimensional inverse-dynamics model of the lower-extremity Fig. 1. Three-dimensional coordinates of lower-extremity ALs and ground reaction force data during gait stance phase, as well as masses, moments of inertia and locations of segment centers of mass were taken from a benchmark data set Lowerextremity segments were assumed to be rigid bodies with local coordinate systems constructed from ALs 18. Lower limb six degrees of freedom kinematics, described with Euler angles for each joint and translations for each segment, were calculated from the motion of the local coordinate systems Fig. 1 a, and the ground reaction data accounted for the boundary conditions between the foot and the ground Fig. 1 b. Intersegmental forces and moments were calculated with Newton s and Euler s equations, respectively 17. Forces were calculated in the anteriorposterior, medial-lateral, and proximal-distal directions, and moments describing hip and knee flexion-extension, abductionadduction, and internal-external rotation, and ankle plantar-dorsi 1 flexion, inversion-eversion, and abduction-adduction were also calculated. The inverse-dynamics analysis was performed every 2.5% of the gait cycle. Force and moment components were computed in the proximal, distal, and floating axis systems according to Vaughan et al., and the results were compared to benchmark results 18. Probabilistic analysis was used to simulate how uncertainty in BSPs and ALs affected the calculated intersegmental forces and moments. The BSPs were the mass, moment of inertia, and centerof-mass location ratio, defined as the distance of the center of mass from the proximal joint to the segment length, for the foot, shank, and thigh. For each body segment, the center of mass was determined from the center-of-mass location ratio and the total segment length, which was based on the AL defined joint center locations. In the probabilistic analysis, each of the BSP parameters was represented by independent normal distributions Fig. 1 c, Table 1. Distribution means for the BSPs were equal to the values reported with the benchmark data set anthropometry and determined with regression equations 18. Standard deviations of the BSPs for each limb segment were assigned by applying the coefficients of variation presented by Rao et al. for six different methods: a geometric model, two cadaver based models, and three in vivo scanning studies 6 Table 1. Average coefficients of variation were 0.12, 0.20, and 0.08 for segment masses, moments of inertia, and center-of-mass location ratios, respectively. Similarly, uncertainty in the location of nine ALs on the right lower extremity was simulated by assigning independent normal distributions to the landmark global coordinates 19 Fig. 1 d.a recent study found that identification of AL location is not uniformly distributed but consists of outliers and is approximately normally distributed 19. The landmarks were fifth metatarsal / Vol. 130, FEBRUARY 2008 Transactions of the ASME

3 Table 1 Summary statistics: mean and standard deviation of BSPs modeled with normal distributions. Masses, moments of inertia I and center-of-mass location ratio distance from proximal joint / segment length. BSP Foot Segment shank Thigh Mass kg I-flexion/extension kg m 2 I-abduction/adduction kg m 2 I-internal/external kg m 2 L CM head, heel, lateral malleolus, tibial tubercle, femoral epicondyle, greater trochanter, right and left anterior superior iliac spine, and the sacrum. For the ALs, distribution means were the threedimensional coordinates in the benchmark data set, and standard deviations of 2 mm in each direction were assumed. This standard deviation level is at the midrange of reported intra-rater variability in digitizing ALs 11,13,14,19. For a given trial and landmark, the perturbation to that landmark s coordinates was applied to the landmark at each temporal point of the gait cycle. In this manner, the simulation modeled intra-rater uncertainty or subjectivity in identifying and digitizing each landmark. The probabilistic model linked the NESSUS probabilistic modeling software Southwest Research Institute, San Antonio, TX with a custom inverse-dynamics model Fig. 1 e. The advanced mean value AMV 20 and Monte Carlo methods were used to perform the probabilistic analysis. The AMV is a computationally efficient technique for performing probabilistic modeling. Computational efficiency is achieved by combining optimization and reliability techniques to determine the most probable point MPP, which represents the combination of parameter values that predicts performance at the specified probability level. The AMV method first transforms the original variables, which can have any distribution type, into independent normal variables, then uses multivariable optimization to locate the MPP, and lastly computes the performance value at the desired probability level. In comparison with the Monte Carlo method, the AMV method is discrete in that it predicts performance at specific probability levels e.g., 1%, 99% and specific locations in the gait cycle. Custom scripting was used to perform the analysis at 40 locations through the gait cycle. The efficiency of the method is dependent on the number of performance measures magnitudes or components, input variables, probability levels, and locations through the gait cycle considered. The probabilistic sensitivity factors are measures of how much performance e.g., force or moment component is affected by each landmark or BSP. With AMV, the sensitivities are computed in the standard normal variate space as the unit vector from the origin to the MPP and serve as relative indicators of the contributions of variability in the parameters to variability in each performance measure. For a given performance measure, the sum of squares of all sensitivities to each parameter will equal unity. It is important to note that the sensitivities are relative and should be evaluated in the context of the predicted variability in performance. If small variability in performance is predicted, then the sensitivities are not as meaningful. In the probabilistic analysis, the range between the 1 and 99 percentiles of forces and moments Fig. 1 f, and the sensitivities of each force and moment to BSPs and ALs were calculated Fig. 1 g. To provide a relative ranking of importance through the gait cycle, the absolute values of sensitivities averaged over the gait stance phase are reported. Results Loading results from the deterministic model were identical to reported joint loadings 18. As in Ref. 17, the coefficient of variation of forces and moments were within 1% of the final coefficient of variation for the last 10% of Monte Carlo iterations, thus demonstrating convergence of the Monte Carlo method. Excellent agreement was also found between the probabilistic analysis results obtained from AMV and Monte Carlo 1000 trials. The mean difference in the 1 99 percentile ranges were 1.82 N for force and 0.42 N m for moment, less than the sampling errors for the 1000 trial Monte Carlo analysis. The AMV analysis required only 91 trials i.e., 91 inverse-dynamics analyses to predict the bounds for an output measure over the gait cycle. All of the results presented were from the AMV method. The effect of BSP and AL uncertainties on the predicted forces and moments varied depending on the joint and the force or moment component Figs. 2 and 3, Table 2. Variability in the magnitude of the forces and moments increased when moving from the ankle to the knee and hip Figs. 2 and 3. The largest ranges, between the 1 and 99 percentile bounds, were observed in hip force magnitude 53.6 N, with more even contributions from the components Fig. 2 and Table 2. Although the mean range of ankle force magnitude was small 4.1 N, when considering individual components, ankle proximal-distal force had the largest range 64.6 N. For the predicted moments, the hip moment was largest with a magnitude of 8.9 N m and was dominated by variability in the abduction-adduction component Fig. 3, Table 2. While ranges are presented here to show the impact, the absolute minimum or maximum bounds of joint loading Figs. 2 and 3 may prove useful in some clinical and design applications. The relative sensitivities of joint forces and moments to BSPs and ALs also varied between joints and between force and moment components Figs. 4 and 5. The proximal-distal joint forces for the hip and knee were generally more sensitive to masses of the distal segments Fig. 4, while proximal-distal forces for the ankle were more sensitive to ALs on the foot. The landmarks used to determine the locations of the joint centers and define the orientation of the coordinate systems were identified as most important to the medial-lateral forces. Specifically, the medial-lateral ankle force was most sensitive to the location of the femoral epicondyle and tibial tubercle Fig. 4 a, while the medial-lateral knee force was most sensitive to the locations of the femoral epicondyle and sacrum Fig. 4 b, since these landmarks are used to determine locations of the hip and knee joint centers. Similarly, knee anterior-posterior force was most affected by the locations of the lateral malleolus and the tibial tubercle. Proximal-distal knee force was relatively sensitive to shank mass and foot mass. Hip proximal-distal force was most sensitive to thigh mass Fig. 4 c ; hip anterior-posterior force was also affected by thigh mass, and the locations of the sacrum and tibial tubercle, whereas mediallateral force was most affected by the location of the right anterior-super iliac spine. While anterior-posterior force in the ankle was shown to be most sensitive to the mass of the foot, this finding must be considered in the context of the small variability predicted in the force component. According to the sensitivity results, with the exception of anterior-posterior ankle force, the locations of segment centers of mass were relatively unimportant. Joint moments were also most sensitive to ALs near the joint of interest Fig. 5. All ankle moments were sensitive to location of the lateral malleolus, and less sensitive to other landmarks near the ankle Fig. 5 a. Femoral epicondyle location was important to all knee moment components, with lateral malleolus and tibial tubercle also affecting knee internal-external and flexionextension moments, respectively Fig. 5 b. In addition, hip internal-external moment was sensitive to locations of the tibial tubercule and femoral epicondyle, with flexion-extension moment sensitive to lateral malleolus and greater trochanter and sacrum, and abduction-adduction moment sensitive to greater trochanter, Journal of Biomechanical Engineering FEBRUARY 2008, Vol. 130 /

4 Fig. 2 Bounds 1 99 percentile of intersegmental forces predicted for gait stance phase with probabilistic simulation of ALs locations and BSPs input to the inverse-dynamics model. Results shown for magnitudes and components. right anterior-superior iliac spine, and sacrum Fig. 5 c. Lastly, hip moments were only slightly affected by thigh moment of inertia. Discussion This study demonstrates the use of probabilistic modeling in quantifying the impact of two sources of variability, uncertainties in BSPs and in ALs, on intersegmental loads calculated with inverse-dynamics analysis. Small uncertainties in BSPs i.e., coefficients of variation: 0.12 for mass, 0.20 for moment of inertia, and 0.08 for center-of-mass location ratio and standard deviations of 2 mm for AL locations resulted in large variability in calculated forces and moments of up to % of the nominal values, respectively. In this study, ALs and BSPs were evaluated simultaneously. The substantial variability in predicted joint loading may have implications for estimating joint contact loads and mechanics for the lower extremity and requires the consideration of this variability when developing models incorporating both muscle forces and inverse dynamics to determine net joint contact forces and moments. Sensitivity factors were used to identify the important parameters for the specific joint and component directions. An interesting finding is that segment moments of inertia and centerof-mass locations were not very important parameters to consider when predicting joint forces or moments, perhaps because they were overshadowed by uncertainty in the ALs. ALs affected moments more than BSPs. Additionally, ALs had a larger effect on forces than BSPs, with the exception of segment masses, which were important to the proximal-distal joint forces. These findings highlight a strength of probabilistic modeling, the ability to conduct a holistic analysis of the system under consideration. A similar analysis to identify the important parameters could be possible experimentally by developing subject-specific models 6,21, but this would require relatively large data sets with associated temporal and monetary costs. The results presented here highlight the potential for and importance of probabilistic analysis in gait studies, where it is valuable to understand the potential bounds of loading. This is especially true when analyzing gait of a specific subject who differs greatly from the data sets used to create the regression relationships for estimating BSPs from gross anthropometry, and joint center locations from ALs. Particularly for obese or overly muscular subjects, identification of ALs, especially those covered by substantial adipose or soft tissue, can be more difficult. Gait analysis systems can be developed to incorporate probabilistic factors, which would help researchers and clinicians better understand the underlying uncertainties present in the results of gait studies. Uncertainty in the location of ALs simulated in this study translates into uncertainties in the identification of functional joint center locations, which are important since joint center locations define the coordinate system origins and segment lengths in the kinematic model. Even when AL locations are accurately identified, there is no guarantee that the resulting segment coordinate systems will accurately identify functional joint center locations / Vol. 130, FEBRUARY 2008 Transactions of the ASME

5 Fig. 3 Bounds 1 99 percentile of intersegmental moments predicted for gait stance phase with probabilistic simulation of AL locations and BSPs input to the inverse-dynamics model. Results shown for magnitudes and components. 4. Furthermore, if the segment coordinate systems are altered due to the improper definition of ALs, the resulting kinematics may not be incorrect, but the interpretation may be affected since kinematics will be calculated with respect to different functional Table 2 Average ranges width of 1 99 percentile of forces and moments during gait stance phase. For ankle moments, internal-external represents inversion-eversion moment, flexion-extension is plantar-dorsi flexion moment, and abduction-adduction is varus-valgus moment. Joint Magnitude Force N Ankle Knee Hip Joint Magnitude Moment Nm Proximaldistal Mediallateral Anteriorposterior Internalexternal Flexionextension Abductionadduction Ankle Knee Hip axes. Similarly, the associated inverse-dynamics intersegmental forces and moments may also be correct, but projected along a different set of mutually orthogonal axes. A recent probabilistic analysis was conducted with Monte Carlo simulation to understand how uncertainties in joint parameters, i.e., uniformly distributed joint locations and orientations and BSPs, affected lower-extremity inverse-dynamics torques 17. In the prior study, uncertainty was considered in joint parameters and BSPs first separate and then together. By performing the analysis in this manner, the separate effects of each parameter type could be determined with the Monte Carlo approach; however, sensitivity factors were not reported. It was found that errors in joint parameters, but not BSPs, had a large effect on inverse-dynamics torques. In agreement with the previous findings, the sensitivity factors determined in the current study revealed that ALs utilized to calculate the positions of joint and orientations of segments affected inverse-dynamics moments to a larger extent than BSPs Fig. 5. Similarly, for inverse-dynamics forces, the current study found ALs to have a larger effect than BSPs, with the exception of segment masses, which were important to the proximal-distal joint forces Fig. 4. Comparison of forces to the recent prior study is not possible since inverse-dynamics forces were not presented. In comparison with the previous study 17, a strength of the sensitivity-based approach is the ability to isolate specific variables that affected the forces and moments. The important contributions of BSP parameters have been highlighted in several cases Figs. 4 and 5. Journal of Biomechanical Engineering FEBRUARY 2008, Vol. 130 /

6 Fig. 4 Sensitivities of intersegmental forces at the a ankle, b knee, and c hip to BSPs: mass of foot M foot, mass of shank M shank, mass of thigh M thigh, and location ratio of the foot center of mass L c.m. foot and ALs: fifth metatarsal head Metatarsal Head, heel Heel, lateral malleolus Lat. Malleolus, tibial tubercle Tib. Tubercle, femoral epicondyle Fem. Epi., greater trochanter Great. Troch., right anterior superior iliac spine R.ASIS, and the sacrum Sacrum. The levels of sensitivities are only presented when any one component demonstrates a sensitivity of greater than 0.20 to a given input parameter. The AMV method is a computationally efficient probabilistic analysis technique requiring fewer function evaluations than a traditional Monte Carlo analysis, which makes AMV attractive for probabilistic analysis of implicit functions, such as finite element models, with high computational costs 22. A limitation of the AMV method is that it requires a well behaved monotonic system; when multiple combinations of parameters result in the same output, the method has difficulty converging to a meaningful solution. In closing, this study combined probabilistic methods with inverse-dynamics analysis of kinematic and ground reaction data to develop a tool for understanding how uncertainty in BSPs and landmark locations affect predictions of intersegmental joint loadings. The probabilistic model provides an efficient platform to understand the limitations of current measurement techniques and Fig. 5 Sensitivities of inter-segmental moments at the a ankle, b knee, and c hip to BSPs: thigh moment of inertia I thigh and ALs: fifth metatarsal head Metatarsal Head, heel Heel, lateral malleolus Lat. Malleolus, tibial tubercle Tib. Tubercle, femoral epicondyle Fem. Epi., greater trochanter Great. Troch., right anterior superior iliac spine R.ASIS and the sacrum Sacrum. The levels of sensitivities are only presented when any one component demonstrates a sensitivity of greater than 0.20 to a given input parameter. to assess potential improvements. The deterministic results of a typical gait analysis may present a sense of certainty to a clinician or engineer; predicted bounds of results from probabilistic analysis can more accurately convey the inherent uncertainty and its potential sources. A better understanding of joint loading variability can improve clinical diagnoses and provide insight into the evaluation of healthy normal and pathologic gait. Acknowledgment This research was sponsored in part by DePuy, a Johnson & Johnson company. References 1 Lu, T. W., O Conor, J. J., Taylor, S. J. G., and Walker, P. S., 1998, Validation of a Lower Limb Model With in Vivo Femoral Forces Telemetered From Two Subjects, J. Biomech., 31, pp Heller, M. O., Bergmann, G., Deuretzbacher, G., Durselen, L., Pohl, M., Claes, L., Haas, N. P., and Duda, G. N., 2001, Musculo-Skeletal Loading Conditions / Vol. 130, FEBRUARY 2008 Transactions of the ASME

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