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Experimental Evaluation of a Natural Knee Contact Model Using Response Surface Optimization

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PAGE 1

EXPERIMENTAL EVALUATION OF A NATURAL KNEE CONTACT MODEL USING RESPONSE SURFACE OPTIMIZATION By YI-CHUNG LIN A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2004

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Copyright 2004 by Yi-Chung Lin

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iii ACKNOWLEDGMENTS First, I would like to sincerely thank Dr. Benjamin J. Fregly, my advisor, for the direction he has given me throughout my re search. His encouragement and dedication have greatly contributed to the accomplishment of this thesis. I would also like to thank Dr. Raphael T. Haftka and Dr. Andrew Kurd ila for being my committee members. It was a great honor to have them both. I would also like to thank all the Com putational Biomechanics Lab members for their support in my past and present wor k, with special thanks to Dong, Yanhong, Jaco and Jeff for their assistance and suggestion.

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iv TABLE OF CONTENTS Page ACKNOWLEDGMENTS.................................................................................................iii LIST OF TABLES...............................................................................................................v LIST OF FIGURES...........................................................................................................vi ABSTRACT......................................................................................................................v ii CHAPTER 1 INTRODUCTION........................................................................................................1 1.1 Need for Accurate Contact Model..........................................................................1 1.2 Need for Efficient Contact Model Evaluation........................................................2 1.3 Approach.................................................................................................................3 2 METHODS...................................................................................................................5 2.1 Response Surface Optimization..............................................................................5 2.2 Contact Pressure Experiments..............................................................................10 2.3 Knee Model Creation............................................................................................12 2.4 Knee Model Evaluation........................................................................................16 3 RESULTS...................................................................................................................18 4 DISCUSSION.............................................................................................................23 5 SUMMARY AND FUTURE STUDY.......................................................................28 5.1 Summary...............................................................................................................28 5.2 Future Study..........................................................................................................28 LIST OF REFERENCES...................................................................................................30 BIOGRAPHICAL SKETCH.............................................................................................35

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v LIST OF TABLES Table page 2-1. The averaged experimental measuremen ts were collected from three compression trials for both loads processing on the sa me cadaver knee via servohydraulic test machine....................................................................................................................12 3-1. Comparison of response surf ace predictions to data poin ts sampled from large and small strain contact models......................................................................................20

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vi LIST OF FIGURES Figure page 2-1. A human cadaver knee st atic experiment setup..........................................................11 2-2. Original and segmented medical images....................................................................13 2-3. The anterior and posterior views of th e A) 3D point clouds and B) 3D NURBS model with six screws created from CT and MRI images.......................................14 3-1. Percent errors between response surfaces and predicted results from both contact models......................................................................................................................21 3-2. Percent errors between experimental a nd predicted results from both contact models......................................................................................................................22

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vii Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science EXPERIMENTAL EVALUATION OF THE NATURAL KNEE CONTACT MODEL USING RESPONSE SURFACE OPTIMIZATION By Yi-Chung Lin August 2004 Chair: Benjamin Jon Fregly Major Department: Mechanic al and Aerospace Engineering Finite element, boundary element, and discrete element models have been employed to predict contact conditions in human joints. When optimization is used to evaluate the ability of such models to reproduce expe rimental measurements, the high computational cost of repeated contact anal ysis can be a limiting factor. This thesis presents a computationally-efficient re sponse surface optimization methodology to address this limitation. Quadratic response surf aces are fit to cont act quantities (peak pressure, average pressure, contact area, and contact force) predicted by a joint contact model for various combinations of material modulus and relative pose (i.e., position and orientation) of the contacting bodies. The response surfaces are used as surrogates for costly contact analyses in an optimization that minimizes differences between measured and predicted contact quantities. The appro ach is demonstrated by evaluating a linear elastic discrete element contact model of th e tibiofemoral joint, where the model was created using CT and MRI data from the same cadaveric specimen used in static pressure

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viii experiments. For variations in material modulus and relative bone pose within the envelope of experimental uncertainty ( 1 mm and 1), quadratic response surfaces accurately predicted contact quantities com puted by the discrete element model. Using these response surfaces, 500 optimizations with different initial gue sses were performed in less than 90 seconds. For a flexion angle of 30 and axial loads of 500 and 1000 N, the optimizations demonstrated that small and la rge strain versions of the contact model could match all experimentally measured cont act quantities to within 10% error with the exception of peak contact pressure, which was in error by as much as 85%. Thus, discrete element models of natural joints may be best suited for predicting contact quantities that involve averaging across the su rface but not quantities associat ed with specific locations on the surface.

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1 CHAPTER 1 INTRODUCTION 1.1 Need for Accurate Contact Model According to the Arthritis Foundation, there were nearly 43 million Americans with arthritis or chronic jo int symptoms in 1998. The number went up to 70 million in 2001 and most likely will keep climbing due to the rising numb er of aging Baby boomers. This foundation also reported th at arthritis limits daily activ ities such as walking, running and dressing for more than seven million Americans. There are many different types of arthritis including osteoarthritis, rheumatoid arthritis, gout, and juvenile arthritis. Among them, osteoarthritis is the most prevalent form of arthritis affecting more than 20.7 million Americans. Osteoarthritis, or degenerative joint disease, is multifactorial with genetic, biologic, and mechanical factors all playing a role. Of the mechanical factors involved, contact pressure within the joint has been shown to have an interactive effect on developing this disease (Hasler et al ., 1998; Herzog et al ., 2003). Thus, knowledge of in vivo contact forces and pressures in human joints would be valuable for improving the prevention and treatment of joint arthritis. Although dynamic imagi ng advances now is capable collecting accurate measurement of in vivo joint kinematics (Komistek et al. 2003; Tashman and Anderst, 2003), joint contact fo rces and pressures are difficult to measure in vivo (Kaufman et al. 1996), therefore, necessitating model-based analyses to develop predictions as well as static in vitro testing to evaluate thes e predictions. A variety of joint contact modeling methods have been used for this purpose, incl uding finite element

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2 (Bendjaballah et al., 1995, 1997, 1998; Donzelli a nd Spilker, 1998; Donahue et al., 2002; Stolk et al., 2002), boundary element (Kuo and Keer 1993; Haider and Guilak, 2000), and discrete element methods (Li et al. 1999, 2001; Pandy and Sasaki, 1998; Piazza and Delp, 2001; Dhaher and Kahn, 2002). 1.2 Need for Efficient Co ntact Model Evaluation Once a joint contact model has been created that represents an in vitro testing situation, its ability to reproduce experimentally measured contact quantities (e.g., peak pressure, average pressure, contact area, and co ntact force) must be evaluated before any further application. At least two factors comp licate the process of evaluation. The first is uncertainties in the experimental measur ements. These uncertainties can often be estimated and involve quantitie s such as the position and orientation (i.e., pose) of cadaveric bones measured by the test apparatu s, contact pressures and areas recorded by a pressure sensor, and the articular surface ge ometry determined from medical imaging data. Unknown model parameters, such as mate rial parameters in the contact model, present additional sources of uncertainty. A second complicating factor is the high co mputational cost of repeated contact analysis. Given an estimated envelope of un certainty, optimization methods can be used to determine if a feasible co mbination of model parameters could be used to reproduce all experimental measurements simultaneously (Fregly et al ., 2003). For example, an optimization can vary model material parame ters and the relative pose of the contacting bodies within the envelope of uncertainty un til the model produces the best match to the experimental contact data. The problem with th is approach is that the high computational cost of repeated contact anal ysis can make such optimizations extremely time consuming and in some cases even impractical.

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3 Response surface (RS) methods have been ut ilized successfully in other situations to eliminate computational bottlenecks in optimization studies. Response surfaces are simply multi-dimensional linear regression curve fits to quantities of interest predicted by an engineering model. Once the mathematical form of the RS is specified, linear least squares is typically used to determine the coefficients that provide the best fit to each predicted quantity of interest (i.e., each response) as a function of the specified design variables. These surface approximations ar e then used as surrogates for costly engineering analyses when the optimization is performed. Outside of the biomechanics community, RS optimization methods have been used for structural design applications (Jansson et al ., 2003; Liu et al ., 2000; Rikards et al ., 2004; Roux et al ., 1998), aerodynamic designs (Ahn and Kim, 2003; Papila et al ., 2002; Sevant et al ., 2000), and fluid dynamics (Burman and Ge bart, 2001; Keane, 2003; Leary et al ., 2004). Within the biomechanics community, little work has been performed using RS optimization methods with the exception of recent stud ies by Jung and Choe (1996), Chang et al (1999), and Hong et al. (2001). To our knowledge, no studies in the literature have used RS methods to perform optimizations of contact problems. 1.3 Approach The two goals of this thesis are first, to develop a computationally efficient RS optimization approach for evaluating a joint contact model’s ability to reproduce static experimental contact measurements, and second, to apply this approach to the evaluation of a discrete element contact model of th e tibiofemoral joint. Our specific hypotheses were that 1) quadratic RSs can accurately predict contact quantities (peak pressure, average pressure, contact area, and contact forces) computed by a discrete element contact model for small variations material modulus and relative pose of the contacting

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4 bodies, and 2) a discrete element contact m odel of the tibiofemoral joint can reproduce experimentally measured contact quantities th at involve averaging ac ross the surface (i.e., average pressure, contact area, contact force) but not quantities associ ated with specific locations on the surface (i.e., peak pressure ). Our study provides a new computationallyefficient approach for contact model evalua tion as well as a better understanding of the capabilities and limitations of discrete element contact models of natural human joints.

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5 CHAPTER 2 METHODS 2.1 Response Surface Optimization This section provides a general overview of RS approximation methods as well as specific modifications required to apply th ese methods to contact analyses. The RS method can be defined as a collection of sta tistical and mathematical techniques useful for constructing smooth approximations to functions in a multi-dimensional design space. Once a mathematical form has been selected, the coefficients of the approximate function (responses surface) are determined using da ta from either physical experiments or numerical simulations. The most common mathematical form for a RS is a low-degree polynomial. For example, a quadratic response surface with design variable inputs 1x and 2x and output y is formulated as 2 1 5 2 2 4 2 1 3 2 2 1 1 0x x x x x x y (1) where the i (i = 0,…,5) are the unknown coefficients to be fitted. A low degree polynomial minimizes the number of unknown coefficients and tends to smooth out noise in the function. Response surface approximations work best when the number of design variable inputs is small (< 10), since a la rge number of design va riables results in a complicated design space that is difficu lt to fit with low-degree polynomials To develop RS approximations for contac t problems, one must identify the design variable (DV) inputs, the output s to be predicted, and the mathematical form of the RS relating them. For contact model evaluation w ith static experimental data, the design

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6 variables are the six relative pose parameters (i .e., three translations and three rotations 6 DVs) and material modulus (1 DV) of the contacting bodies. The experimental uncertainty of these quantities can be estima ted and their values can be changed in the contact model. The RS outputs are peak pre ssure, average pressure, contact area, and contact force. These quantities can be cal culated by the contact model and measured experimentally for comparison. The hypothesize d mathematical form is a quadratic RS with one modification. For linear ly elastic Hertzian point contact, the peak pressure, average pressure, contact force, and contact area are all functions of interpenetration (vertical translation) to a pow er less than two, while the material modulus (assumed to be the same for both bodies) linearly scales e ach quantity except fo r area (Johnson, 1985). Thus, data for the RSs are generated using a material modulus of one, only the six pose parameters are used as RS inputs, and th e RS outputs (except ar ea) are scaled by the desired modulus value. With the RS formulation specified, the next step is to determine a sampling scheme within the design space to provide data for f itting the RS. Since this sampling process is only preformed once to generate the RSs, the computational cost of the contact analyses is paid only once up front. A quadratic response surface using k design variables will possess 2 / ) 2 )( 1 ( k k p unknown coefficients, where 6 ksince only pose parameters are used as RS inputs. Consequently, a minimum of 28 p data points must be sampled to perform the linear least-squares fit. However, to cover the design space in a systematic manner, we select a larger number of sample points using design of experiments (DOE) theory. Several DOE samp le criteria are avai lable, including the factorial design, face centered central composite design (FCCCD), and the D-optimality

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7 design. We choose the FCCCD criteria for its abil ity to sample all regions of the design space. For a quadratic RS, this approach utilizes 77 1 2 2 kk sample points, where the samples are taken at the center, th e corners, and the face centers of a k dimensional hypercube. For contact analyses, we ma ke two modifications to the FCCCD sampling scheme to improve the quality of the fit. The first modification accounts for infeasible points. A sampled point is deemed to be infeasible and is therefore omitted if the contact force and area predicted by the contact model are zero. Th is modification avoids fitting regions of the design space where no contact is occurr ing. The second modification accounts for outlier points. Once a RS is generated from f easible points, the RS output is compared to the computed value from the contact model fo r every sample point. The point with the largest absolute percent error above a pre-selected cut-off va lue of 10% (a typical value for engineering analyses) is omitted and the RS re-generated from the remaining sample points. The procedure is itera ted until all sample points are below 10% error. This modification avoids fitting regions of th e design space where only light contact is occurring, thereby providing a be tter fit in the regions of interest where the contact force is large. Omission of several sample points does not pose a problem to the fitting process since the FCCCD sampling scheme is highly over determined. After a RS is generated, the quality of th e resulting fit must be assessed, since a poor quality fit indicates that a different mathematical form for the RS should be considered. We use three common error measures for this purpose. All of these measures make use of the sum of th e squares of the errors SSE between predicted responses ˆiy from

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8 the RS and actual responses yi computed by the contact model, where n (28 n and 77 n) is the number of sample points used to generate the RS: 2 1) ˆ (i n i iy y SSE (2) The first measure of fit quality is the adjusted root-mean-square error (adjRMSE). Given the SSE, the root-mean square error (RMSE) can be calculated from n SSE RMSE (3) However, this measure will be zero if p n (i.e., no redundant points), even though the errors would not necessarily be ze ro at non-sampled points. To address this limitation, we choose a more conservative adjusted RMSE that uses p n (i.e., the number of degrees of freedom remaining in the fitting process) rather than n in the denominator of Eq. (3): p n SSE RMSEadj (4) To provide a relative measure of fit quality, we also compute the percent adjusted RMSE using p n SSE y RMSEadj ~ 100 % (5) where y ~ represents the magnitude of the fitted quantity: n i iy n y11 ~ (6) The second measure of fit quality is the adjusted coefficient of determination (adjR2). The coefficient of determination 2 R suffers from a similar problem to RMSE in

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9 that a perfect fit wi ll be indicated if p n Consequently, we used the adjusted 2 R value to account for the degrees of freedom p n remaining in the fit: 2 2 1/ 1 /1adj n i iSSEnp R yyn (7) where y is the mean of the actual responses. The final measure of fit quality is the RMSE calculated from the prediction error sum of squares ( PRESS ) statistic. To evaluate the pr edictive capability of a RS, one should ideally sample additional points distin ct from those used to generate the RS. However, this approach would require a sign ificant number of additional costly contact analyses. To circumvent this issue, the PRESS analysis excludes one sample point at a time from the set used to generate each RS The RS is regenerated using the remaining 1 n sample points and the prediction error at the omitted sample point calculated. This process is repeated for all n sample points, and the resulti ng errors are used to compute a PRESS -based SSE called the PRESS statistic. From there, a PRESS -based RMSE can be calculated from Eq. (3), where n rather than p n is used in the denominator since each error is calculated from a RS that omits that point. Once accurate RSs are generated for the outpu t quantities of interest, they are used in an optimization to evaluate the contact model’s ability to reproduce experimental measurements. Each time the optimization re quires a peak pressure, average pressure, contact force, or contact area from the contac t model, a response surface is used in place of a contact analysis to provide the value. By fitting quantities computed by the contact model, one can create any cost function that can be built up from the basic quantities. If the cost function was fitted directly usi ng its own response surface, then additional

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10 contact analyses would be required to genera te a new response surface each time the cost function was modified. With our approach, a wide variety of cost functions can be constructed without the need for any additional contact analyses. 2.2 Contact Pressure Experiments The response surface methodology described ab ove was used to evaluate a natural knee contact model’s ability to reproduce experimental contact measurements. The experiments were performed on a single cadav eric knee specimen cut approximately 15 cm above and below the joint line and showing no visible si gns of degenerative joint disease. Institutional review board approval was obtained for the te sting and subsequent modeling efforts. The menisci, fibula, and pa tella were removed from the specimen, and three titanium bone screws were inserted into the tibia and femur as landmarks for contact model alignment. The tibia and femur were potted in neutral alignment (MacWillams et al ., 1998) and mounted in a MTS Mi niBionix 858 servohydraulic test machine. The position and orientation of the femur were constrained using custom fixturing that allowed adjustment of the sagittal plane rotation and medial-lateral translation relative to the ram of the test machine. The axial plane position and orientation of the tibia were unconstrained usi ng a ball plate, thereby allowing the tibia to self-align with the femur once an axial load was applied. Using this set-up (Fig. 2-1), we collected four experimental quantities of interest from the medial and lateral compartments of the knee: contact force, peak pressure, average pressure, and contact area. The knee was fixed at a flexion angle of 30 and a Tekscan K-scan sensor (Tekscan, South Boston, MA) inserted anteriorly into the medial and lateral joint space. The medial-lateral po sition of the femur was adjusted to produce an approximately 70% medial-30% lateral load split between the two sides (Hurwitz et

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11 al. 1998; Schipplein and Andriacchi, 1991). The specimen was subjected to three trials of a 4 second ramp load from 200 to 1000 N. At the end of each ramp, the locations of the six screw heads were digitized using a Microscribe 3DX digitizer (Immersion Corp., San Jose, CA) possessing an accuracy of 0.23 mm. Figure 2-1. A human cadaver kn ee static experiment setup. A) The knee was potted in neutral alignment with six screws. B) The Tekscan K-scan sensor. C) The knee was mounted with fixed 30o flexion angle in a servohydraulic test machine with a sensor to measure intr a-articular contact quantities. D) The close-up view of the contact area. Drift in the Tekscan sensor (Otto et al. 1999) was eliminated by post-calibrating each trial with the manufacturer-suggested two-point calibration procedure using the known loads at the start and end of the ramp. Crinkling of the sensor (Harris et al. 1999), which introduces erroneous pressures on se nsels outside the true contact area, was accounted for by determining the pressure cu t-off value (0.05 MPa) above which little

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12 additional drop in contact area occurs when pressures below this value are set to zero (Fregly et al ., 2003). Contact quantities measured with the Tekscan sensor were therefore calculated by ignoring all sensels with a pressure below 0.05 MPa. Following Tekscan sensor calibration and pressure cut-off determination, the four experimental quantities of interest were calculated on each side for a pplied loads of 500 and 1000 N. Peak pressure was calculated using the aver aging function in the Tekscan software, thereby reducing the effect of local sensor “hot spots” on th is quantity. The resulting data from the K-scan sensor (Table 2-1) and the digitizer were av eraged over the three trials to facilitate contact model evaluation under two loading conditions. Table 2-1. The averaged experimental measurements were collected from three compression trials for both loads processing on the same cadaver knee via servohydraulic test machine Experimental Load Experimental Quantity Side 500 N 1000 N Force (N) Medial317 4 658 5 Max Pressure (MPa) 4.10 0.05 7.94 0.14 Ave Pressure (MPa) 1.11 0.03 2.07 0.02 Area (mm2) 287 10 318 4 Force (N) Lateral183 4 337 5 Max Pressure (MPa) 1.51 0.03 2.63 0.04 Ave Pressure (MPa) 0.79 0.01 1.33 0.03 Area (mm2) 229 4 252 5 2.3 Knee Model Creation Prior to experimental contact testing, MRI (magnetic resonance imaging) and CT (computed tomography) data were collected from the same cadaveric specimen for purposes of contact model creati on. Sagittal plane MRI data were collected using a 3.0-T GE Signa Horizon LX scanner with a quadr ature knee coil. A T2-weighted 3D FastGRE sequence was used with a 1 mm slice th ickness, 256 x 256 image matrix (0.625 0.625

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13 mm pixel size), and 160 x 160 mm field of view. Axial CT data were collected from the same specimen using a GE LightSpeed QX/i scanner in helical mode. The scanning parameters were a 1.25 mm overlapping sli ce thickness, 512 x 512 image matrix (0.313 0.313 mm pixel size), and 160 mm x 160 mm fi eld of view. The tibia, femur, and bone screws in both data sets were segmented (Fig. 2-2) using commercial image processing software (SliceOmatic, Tomovision, Montreal, CA). Figure 2-2. Original and segmented medical images. A) Original CT slice. B) Segmented CT slice. C) Original MRI slice. D) Segmented MRI slice. The menisci were not segmented and were omitted from the model. Articular cartilage and subchondral bone surfaces were segmented manually from the MRI data,

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14 while cortical bone and bone screw surfaces we re segmented semi-automatically from the CT data using a watershed algorithm. The point clouds from both scans were exported for subsequent surface creation. Commercial reverse engineering software (Geomagic Studio, Raindrop Geomagic, Research Triangle Park, NC) was used to conve rt the MRI and CT point cloud data into a combined geometric model for contact analysis. Point clouds from each imaging modality were imported sepa rately and converted to polygonal surface models. The subchondral bone surfaces from MRI were regi stered automatically to the corresponding cortical bone surfaces from CT, creating a composite geometric model with articular cartilage surfaces from MRI and cortical bone and bone screw surfaces from CT. NURBS (Non-Uniform Rational B-Spline) surfaces were fitted to the polygonal models, with the tolerance (mean standard de viation) between the origin al point clouds from MRI and the final NURBS surfaces being 18 0 18 0 mm for the femur and 29 0 20 0 mm for the tibia (Fig. 2-3). Figure 2-3. The anterior and posterior view s of the A) 3D point clouds and B) 3D NURBS model with six screws cr eated from CT and MRI images The NURBS surfaces for the tibia and femur (articular cartilage, cortical bone, and bone screws) were imported into Pro/ME CHANICA MOTION (Parametric Technology

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15 Corporation, Waltham, MA) to construct a multibody contact model. The mean digitized bone screw locations were also imported to determine the nominal alignment of the tibia and femur. For both bones, a stiff linear sp ring was placed between each screw head and its mean experimental location and a static an alysis performed to de termine the pose that best matched the experiments. Differences between the digitized and nominal bone screw locations were on the order of 1 mm. Starting from these nominal poses, the tibia was fixed to ground and the femur connected to it via a 6 degree-of-freedom (DOF) joint. Custom contact code was incorporated into the multibody model and used to solve for the medial and lateral contact conditions as a function of the 6 DOFs between the two bones (Bei and Fregly, 2004). The contact code implemente d two versions of a linear elastic discrete element contact model. The first was a small strain version, where the contact pressure p for each contact element on the tib ial articular surfaces was calculated from (An et al. 1990; Blankevoort et al ., 1991; Li et al., 1997) h d E p ) 2 1 )( 1 ( ) 1 ( (8) where E is Young’s modulus of the articular cartilage, v is Poisson’s ratio, h is the combined thickness of the femoral and tibial articular cartilage, and d is the interpenetration of the undefo rmed contact surfaces. Both h and d were calculated on an element-by-element basis using the ACIS 3D Toolkit (Spatial Corporation, Wesminster, CO). For large strains, a s econd version of the model was implemented that accounted for geometric nonlinear behavior (Blankevoort et al ., 1991): (1) ln1 (1)(12) Ed p h (9)

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16 For both versions, a dense contact element grid of 50 x 50 was used for the medial and laterial articular surfaces of the tibia. Th e femoral and tibial articular cartilage were assumed to be linear elastic and isotropi c with Poisson’s ratio = 0.45 (Blankevoort et al. 1991). Young’s modulus was set to 1 MPa to faci litate its use as a de sign variable in the response surface optimizations. 2.4 Knee Model Evaluation Seventy seven contact analyses were pe rformed with the model to provide the sample points necessary to generate respons e surfaces using the FCCCD. Each sample point represented a different pose of the femur relative to the tibia within the neighborhood of the nominal pose. This nei ghborhood was defined to be 1 mm and 1 based on the estimated envelope of e xperimental pose uncertainty. Though this envelope appears small, it corre sponds to large changes in co ntact conditions. Within this envelope, response surfaces were generated as described above for the medial and lateral contact force, peak pressure, average pressu re, and contact area computed by the contact model. The optimization cost function ) (xg constructed from these response surfaces sought to match experime ntal average pressures avep in both compartments simultaneously with a penalty term to ensure the experimental contact forces F were reproduced: 2 2 2 2)) ( ˆ ( )) ( ˆ ( )) ( ˆ ( )) ( ˆ ( ) (lateral medial lateral ave ave medial ave aveF E F F E F w p E p p E p E g x x x x x (10) where ˆ F x and ˆavepx are the force and average pr essure predicted by response surfaces.

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17 In this equation, x represents the 6 pose parameter design variables, E is the material modulus seventh design variable, and w = 1000 is the weight of the penalty term. This cost function mimics the results of a st atic analysis, since contact force is matched closely while minimizing errors in the most reliable experimental measure of interest. The form of Eq. (10) was specified by follo wing a trial-and-error a pproach that included different quantities in the cost function. A larger value of w was not used since it resulted in a poor match to the other contact quantities. To seek the global minimum, we pe rformed 500 nonlinear least-squares optimizations using the response surfaces from each contact model and the Matlab Optimization Toolbox (The Mathworks, Natic k, MA). Uniformly distributed random initial guesses were selected within the bounds 1 for the firs t six design variables and 1 to 10 for the seventh. The best set of design variables was selected based on the smallest cost function value from the 500 optimi zations. A final Pro/MECHANICA contact analysis was performed for both contact mode ls using the optimized design variables to verify the accuracy of the response surface approximations.

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18 CHAPTER 3 RESULTS All adjRMSE % and pressRMSE % results were less than 5% while all adjR2 values were greater than or equal to 0.995. Error measures from the small and large strain versions of the contact model were generally close. pressRMSE % was always larger than adjRMSE% but not dramatically so. Furthermore, adjRMSE% and pressRMSE% were of comparable magnitude to the variability in the corresponding experimental measurements (Table 3-1). For each RS, a minimum of 7 a nd maximum of 26 infeasible/outlier points were eliminated from the original FCCCD set of 77 sample points, and all of these points involved a superior transl ation of +1 mm (i.e., no c ontact or light contact). Based on the best results from the 500 RS optimizations, the contact quantities computed from response surfaces were compar ed to both versions of the contact model (Fig. 3-1). The relative errors from all cont act quantities were below 10%. Both versions of the contact model could reproduce all expe rimentally measured contact quantities to within 10% error at both loads (Fig. 3-2) The one exception was peak pressure, which was only matched to within 50% error. Contact force was matched to within 1% error, consistent with the use of a penalty term on this quantity in the cost function. Average pressure and contact area errors were always in opposite directions, consistent with matching contact force closely. Peak pressu re was over-predicted by the contact models on the lateral side and under-predicted on th e medial side (worst error). The optimal value of Young’s modulus was 2.6 MPa for th e small strain model and 2.3 MPa for the

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19 large strain one. The best result from 500 optimizations never hit the upper or lower bounds on the pose parameter design variables. Each set of 500 optimizations required approximately 90 seconds of CPU time on a 2.8 GHz Pentium 4 PC.

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20Table 3-1. Comparison of response surface predictions to data points sampled from large and small strain contact models. adjRMSE adj%RMSE pressRMSE press%RMSE adj 2R Predicted Quantity Side LargeSmall LargeSmall Large Small LargeSmall LargeSmall Force (N) Medial 4.67 4.24 0.601 0.674 10.0 6.13 1.29 0.976 0.999 0.999 Max Pressure (MPa) 0.065 0.035 1. 53 1.16 0.090 0.048 2.13 1.58 0.999 0.999 Ave Pressure (MPa) 0.059 0.048 3. 31 3.19 0.078 0.064 4.40 4.25 0.999 0.999 Area (mm2) 10.8 10.8 3.50 3.50 14. 3 14.3 4.64 4.64 0.995 0.995 Force (N) Lateral 2.12 2.12 0.788 0.886 2.92 2.85 1.09 1.19 0.999 0.999 Max Pressure (MPa) 0.030 0.018 1. 15 0.815 0.039 0.023 1.49 1.06 0.999 0.999 Ave Pressure (MPa) 0.026 0.024 2. 24 2.27 0.033 0.030 2.91 2.97 0.998 0.998 Area (mm2) 4.54 4.54 2.58 2.58 6. 00 6.00 3.41 3.41 0.996 0.996

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21 Figure 3-1. Percent errors between respons e surfaces and predicted results from both contact models. They were calculated fo r A) contact force, B) contact peak pressure, C) contact averaged pressure and D) contact area. Pose parameters and Youngs modulus were defined ba sed on the best result from 500 optimization runs.

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22 Figure 3-2. Percent errors between experiment al and predicted resu lts from both contact models. They were calculated for A) cont act force, B) contact peak pressure, C) contact averaged pressure and D) contact area. Pose parameters and Young’s modulus were defined based on th e best result from 500 optimization runs.

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23 CHAPTER 4 DISCUSSION This thesis has presented a novel resp onse surface optimiza tion methodology for evaluating a joint contact model’s ability to reproduce static experimental contact measurements. The approach modifies tr aditional RS approximation methods for application to contact analyses. By replaci ng computationally-costly contact analyses with quadratic RS approximations, optimizations that vary the relative pose of the contacting bodies can be performed rapidl y to minimize differences between model predictions and experimental measurements. Evaluation of a discrete element contact model of the tibiofemoral joint constructed from CT and MRI data revealed that quadratic RSs can accurately approximate m odel outputs within a small envelope of relative pose uncertainty. Furthermore, optimization studies utilizing these RSs demonstrated that the model could reproduce the contact force, average pressure, and contact area. However, the peak pressure measured experimentally on the medial and lateral sides could not be repr oduced. This finding suggests th at discrete element contact models of natural human joints that util ize homogeneous, isotropic material properties may be best suited for analyses such as multibody dynamic simulations where obtaining correct contact forces by integrating over th e surfaces is more critical than obtaining correct contact pressures at spec ific locations on the surfaces. Use of RSs to replace repeated contac t analyses in optimization studies is worthwhile for several reasons. First, RS te nds to smooth out noise in the design space. During optimization, the risk of entrapment in a local minimum is therefore reduced.

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24 Second, RS approximations are computationally efficient. Rather th en repeating costly contact analyses during an optimization, a si ngle set of contact an alyses is performed once up front to generate the necessary RS approximations for output quantities of interest. Extremely fast function evaluations allow one to search for the global optimum using either repeated gradient-based optimi zation starting from multiple initial guesses, as performed in our study, or global op timization employing a large population size. Third, RS optimizations are convenient to im plement. Optimized contact solutions can be founded utilizing any off-the-shelf optimi zation algorithm once the RS approximations are constructed. Fourth, a vari ety of optimization problem formulations can be evaluated quickly. Each contact quantity that could pot entially appear in the cost function or constraints can be fitted with its own RS. A wi de variety of cost functions can then be constructed by weighting contributions from th e different RSs. Fifth, RS approximations facilitate the calculation of an alytical derivatives for gradient-based optimization. This benefit is the direct result of having analytical representations for the contact quantities of interest as a function of the design variables. The caveat to these benefits is that an appropriate mathematical form (polynomial or otherwise) must be identified to represent the responses of interest as a function of the design variable inputs, which is not always possible. The primary deficiency of the discrete el ement contact model was its inability to match the peak contact pressures measured experimentally. Peak pressures were not included in the cost function since they are se nsitive to local inhomogeneities in cartilage material properties and Tekscan sensor re sponse. When we replaced average pressures with peak pressures in the co st function for curiosity, we f ound that peak pressure errors

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25 decreased on the medial side by about 25% but increased on the la teral side by roughly 50%, while average pressure and contact area errors increased by approximately 30% and 20%, respectively. Thus, a variety of other f actors likely contributed to the large peak pressure errors, including insertion of a rela tively stiff sensor into the joint space (Wu et al., 1998), small inaccuracies in the surface geom etry, and local variations in material properties. Underprediction of peak pressure by the model on the moderately conformal medial side and overpredicti on on the non-conformal lateral side is consistent with observations made by Wu et al. (1998) on how insertion of a relatively stiff sensor into the joint space affects peak pressure measur ements as a function of contact conformity. Contact model limitations may have also contributed to the poor match of the peak pressure data. Though a strength of our model is that it accounts for local variations in cartilage thickness, a weakness is that it does not account for lo cal variations in cartilage material properties. Though homoge neous, isotropic cartilage material models have been used in the literature (Bendjaballah et al., 1995; Blankevoort et al., 1991; Haut et al., 2002; Pri and Hobatho, 1998), recent studies have reporte d significant local variations in Young’s modulus Poisson’s ratio and thickne ss for articular cartilage (Jurvelin et al., 2000; Laasanen et al., 2003). Mukherjee and Wayne (1998) suggested that regions with the highest Young’s modul us correspond to regions with the highest contact pressure and cartilage thickness. This observati on fits our under-prediction of peak pressure on the medial side, where an increase in Young’s modulus would produce improved agreement with the large measured peak pressures. In addition, our elastic contact model does not account for the effect of time-dependent fluid flow on contact pressures or areas as captured by bi phasic models of cartilage (Han et al., 2004; Mow et

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26 al., 1980; Mow et al., 1982). However, for loading over a short time period, an elastic model may still provide a r easonable approximation of the in vivo situation depending on the intended application of the model (Donzelli et al., 1999; Mow et al., 1982; Shepherd and Seedhom, 1997;). The value of Young’s modulus predicted by our optimizations is consistent with studies found in the literatur e. Experimental studies of the compressive modulus of articular cartilage have reported values as low as 2.0 MPa for short time frame loading (Setton et al., 1999; Shepherd and Seedhom, 1997). Other studies have reported the modulus of the solid phase alone to be in the neighborhood of 0.3 MPa (Hasler et al., 1999). Thus, the best-fit modulus values of 2.3 and 2.6 MPa found in our study are consistent with these numbers. Other discrete element knee studies have utilized an elastic modulus of 4 MPa (Cohen et al., 2003; Kwak et al., 2000) which again is close to our optimized modulus values. Furthermore, if our estimate of Poisson’s ratio is decreased slightly from 0.45 to 0.4, our optimized value of Young’s modulus will increase from 2.6 to 4.6 MPa, closer to the middle of the range commonly reported (Setton et al., 1999; Shephard and Seedhom, 1997). A weakness of our study is that only a single knee specimen was evaluated. Significant modeling effort is required to cr eate a knee model with ar ticular cartilage and subchondral bone geometry that matches a par ticlar cadaver specimen. This may explain why other studies that utili zed specimen-specific geometric models only analyzed a single specimen as well (Bendjaballah et al., 1995; Haut et al., 2002; Li et al., 1999). The knee used in our study was part of a la rger experimental study involving 20 knees. However, we were able to collect CT and MRI data from only two of those knees. Since

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27 the contact area for the second specimen wa s not completely contained within the boundary of the Tekscan sensor at the maximu m load of 1000 N, we were not able to post-calibrate those experimental contact data and use them for a second model evaluation. Comparison of m odel predictions with experimental measurements from additional knee specimens using a range of flexion angles and loads would provide further verification of the cap abilities and limitations of discrete element models of natural joints.

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28 CHAPTER 5 SUMMARY AND FUTURE STUDY 5.1 Summary In summary, this thesis has presented a computationally efficient RS optimization for predicting knee contact measurements using a 3D simulation model. The more computational costly is each contact analys is; the more beneficial is the use of RS optimization method. The present implementatio n works for the tibiofemoral joint of the natural knee with either large or small stra in contact model. The RS optimization can predict contact forces, areas and average pressures well with the exception of peak pressures. Our ultimate goal is to incorporate the elastic foundation contact model into a full-body dynamic musculoskeletal model. Th e resulting full-body model can then be utilized to study joint contact pressures during motion. Futu re research is required to evaluate whether the peak pressure errors are due to experiment al or contact model inaccuracies. 5.2 Future Study While our results showed great potential to predict cont act quantities that involve averaging across the surface, there are some ex tra steps that could be taken to improve the accuracy of RS. Instead of using two modi fications in this thesis to remove the outliers ,there is another more robust proce dure called iterativ ely re-weighted least square (IRLS) fitting (Holland and Welsch, 1977). It can be utili zed to remove or weight down outliers. Additionally, varying numbers of insignificant coefficients for each RS was found based on t-statistics analysis. While it wa s not performed in this study, there are

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29 several statistical methods, (e.g., a stepwise regression procedure) that can be used to discard those insignificant coefficien ts to improve the prediction accuracy. In addition to 500 optimizations, another 50 groups of optimizations were run. The best result was selected from each group containing 100 optimizations with different initial guesses. Within 50 sets of optimized design variables, the maximum variations in translation and rotation were 1.90 mm and 1.95 respectively. Compared to the range of design space, these variations suggested that the final result from 500 optimizations may not be unique. Thus an alternative cost f unction or type of different RS method (e.g., neural networks and kriging) could be another candidate for future improvement. Given how well RSs could approximate cont act model outputs for a small range of relative pose variations, our next step will be to investigate whether RSs can be used to generate extremely fast forward dynamic contact simulations that util ize a wider range of relative pose variations. The co mputational cost of repeated geometry evaluations is the current limiting factor to th e incorporation of deformable contact models into multibody dynamic simulations of human joints (Bei and Fregly, 2004). However, if accurate response surfaces could be generated to repr esent the net force and torque calculated about a pre-selected point on the tibia (e.g., th e origin of the tibial coordinate system), then extremely fast contact solutions could be produced for any given set of relative pose parameters. Though additional design variables would need to be included to account for friction or damping effects, these effects ar e expected to be small for most modeling situations involving human joints. Higher degree polynomials or some other mathematical formulation would likely be requ ired to represent a wider range of relative pose variations.

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30 LIST OF REFERENCES Ahn, C.-S. & Kim, K.-Y. (2003). Aerodynamic design optimization of a compressor rotor with Navier-Stokes analysis. Proceedings of the Inst itution of Mechanical Engineers, Part A: J ournal of Power and Energy, 217, 179-18. An, K.N., Himeno, H., Tsumura, H., Kawa i, T., & Chao, E.Y.S. (1990). Pressure distribution on articular surfaces: applic ation to joint stability evaluation. Journal of Biomechanics, 23, 1013-1020. Bei, Y. & Fregly, B. J. (2004). Multibody dynamic simulation of knee contact mechanics. Medical Engineering & Physics (submitted). Bendjaballah, M.Z., Shirazi-Adl, A., & Zukor D.J. (1995). Biomechanics of the human knee joint in compression: Reconstruction, mesh generation and Finite element analysis. The Knee, 2, 69-79. Bendjaballah, M.Z., Shirazi-Adl, A., & Zukor D.J. (1997). Finite element analysis of human knee joint in varus-valgus. Clinical Biomechanics, 12, 139-148. Bendjaballah, M.Z., Shirazi-Adl, A., & Zukor D.J. (1998). Biomechanical response of the passive human knee joint und er anterior-posterior forces. Clinical Biomechanics, 13, 625-633. Blankevoort, L., Kuiper, J.H., Huiskes, R., & Grootenboer, H.J. ( 1991). Articular contact in a three-dimensional model of the knee. Journal of Biomechanics, 24, 1019-1031. Burman, J. & Gebart, B.R. (2001). Influen ce from numerical noise in the objective function for flow design optimization. International Journal of Numerical Methods for Heat and Fluid Flow, 11, 6-19. Chang, P.B., Williams, B.J., Santer, T.J., Notz, W.I., & Bartel, D.L. (1999). Robust optimization of total joint replacements incorporating environmental variables. Journal of Biomec hanical Engineering, 121, 304-310. Cohen, Z.A., Henry, J.H., McCarthy, D.M ., Mow, V.C., & Ateshian, G.A. (2003). Computer simulations of patellofemoral joint surgery. The American Journal of Sports Medicine, 31, 87-98. Dhaher, Y.Y. & Kahn, L.E. (2002). The effect of vastus medialis forces on patellofemoralcontact: a model-based study. Journal of Biomechanical Engineering, 124, 758-767.

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31 Donahue, T.L.H., Rashid, M.M., Jacobs, C.R ., & Hull, M.L. (2002). A finite element model of the human knee joint for the study of tib io-femoral contact. Journal of Biomechanical Engineering, 124, 273-280. Donzelli, P. S., & Spilker, R. L. (1998). A contact finite element formulation for biological soft hydrated tissues. Computer Methods in Applied Mechanics and Engineerin, 153, 63–79. Donzelli, P.S., Spilker, R.L., Ateshian, G.A. & Mow, V.C. (1999). Contact analysis of biphasic transversely isotropi c cartilage layers and correlations with tissue failure. Journal of Biomechanics, 32, 1037-1047. Fregly, B.J., Bei, Y., & Sylvester, M.E. (2003). Experimental evaluation of a multibody dynamic model to predict contact pressures in knee replacements. Journal of Biomechanics, 36, 1658-1668. Haider, M.A., & Guilak, F. (2000). Axis ymmetric boundary in tegral model for incompressible linear viscoelasticity: A pplication to the micropipette aspiration contact problem. Journal of Biomechanical Engineering, 122, 236-244. Han, S.-K., Federico, S., Epstein, M., & Herz on, W. (2004). An articul ar cartilage contact model based on real surface geometry. Journal of Biomechanics (in press). Harris, M.L., Morberg, P., Bruce, W.J.M ., & Walsh, W.R. (1999). An improved method for measuring tibiofemoral contact areas in total knee arthroplasty: A comparison of K-scan sensor and Fuji film. Journal of Biomechanics, 32, 951-958. Hasler, E.M., Herzog, W., Wu, J.Z., Muller, W., & Wyss, U. (1998). Articular cartilage biomechanics: theoretical models, material properties, and biosynthetic response. Critical Reviews in Biomedical Engineering, 27, 415-488 Haut, T.L., Hull, M.L., Rashid, M.M., & Jacobs C.R. (2002). A finite element model of the human knee joint for the study of tibio-femoral contact. Journal of Biomechanical Engineering, 124, 273-280. Herzog, W., Longino, D., & Clark, A. (2003). The role of muscles in joint adaptation anddegeneration. Langenbeck’s Archives of Surgery, 388, 305-315. Holland, P.W., & Welsch, R.E. (1977). Robust regression using iter atively reweighted least-squares. Communications in Statistics: Theory and Methods, 6, 813-827 Hong, J.H., Mun, M.S., & Song, S.H. (2001). An optimum design methodology development using a statistical tech nique for vehicle occupant safety. Proceedings of the Institution of Mec hanical Engineers, Part D: Journal of Automobile Engineering, 215, 795-801.

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32 Hurwitz, D.E., Sumer, D.R., Andriacchi, T.P., & Sugar, D.A. (1998). Dynamic knee loads during gait predict proximal tibia l bone distribution. Journal of Biomechanics, 31, 423-430. Jansson, T., Nilsson, L., & Redhe, M. ( 2003). Using surrogate models and response surfaces in structural optimization With application to crashworthiness design and sheet metal forming. Structural and Multidis ciplinary Optimization, 25, 129-140. Johnson, K.L. (1985). Contact Mechanics. Cambridge University Press, New York. Jung, E.S. & Choe, J. (1996). Human reach posture prediction based on psychophysical discomfort. International Journal of Industrial Ergonomics, 18, 173-179. Jurvelin, J.S., Arokoski, J.P., Hunziker, E. B., & Helminen, H.J. (2000). Topographical variation of the elastic properties of articular cartilage in canine knee. Journal of Biomechanics, 33, 669-75. Kaufman, K.R., Kovacevic, N., Irby, S.E., & Colwell, C.W. (1996). Instrumented implant for measuring tibiofemoral forces. Journal of Biomechanics, 29, 667-671. Keane, A.J. (2003). Wing optimization using de sign of experiment, response surface, and data fusion methods. Journal of Aircraft, 40, 741-750. Komistek, R. D., Dennis, D. A., & Mahfouz, M. (2003). In vi vo fluoroscopic analysis of the normal human knee. Clinical Orthopaedics and Related Research, 410, 69-81. Kuo, C.H. & Keer, L.M. (1993). Contact stress and fracture anal ysis of articular cartilage. Biomedical Engineering App lications Basis Communications, 5, 515-521. Kwak, S.D., Blankevoort, L., & Ateshian, G. A. (1999). A mathematical formulation for 3D quasi-static multibody models of diarthrodial joints. Computer Methods in Biomedical Engineering, 3, 41-64. Laasanen, M.S., Toyras, J., Korhonen, R.K., Rieppo, J., Saarakkala, S., Nieminen, M.T., Hirvonen, J., & Jurvelin, J.S. (2003). Bi omechanical properties of knee articular cartilage, Biorheology, 40, 133-140. Leary, S.J., Bhaskar, A., & Keane A.J. (2004). Global approxima tion and optimization using adjoint computationa l fluid dynamics codes. AIAA Journal, 42, 631-641. Li, G., Sakamoto, M., & Chao, E.Y.S. (1997) A comparison of different methods in predicting static pressure distribution in articulating joints. Journal of Biomechanics, 30, 635-638. Li, G., Gil, J., Kanamori, A., & Woo, S.L.-Y. (1999). A validated 3D computational model of a human knee joint. Journal of Biomech anical Engineering, 121, 657-662.

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33 Li, G., Lopez, O., & Rubash, H. (2001). Va riability of a 3D fi nite element model constructed using MR Images of a kn ee for joint contact stress analysis. Journal of Biomechanical Engineering, 123, 341-346. Liu, B., Haftka, R.T., & Akgun, M.A. (2000 ). Two-level composite wing structural optimization using response surfaces. Structural and Multidisciplinary Optimization, 20, 87-96. MacWilliams, B.A., DesJardins, J.D., Wilson, D.R., Romero, J., & Chao, E.Y.S. (1998). A repeatable alignment method and local coordinate description for knee joint testing and kinematic measurement. Journal of Biomechanics, 31, 947-950. Mow, V.C., Kuei, S.C., Lai, W.M., & Armstr ong, C.G. (1980). Biphasic creep and stress relaxation of articular cartilage in compression: Theory and experiments. Journal of Biomechanical Engineering, 102, 73-84. Mow, V.C., Lai, W.M., & Holmes, M.H. (1982). Advanced theoretical and experimental techniques in cartilage research. In Biomechanics: Principles and Applications (edited by R. Huiskes, D. van Campen, and J. DeVijn), Martinus Nijhoff Publishers, Boston, 63-69 Mukherjee, N. & Wayne, J.S. (1998). Load sharing between solid and fluid phases in articular cartilage: I--Experimental determina tion of in situ mechanical conditions in a porcine knee. Journal of Biomechan ical Engineering, 120, 614–619. Otto, J.K., Brown, T.D., & Callaghan, J. J. (1999). Static and dynamic response of a multiplexed-array piezoresistive contact sensor. Experimental Mechanics, 39, 317323. Pandy, M.G. & Sasaki, K. (1998). A three-di mensional musculoskeletal model of the human knee joint. Part 2: An alysis of ligament function. Computer Methods in Biomechanics and Biomedical Engineering, 1, 265-283. Papila, N., Shyy, W., Griffin, L., & Do rney, D.J. (2002). Shape optimization of supersonic turbines using global approximation methods. Journal of Propulsion and Power, 18, 509-518. Pri, D. & Hobatho, M.C. (1998). In vivo dete rmination of contact ar eas and pressure of the femorotibial joint using non-linear finite element analysis. Clinical Biomechanics, 13, 394-402. Piazza, S.J. & Delp, S.L. (2001). Three-dimensional dynamic simulation of total knee replacement motion dur ing a step-up task. Journal of Biomec hanical Engineering, 123, 599-606. Rikards, R., Abramovich, H., Auzins, J., Ko rjakins, A., Ozolinsh, O., Kalnins, K., & Green, T. (2004). Surrogate models for optimum design of stiffened composite shells. Composite Structures, 63, 243-251.

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34 Roux, W.J., Stander, N., & Haftka, R.T. (1998). Response surface approximations for structural optimization. International Journal fo r Numerical Methods in Engineering, 42, 517-534. Sevant, N.E., Bloor, M.I.G., & Wilson, M.J. (2000). Aerodynamic design of a flying wing using response surface methodology. Journal of Aircraft, 37, 562-569. Schipplein, O.D. & Andriacchi, T.P. (1991). Interaction between active and passive knee stabilizers during level walking. Journal of Orthopaedic Research, 9, 113-119. Setton, L.A., Elliot, D.M., & Mow, V.C. (1 999). Altered mechanics of cartilage with osteoarthritis: human osteoarthritis and an experimental model of joint degeneration. Osteoarthritis and Cartilage, 7, 2-14. Shepherd, D.E.T. & Seedhom B.B. (1997). A technique for measuring the compressive modulus of articular cartilage under physio logical loading rate s with preliminary results. Journal of Engin eering in Medicine, 211, 155-165. Stolk, J., Verdonschot, N., Cristofolini, L., T oni, A., & Huiskes, R. (2002). Finite element and experimental models of cemented hip joint reconstructions can produce similar bone and cement strains in pre-clinical tests. Journal of Biomechanics, 35, 499-510. Tashman, S. & Anderst, W. (2003). In-vivo measurement of dynamic joint motion using high speed biplane radiography and CTL: ap plication to canine ACL deficiency. Journal of Biomec hanical Engineering, 125, 238-245. Wu, J.Z., Herzog, W., & Epstein, M. (1998). Ef fects of inserting a pressensor film into articular joints on the actual contact mechanics. Journal of Biomechanical Engineering, 120, 655-659.

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35 BIOGRAPHICAL SKETCH Yi-Chung Lin was born on August 14, 1976, in Kaohsiung, Taiwan. In 1998, he received the bachelors degree in the Depa rtment of Mechanical Engineering of the National Cheng Kung University, Taiwan He joined the Computational and Biomechanics Laboratory of Assistant Professo r B. J. Fregly in the fall of 2002. He is planning to enter the doctoral program in mechanical and aerospace engineering after finishing the master's program.


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Title: Experimental Evaluation of a Natural Knee Contact Model Using Response Surface Optimization
Physical Description: Mixed Material
Copyright Date: 2008

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EXPERIMENTAL EVALUATION OF A NATURAL KNEE CONTACT MODEL
USINTG RESPONSE SURFACE OPTIMIZATION



















By

YI-CHIUNG LIN


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2004
































Copyright 2004

by

Yi-Chung Lin
















ACKNOWLEDGMENTS

First, I would like to sincerely thank Dr. Benj amin J. Fregly, my advisor, for the

direction he has given me throughout my research. His encouragement and dedication

have greatly contributed to the accomplishment of this thesis. I would also like to thank

Dr. Raphael T. Haftka and Dr. Andrew Kurdila for being my committee members. It was

a great honor to have them both.

I would also like to thank all the Computational Biomechanics Lab members for

their support in my past and present work, with special thanks to Dong, Yanhong, Jaco

and Jeff for their assistance and suggestion.





















TABLE OF CONTENTS

Page


ACKNOWLEDGMENT S .........__.. ..... .__. .............._ iii..


LIST OF TABLES ........._.___..... .__. ...............v....


LIST OF FIGURES .............. ....................vi


AB STRAC T ................ .............. vii


CHAPTER


1 INTRODUCTION ................. ...............1.......... ......


1.1 Need for Accurate Contact Model ................ ...............1..............
1.2 Need for Efficient Contact Model Evaluation ................. .......... ................2
1.3 Approach............... ...............3


2 M ETHODS .............. ...............5.....


2.1 Response Surface Optimization............... ..............
2.2 Contact Pressure Experiments ................ ...............10................
2.3 Knee Model Creation............... ...............12
2.4 Knee Model Evaluation ................ ...............16................


3 RE SULT S ................. ...............18.......... .....


4 DI SCUS SSION ................. ...............23................


5 SUMMARY AND FUTURE STUDY ................. ...............28........... ...


5.1 Summary ................. ...............28................
5.2 Future Study ................. ...............28........... ...


LIST OF REFERENCES ................. ...............30........... ....


BIOGRAPHICAL SKETCH .............. ...............35....










1V

















LIST OF TABLES


Table pg

2-1. The averaged experimental measurements were collected from three compression
trials for both loads processing on the same cadaver knee via servohydraulic test
m machine .............. ...............12....

3-1. Comparison of response surface predictions to data points sampled from large and
small strain contact model s. ............. ...............20.....


















LIST OF FIGURES

Figure pg

2-1. A human cadaver knee static experiment setup .........._._ ....... ..._ .............._.11

2-2. Original and segmented medical images ..........._..__......_. ......._._..........1

2-3. The anterior and posterior views of the A) 3D point clouds and B) 3D NURBS
model with six screws created from CT and MRI images .............. ....................14

3-1. Percent errors between response surfaces and predicted results from both contact
m odels .............. ...............21....

3-2. Percent errors between experimental and predicted results from both contact
m odels.. ............ ...............22.....
















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

EXPERIMENTAL EVALUATION OF THE NATURAL KNEE CONTACT MODEL
USINTG RESPONSE SURFACE OPTIMIZATION

By

Yi-Chung Lin

August 2004

Chair: Benjamin Jon Fregly
Major Department: Mechanical and Aerospace Engineering

Finite element, boundary element, and discrete element models have been

employed to predict contact conditions in human j points. When optimization is used to

evaluate the ability of such models to reproduce experimental measurements, the high

computational cost of repeated contact analysis can be a limiting factor. This thesis

presents a computationally-efficient response surface optimization methodology to

address this limitation. Quadratic response surfaces are fit to contact quantities (peak

pressure, average pressure, contact area, and contact force) predicted by a joint contact

model for various combinations of material modulus and relative pose (i.e., position and

orientation) of the contacting bodies. The response surfaces are used as surrogates for

costly contact analyses in an optimization that minimizes differences between measured

and predicted contact quantities. The approach is demonstrated by evaluating a linear

elastic discrete element contact model of the tibiofemoral j oint, where the model was

created using CT and MRI data from the same cadaveric specimen used in static pressure










experiments. For variations in material modulus and relative bone pose within the

envelope of experimental uncertainty (+ 1 mm and lo), quadratic response surfaces

accurately predicted contact quantities computed by the discrete element model. Using

these response surfaces, 500 optimizations with different initial guesses were performed

in less than 90 seconds. For a flexion angle of 300 and axial loads of 500 and 1000 N, the

optimizations demonstrated that small and large strain versions of the contact model

could match all experimentally measured contact quantities to within 10% error with the

exception of peak contact pressure, which was in error by as much as 85%. Thus, discrete

element models of natural j points may be best suited for predicting contact quantities that

involve averaging across the surface but not quantities associated with specific locations

on the surface.















CHAPTER 1
INTTRODUCTION

1.1 Need for Accurate Contact Model

According to the Arthritis Foundation, there were nearly 43 million Americans

with arthritis or chronic j oint symptoms in 1998. The number went up to 70 million in

2001 and most likely will keep climbing due to the rising number of aging Baby boomers.

This foundation also reported that arthritis limits daily activities such as walking, running

and dressing for more than seven million Americans. There are many different types of

arthritis including osteoarthritis, rheumatoid arthritis, gout, and juvenile arthritis. Among

them, osteoarthritis is the most prevalent form of arthritis affecting more than 20.7

million Americans.

Osteoarthritis, or degenerative joint disease, is multifactorial with genetic, biologic,

and mechanical factors all playing a role. Of the mechanical factors involved, contact

pressure within the joint has been shown to have an interactive effect on developing this

disease (Hasler et al., 1998; Herzog et al., 2003). Thus, knowledge of in vivo contact

forces and pressures in human joints would be valuable for improving the prevention and

treatment of joint arthritis. Although dynamic imaging advances now is capable

collecting accurate measurement of in vivo j oint kinematics (Komistek et al., 2003;

Tashman and Anderst, 2003), joint contact forces and pressures are difficult to measure

in vivo (Kaufman et al., 1996), therefore, necessitating model-based analyses to develop

predictions as well as static in vitro testing to evaluate these predictions. A variety of

joint contact modeling methods have been used for this purpose, including finite element










(Bendjaballah et al., 1995, 1997, 1998; Donzelli and Spilker, 1998; Donahue et al., 2002;

Stolk et al., 2002), boundary element (Kuo and Keer, 1993; Haider and Guilak, 2000),

and discrete element methods (Li et al., 1999, 2001; Pandy and Sasaki, 1998; Piazza and

Delp, 2001; Dhaher and Kahn, 2002).

1.2 Need for Efficient Contact Model Evaluation

Once a joint contact model has been created that represents an in vitro testing

situation, its ability to reproduce experimentally measured contact quantities (e.g., peak

pressure, average pressure, contact area, and contact force) must be evaluated before any

further application. At least two factors complicate the process of evaluation. The first is

uncertainties in the experimental measurements. These uncertainties can often be

estimated and involve quantities such as the position and orientation (i.e., pose) of

cadaveric bones measured by the test apparatus, contact pressures and areas recorded by a

pressure sensor, and the articular surface geometry determined from medical imaging

data. Unknown model parameters, such as material parameters in the contact model,

present additional sources of uncertainty.

A second complicating factor is the high computational cost of repeated contact

analysis. Given an estimated envelope of uncertainty, optimization methods can be used

to determine if a feasible combination of model parameters could be used to reproduce all

experimental measurements simultaneously (Fregly et al., 2003). For example, an

optimization can vary model material parameters and the relative pose of the contacting

bodies within the envelope of uncertainty until the model produces the best match to the

experimental contact data. The problem with this approach is that the high computational

cost of repeated contact analysis can make such optimizations extremely time consuming

and in some cases even impractical.










Response surface (RS) methods have been utilized successfully in other situations

to eliminate computational bottlenecks in optimization studies. Response surfaces are

simply multi-dimensional linear regression curve fits to quantities of interest predicted by

an engineering model. Once the mathematical form of the RS is specified, linear least

squares is typically used to determine the coefficients that provide the best fit to each

predicted quantity of interest (i.e., each response) as a function of the specified design

variables. These surface approximations are then used as surrogates for costly

engineering analyses when the optimization is performed. Outside of the biomechanics

community, RS optimization methods have been used for structural design applications

(Jansson et al., 2003; Liu et al., 2000; Rikards et al., 2004; Roux et al., 1998),

aerodynamic designs (Ahn and Kim, 2003; Papila et al., 2002; Sevant et al., 2000), and

fluid dynamics (Burman and Gebart, 2001; Keane, 2003; Leary et al., 2004). Within the

biomechanics community, little work has been performed using RS optimization methods

with the exception of recent studies by Jung and Choe (1996), Chang et al. (1999), and

Hong et al. (2001). To our knowledge, no studies in the literature have used RS methods

to perform optimizations of contact problems.

1.3 Approach

The two goals of this thesis are first, to develop a computationally efficient RS

optimization approach for evaluating a joint contact model's ability to reproduce static

experimental contact measurements, and second, to apply this approach to the evaluation

of a discrete element contact model of the tibiofemoral joint. Our specific hypotheses

were that 1) quadratic RSs can accurately predict contact quantities (peak pressure,

average pressure, contact area, and contact forces) computed by a discrete element

contact model for small variations material modulus and relative pose of the contacting









bodies, and 2) a discrete element contact model of the tibiofemoral j oint can reproduce

experimentally measured contact quantities that involve averaging across the surface (i.e.,

average pressure, contact area, contact force) but not quantities associated with specific

locations on the surface (i.e., peak pressure). Our study provides a new computationally-

efficient approach for contact model evaluation as well as a better understanding of the

capabilities and limitations of discrete element contact models of natural human j points.















CHAPTER 2
IVETHOD S

2.1 Response Surface Optimization

This section provides a general overview of RS approximation methods as well as

specific modifications required to apply these methods to contact analyses. The RS

method can be defined as a collection of statistical and mathematical techniques useful

for constructing smooth approximations to functions in a multi-dimensional design space.

Once a mathematical form has been selected, the coefficients of the approximate function

(responses surface) are determined using data from either physical experiments or

numerical simulations. The most common mathematical form for a RS is a low-degree

polynomial. For example, a quadratic response surface with design variable inputs x, and

x, and output y is formulated as


y = Po + P x1 + P x2 + P x12 + Pqx2 + P x, x2 (1)

where the p, (i = 0,...,5) are the unknown coefficients to be fitted. A low degree

polynomial minimizes the number of unknown coefficients and tends to smooth out noise

in the function. Response surface approximations work best when the number of design

variable inputs is small (< 10), since a large number of design variables results in a

complicated design space that is difficult to fit with low-degree polynomials

To develop RS approximations for contact problems, one must identify the design

variable (DV) inputs, the outputs to be predicted, and the mathematical form of the RS

relating them. For contact model evaluation with static experimental data, the design









variables are the six relative pose parameters (i.e., three translations and three rotations -

6 DVs) and material modulus (1 DV) of the contacting bodies. The experimental

uncertainty of these quantities can be estimated and their values can be changed in the

contact model. The RS outputs are peak pressure, average pressure, contact area, and

contact force. These quantities can be calculated by the contact model and measured

experimentally for comparison. The hypothesized mathematical form is a quadratic RS

with one modification. For linearly elastic Hertzian point contact, the peak pressure,

average pressure, contact force, and contact area are all functions of interpenetration

(vertical translation) to a power less than two, while the material modulus (assumed to be

the same for both bodies) linearly scales each quantity except for area (Johnson, 1985).

Thus, data for the RSs are generated using a material modulus of one, only the six pose

parameters are used as RS inputs, and the RS outputs (except area) are scaled by the

desired modulus value.

With the RS formulation specified, the next step is to determine a sampling scheme

within the design space to provide data for fitting the RS. Since this sampling process is

only preformed once to generate the RSs, the computational cost of the contact analyses

is paid only once up front. A quadratic response surface using k design variables will

possess p = (k +1l)(k + 2) /2 unknown coefficients, where k = 6 since only pose

parameters are used as RS inputs. Consequently, a minimum of p = 28 data points must

be sampled to perform the linear least-squares fit. However, to cover the design space in

a systematic manner, we select a larger number of sample points using design of

experiments (DOE) theory. Several DOE sample criteria are available, including the

factorial design, face centered central composite design (FCCCD), and the D-optimality









design. We choose the FCCCD criteria for its ability to sample all regions of the design

space. For a quadratic RS, this approach utilizes 2k + 2k +1 = 77 sample points, where

the samples are taken at the center, the corners, and the face centers of a k dimensional

hypercube.

For contact analyses, we make two modifications to the FCCCD sampling scheme

to improve the quality of the fit. The first modification accounts for infeasible points. A

sampled point is deemed to be infeasible and is therefore omitted if the contact force and

area predicted by the contact model are zero. This modification avoids fitting regions of

the design space where no contact is occurring. The second modification accounts for

outlier points. Once a RS is generated from feasible points, the RS output is compared to

the computed value from the contact model for every sample point. The point with the

largest absolute percent error above a pre-selected cut-off value of 10% (a typical value

for engineering analyses) is omitted and the RS re-generated from the remaining sample

points. The procedure is iterated until all sample points are below 10% error. This

modification avoids fitting regions of the design space where only light contact is

occurring, thereby providing a better fit in the regions of interest where the contact force

is large. Omission of several sample points does not pose a problem to the fitting process

since the FCCCD sampling scheme is highly over determined.

After a RS is generated, the quality of the resulting fit must be assessed, since a

poor quality fit indicates that a different mathematical form for the RS should be

considered. We use three common error measures for this purpose. All of these measures

make use of the sum of the squares of the errors SSE between predicted responses A from









the RS and actual responses y, computed by the contact model, where n (n >> 28 and

n < 77) is the number of sample points used to generate the RS:





The first measure of fit quality is the adjusted root-mean-square error ( RMSEa4 ).

Given the SSE, the root-mean square error (RM~SE) can be calculated from


RM~SE = (3)


However, this measure will be zero if n = p (i.e., no redundant points), even

though the errors would not necessarily be zero at non-sampled points. To address this

limitation, we choose a more conservative adjusted RM~SE that uses n p (i.e., the

number of degrees of freedom remaining in the fitting process) rather than n in the

denominator of Eq. (3):

ISSE
RM~SE,, =,1 (4)
Vn-p

To provide a relative measure of fit quality, we also compute the percent adjusted RM\~SE

usmng

100 rSSE
%RM~SEad =-1 (5)


where y~ represents the magnitude of the fitted quantity:

1 "6



The second measure of fit quality is the adjusted coefficient of determination

(R2 aq ). The coefficient of determination R2 SUfferS from a, similar problem to RMSE in










that a perfect fit will be indicated if n = p Consequently, we used the adjusted R2 value

to account for the degrees of freedom n p remaining in the fit:

SSE / (n p)
RZ = 1 (7)



where ]7 is the mean of the actual responses.

The Einal measure of fit quality is the RM~SE calculated from the prediction error

sum of squares (PRESS) statistic. To evaluate the predictive capability of a RS, one

should ideally sample additional points distinct from those used to generate the RS.

However, this approach would require a significant number of additional costly contact

analyses. To circumvent this issue, the PRESS analysis excludes one sample point at a

time from the set used to generate each RS. The RS is regenerated using the remaining

n 1 sample points and the prediction error at the omitted sample point calculated. This

process is repeated for all n sample points, and the resulting errors are used to compute a

PRESS-based SSE called the PRESS statistic. From there, a PRESS-based RM~SE can be

calculated from Eq. (3), where n rather than n p is used in the denominator since each

error is calculated from a RS that omits that point.

Once accurate RSs are generated for the output quantities of interest, they are used

in an optimization to evaluate the contact model's ability to reproduce experimental

measurements. Each time the optimization requires a peak pressure, average pressure,

contact force, or contact area from the contact model, a response surface is used in place

of a contact analysis to provide the value. By fitting quantities computed by the contact

model, one can create any cost function that can be built up from the basic quantities. If

the cost function was fitted directly using its own response surface, then additional









contact analyses would be required to generate a new response surface each time the cost

function was modified. With our approach, a wide variety of cost functions can be

constructed without the need for any additional contact analyses.

2.2 Contact Pressure Experiments

The response surface methodology described above was used to evaluate a natural

knee contact model's ability to reproduce experimental contact measurements. The

experiments were performed on a single cadaveric knee specimen cut approximately 15

cm above and below the joint line and showing no visible signs of degenerative joint

disease. Institutional review board approval was obtained for the testing and subsequent

modeling efforts. The menisci, fibula, and patella were removed from the specimen, and

three titanium bone screws were inserted into the tibia and femur as landmarks for

contact model alignment. The tibia and femur were potted in neutral alignment

(MacWillams et al., 1998) and mounted in a MTS MiniBionix 858 servohydraulic test

machine. The position and orientation of the femur were constrained using custom

fixturing that allowed adjustment of the sagittal plane rotation and medial-lateral

translation relative to the ram of the test machine. The axial plane position and

orientation of the tibia were unconstrained using a ball plate, thereby allowing the tibia to

self-align with the femur once an axial load was applied.

Using this set-up (Fig. 2-1), we collected four experimental quantities of interest

from the medial and lateral compartments of the knee: contact force, peak pressure,

average pressure, and contact area. The knee was fixed at a flexion angle of 30o and a

Tekscan K-scan sensor (Tekscan, South Boston, MA) inserted anteriorly into the medial

and lateral j oint space. The medial-lateral position of the femur was adjusted to produce

an approximately 70% medial-30% lateral load split between the two sides (Hurwitz et










al., 1998; Schipplein and Andriacchi, 1991). The specimen was subjected to three trials

of a 4 second ramp load from 200 to 1000 N. At the end of each ramp, the locations of

the six screw heads were digitized using a Microscribe 3DX digitizer (Immersion Corp.,

San Jose, CA) possessing an accuracy of 0.23 mm.


Figure 2-1. A human cadaver knee static experiment setup. A) The knee was potted in
neutral alignment with six screws. B) The Tekscan K-scan sensor. C) The
knee was mounted with fixed 300 flexion angle in a servohydraulic test
machine with a sensor to measure intra-articular contact quantities. D) The
close-up view of the contact area.

Drift in the Tekscan sensor (Otto et al., 1999) was eliminated by post-calibrating

each trial with the manufacturer-suggested two-point calibration procedure using the

known loads at the start and end of the ramp. Crinkling of the sensor (Harris et al., 1999),

which introduces erroneous pressures on sensels outside the true contact area, was

accounted for by determining the pressure cut-off value (0.05 MPa) above which little










additional drop in contact area occurs when pressures below this value are set to zero

(Fregly et al., 2003). Contact quantities measured with the Tekscan sensor were therefore

calculated by ignoring all sensels with a pressure below 0.05 MPa. Following Tekscan

sensor calibration and pressure cut-off determination, the four experimental quantities of

interest were calculated on each side for applied loads of 500 and 1000 N. Peak pressure

was calculated using the averaging function in the Tekscan software, thereby reducing

the effect of local sensor "hot spots" on this quantity. The resulting data from the K-scan

sensor (Table 2-1) and the digitizer were averaged over the three trials to facilitate

contact model evaluation under two loading conditions.

Table 2-1. The averaged experimental measurements were collected from three
compression trials for both loads processing on the same cadaver knee via
servohydraulic test machine
Experimental Load
Experimental Quantity Side 500 N 1000 N
Force (N) Medial 317 & 4 658 & 5
Max Pressure (MPa) 4.10 & 0.05 7.94 & 0.14
Ave Pressure (MPa) 1.11 & 0.03 2.07 & 0.02
Area (mm2) 287 & 10 318 & 4
Force (N) Lateral 183 & 4 337 & 5
Max Pressure (MPa) 1.51 & 0.03 2.63 & 0.04
Ave Pressure (MPa) 0.79 & 0.01 1.33 & 0.03
Area (mm2) 229 & 4 252 & 5

2.3 Knee Model Creation

Prior to experimental contact testing, MRI (magnetic resonance imaging) and CT

(computed tomography) data were collected from the same cadaveric specimen for

purposes of contact model creation. Sagittal plane MRI data were collected using a 3.0-T

GE Signa Horizon LX scanner with a quadrature knee coil. A T2-weighted 3D FastGRE

sequence was used with a 1 mm slice thickness, 256 x 256 image matrix (0.625 x 0.625









mm pixel size), and 160 x 160 mm field of view. Axial CT data were collected from the

same specimen using a GE LightSpeed QX/i scanner in helical mode. The scanning

parameters were a 1.25 mm overlapping slice thickness, 512 x 512 image matrix (0.313 x

0.313 mm pixel size), and 160 mm x 160 mm field of view. The tibia, femur, and bone

screws in both data sets were segmented (Fig. 2-2) using commercial image processing

software (SliceOmatic, Tomovision, Montreal, CA).


Figure 2-2. Original and segmented medical images. A) Original CT slice. B) Segmented
CT slice. C) Original MRI slice. D) Segmented MRI slice.

The menisci were not segmented and were omitted from the model. Articular

cartilage and subchondral bone surfaces were segmented manually from the MRI data,









while cortical bone and bone screw surfaces were segmented semi-automatically from the

CT data using a watershed algorithm. The point clouds from both scans were exported for

subsequent surface creation.

Commercial reverse engineering software (Geomagic Studio, Raindrop Geomagic,

Research Triangle Park, NC) was used to convert the MRI and CT point cloud data into a

combined geometric model for contact analysis. Point clouds from each imaging

modality were imported separately and converted to polygonal surface models. The

subchondral bone surfaces from MRI were registered automatically to the corresponding

cortical bone surfaces from CT, creating a composite geometric model with articular

cartilage surfaces from MRI and cortical bone and bone screw surfaces from CT. NURBS

(Non-Uniform Rational B-Spline) surfaces were fitted to the polygonal models, with the

tolerance (mean + standard deviation) between the original point clouds from MRI and

the final NURBS surfaces being 0. 18 f 0. 18 mm for the femur and 0.20 f 0.29 mm for

the tibia (Fig. 2-3).
















Figure 2-3. The anterior and posterior views of the A) 3D point clouds and B) 3D
NURBS model with six screws created from CT and MRI images

The NURBS surfaces for the tibia and femur articularr cartilage, cortical bone, and

bone screws) were imported into Pro/MECHANICA MOTION (Parametric Technology









Corporation, Waltham, MA) to construct a multibody contact model. The mean digitized

bone screw locations were also imported to determine the nominal alignment of the tibia

and femur. For both bones, a stiff linear spring was placed between each screw head and

its mean experimental location and a static analysis performed to determine the pose that

best matched the experiments. Differences between the digitized and nominal bone screw

locations were on the order of 1 mm. Starting from these nominal poses, the tibia was

fixed to ground and the femur connected to it via a 6 degree-of-freedom (DOF) joint.

Custom contact code was incorporated into the multibody model and used to solve

for the medial and lateral contact conditions as a function of the 6 DOFs between the two

bones (Bei and Fregly, 2004). The contact code implemented two versions of a linear

elastic discrete element contact model. The first was a small strain version, where the

contact pressure p for each contact element on the tibial articular surfaces was calculated

from (An et al., 1990; Blankevoort et al., 1991; Li et al., 1997)

(1 v)E d
p = (8)
(1+ v)(1- 2v) h

where E is Young' s modulus of the articular cartilage, v is Poisson' s ratio, h is the

combined thickness of the femoral and tibial articular cartilage, and d is the

interpenetration of the undeformed contact surfaces. Both h and d were calculated on an

element-by-element basis using the ACIS 3D Toolkit (Spatial Corporation, Wesminster,

CO). For large strains, a second version of the model was implemented that accounted for

geometric nonlinear behavior (Blankevoort et al., 1991):

-(1- v)E I'dl
p = In1 (9)
(1 + v)(1 2v) hi









For both versions, a dense contact element grid of 50 x 50 was used for the medial

and material articular surfaces of the tibia. The femoral and tibial articular cartilage were

assumed to be linear elastic and isotropic with Poisson's ratio = 0.45 (Blankevoort et al.,

1991). Young's modulus was set to 1 MPa to facilitate its use as a design variable in the

response surface optimizations.

2.4 Knee Model Evaluation

Seventy seven contact analyses were performed with the model to provide the

sample points necessary to generate response surfaces using the FCCCD. Each sample

point represented a different pose of the femur relative to the tibia within the

neighborhood of the nominal pose. This neighborhood was defined to be & 1 mm and a

10 based on the estimated envelope of experimental pose uncertainty. Though this

envelope appears small, it corresponds to large changes in contact conditions. Within this

envelope, response surfaces were generated as described above for the medial and lateral

contact force, peak pressure, average pressure, and contact area computed by the contact

model. The optimization cost function g(x) constructed from these response surfaces

sought to match experimental average pressures pospin both compartments

simultaneously with a penalty term to ensure the experimental contact forces F were

reproduced:

g(x, E) = ( pm, E pove (x)) tedral (are E pove (x)) ar,,w; +
w ~F -EF(x)) medral + (F EF(x)) ateral1 (10)

where ~(x) and pore, (x) are the force and average pressure predicted by response

surfaces.









In this equation, x represents the 6 pose parameter design variables, E is the

material modulus seventh design variable, and w = 1000 is the weight of the penalty term.

This cost function mimics the results of a static analysis, since contact force is matched

closely while minimizing errors in the most reliable experimental measure of interest.

The form of Eq. (10) was specified by following a trial-and-error approach that included

different quantities in the cost function. A larger value of w was not used since it resulted

in a poor match to the other contact quantities.

To seek the global minimum, we performed 500 nonlinear least-squares

optimizations using the response surfaces from each contact model and the Matlab

Optimization Toolbox (The Mathworks, Natick, MA). Uniformly distributed random

initial guesses were selected within the bounds + 1 for the first six design variables and 1

to 10 for the seventh. The best set of design variables was selected based on the smallest

cost function value from the 500 optimizations. A final Pro/MECHANICA contact

analysis was performed for both contact models using the optimized design variables to

verify the accuracy of the response surface approximations.















CHAPTER 3
RESULTS

All %RMiS~ad and %RMSFvress results were less than 5% while all R2, vadV8UeS

were greater than or equal to 0.995. Error measures from the small and large strain

versions of the contact model were generally close. %RM~SEvress was always larger than

%RM~SEad but not dramatically so. Furthermore, %RM~SEady and %RM~SEvress were of

comparable magnitude to the variability in the corresponding experimental measurements

(Table 3-1). For each RS, a minimum of 7 and maximum of 26 infeasible/outlier points

were eliminated from the original FCCCD set of 77 sample points, and all of these points

involved a superior translation of +1 mm (i.e., no contact or light contact).

Based on the best results from the 500 RS optimizations, the contact quantities

computed from response surfaces were compared to both versions of the contact model

(Fig. 3-1). The relative errors from all contact quantities were below 10%. Both versions

of the contact model could reproduce all experimentally measured contact quantities to

within 10% error at both loads (Fig. 3-2). The one exception was peak pressure, which

was only matched to within 50% error. Contact force was matched to within 1% error,

consistent with the use of a penalty term on this quantity in the cost function. Average

pressure and contact area errors were always in opposite directions, consistent with

matching contact force closely. Peak pressure was over-predicted by the contact models

on the lateral side and under-predicted on the medial side (worst error). The optimal

value of Young' s modulus was 2.6 MPa for the small strain model and 2.3 MPa for the










large strain one. The best result from 500 optimizations never hit the upper or lower

bounds on the pose parameter design variables. Each set of 500 optimizations required

approximately 90 seconds of CPU time on a 2.8 GHz Pentium 4 PC.














Table 3-1. Comparison of response surface predictions to data points sampled from large and small strain contact models.


RMSE,
Side Large Small
Medial 4.67 4.24

0.065 0.035

0.059 0.048

10.8 10.8

Lateral 2.12 2.12

0.030 0.018

0.026 0.024

4.54 4.54


%RMSEag


RMSE,,


%RMSE .


Predicted Quantity

Force (N)

Max Pressure (MPa)

Ave Pressure (MPa)

Area (mm2)

Force (N)

Max Pressure (MPa)

Ave Pressure (MPa)

Area (mm2)


Large
0.601

1.53

3.31

3.50

0.788

1.15

2.24

2.58


Small

0.674

1.16

3.19

3.50

0.886

0.815

2.27

2.58


Large
10.0

0.090

0.078

14.3

2.92

0.039

0.033

6.00


Small

6.13

0.048

0.064

14.3

2.85

0.023

0.030

6.00


Large
1.29

2.13

4.40

4.64

1.09

1.49

2.91

3.41


Small

0.976

1.58

4.25

4.64

1.19

1.06

2.97

3.41


Large
0.999

0.999

0.999

0.995

0.999

0.999

0.998

0.996


Small

0.999

0.999

0.999

0.995

0.999

0.999

0.998

0.996












Pmax

B


S500N Large strain
S500N Small strain
~i1000N Large strain
O 1000N Small strain


Medial Lateral

Pave


Force


20










E 20





20



-20


-40


Medial


Lateral


Medial


Lateral


Figure 3-1. Percent errors between response surfaces and predicted results from both
contact models. They were calculated for A) contact force, B) contact peak
pressure, C) contact averaged pressure and D) contact area. Pose parameters
and Young's modulus were defined based on the best result from 500
optimization runs.












Force

20 A




S500N Large strain
2 20 500N Small strain
1000N Large strain
a- 1000N Small strain
-40

Medial Lateral


Pave




20





-40


Pmax


c.B

o-


o


0

Medial Lateral


Area

01


D

1-


Medial


Lateral


Medial


Lateral


Figure 3-2. Percent errors between experimental and predicted results from both contact
models. They were calculated for A) contact force, B) contact peak pressure,
C) contact averaged pressure and D) contact area. Pose parameters and
Young's modulus were defined based on the best result from 500 optimization
runs.















CHAPTER 4
DISCUSSION

This thesis has presented a novel response surface optimization methodology for

evaluating a joint contact model's ability to reproduce static experimental contact

measurements. The approach modifies traditional RS approximation methods for

application to contact analyses. By replacing computationally-costly contact analyses

with quadratic RS approximations, optimizations that vary the relative pose of the

contacting bodies can be performed rapidly to minimize differences between model

predictions and experimental measurements. Evaluation of a discrete element contact

model of the tibiofemoral j oint constructed from CT and MRI data revealed that

quadratic RSs can accurately approximate model outputs within a small envelope of

relative pose uncertainty. Furthermore, optimization studies utilizing these RSs

demonstrated that the model could reproduce the contact force, average pressure, and

contact area. However, the peak pressure measured experimentally on the medial and

lateral sides could not be reproduced. This finding suggests that discrete element contact

models of natural human j points that utilize homogeneous, isotropic material properties

may be best suited for analyses such as multibody dynamic simulations where obtaining

correct contact forces by integrating over the surfaces is more critical than obtaining

correct contact pressures at specific locations on the surfaces.

Use of RSs to replace repeated contact analyses in optimization studies is

worthwhile for several reasons. First, RS tends to smooth out noise in the design space.

During optimization, the risk of entrapment in a local minimum is therefore reduced.









Second, RS approximations are computationally efficient. Rather then repeating costly

contact analyses during an optimization, a single set of contact analyses is performed

once up front to generate the necessary RS approximations for output quantities of

interest. Extremely fast function evaluations allow one to search for the global optimum

using either repeated gradient-based optimization starting from multiple initial guesses,

as performed in our study, or global optimization employing a large population size.

Third, RS optimizations are convenient to implement. Optimized contact solutions can be

founded utilizing any off-the-shelf optimization algorithm once the RS approximations

are constructed. Fourth, a variety of optimization problem formulations can be evaluated

quickly. Each contact quantity that could potentially appear in the cost function or

constraints can be fitted with its own RS. A wide variety of cost functions can then be

constructed by weighting contributions from the different RSs. Fifth, RS approximations

facilitate the calculation of analytical derivatives for gradient-based optimization. This

benefit is the direct result of having analytical representations for the contact quantities of

interest as a function of the design variables. The caveat to these benefits is that an

appropriate mathematical form (polynomial or otherwise) must be identified to represent

the responses of interest as a function of the design variable inputs, which is not always

possible.

The primary deficiency of the discrete element contact model was its inability to

match the peak contact pressures measured experimentally. Peak pressures were not

included in the cost function since they are sensitive to local inhomogeneities in cartilage

material properties and Tekscan sensor response. When we replaced average pressures

with peak pressures in the cost function for curiosity, we found that peak pressure errors









decreased on the medial side by about 25% but increased on the lateral side by roughly

50%, while average pressure and contact area errors increased by approximately 30% and

20%, respectively. Thus, a variety of other factors likely contributed to the large peak

pressure errors, including insertion of a relatively stiff sensor into the joint space (Wu et

al., 1998), small inaccuracies in the surface geometry, and local variations in material

properties. Underprediction of peak pressure by the model on the moderately conformal

medial side and overprediction on the non-conformal lateral side is consistent with

observations made by Wu et al. (1998) on how insertion of a relatively stiff sensor into

the joint space affects peak pressure measurements as a function of contact conformity.

Contact model limitations may have also contributed to the poor match of the

peak pressure data. Though a strength of our model is that it accounts for local variations

in cartilage thickness, a weakness is that it does not account for local variations in

cartilage material properties. Though homogeneous, isotropic cartilage material models

have been used in the literature (Bendj aballah et al., 1995; Blankevoort et al., 1991; Haut

et al., 2002; Perie and Hobatho, 1998), recent studies have reported significant local

variations in Young's modulus, Poisson's ratio and thickness for articular cartilage

(Jurvelin et al., 2000; Laasanen et al., 2003). Mukherj ee and Wayne (1998) suggested

that regions with the highest Young's modulus correspond to regions with the highest

contact pressure and cartilage thickness. This observation fits our under-prediction of

peak pressure on the medial side, where an increase in Young's modulus would produce

improved agreement with the large measured peak pressures. In addition, our elastic

contact model does not account for the effect of time-dependent fluid flow on contact

pressures or areas as captured by biphasic models of cartilage (Han et al., 2004; Mow et










al., 1980; Mow et al., 1982). However, for loading over a short time period, an elastic

model may still provide a reasonable approximation of the in vivo situation depending on

the intended application of the model (Donzelli et al., 1999; Mow et al., 1982; Shepherd

and Seedhom, 1997;).

The value of Young' s modulus predicted by our optimizations is consistent with

studies found in the literature. Experimental studies of the compressive modulus of

articular cartilage have reported values as low as 2.0 MPa for short time frame loading

(Setton et al., 1999; Shepherd and Seedhom, 1997). Other studies have reported the

modulus of the solid phase alone to be in the neighborhood of 0.3 MPa (Hasler et al.,

1999). Thus, the best-fit modulus values of2.3 and 2.6 MPa found in our study are

consistent with these numbers. Other discrete element knee studies have utilized an

elastic modulus of 4 MPa (Cohen et al., 2003; Kwak et al., 2000) which again is close to

our optimized modulus values. Furthermore, if our estimate of Poisson' s ratio is

decreased slightly from 0.45 to 0.4, our optimized value of Young' s modulus will

increase from 2.6 to 4.6 MPa, closer to the middle of the range commonly reported

(Setton et al., 1999; Shephard and Seedhom, 1997).

A weakness of our study is that only a single knee specimen was evaluated.

Significant modeling effort is required to create a knee model with articular cartilage and

subchondral bone geometry that matches a particular cadaver specimen. This may explain

why other studies that utilized specimen-specific geometric models only analyzed a

single specimen as well (Bendjaballah et al., 1995; Haut et al., 2002; Li et al., 1999). The

knee used in our study was part of a larger experimental study involving 20 knees.

However, we were able to collect CT and MRI data from only two of those knees. Since









the contact area for the second specimen was not completely contained within the

boundary of the Tekscan sensor at the maximum load of 1000 N, we were not able to

post-calibrate those experimental contact data and use them for a second model

evaluation. Comparison of model predictions with experimental measurements from

additional knee specimens using a range of flexion angles and loads would provide

further verification of the capabilities and limitations of discrete element models of

natural joints.















CHAPTER 5
SUMMARY AND FUTURE STUDY

5.1 Summary

In summary, this thesis has presented a computationally efficient RS optimization

for predicting knee contact measurements using a 3D simulation model. The more

computational costly is each contact analysis; the more beneficial is the use of RS

optimization method. The present implementation works for the tibiofemoral j oint of the

natural knee with either large or small strain contact model. The RS optimization can

predict contact forces, areas and average pressures well with the exception of peak

pressures. Our ultimate goal is to incorporate the elastic foundation contact model into a

full-body dynamic musculoskeletal model. The resulting full-body model can then be

utilized to study joint contact pressures during motion. Future research is required to

evaluate whether the peak pressure errors are due to experimental or contact model

inaccuracies.

5.2 Future Study

While our results showed great potential to predict contact quantities that involve

averaging across the surface, there are some extra steps that could be taken to improve

the accuracy of RS. Instead of using two modifications in this thesis to remove the

outliers ,there is another more robust procedure called iteratively re-weighted least square

(IRLS) fitting (Holland and Welsch, 1977). It can be utilized to remove or weight down

outliers. Additionally, varying numbers of insignificant coefficients for each RS was

found based on t-statistics analysis. While it was not performed in this study, there are









several statistical methods, (e.g., a stepwise regression procedure) that can be used to

discard those insignificant coefficients to improve the prediction accuracy.

In addition to 500 optimizations, another 50 groups of optimizations were run. The

best result was selected from each group containing 100 optimizations with different

initial guesses. Within 50 sets of optimized design variables, the maximum variations in

translation and rotation were 1.90 mm and 1.950 respectively. Compared to the range of

design space, these variations suggested that the final result from 500 optimizations may

not be unique. Thus an alternative cost function or type of different RS method (e.g.,

neural networks and kriging) could be another candidate for future improvement.

Given how well RSs could approximate contact model outputs for a small range of

relative pose variations, our next step will be to investigate whether RSs can be used to

generate extremely fast forward dynamic contact simulations that utilize a wider range of

relative pose variations. The computational cost of repeated geometry evaluations is the

current limiting factor to the incorporation of deformable contact models into multibody

dynamic simulations of human j points (Bei and Fregly, 2004). However, if accurate

response surfaces could be generated to represent the net force and torque calculated

about a pre-selected point on the tibia (e.g., the origin of the tibial coordinate system),

then extremely fast contact solutions could be produced for any given set of relative pose

parameters. Though additional design variables would need to be included to account for

friction or damping effects, these effects are expected to be small for most modeling

situations involving human joints. Higher degree polynomials or some other

mathematical formulation would likely be required to represent a wider range of relative

pose variations.
















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BIOGRAPHICAL SKETCH

Yi-Chung Lin was born on August 14, 1976, in Kaohsiung, Taiwan. In 1998, he

received the bachelor' s degree in the Department of Mechanical Engineering of the

National Cheng Kung University, Taiwan. He j oined the Computational and

Biomechanics Laboratory of Assistant Professor B. J. Fregly in the fall of 2002. He is

planning to enter the doctoral program in mechanical and aerospace engineering after

finishing the master's program.