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Computational modeling: the effects of motion on total knee replacements

University of Florida Institutional Repository

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COMPUTATIONAL MODELING: THE EFFECTS OF MOTION ON TOTAL KNEE REPLACEMENTS By MATTHEW A. HAMILTON 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 2003

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ii ACKNOWLEDGMENTS I wish to thank Dr. Greg Sawyer for his guidance and support on not only the present project, but past and future ones as well. I also appreciate the help of Dr. Ziegert and Dr. Arakere throughout my time at the University of Florida and for agreeing to review this work. Special thanks go to all the members of the Tribology Lab (Aaron Ison, Jason Action, Dan and Pam Dickrell, Darren McGuire, Brian Micklos and Dave Burris) for help provided in their various fields of expertise. I would lik e to thank my parents for their support through all my years in school. I would like to thank my wife, Melissa. Her encouragement throughout this endeavor has helped me tremendously. Without her I could not have achieved this goal.

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iii TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ ... ii LIST OF TABLES ................................ ................................ ................................ .............. iv LIST OF FIGURES ................................ ................................ ................................ ............. v ABSTRACT ................................ ................................ ................................ ....................... viii Chapter 1 INTRODUCTION ................................ ................................ ................................ ........... 1 2 BACKGROUND AND LITERATURE REVIEW ................................ ......................... 5 3 ENGINEERING APPROACH ................................ ................................ ...................... 18 Experimental Dat a ................................ ................................ ................................ ...... 18 Mathematics ................................ ................................ ................................ ................ 23 4 RESULTS AND DISCUSSION ................................ ................................ .................... 37 Wear Analysis ................................ ................................ ................................ ............. 37 Crossing Analysis ................................ ................................ ................................ ....... 43 Thermal Analysis ................................ ................................ ................................ ........ 47 5 SUMMARY AND CLOSURE ................................ ................................ ...................... 54 A CLASS HEIRARCHY FOR SOFTWARE DEVELOPED ................................ .......... 55 B DERIVATION OF ALGORITHMS ................................ ................................ ............. 58 Temperature Rise Derivation ................................ ................................ ...................... 58 Crossing Motion Derivation ................................ ................................ ....................... 60 LIST OF REFERENCES ................................ ................................ ................................ ... 64 BIOGRAPHICAL SKETCH ................................ ................................ ............................. 70

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iv LIST OF TABLES Table page 2 1 Causes of TKR failure. ................................ ................................ ................................ 9 4 1 Quantitative summary of damage results predicted by the computer simulations for gait and stair activities with 70 30 and 50 50 load splits. Maximum wear, creep, and total damage may occur at different locations on the surface. .......................... 40 4 2 Quantitative comparison between retrieval damage and simulation damage predicted by an activity partition of 70% gait, 30% stair with a 70 30 load split. ................... 41 4 3 List of standard conditions ................................ ................................ ......................... 47 4 4 Conditions run off of the standard condition (bold) and corresponding maximum temperature rise in degrees Cels ius for the gait and stair activities accordingly. ..... 51

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v LIST OF FIGURES Figure page 2 1: Normal knee vs. pro sthetic knee. ................................ ................................ ................. 6 2 2: Percentage of failures in TKRs vs. years implanted. ................................ ................. 10 3 1: Component design and film strips of the kinematics for the gait and stair rise acti vities, the corresponding locus plots of cont act pressure centroid is shown to the far right. ................................ ................................ ................................ .................... 18 3 2: In vivo experimental data used as inputs to the dynamic contact model. a)Anterior posterior (AP) translation. b) Internal external (IE) rotation. c) Flexion. d) Axial force. Kinematic data are from pre retrieval video fluoroscopy gait and stair experiments with the femur movi n g with respect to the tibia. Anterior translation and external rotation are positive. Axial force da ta are scaled vertical ground reaction force data from a patient of similar age, height, weight, and knee flexion characteristics. ................................ ................................ ................................ .......... 21 3 3: S chematic of the knee joint showing the partitioning of energy into the femoral and tibial component. ................................ ................................ ................................ ...... 26 3 4: Schematic of the solution approach for temperature rise of element l with neighboring elements treated as point so urces of heat (both frictional heating and convective cooling). ................................ ................................ ................................ 27 3 5: (a) Vector plot of the counterface slip velocities for 17 different instan ces during contact. This particular location in the lateral compartment duri ng a stair rise activity; this location showed significant crossing motion. ( b) The motion path of the femoral component over this location, notice the x axis is exaggerated by 2 0 times, a 1 to 1 motion path is shown to the right. ................................ .................... 30 3 6: (a) Vector plot for tribological intensities shown with the angular coordinate convention ( b) Overlaying plot with angular coordinate c) Scatter plot of tribological intensity versus angular coordinate compared with uni directional motion (delta function) and vari d irectional motion (step function). ....................... 32 3 7: A series of simple intensity maps, with corresponding counterface motion and normalized crossing severity index. ................................ ................................ ......... 35

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vi 3 8: Overlaying plots of velocity vectors, normalized by the largest velocity vector in the simulation, ont o the corresponding elements on the tibial mesh. These vectors are from a stair rise activity, which shows the greatest degree of crossing motion. ...... 36 4 1: Damage visualization of the retrieved tibial insert. a) Laser scan showing damage regions visible to the naked eye. b) Contour map indicating depth of damage zones. The color bar indicates depth in mm. Stars indicate loc ation of maximum damage on each side. ................................ ................................ ................................ ............. 39 4 2: Damage contour maps predicted by the computer simulations. a) Gait with 70 30 load split. b) Stair with 70 30 load split. c) Gait with 50 50 load split. d) Combined activity ass uming 70% gait, 30% stair with 70 30 load split. ................................ .. 40 4 3: Visualization of the static contact pressures predicted by the dynamic model for an axial load of 3 BW. a) Linear mater ial model. b) Nonlinear material model. Color b ar indicates pressure in MPa. The element grid is 50 x 50, the same as that used in the damage predictions. ................................ ................................ ............................ 43 4 4: Contour maps of a) tribological intensity and b) normalized crossing severity index for elements in contact during gait. ................................ ................................ .......... 45 4 5: Contour maps of a) tribological intensity and b) normalize d crossing severity index for elements in contact during stair rise. ................................ ................................ .. 46 4 6: Contour maps for continuous stair activity for the standard condition of a) the average frictional heat flux and b) the temperature rise in degrees Celsius. ( 1Hz, 13 W/( m K) f k = 0.06 m = 2 30 (W/( m K)) h = ................................ .................. 49 4 7: Contour maps for continuous stair activity for the standard condition of a) the average frictional heat flux and b) the temperature rise in degrees Celsius. (1Hz, 13 W/( m K) f k = 0.06 m = 2 30 (W/( m K)) h = ................................ .................. 50 4 8: Plots of maximum temperature for variations in a) activity rate, b) thermal conductivity of the femoral component, and c) convection heat transfer coefficient. 53 A 1: Class hierarchy of java packages written for this project Part 1. (ArrayManipulation, DoubleFormat, Encryption, ColorWindow, Library_Files). .. 55 A 2: Class hierarchy for java packag es written for this project Part 2. (Library_Files, Eng_Math, threads). ................................ ................................ ................................ 56 A 3: Class hierarchy for java packages written for this project Part 3. (Graphing, ColorWheel, File_IO) ................................ ................................ .............................. 57 B 1: Results of crossing intensity analysis with motion paths which are elliptical in nature. ................................ ................................ ................................ ....................... 62

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vii B 2: Results of crossing intensity analysis with motion paths described by a pin rotatin g about its central axis while sliding linearly across a surface. ................................ ... 63

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viii Abstract of Thesis Presented to the Graduate School of the University of Florida i n Partial Fulfillment of the Requirements for the Degree of Master of Science COMPUTATIONAL MODELING: THE EFFECTS OF MOTION ON TOTAL KNEE REPLACEMENTS By Matthew A. Hamilton May 2003 Chair: W. Gregory Sawyer Department: Mechanical and Aerospace Engineer ing Total knee replacements have evolved significantly in the past half century. A major development, the use of Ultra High Molecular Weight Polyethylene (UHMWPE), has increased functional life to more than a decade. Although the occurrence of failures i n current TKRs is low, wear and its results still plague the total joint replacement. Wearing of the UHMWPE tibial bearing changes its geometry allowing the femoral component to move in ways not intended by manufacturers. This loosening results in insta bility and poor functionality in the joint. When the joint no longer functions as intended, the recourse is revision surgery. Due to the loss of bone surrounding the joint from osteolysis and the original implant surgery, doctors are left with little to work with when revising a TKR. This results in success rates of revision surgery is below 70% causing major concern to patients at higher risk for component failure. The focus of current research is to increase the life of the components by developing a t ool capable of aiding researchers in the design stage of TKR production. By allowing designers to identify possible design flaws prior to prototyping, the testing of new designs would be expedited. A tool of this nature would help reduce the workload

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ix for knee simulator testing. Testing in a knee simulator is not only time consuming (each test could take up to four months), but it is also expensive (average cost $40,000). Thus, new ideas could get from the drawing board to the doctors office more quickl y and at a lower cost. The purpose of this project was to develop a tool, which could use fluoroscopic data from a patient performing a set of exercises, to predict the effect of these motions on a TKR over a finite period of time. The motions experienc ed by the surface of the bearing were evaluated to establish a method of categorizing motions based on their ability to damage the surface. Wear and creep damage predictions were made based on a variety of motion inputs. These damage predictions, made us ing the software, were compared to actual data from a retrieved tibial bearing component. Finally, the thermal effects created by motions in the knee were evaluated. Although it is not likely that thermal effects created by everyday activity could be a f actor, those experienced in knee simulators may be sufficient to affect results. The tool developed accurately predicted wear geography, which is a first step toward predicting total damage. Also tools for quantifying multidirectional motion and temperature rise due to frictional heating were developed. The crossing motion results gi ve an indication of the severity of crossing in a particular motion. Thermal results from frictional heating were from 10 to 15 o C

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1 CHAPTER 1 INTRODUCTION The evolution of the Total Knee Replacement (TKR) began in the early 1940s. The first generation of TKR consisted of a hinge joint, which connected the femur to the tibia. Although the mechanism provided short term pain relief, its motion was limited and had severe problems with loosening and infection. As a result, the design was abandoned after a few years. The second generation prosthesis, the McKeever developed in1957 and the MacIntosh developed in 1958, were simply metal plates separating the tibia and femur. The theory behind the plate was to prevent rubbing of irregular surfaces. These devices had mild success, but were unpredictable and did not relieve symptoms in many patients. The next generation of design came from Massachusetts General Hospital. It was a mold type arthroplasty designed to replace the femoral half of the knee joint, but like its predecessors it was unpredictable and failed after short periods of use. A new era of knee replacement came about in the 1960s, when Frank Gunston, an orthopedist from Sir John Charnleys Hip Center, developed a metal on plastic knee replacement. The prosthesis was cemented directly to the bone, and was the first metal and plastic knee (introduced in 1968). In 1972, Dr. John Insall developed a prototype for the current total knee replacement. To replace all the surfaces in the knee the prosthesis had three components the femur, tibia and patella. The components were attached to the bones using bone cement. Since the development of the metal and plastic replacement researchers have been testing new designs and materia ls to increase the versatility and durability of the arthroplasty. A major development was the polymer

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2 Ultra High Molecular Weight Polyethylene (UHMWPE). The polymer has been the choice for bearing material for the past 30 years, replacing the original bearing choice Polytetrafluoroethylene (Teflon)(Oonishi, Ishimaru, and Kato, 1996). UHMWPE and Teflon are both biocompatible, however, UHMWPE has a much higher wear resistance. This plastic has increased the useful life of TKRs. Current replacements hav e success rates above 90% ten years after implantation, but the need for increased function and life in the replacements still challenges designers and researchers. Although the occurrence of failures in current TKRs is low, wear and its results still pl ague the total joint replacement. Wear is defined as damage to a solid surface, generally involving progressive loss of material, due to relative motion between that surface and a contacting substance or substances. Factors influencing wear are normal lo ad, sliding distance, friction coefficient, nominal temperature and environment. The wearing of the components in a TKR has a dual effect on the life of the device. Wearing of the UHMWPE tibial bearing changes its geometry allowing the femoral component to move in ways not intended by manufacturers. Loosening, as a result of wear, causes instability and poor functionality in the joint. Another issue troubling designers of these prostheses is the effect of wear debris generated from the bearing. Althoug h the volume of material worn away by use has decreased with the advent of UHMWPE, the size of the particles itself has become an issue. The average size of a UHMWPE wear particle in knee joint replacements is sub micron, particles of this size have been labeled the most biologically active. The body identifies these particles as foreign bodies and triggers defense mechanisms to eliminate them. These defense mechanisms are the source of bone resorption, also called osteolysis, a condition where the bon es surrounding the joint

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3 are broken down. This softening of tissue surrounding the TKR causes the components to become unstable, and eventually failure of the TKR. When the joint no longer functions as intended the recourse is revision surgery. Due to t he loss of bone surrounding the joint from osteolysis and the original implant surgery, doctors are left with little to work with when revising a TKR. This results in success rates of revision surgery below 70%, causing major concern to patients at higher risk for component failure. The focus of current research is to increase the life of the components by developing a tool capable of aiding researchers in the design stage of TKR production. By allowing designers to identify possible design flaws prior to prototyping, the testing of new designs would be expedited. A tool of this nature would help reduce the workload for knee simulator testing. Testing in a knee simulator is not only time consuming (each test could take up to four months), but it is also expensive (average cost $40,000). Thus, new ideas could get from the drawing board to the doctors office more quickly and at a lower cost. The purpose of this project was to develop a tool, which could use fluoroscopic data from a patient performing a se t of exercises, to predict the effect of these motions on a TKR over a finite period of time. The motions experienced by the surface of the bearing were evaluated to establish a method of categorizing motions based on their ability to damage the surface. Wear and creep damage predictions were made based on a variety of motion inputs. These damage predictions, made using the software, were compared to actual data from a retrieved tibial bearing component. Finally, the thermal effects created by motions i n the knee were evaluated. Although it is not likely that

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4 thermal effects created by everyday activity could be a factor, those experienced in knee simulators may be sufficient to affect results.

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5 CHAPTER 2 BACKGROUND AND LITERATURE REVIEW From the beginning, the durability of TKRs has been a major focus for researchers studying the prostheses. Even the first replacements were studied from the time of implantation to see if the solution was viable for a long term use. Although the first component designs were not sufficient to sustain activity for long periods of time, they were a valuable first step toward successful replacements. As with any new technology, fledgling attempts are often failures, but the abilities of designers to recognize problems and create new solutions leads to success. The evolution from a simple hinge joint or a metal plate inserted between two sides of an arthritic joint, to molded metal replacements of the femur with a pl astic bearing component cemented directly to the tibia, and finally to replacing the ends of the femur and tibia with anatomically correct metal replacements separated by a molded plastic insert has taken over 50 years. While no standard design has been s et, most have the same basic components with subtle variations The typical TKR consists of three major parts. The femoral condyle, typically made of cobalt chrome, is connected to the femur using rods protruding from the backside of the component. The t ibial plateau is attached to the tibia via a long rod extending down from the bottom of flat metal plate. Finally, the tibial bearing, a molded piece of ultra high molecular weight polyethylene, is snapped into the tibial plateau providing a place for the femoral condyle to rest. The way these parts are attached to the bones, and their shapes vary from one design to the next, but these are generally the parts that make up the TKR.

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6 The procedure to insert a total knee replacement involves surgically removi ng damaged or injured parts of the knee, and replacing them with the components mentioned above. Initially the muscles and ligaments surrounding the knee are separated from the bone, leaving the joint exposed. The ends of the femur and tibia are surgical ly removed so the pieces of the TKR can be easily attached. Metal rods attached to the backside of the femoral condyle and tibial plateau are forced into the femur and tibia respectively. In some cases the two components are cemented to the bones using b one cement. The bearing is attached to the tibial plateau, usually it snaps into the surface of the plateau. Some varieties of TKR require the backside of the patella be removed and replaced with a UHMWPE piece as well. Figure 2 1: Normal knee vs. pro sthetic knee.

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7 In an effort to improve the versatility of the TKR, researchers have set out to determine what causes the prostheses to fail. The study of knee replacement survivorship is an active topic demonstrated by Schmalzried and Callaghan (1999) who referenced over 200 articles. Although many designs for knee replacements exist, the results of the studies are very similar. Sextro et al. (2001), Pavone et al. (2001), Diduch et al. (1997), Beuchel et al. (2001) and Meding et al. (2001) all reported les s than 10% failure after a ten year period. Table 2 1 gives an overview of studies of knee replacement survivorship published over the last 15 years. Figure 2.2 shows a graph of survivorship rates versus years in vivo created from the results of those s tudies. While the results of these studies indicated that success rates for the implants are high, a variety of observations were made in the studies. Bearing thickness was listed as a variable affecting the success of total knee replacements in multip le studies. Sextro et al. (2001) noted that a bearing thickness less than 8 mm significantly increased the chance of failure in the component. Meding et al. (2001) also mentioned bearing thickness, although no failures were shown as a result of bearing t hickness, the study noted a large increase in knee pain scores for bearings with a thickness of less than 6 mm. Wright et al. (1992) found that 11 of the 12 failures found in their study occurred in bearings having a thickness of 6 mm or less. The author s concluded the smaller thickness increased the stresses on and within the polyethylene.(131) The less conforming bearing and smaller thickness combined to produce higher stresses, which are emphasized by the predominance and the severity of the delami nation and surface deformation . on the retrieved components.(131) Higher stresses are associated with an increase in wear debris which is a contributory factor . to long term complications, especially component loosening.(132) Diduch, Insall, Scott,

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8 Scuderi and Font Rodriguez (1997) commented on the importance of affixing the components to the bones using cement. The report stated that cement fixation has also been shown to be an effective barrier against penetration of polyethylene particul ate debris.(60) This debris is a major cause of aseptic loosening in the TKR components. Four of the studies (Wright et al. 1992; Sextro et al. 2001; Beuchel et al. 2001; Pavone et al. 2001) noted osteolysis and wear specifically as causes of failur e in TKRs studied. Osteolysis or bone resorption is the result of the body trying to eliminate wear particles building up in the tissue and fluid surrounding the knee joint. The failure rates ranged from 2.2% to 4.4% of knees studied failing due to wear or osteolysis. While these percentages seem small, over 266,000 total knee replacements are performed in the United States annually, giving rise to the potential for hundreds of thousands of knees to be affected by wear and its results within the next dec ade. Wear is defined as damage to a solid surface, generally involving progressive loss of material, due to relative motion between that surface and a contacting substance or sub stances. Historically three types have been identified in retrieved total jo int replacements: abrasive wear, adhesive wear and fatigue wear. Abrasive wear is a result of plowing of UHMWPE surface or scratching of the metallic femoral head caused by loose third body particles. The third body particles could be bone chips, bone c ement or metallic debris trapped between the surfaces of the bearing and the femoral condyle (Wang, 2001). Adhesive wear occurs when the asperities on the articulating surface (typically the UHMWPE) are sheared off. In order for this to occur, the interf acial shear strength must be greater than the bulk shear strength of the material (Bragdon et al.

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9 1996). Fatigue wear occurs through the initiation and propagation of surface or subsurface cracks under cyclic loading conditions (Wang, 2001). Table 2 1. Causes of TKR failure.

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10 Figure 2 2: Percentage of failures in TKRs vs. years implanted. Although wear can occur in many ways, the results of wear are similar, namely, debris is released from the articulating surface into the surrounding environment. The effect of the debris in vivo has been studied extensively (Hahn et al., 1996; Howling et al., 2001). Both of these studies linked UHMWPE wear debris found in the fluid and tissue surrounding the joint to osteolysis. Wear debris in TKRs ranges in si ze from sub micron to tens of microns, but Howling et al. characterized particles less than one micron in size as the most biologically active. Wolfarth et al. also made a connection between sub micron sized particles and osteolysis in their 1996 study of UHMWPE particles in the synovia and tissue surrounding the knee. The search for a polymer capable of withstanding the harsh conditions of the knee joint resulted in the discovery of ultra high molecular weight polyethylene. UHMWPE is a polymer characte rized by long molecular chains, resulting in an ultra high molecular weight. The chains in the polymer are not typically interconnected with extremely strong bonds allowing the polymer chains to orient themselves in the direction of sliding. By

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11 doing t his, the wear resistance of the material increases and makes the polymer a valuable solid lubricant. This characteristic also creates a problem for the material when exposed to multidirectional sliding. When the chains orient themselves in the direction of sliding it makes them more susceptible to breaking from motion perpendicular to the sliding direction. The issue of multidirectional sliding was addressed by Bragdon et al. (1996) and Wang (2001). In Wangs unified theory, he compared wear rates for UHMWPE created by linear reciprocating to those from clinically retrieved prostheses. The clinical retrievals displayed wear rates one to two orders of magnitude greater than those predicted by linear reciprocating. An innovation in the processing of U HMWPE which came about specifically to combat the effects of multidirectional sliding was improved cross linking. Cross linking in UHMWPE is the connection between two adjacent polymer chains via covalent C C bonds. Increasing the density of these covale nt bonds between chains prevents chain mobility and orientation as well as increasing the difficulty of breaking one chain from another. Higher cross link density has been shown to decrease wear rates in multidirectional sliding (Wang, 2001). There are s everal ways to control the cross link density, mainly irradiation followed by heat treatment is used. By irradiating the material carbon bonds are broken leaving free radicals dispersed throughout the polymer. Heat treating serves to ensure uniform cross link density (Wang et al. 1998). After the free radicals are created, the carbon is available to bond with other molecules. One issue with cross linking is preventing other molecules from occupying the open carbon bonds. If oxygen is present during or after the polymer is processed it can diffuse through the material and bond to the newly formed free radicals. Exposure to oxygen after

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12 processing dramatically decreases the number of linking reactions (Premmath et al. 1996) effectively weakening the in ter molecule strength. This oxidation can greatly reduce the wear resistance of the material. Storage of the material after treatment can also influence the degree of oxidation. If the UHMWPE is stored in an open air environment it allows oxygen time to diffuse throughout the material preventing cross linking from occurring. The problems with oxidation lead to processing and storing UHMWPE in an inert environment leaving many free radicals in the polymer un bonded. While oxidation can occur in the body (Premnath et al., 1996), the degree of oxidation is greatly reduced by preventing oxidation until the component is implanted. Debris particle generation of UHMWPE remains a clinical issue in total joint replacements such as knees and hips. Significant advances in modeling and understanding the origin of wear debris liberation in total hip replacements have been made over the past decade. One aspect recently discovered is the importance of sliding direction on wear in UHMWPE (Bragdon et al. 1996; Murat oglu et al. 1999; Burroughs and Blanchet, 2001; Wang, 2001). Orders of magnitude changes in wear rate with increasing degrees of crossing motion have been reported. Linear reciprocation is motion in one direction. Since UHMWPE has the ability to orient itself in a particular sliding direction, it can accommodate unidirectional motion very effectively. Conversely, when sliding direction is varied, UHMWPE can exhibit wear rates two to three orders of magnitude greater than linear tests find (Wang, 2001). As mentioned above, several groups have researched the effects of multidirectional sliding. The focus of these papers was motion in hip replacements, because of the hip replacements conforming design is conducive to multidirectional motion. In the stud y

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13 by Bragdon (1996) they found that an increase in the variation of sliding direction increased the wear rates demonstrated by the UHMWPE. Although Wangs study was similar, he went a step further and tried to quantify the degree of multidirectional motio n by defining an angle alpha. Wang used this angle to measure degree of multidirectionality and showed increase in the crossing angle led to an increase in wear. The conforming and axis symmetric nature of hip joint replacements has facilitated the study of their relative motion for fluid mechanics and lubrication (Meyer and Tichy, 1999) and contact studies (Bragdon et al. 1996; Maxian et al. 1996; Wang, 2001; Kurtz et al. 1999). To date, the complexity of the knee motions has prevented detailed study of the tibial component. Locus plots, which seek to represent the kinematics by following the most probable locations of contact on the tibial component, do not capture the relative motions experienced by a particular location on the surface. To capture this information it is critical to model the pressure distribution (which spans multiple surface elements) and the slip velocity over an entire activity cycle tracking the contact conditions for each surface element. In order to accurately depict the moti ons in a knee joint during patient activity Scott Banks and Andrew Hodge (1996) developed a method for imaging the components in vivo known as video fluoroscopy. This method is used to determine the kinematics of TKR components throughout a variety of exe rcises. To accomplish this, Banks and Hodge took x rays of the patients at discrete time intervals during the motion, and compared the component outlines with predefined libraries of component positions. The image comparison was made using Fourier transf orms of the shapes. The results were data files containing the relative position of the origin of the femoral condyle with respect

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14 to the tibial bearings origin, as well as, the orientation of the axes of the femoral condyles coordinate system with resp ect to the axes of the bearings system. This technique has since been used in other studies of knee positioning (Asano et al. 2001; Hoff et al. 1998) Other groups have tried to accurately reproduce the motions of the knee. Chen et al. (2001) developed a computational technique to determine preoperatively the kinematics of a knee replacement design. The program was validated using a theoretical symmetric knee, so any motion in the medial lateral direction would be shown as erroneous. This was not the only attempt to use technology to estimate in vivo motions. Godest et al. (2002) used computational models to estimate the motions in a knee using models of muscles and ligaments as well as Finite Element Analysis (FEA) to determine feasible pressure dist ributions. The result was pressures ranging from 17 to 22 MPa depending on the mesh size and angles accurate to 15% The ability to accurately predict pressure distributions on the surface of the tibial component depends on the s imulations capability of reproducing muscle and ligament forces in the knee accurately. Forces experienced by the knee joint during motions are not clearly defined to date. While many studies have focused on estimating the forces in the muscles, tendons ligaments etc., there is yet to be a perfect solution. The reason for this is the complexity of the tissue surrounding the knee joint. Not only the muscles which move the legs, but the tendons and ligaments which hold the joint together must be account ed for in modeling of the system. Costigan et al (2002) studied the gait and stair cycles of 35 patients to estimate the kinetics of the knee during these motions. The study reported maximum contact forces in the distal proximal direction ranging from 3 to 6

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15 times bodyweight. Because the instrumentation of a knee in vivo is not feasible at this time many studies have focused on developing models for the various tissues surrounding the knee (Seireg and Arvikar, 2001; Mikosz et al., 2000; Komistek et al., 2000). Seireg and Arvikar used a linear springs method to model the muscle and bone forces in the knee and found maximum force values of six times body weight. This result is supported by the study by Costigan et al. (2002). Another solution incorporati ng the springs method was Mikosz et al. who modeled 13 muscles and various tendons and ligaments to find a solution for knee forces. Their solution was compared to EMG readings, which approximate muscle forces, and was validated as a first order approxima tion. Komistek et al incorporated Kanes method of dynamics into a model of the lower extremity. This study modeled forces from muscles, ligaments and tendons in the ankle, knee and hip joints. The solution required solving 30 equations with as many unk nowns, for several predetermined positions. The results were compared to available video fluoroscopy data. Temperature rise in total hip joint replacements as a result of frictional energy being dissipated at the contact has been measured in vivo (Bergman n et al., 2001a), in vitro (Liao et al., 2002), and has been modeled using 2 d and 3 d finite element techniques (Hu et al., 2001, and Bergmann et al., 2001b, respectively). In the main, the studies conclude that frictional heating may cause temperature r ises on the order of 1 10 C, and that low friction implant materials are the best way to reduce frictional heating. Kurtz et al (2002) reviewed the thermomechanical behavior of UHMWPE. It was shown that thermal processing, primarily used to increase oxid ation resistance, significantly changed the mechanical behavior of the UHMWPE. Kurtz also determined that environmental temperatures between 20 C o and 60 C o do affect mechanical

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16 properties of the polymer, but th ey can be accurately predicted using an Arrhenius model. . . Young et al. investigated the effect of heat generated by friction in the knee joint. It was determined that frictional heating caused temperature rise at the surface, but due to cooling pro vided in vivo by synovial fluid the subsurface experienced the highest temperature rise. Modeling of the bearing yielded a maximum temperature rise below the surface 1 to 2 mm. This distance corresponds to a maximum subsurface shear stress, and the combi nation of the two could result in higher wear for the prosthesis. The authors conclude that reduction in frictional heating would increase the functional life of a TKR. Bergmann, Graichen, Rohlmann, Verdonschot and van Lenthe (2001) studied temperature ri se in hip replacements in vivo Data was recorded from instrumented components using telemetry. The patients were studied during gait and cycling activities. Maximum recorded temperature was 43.1 C o after one hour of walking. The te mperatures for cycling were lower than gait indicating that normal load had a large influence on the heat produced. This conclusion is supported by the equation for power produced from frictional heating: N P FV m 2.1 The group al so published a follow up paper using FEA to determine the sensitivity of temperature rises to materials considered for the prostheses. Material properties of the femoral head in the replacement were shown to affect the thermal state e.g. a cobalt chromium head resulted in temperatures in the synovia of 46 C o Hu et al. (2001) studied the affect of temperature rise in total hip replacements. The thermomechanical finite element simulation showed temperature rise in the hip to be

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17 significant in changing the properties of the polymer. The maximum temperature seen in the simulations was ( ) 42. 9 110 CF oo this occurred at a friction coefficient of 0.1. The results agree with experimental results produced by Bergmann et al. The study noted that temperature can affect wear rate, creep, fatigue and oxidati ve degradation. All of these factors lead to aseptic loosening of the parts and early revision surgery.

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18 CHAPTER 3 ENGINEERING APPROACH 3.1 Experimental Data Fluoroscopic kinematic data were collected from one total knee arthroplasty patient: female, age 65 at time of surgery, height 170 cm, mass 70 kg ( Harman et al. 2001) The implanted component was a cemented posterior cruciate ligament retaining prosthesis (Series 7000, Stryker Howmedica Osteonics, Inc, Allendale, NJ) with a 6.8 mm thick insert. The Knee Society Clinical Rating System scores for this patient ( Insall et al. 1989) were 97 knee and 80 function after one year and 99 knee and 100 function after two years. The collected kinematics of this implant and patient are shown in Figure 3 1, with a locus plot describing the path of the contact pressure centroid over the insert for the various activities. Figure 3 1: Component design and film strips of the kinematics for the gait and stair rise activities, the corresponding locus plots of cont act pressure centroid is shown to the far right.

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19 The patient performed treadmill gait and stair rise/descent activities during fluoroscopic motion analysis (Banks, 1992; Banks and Hodge, 1996; Banks et al. 1997a, Banks et al. 1997b) 21 months after surge ry. Fluoroscopic analysis matches three dimensional geometric models of the prosthetic components to the two dimensional fluoroscopic images. The technique is accurate to approximately 1 for all rotations and 0.5 mm for translations in the sagittal plan e ( Banks and Hodge, 1996). Kinematic data from one representative cycle of each activity was averaged in 5 increments of knee flexion for stair and 1% increments for gait including stance and swing phases. A dynamic simulation of the patients in vivo kn ee mechanics was created by incorporating an elastic contact model into a commercial multibody dynamics software program (Fregly, Sawyer, Banks and Harman, 2002)(Pro/MECHANICA MOTION, Parametric Technology Corporation, Waltham, MA). This approach was taken to predict joint kinematics and contact pressures in a fraction of the time possible with current dynamic finite element methods (Giddings et al. 2001; Godest et al. 2002). The contact model uses elastic foundation theory (Johnson, 1985; An et al. 1990 ; Blankevoort et al. 1991; Li et al. 1997) and treats the tibial insert as an elastic layer contacting a rigid femoral component. This model formulation accommodates the finite thickness and dimensions of the tibial insert, conformal or non conformal con tact situations, and linear or nonlinear polyethylene material properties. The elastic contact model was implemented as a dynamic link library that can be incorporated into any multibody dynamics software. The elastic foundation approach calculates contact pressures on a grid of elements covering the tibial insert contact surfaces. The elements define a bed of springs where

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20 each spring is independent from its neighbors (Johnson, 1985). This approximation eliminates the integral nature of contact problems, thereby greatly simplifying the analysis of conformal geometry or nonlinear materials. For any element, given the interpenetration d between the undeformed tibial and femoral surfaces in the direction of the local surface normal, the contact pressure p acting on the element can be calculated from (Johnson, 1985; An et al. 1990; Blankevoort et al. 1991). ( 1) (( 1 )( 12 )) E pd h n nn = +3.1 where E is Youngs modulus of the elastic layer, n is Poissons ratio of the elastic layer, and h is the layer thickness at that location. The interpenetration d for each element is calculated using the ACIS 3D Toolkit (Spatial Corporation, Westminster, CO). For a no nlinear material, E can be defined as a nonlinear function of p (Cripton, 1993), in which case Eqn 3.1 becomes a nonlinear equation in p that can be solved using standard root finding methods (Nuo and Ahmed, 2001). To produce a set of point forces, the re sulting element pressures are multiplied by their corresponding areas. These forces are replaced with a single equivalent force and torque applied to both bodies for purposes of dynamic simulation (Kane and Levinson, 1985). The dynamic contact model used i n vivo fluoroscopic measurements (anterior posterior translation, internal external rotation, and flexion; Fig. 3 2 a, b, and c, respectively) as prescribed kinematic inputs. The model predicted the remaining degrees of freedom (axial translation, varus va lgus rotation, and medial lateral translation) via forward dynamic simulation to ensure compatibility with the applied loads. All prescribed and predicted motions were for the femur moving with respect to a fixed tibia.

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21 Figure 3 2: In vivo experimental data used as inputs to the dynamic contact model. a)Anterior posterior (AP) translation. b) Internal external (IE) rotation. c) Flexion. d) Axial force. Kinematic data are from pre retrieval video fluoroscopy gait and stair experiments with the femur movin g with respect to the tibia. Anterior translation and external rotation are positive. Axial force data are scaled vertical ground reaction force data from a patient of similar age, height, weight, and knee flexion characteristics. Calculation of accurate s lip velocities and contact pressures on individual surface elements requires the dimensions of the simulated contact patch to match in vivo conditions. The ability of the elastic foundation contact model to predict contact areas and pressures was evaluated experimentally using 16 different static loading conditions (loads of 750, 1500, 2250, and 3000 N and flexion angles of 0, 30, 60, and 90). For each condition, experimental pressure measurements were made with a Tekscan K Scan sensor. Using linear polyeth ylene material properties, the model was able to predict experimentally measured average contact pressures to within 0.3 0.5 MPa, indicating that the contact areas were also well predicted. Predictions made using nonlinear material

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22 properties were less a ccurate. Consequently, a linear rather than nonlinear polyethylene material model was used in the present study. Dynamic simulations, representing patient specific in vivo conditions, were generated by combining in vivo fluoroscopic measurements with assum ed loading conditions. In the dynamic contact model, the femoral component possessed six degrees of freedom (DOFs) relative to the tibial insert. Three DOFs (anterior posterior translation, internal external rotation, and flexion) were defined to match flu oroscopically measured gait and stair kinematics (two separate simulations). The remaining three DOFs (axial translation, varus valgus rotation, and medial lateral translation) were numerically integrated to predict their motion. An axial force was applied vertically downward to the femoral component to produce a 70% medial 30% lateral load split at 0 flexion (Johnson et al. 1981; Schipplein and Andriacchi, 1991; Hurwitz et al. 1998). The axial force curve for each activity was defined by scaling a verti cal ground reaction force curve (Lu et al. 1997; Taylor et al. 1998, Taylor and Walker, 2001) to be between 0.25 and 3.0 BW (Schipplein and Andriacchi, 1991; Lu et al. 1997; Taylor et al. 1998, Taylor and Walker, 2001). Ground reaction force data were taken from a patient of similar age, height, weight, and knee flexion characteristics. The dynamic contact model generated contact pressures and slip velocities in two steps. First, a forward dynamics simulation calculated contact forces and kinematics. T hen an inverse dynamics analysis used the results from the forward dynamics simulation to calculate contact pressures and slip velocities over a 50x50 element grid on each tibial contact surface.

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23 3.2 Mathematics A computational wear model was developed to produce element by element damage predictions given the calculated time history of contact pressures and slip velocities experienced by each element. The model computes total damage depth for each element as the sum of material removal due to mild wear and surface deformation due to compressive creep: Damag e Wea r Creep N d dd = + Eqn 3.2 where Damage d is the total damage, Wear d is the damage per cycle due to mild wear, N is the total number of cycles, and Creep d is the damage due to creep. The number of cycles was based on the number of months implanted assuming 1 million cycles per year of gait or stair (Schmalzried et al. 1998). Wear predictions were made in the software using Archards wear model for mild wear (Archard and Hirst, 1956). The model predicts the wear depth at a point on the surface of a sliding contact based on the contact pressure, sliding distance and the wear rate: 11 nn Wea r i i ii ii k p dk p vt d == ==D Eqn 3.3 where k is the wear rate ( ) 3 m m Nm i p is the contact pressure (MPa) and the sliding distance is calculated as the product of the slip velocity magnitude i v (m/s) multiplied by t he time increment t D (s). The appropriate value of wear rate was based on surface roughness values measured from contacting areas of the retrieved femoral component. The measurements were made using a white light optical interfero meter

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24 (Wyko NT1000, Veeco Instruments, Woodbury, NY). Multiple locations on the surface yielded results varying from 46 to 275 nm with an average value of 131 nm. Fisher et al. (1994) published wear rates for UHMWPE, subjected to similar contact conditio ns, as a function of a R Based on the average a R value, a wear rate of 93 22 0 10 k m m Nm = was chosen. Ultra high molecular weight polyethylene is a viscoelastic and viscoplastic material, as such it deforms in a time dependent manner when stressed (Waldman and Bryant, 1997; Lee and Pienkowski, 1998). This characteristic of the polymer is responsible for some portion of surface deformation and must be accounted for. Creep characteristics for medi cal grade UHMWPE have been published by Lee and Pienkowski (1998). According to their research, compressive creep can be estimated as a function of time of compression and contact pressure using the following model: 34 1 1 1 3.49 1 1 0 7.99 6 1 04 n ci n i Cree p ci n i ci i p Lo g N th t d -= = = = + DD 3.4 wher e all notation is previously defined with the exception of the subscript c indicating instances in time when the surface pressure i p is non zero, and h the initial thickness of th e tibial insert. The model assumes pressures are in MPa and time has units of minutes. The resulting deformation has units matching the units of bearing thickness e.g. bearing thickness in mm yields deformations in mm. The thermal modeling begins by as suming a particular partitioning of energy Q (J) into the femoral component f Q and tibial component t Q respectively. Eqn 3.5 gives an expression for the energy dissipated in the co ntact, where m is the friction coefficient, F

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25 is the normal load, V is the slip speed, T is the total activity time, and t is an instant in time during the activity. 0 T ft Q Q Q FVd tt mt =+= 3.5 The energy is partitioned into the two bodies as shown in Figure 3 3 and given by Eqn 3.6. The energy partition rule is based on matching the interface temperature for: (a) stat ionary contact area of two half spaces, and (b) femoral component moving relative to a stationary tibial component. Both the femoral component and tibial component are modeled as half spaces of thermal conductivities f k and t k respectively. Partitioning heat based on a moving heat source (b) is generally preferred when the Peclet number, Pe is greater than 10 (Williams et al. 1994; Bhushan, 1999). 1 1 11 10 1 1 1.6 10 1 1 1.6 f t tf tf f f t tf tf f k k P e Q Q QQ kk Pe k k P e Q Q QQ kk Pe -< = + =+ > = + =+ 3 .6 The Peclet number for the femoral component is defined as f f f Vl Pe a = where f V is a characteristic sliding speed of the femoral component, l is a characteristic half width of the contact patch, and f a is the thermal diffusivity of the femoral component. From Eqn 3.6 it can readily be seen that the moving source partitions greater heat into the femoral component for 10 f Pe > A stationary parti tioning of heat is assumed for the remainder of the modeling and analysis, in part because partitioning greater amounts of heat into the

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26 tibial component should provide an upper bound on the frictional heating likely to be seen in service. Figure 3 3: S chematic of the knee joint showing the partitioning of energy into the femoral and tibial component. The heat flux 2 ( /) i qWm & is found for a tibial element i by partitioning the average energy dissipated over tha t element and dividing by the activity time. This is shown in Eqn 3.7, where P t is the pressure on element i at time t 0 1 T i f t q PVd k T k tt m t = + & 3.7 The temperature rise l q at a particular element l is found using the algorithm shown schematically in Figure 3 4 and given by Eqn 3.8. This algorithm is for square

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27 elements of area A half width a and the vector from the origin element l to element i is i r l r ; 0 1.12 2 ni ii i t ti q a qA k kr l ll l l q p = =+ && r 3.8 The first group in the algorithm is the central temper ature rise of the square element under uniform heat flux q l & The second group in the algorithm is the summation of all of the contributions of the remaining surface elements to the temperature rise. This algorithm treats all of t he contributing elements as a single point source of heat, i ii Q qA = & & and uses the classic point source solution discussed in detail in Carslaw and Jaeger (1959). This algorithm is used for all elements regardless of whether or not they experience a frictional heat flux. Figure 3 4: Schematic of the solution approach for temperature rise of element l with neighboring elements treated as point sources of heat (both frictional heating and convective cooling).

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28 The elements that do not experience a frictional heat flux during the activity are exposed to the synovial fluid, and convective cooling of the component occurs through this surface/fluid interface. The heat flux of these elements not in contact is assume d to follow Newtons law of cooling, as shown in Eqn 3.9, where i q is the temperature rise of the element above ambient temperature is amb TT q =and h is the convection heat transfer coeffic ient with units of ( ) 2 / WmK ii qh q =& 3.9 The solution for the surface temperature map of the tibial component requires an iterative solver due to the coupling between the convective cooling heat flux and the s urface temperature. The solution approach is similar to a fixed point iteration with two constraints on the solution method: 1) the convective cooling power cannot exceed the frictional heat, and 2) surface temperatures below ambient are not permitted. An alysis of crossing motion occurs at the level of the individual surface elements on the tibial insert and observes the motion of the femoral component relative to each element. The only motions of interest are those that occur when the element is in conta ct with the femoral component, a condition of non zero contact pressure. Following the style of presentation previously shown by (Wang, 2001) on crossing motions in hip contacts, a trimetric view of the tibial surfaces with over laid velocity vectors for 10 representative elements is shown in Figure 2. This qualitative plot is generated from stair rise data, which shows the greatest degree of crossing. It is shown here to illustrate the very limited degree of crossing motion seen in this design, and to i llustrate the need to develop a formulaic quantitative measure of crossing intensity that can be used to visualize crossing on all the elements simultaneously.

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29 Figure 3 5 (a) gives a vector plot of the slip velocities for 17 evenly spaced increments in ti me, for an element in the lateral compartment of the tibial bearing during stair rise. As can be seen from the locus plot shown in Figure 3.1 this patient showed lateral pivoting. The elements in the lateral compartment are in contact for the most number of time steps; thus, they provide the most interesting vector plots. In Figure 3 5 (b) the x axis, which is the medial lateral axis, of the slip velocity plot is exaggerated by almost an order of magnitude. In the motion path plot, these slip velocities are multiplied by the time increment to give incremental displacements of the femoral component over the element. These incremental motion vectors are strung head to tail to give a motion path. In this plot the medial lateral axis is exaggerated 20 time s, to the right an un exaggerated motion path is shown. The arrow heads on the motion path plot are evenly spaced by time, and some information about sliding speed at various locations along the path can be gleaned. Instinctively, one knows that motion in the absence of load is not damaging. Similarly, motion during lightly loaded contact imparts less energy to the surface than identical motion during a highly loaded contact. To account for this, a tribological intensity vector is defined as the product of the contact pressure ( P ) and the slip vector ( t = D v v dV ), as shown by Eqn 3.10. tribological intensity = P v d 3.10

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30 Figure 3 5: (a) Vector plot of the counterface slip velocities for 17 different instan ces during contact. This particular location in the lateral compartment during a stair rise activity; this location showed significant crossing motion. ( b) The motion path of the femoral component over this location, notice the x axis is exaggerated by 2 0 times, a 1 to 1 motion path is shown to the right.

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31 Figure 3 6 (a) shows a vector plot of tribological intensity, with an angular coordinate ( q ) defined from the positive medial lateral axis for a vector moving in the anterior di rection (solid) and from the negative medial lateral axis for a vector moving in the posterior direction (dashed). This allows all vectors to be placed on a single plot where the orientation of slip has the same angular coordinate regardless of direction, such an orientation plot is show in figure 3 6 (b). A scatter plot of tribological intensity versus angular coordinate is shown in Figure 3 6 (c). It is hypothesized that the most probable direction of polymer orientation on a particular element coincid es with the dominant orientation of tribological intensity ( q ) for that element; this is given by Eqn 3.11, where the subscript ( i ) denotes a particular time step. 1 1 n i i ii i n i ii i Pt Pt q q = = D = D v v V V 3.11

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32 Figure 3 6: (a) Vector plot for tribological intensities shown with the angular coordinate convention ( b) Overlaying plot with angular coordinate c) Scatter plot of tribological intensity versus angular coordinate compared with uni directional motion (delta function) and vari d irectional motion (step function).

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33 As observed in laboratory tests (Burroughs and Blanchet, 2001) hip joint simulator tests (Bragdon et al. 1996), and modeled (Wang, 2001; Muratoglu et al., 1999) unidirectional motions produce significantly less wear than multi directional motions. In hip and knee joints multidirectional motion arises from abduction/adduction and internal/external rotation respectively. A crossing intensity parameter ( s ) is defined by Eqn 3.12, and describes the spread of tribological int ensity vectors about the dominant orientation direction ( q ). This is a statistical formulation for crossing motion intensity as opposed to an extreme value parameter proposed earlier by Wang, 2001. ( ) ( ) 2 1 1 n i i ii i Pt n s qq = = Dv V 3.12 The crossing intens ity of a circular counterface motion gives continuously varying and evenly distributed (vari directional) tribological intensity vector orientation and is defined as o s This is given by Eqn 3.13, where the subscript ( o ) denotes v ari directional motion parameters. 2 1 1 2 n o o o oi i Pt n p sq = = Dv V 3.13 The products of slip distance and contact pressure for this vari directional motion are constant and the summation must give the same overall tribological intensity as shown by Eqn 3.14. The normalized crossing intensity ( s ) is given by dividing the element crossing intensity (Eqn 3.12) by the equivalent vari directional crossing intensity (Eqn 3.13), which is shown in Eqn 3.15 in terms of only element para meters. 1 1 n o o o i ii i P t Pt n = D =D vv VV 3.14

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34 ( ) ( ) 2 1 2 11 1 2 n i i ii i nn o i i ii ii Pt Pt n qq s s s p q = == D== Dv v V V 3.15 Using Eqns 3.11 and 3.15 both the orientation direction and normalized spread of crossing motion can be found (the crossing motion intensity is defined as 0 s = for 1 n = ). Figure 3 7 shows the algorithm visually overlaying three different motions: (a) delta function that corresponds to uni directional motion 0 s = (b) multi directional motion distributed about 0 s = .09, and (c) vari directional motion distributed uniformly along the angular coordinate axis 1 s = Crossing motion is often conceptualized using a bi directional diamond pattern, where a square has a crossing angle of 90. Figure 3 8 shows a series of simple counterface motions. The usefulness of this normalized crossing intensity parameter is immediately obvious; the circular motion gives 1 s = the reciprocating motion gives 0 s = and the narrowing diamond patterns span the range sm oothly.

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35 Figure 3 7: A series of simple intensity maps, with corresponding counterface motion and normalized crossing severity index.

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3 6 Figure 3 8: Overlaying plots of velocity vectors, normalized by the largest velocity vector in the simulation, ont o the corresponding elements on the tibial mesh. These vectors are from a stair rise activity, which shows the greatest degree of crossing motion.

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37 CHAPTER 4 RESULTS AND DISCUSSION 4.1 Wear Analysis Five computational wear predictions (two activities with two load splits, and one partition of activities; 70% gait and 30% stair) were compared to the actual wear depths and patterns measured on the tibi al insert retrieved from the patient post mortem The total time of implantation at retrieval was 51 months. For both the predictions and the retrieval, visualizations of the wear contours were generated using commercial automatic inspection software (Geom agic Qualify, Raindrop Geomagic, Research Triangle Park, NC). Using the wear predictions, the center of each contact element on the tibial insert surface was displaced by the calculated damage depth Total d in the direction of the loca l surface normal. A worn polygonal surface model was created from these points, and the software generated a contour plot of the deviations between the original and worn surfaces. The retrieval showed scratching, burnishing, and tractive striations on the articular surfaces (Harman et al. 2001). Pitting and delamination were not observed. A three dimensional scan was obtained of the worn insert (Fig. 4.1a) and a matched unworn insert using a laser scanner (Vivid 900, Minolta Corporation, Ramsey, NJ) po ssessing a manufacturer reported accuracy of 0.04 mm Once the point clouds generated by the laser scans were converted to polygonal surface models and aligned by the software, a retrieval wear contour plot was also generated (Fig. 4.1b ). To determine a threshold for

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38 reporting retrieval wear, the unworn insert was aligned with the insert CAD model and the maximum deviation between contact surfaces (0.25 mm) determined. Qualitatively, the damage regions predicted by the computer simulatio ns were in good agreement with the clinical wear regions (compare Fig. 4.2 with Fig. 4.1). The medial wear scars for the 70 30 gait case (Fig. 4.2a) extended along a focalized track to the anterior medial corner of the insert, similar to the retrieval. In contrast, the medial wear scars for the 70 30 (Fig. 4.2b) stair case extended broadly to the posterior rim of the insert, enlarging the region predicted by the gait cases. The lateral wear scars for the 70 30 gait case extended more anteriorly than in the retrieval, but the anterior border on the lateral side in the 70 30 stair case corresponded well with the retrieval. Altering the load split to 50 50 decreased medial damage while increasing lateral damage for both gait (Fig. 4.2c) and stair (not shown). F or a 70% gait, 30% stair partitioning of activities based on linear rules of mixture, the damage area for a 70 30 load split (Fig. 4.2d) was a combination of the gait (Fig. 4.2a) and stair (Fig. 4.2b) damage areas. For both gait and stair, the lateral wear regions were more central in the anterior posterior direction than were the medial regions, similar to the retrieval, and possessed a posterior border of extremely similar shape and location to the retrieval. The predicted locations of maximum damage wer e in good agreement with the retrieval (stars in Fig. 4.1b and Fig 4.2). On the lateral side, the location of maximum damage was the same in all four simulations and was consistent with the retrieval. On the medial side, the maximum damage location was shi fted toward the tibial eminence for the gait simulations (Fig. 4a and c) and posteriorly for the stair simulations (Fig. 4b). However, when a 70% gait, 30% stair partitioning of activities was considered (Fig. 4d),

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39 the predicted maximum damage location on the medial side also became consistent with the retrieval. Quantitatively, the simulations predicted maximum total damage depths on the same order of magnitude as those measured from the retrieved insert (Table 4 1). The predicted maximum damage depths ran ged from 0.9 mm to 1.8 mm. The predicted creep deformation was approximately one third to one half of the total damage. The 70 30 load split for gait and stair activities exhibited approximately equal medial and lateral maximum damage depths, whereas the 5 0 50 load cases produced about 1.5 times deeper damage on the lateral side. Total damage area was greater for gait than for stair, while total damage volume was approximately 50% larger for stair than for gait. Smaller medial loads (50 50 split) decreased the damage volume in the medial compartment and increased damage volume in the lateral compartment such that the total damage volume was unaffected by load split. Figure 4 1: Damage visualization of the retrieved tibial insert. a) Laser scan showing damage regions visible to the naked eye. b) Contour map indicating depth of damage zones. The color bar indicates depth in mm. Stars indicate location of maximum damage on each side.

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40 Figure 4 2: Damage contour maps predicted by the computer simulations. a) Gait with 70 30 load split. b) Stair with 70 30 load split. c) Gait with 50 50 load split. d) Combined activity assuming 70% gait, 30% stair with 70 30 load split. Table 4 1: Quantitative summary of damage results predicted by the computer simulations for gait and stair activities with 70 30 and 50 50 load splits. Maximum wear, creep, and total damage may occur at different locations on the surface.

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41 Combining damage predictions from the two activities (70% gait, 30% stair) resulted in damage similar to the retrieved implant (Table 4 2). The predicted locations of maximum damage depth were the same as on the retrieved insert (Fig. 4 2 (d)). Maximum damage depths for the retrieval were 0.7 mm medial and 0.8 mm lateral versus 1.0 mm and 1.1 mm for the s imulation. The combined case predicted 87% of the total damage area on the retrieval, 84% medially and 91% laterally. The medial lateral ratio for damage depth was 0.88 for the retrieval and 0.89 for the simulation while for damage area it was 1.38 for the retrieval and 1.28 for the simulation. Table 4 2: Quantitative comparison between retrieval damage and simulation damage predicted by an activity partition of 70% gait, 30% stair with a 70 30 load split. Damage results predicted by the software were ma de using a mixture of stair and gait kinematics. Although the average person does not spend 30% of their time on stairs, the flexed knee under high load motion performed when climbing stairs can be likened to a variety of everyday activities e.g. sitting in or rising from a chair, kneeling to tie a shoe, getting into or out of bed. While these activities are less frequent the summation of all the motions is accounted for in the 30% estimate.

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42 Despite its computational advantages, the current contact model formulation has limitations. It does not account for viscoelastic material properties (Waldman and Bryant, 1994; Waldman and Bryant, 1997), friction (Sathasivam and Walker, 1997), or how pressure applied at one location affects the displacement of other lo cations (Johnson, 1985). However, the most significant issue is the use a linear material model, which explains the more focalized damage regions seen in the predictions compared to the retrieval. This material model was chosen over a nonlinear model for t wo reasons. First, a linear model is more in line with the guiding concept of using models with previously published, well established parameter values. Second, in recent simulations of a different knee implant using the same dynamic contact model, a linea r model matched static contact pressure measurements better than did a nonlinear material model (Cripton, 1993) for 16 different loading conditions (loads of 750, 1500, 2250, and 3000 N and flexion angles of 0, 30, 60, and 90; Fregly et al. 2002). The va lue of Youngs modulus that reproduced the experimental data (400 MPa) was the same as the value reported by Kurtz et al. (2002). Use of a nonlinear material model (Cripton, 1993) in the simulations produces the same dog bone contact patch (due to end ef fects) as the linear material model but with more uniform contact pressures across the patch (Fig. 4 3). Thus, a nonlinear material model with well established parameter values would produce more uniform damage predictions in the medial lateral direction, similar to the retrieval, but would not likely cause dramatic changes in the depth or distribution of predicted damage.

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43 Figure 4 3: Visualization of the static contact pressures predicted by the dynamic model for an axial load of 3 BW. a) Linear mater ial model. b) Nonlinear material model. Color bar indicates pressure in MPa. The element grid is 50 x 50, the same as that used in the damage predictions. 4.2 Crossing Analysis The algorithms discussed in the mathematics section of Chapter 3 were applied to the patient kinematic data for two activities: gait and stair rise, and contour plots of tribological intensity and normalized crossing motion intensity were generated (figures 4 4 and 4 5). Both gait and stair activities showed limited crossing inte nsity, with maximum 004 s = *. and 009 s = *. respectively. The coincidence of highest crossing intensity with greatest tribological intensity occurs on the lateral compartments for both activities. This is believed to b e due to the pivoting nature of this patients kinematics. The crossing motions observed for this patient suggest that uniform bi directional patterns with 10 of included angle are a reasonable screening motion for pin on disk

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44 testing. It is unclear how significant this degree of crossing is to the tribological behavior of the implanted UHMWPE. A model previously proposed by Wang for multidirectional sliding of UHMWPE offered a dependence of wear rate on maximum included angle ( 2 a ) as shown in Eqn 4.1. 2 1 2 k a a sin 4.1 In the study by Wang (2001) experiments with pin on disk multidirectional sliding produced one order of magnitude increases in wear rate with as little as 15 of crossing motion. Conforming kn ee prostheses are currently available, with designed lateral or medial pivoting. The challenges in evaluating the potential tribological impact of such designs require evaluation of both tribological intensity and crossing intensity. Perhaps, overall tri bological severity is the product of the two.

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45 Figure 4 4: Contour maps of a) tribological intensity and b) normalized crossing severity index for elements in contact during gait.

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46 Figure 4 5: Contour maps of a) tribological intensity and b) normalize d crossing severity index for elements in contact during stair rise.

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47 4.3 Thermal Analysis The values of the various variables needed in this code that defines the standard condition for both gait and stair rise activities are given in Table 4 3. The standard condition assumes a 1 Hz activity with a Cobalt Chrome femoral component, a low convective heat transfer coefficient, and a low Peclet number partitioning of heat. Table 4 3: List of standard conditions thermal conductivity elastic modulus E tibial component k t Poisson's ratio n convection coefficient h cycle time (stair & gait) T friction coefficient m femoral component k f value 460. 0.46 0.4 13. 30. 1.0 0.06 units MPa W/(m K) sec W/(m K) W/(m 2 K) reference(s) Kurtz et al ., 2002 Hall and Unsworth, 1997 Bartel et al ., 1995 Kakac and Yener, 1994 The heat flux contour plot as described by Eqn 3.7 is shown in figure 4 6 (a) and 4 7 (a) for the gait and stair activities respectively. The resulting temperature rise contour plots are shown in figure 4 6 (b) and 4 7 (b) for the gait and stair activities respectively. These temperature resu lts converge quickly, less than 100 iterations. For both gait and stair rise activity maximum temperature occurs on the medial compartment.

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48 The code was run varying three different parameters off of the standard condition; these were activity frequency, h eat transfer coefficient, and thermal conductivity of the femoral component. Table 4 4 summarizes these data. The choice of femoral component thermal conductivity for the standard condition corresponds to a cobalt chrome component, which is widely used. The maximum contact temperature is extremely sensitive to the choice of femoral component thermal conductivity. At 0.4 f k = half of the frictional heating is carried away by the femoral component and half goes in the tibial component. At 40.0 f k = only 1% of the frictional heating goes into the tibial component. The stationary analysis partitions heat to the respective bodies as a function of their thermal conductivities only. This is considered reasonable for low v alues of Peclet number. However, for the standard condition (Cobalt Chrome femoral component) characteristic sliding speed is 0.100 / V f ms = representative contact half length is 0.002 lm = and the thermal diffusivity is 6 3. 4 10 f x a = thus 60 f Pe The energy partition and corresponding maximum temperature rise for this standard condition is max 3% & 10.5 o t total Q QC q == and max 0.6% & 2.5 o t total Q QC q == for stationary and moving analysis resp ectively. In figure 6b the upper horizontal axis, 2 x gives the ratio of the heat going into the tibial component to the total frictional heat; thus the graph can be used to calculate temperature rise for any partitioning of energ y over the range 1% t total QQ = to 50% t total QQ =

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49 Figure 4 6: Contour maps for continuous stair activity for the standard condition of a) the average frictional heat flux and b) the temperature rise in degrees Celsius. ( 1Hz, 13 W/( m K) f k = 0.06 m = 2 30 (W/( m K)) h =

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50 Figure 4 7: Contour maps for continuous stair activity for the standard condition of a) the average frictional heat flux and b) the temperature rise in degrees Celsius. (1Hz, 13 W/( m K) f k = 0.06 m = 2 30 (W/( m K)) h =

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51 Table 4 4: Conditions run off of the standard condition (bold) and corresponding maximum temperature rise in degrees Celsius for the gait and stair act ivities accordingly. The activity frequencies were varied between 0.5Hz and 2Hz. The nearly linear trend of frictional heating with activity frequency (Fig. 4 8 (a)) is expected. This can be most easily seen in Eqn 3.7, where the frictional heat flux is directly proportional to the sliding speed. The activity period is proportional to the kinematic path, which is essentially constant, divided by the speed. Activity frequency is the reciprocal of the

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52 activity period; thus, activity frequency is direc tly proportional to speed. Similarly, though not shown, linear dependencies with friction coefficient were expected and found. The very weak dependence on convection coefficient (Fig. 4 8 (c)) is somewhat surprising, but perhaps fortuitous. Because the f rictional heat flux was discretized and time averaged for the various elements, the tibial surface has islands of frictional heating surrounded by regions of convective cooling. Any element that experiences contact is precluded from providing cooling. Du ring these activities, elements are continuously exposed to both frictional heating and convective cooling. This mixture of heating and cooling would yield a lower average heat flux than that used by this solution procedure. In vivo the knee does not go for thousands of cycles without a break. As discussed above this steady state analysis takes average heat fluxes to predict the contact temperature. The time required for this system to reach steady state was not explored. Thus, many interrupted activi ties may not reach the temperatures predicted here. The analysis is more similar to joint simulator machines that can run for hours or days without stopping. The feasibility of these results are interesting given that many studies on the temperature ris es for hip replacements find 1 10 C and the present results for the knee replacement are generally in line with those findings. Overall the analysis here aims to provide a simple first model for thermal analysis during continuous activities such as gait and stair for total knee replacements with UHMWPE tibial bearings. The findings from the standard conditions are most likely overestimates of actual steady state temperature rises, primarily because of the heat partitioning and time averaging of the heat flux,

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53 which precludes frictional heating and convective cooling to operate on the same element. Figure 4 8: Plots of maximum temperature for variations in a) activity rate, b) thermal conductivity of the femoral component, and c) convection heat transfer coefficient.

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54 CHAPTER 5 SUMMARY AND CLOSURE Patients specific kinematic data derived from fluoros copy has been a source of inputs for making wear predictions. Unfortunately, these measurements are based on 2 D x ray images, and have uncertainties based on lack of depth perception and inaccuracies of the 3 D image matching. In the sagittal plane the errors can be as much as 0.5 mm, which is significant when wear depths are on the order of 1 mm. Fortunately, another group on campus has been working with multibody dynamics software and was able to import the data and optimize the motion of the two comp onents based on elastic foundation. This group was capable of resolving unknowns which could not be determined through video fluoroscopy. The refined kinematics files were used with a custom software package for producing wear predictions, creep predicti ons, crossing motion analysis and thermal effects of gait and stair rise exercises. The predictions were made using gait motion and stair rise motion separately, but to predict damage for a retrieval a combination of the two motions was considered. Assum ing the gait motion made up 70% of the patients total motion and the remainder was stair rise motion, the area where damage occurred and the depth of the damage coincided with that found from the laser scanned retrieval. Although this analysis has only be en made for one implant and one patient, preliminary results are promising. Future work available for this project is refining the contact model used in the multibody dynamics software and implementing creep recovery models into the damage prediction soft ware. Also validation of the thermal analysis could be performed using a motion which is easily reproduced experimentally.

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55 APPENDIX A CLASS HEIRARCHY FOR SOFTWARE DEVELOPED Figure A 1: Class hierarchy of java packages written for this project Part 1. (ArrayManipulation, DoubleFormat, Encryption, ColorWindow, Library_Files).

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56 Figure A 2: Class hierarchy for java packag es written for this project Part 2. (Library_Files, Eng_Math, threads).

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57 Figure A 3: Class hierarchy for java packages written for this project Part 3. (Graphing, ColorWheel, File_IO).

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58 APPENDIX B DERIVATION OF ALGORITHMS B.1 Temperature Rise Derivation The temperature rise analysis begins with the differential heat conducti on equation: 222 222 1 d xy z dt qq qq a ++= B 1 This equation assumes a homogeneous solid, and is satisfied by the following: ( ) 222 { ( ) ( ) ( )} 4 3 2 8 x x y y zz t p Q e Ct a q r pa -+ += B 2 where Q is the heat released (J), r is the density 3 kg m p C is the specific heat J k gK and a is the thermal diffusivity 2 m s and t is time (s). Knowing that ( ) ( ) ( ) 222 2 ''' r x x y y zz = + +B 3 equation (2) si mplifies to ( ) 2 4 3 2 8 r t p Q e Ct a q r pa = B 4 The above temperature rise is for a point source instantaneously releasing a packet of heat at a distance r from the point in question. The time that has passed since the heat was released is t. This solut ion can be integrated from the initial time corresponding to the first packet being released to the final time to give a temperature rise from a constant heat source at a constant distance producing a fixed flux for a known time period. The integral is de scribed below:

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59 ( ) 2 4 3 0 2 8 T r t p Q e dt Ct a q r pa = & B 5 where q & is the energy released per unit time ( ) W and T is the total time the heat is released from the point source. By substituting tTt =the solution for temperature rise becomes ( ) ( ) 2 4' 3 0 2 8' r T Tt p Q e dt C Tt a q r pa = & B 6 further substituting ( ) 1/2 Tt t =the equation can be rewritten as [ ] 22 4 3 1 2 4 r p T Q ed C t a qt r pa = & B 7 This equation can be evaluated and simplified to 4 4 Qr erfc kr t q p a = & B 8 where k is the thermal conductivity of the substance. If the equation is considered at very long times ( t ), the temperature rise becomes only a function of material properties and distance from the heat source: 4 Q kr q p = & B 9 As mentioned earlier this solution is for a infinite solid or whole space, however, the solution required is for a half space indicating that the temperature rise would be doubled. Therefore, 2 Q kr q p = & B 10

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60 This solution is considered for surface elements which are squares. Finding the solution for the maximum temperature rise on the surface requires integrating over the entire surface. B.2 Crossing Motion Derivation The startin g point for determining the crossing intensity of a motion path is to define a vector whose direction will be used to measure all the directions against. This vector provides a reference for the other direction vectors. In this case, a vector from medial to lateral was chosen. Tribological intensity is a variable defined in this analysis as Pd T = B 11 where P is the pressure at any point and d is the sliding distance over which the pressure is applied. In the software, each e lement has a time history of pressure, slip velocity and time increment. The equation for T can be rewritten as follows: i i ii P Vt T =D B 12 Tribological intensity for each timestep is calculated from the pressure ( P ) at the timestep, t he magnitude of the slip velocity ( i V ) at the timestep and the period of time which the two act ( i t D ). An average direction of sliding is then calculated by using the angle created between the sliding vector and the arbitrary reference vector. The angle was measured, as shown in Figure 3 6, so that a unidirectional motion would yield a delta function. The angular measurement was weighted using tribological intensity as follows:

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61 1 1 n i i ii i n i ii i P Vt P Vt q q = = D = D uv uv B 13 The angle q represents the most probable sliding direction, or the direction receiving the most energy input. This value is used to determine how close to linearity the motion is by using a standard deviation calculation : ( ) ( ) 2 1 1 n i i ii i P Vt n s qq = = Duv B 14 This equations result holds little value alone, but by normalizing it with respect to a motion which is varidirectional (a circular motion) whose tribological intensity is the same as the motion above the normalized v alue gives a result which is useful. The equivalent circular path can be described as follows: B 15 B 16 The normalized crossing motion intensity is defined as ( ) ( ) 2 1 2 1 1 1 2 n i i ii i n o o o oo i P Vt n P Vt n qq s s s p q = = D== Duv uuv B 17 Various cases of motion and their corresponding cros sing motion intensity are described in Figures 3 7, B 1 and B 2. ( ) 2 1 1 1 2 1 n o o o oo i n o o o i ii i P Vt n P V t P Vt n p sq = = = DD =D uuv uu v uv

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62 Figure B 1: Results of crossing intensity analysis with motion paths which are elliptical in nature.

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63 Figure B 2: Results of crossing intensity analysis with motion paths described by a pin rotating about its central axis while sliding linearly across a surface.

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64 LIST OF REFERENCES An, K.N., Himenso, S., Tsumura, H., Kawai, T., and Chao, E.Y.S. (1990). Pressure distribution on articular surfaces: application to joint stability analysis, Journal of Biomechanics 23 1013 1020. Archard, J.F. and Hirst, W. (1956). The wear of metals under unlubricated conditions, Proceedings of the Royal Society A236 397 410. Asano, T., Akagi, M., Tanaka, K., Tamura, J., and Nakamura, T. (2001). In vivo three dimensional knee kinematics using a biplanar image matching technique Clinical Orthpaedics and Related Research 388 157 166. Banks S.A. (1992). Model based 3D kinematic estimation from 2D perspective silhouettes: Application with total knee prostheses. PhD Dissertation, Massachusetts Institute of Technology, Cambridge MA. Banks, S.A., and Hodge, W.A. (1996). Accurate measurement of three dimensional knee replacement kinematics using single plane fluoroscopy, Transactions on Biomedical Engineering 43 638 649. Banks S.A., Markovich G.D., and Hodge W.A: (1997a). In vivo kinematics of cruciate retaining and substituting knee arthroplasties, Journal of Arthroplasty 12 297 304. Banks S.A., Markovich G.D., and Hodge W.A. (1997b). The mechanics of knee replacements during gait: In vivo fluoroscopic analysis of two de signs. American Journal of Knee Surgery 10 261 267. Bergmann, G., Graichen, F., Rohlmann, A., Verdonschot, N., and Van Lenthe, G. (2001a). Frictional heating of total hip implants. Part 1: Measurements in patients, Journal of Biomechanics 34 421 42 8. Bergmann, G., Graichen, F., Rohlmann, A., Verdonschot, N., and Van Lenthe, G. (2001b). Frictional heating of total hip implants. Part 2: Finite element study, Journal of Biomechanics 34 429 435. Beuchel, F., Beuchel, F., Pappas, M., and DAlessio J. (2001). Twenty year evaluation of meniscal bearing and rotating platform knee replacement, Clinical Orthopaedics and Related Research 388 41 50. Bhushan, B. (1999). Principles and Applications of Tribology John Wiley and Sons, New York.

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65 Blankev oort, L., Kuiper, J.H., Huiskes, R., and Grootenboer, H.J. (1991). Articular contact in a three dimensional model of the knee, Journal of Biomechanics 24 1019 1031. Bragdon, C.R., OConner, D.O., Lowenstein, J.D., Jasty, M., and Syniuta, W.D. (1996) Th e importance of multidirectional motion on the wear of polyethylene. IMechE Part H: Journal of Engineering in Medicine 210 157 165. Burroughs, B.R., and Blanchet, T.A. (2001). Factors affecting the wear of irradiated UHMWPE, Tribology Transactions 44 215 223. Cadambi, A., Engh, G.A., Dwyer, K.A., and Vinh, T.N. (1994). Osteolysis of the distal femur after total knee arthroplasty, Journal of Arthoplasty 9 579 594. Carslaw, H.S., and Jaeger, J.C. (1959). Conduction of Heat in Solids Oxford Univers ity Press, Oxford. Chen, A., Mujtaba, M., Zucherman, J., Jeong, G., Joseph, T., Wright, K., and Di Cesare, P. (2001). Midterm clinical and radiographic results with the genesis I total knee prosthesis, Journal of Arthroplasty 16 1055 1062. Costigan, P ., Deluzio, K., and Wyss, U. (2002). Knee and hip kinetics during normal stair climbing, Gait and Posture 16 31 37. Cripton, P.A. (1993). Compressive characterization of ultra high molecular weight polyethylene with applications to contact stress anal ysis of total knee replacements, Master of Science Thesis. Queens University, Kingston, Ontario. Diduch, D., Insall, J., Scott, W., Scuderi, G., and Font Rodriguez, D. (1997). Total knee replacement in young patients, Journal of Bone and Joint Surgery 34 57 61. Duffy, G., Trousdale, R., and Stuart, M. (1998). Total knee arthroplasty in patients 55 years or younger: 10 17 year results. Clinical Orthopaedics and Related Research 356 22 27. Fisher, J., Dowson, D., Hamdzah, H., and Lee, H.L. (1994). The effect of sliding velocity on the friction and wear of ultra high molecular weight polyethylene for use in total artificial joints, Wear 175 219 225. Font Rodriguez, D., Scuderi, G., and Insall, J. (1997). Survivorship of cemented total knee arth roplasty, Clinical Orthopaedics and Related Research 345 79 86. Godest, A.C., Beaugonin, M., Haug, E., Taylor, M., and Gregson, P.J. (2002). Simulation of a knee joint replacement during a gait cycle using explicit finite element analysis, Journal of B iomechanics 35 267 273.

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66 Godest, A.C., Simonis de Cloke, C., Taylor, M., Gregson, P.J., Keane A.J., Sathasivan, S., and Walker, P.S. (2000). A computational model for the prediction of total knee replacement kinematics in the sagittal plane, Journal of Biomechanics 33 435 442. Giddings, V.L., Kurtz, S.M., and Edidin, A.A. (2001). Total knee replacement polyethylene stresses during loading in a knee simulator, Journal of Tribology 123 842 847. Hahn, D.W., Wolfarth, and D.L., Parks, N.L. (1997). An alysis of polyethylene wear debris using micro Raman spectroscopy: A report on the presence of beta carotene, Journal of Biomedical Materials Research 35 31 37. Harman M.K., Banks S.A., and Hodge W.A: (2001). Polyethylene damage and knee kinematics af ter total knee arthroplasty, Clinical Orthopaedics and Related Research 392 383 393. Hoff, W., Komistek, R., Dennis, D., Gabriel, S., and Walker, S. (1998). Three dimensional determination of femoral tibial contact positions under in vivo conditions us ing fluoroscopy, Clinical Biomechanics 13 455 472. Howling, G.I., Barnett, P.I., Tipper, J.L., Stone, M.H., Fisher, J., and Ingham, E. (2001). Quantitative characterization of polyethylene debris isolated from periprosthetic tissue in early failure kne e implants and early and late failure Charnley hip implants, Journal of Biomedical Materials Research 58 415 420. Hu, C., Liau, J., Lung, C., Huang, C., and Cheng, C. (2001). A two dimensional finite element model for frictional heating analysis of tot al hip prosthesis, Materials Science and Engineering 17 11 18. Hurwitz, D.E., Sumer, D.R., Andriacchi, T.P., and Sugar, D.A. (1998). Dynamic knee loads during gait predict proximal tibial bone distribution, Journal of Biomechanics 31 423 430. Insall J.N., Dorr L.D., Scott R.D., and Scott, W.N. (1989). Rationale of the Knee Society clinical rating system, Clinical Orthopaedics and Related Research 248, 13 14. Johnson, F., Scarrow, P., and Waugh, W. (1981). Assessments of loads in the knee joint. M edical and Biological Engineering and Computing 19 237 243. Johnson, K.L. (1985). Contact Mechanics Cambridge University Press, Cambridge. Kristensen, O., Nefei, A., Kjaersgaard Andersen, P., Hvid, I., and Jensen, J. (1992). Long term results of tot al condylar knee arthroplasty in rheumatoid arthritis, Journal of Bone and Joint Surgery 74 B 803 806.

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67 Laskin, R. (2001). The genesis total knee prosthesis, Clinical Orthopaedics and Related Research 388 95 102. Lee, K., and Pienkowski, D. (1998). Compressive creep characteristics of extruded ultra high molecular weight polyethylene, Journal of Biomedical Materials Research 39 261 265. Li, G., Sakamoto, M., and Chao, E.Y.S. (1997). A comparison of different methods in predicting static pressure distribution in articulating joints, Journal of Biomechanics 30 635 638. Liao, Y.S., Lu, Z., Benya, P.D., and McKellop, H.A. (2002). Frictional heating and forced cooling of polyethylene metal and polyethylene ceramic bearing for hip joint replacement, In Proceedings of the 48 th Annual Meeting of the Orthopaedic Research Society February, Dallas, TX. Lu, T.W., Taylor, S.J.G., OConnor, J.J., and Walker, P.S. (1997). Influence of muscle activity on the forces in the femur: An in vivo study, Journal of Biomechanics 30 1101 1106. Kane, T.R. and Levinson, D.A. (1985). Dynamics: Theory and Applications McGraw Hill, New York. Komistek, R.D., Stiehl, J.B., Dennis, D.A., Paxson, R.D., and Soutas Little, R.W. (1998). Mathematical model of the lower extr emity joint reaction forces using Kanes method of dynamics, Journal of Biomechanics 31 185 189. Kurtz, S., Muratoglu, O., Evans, M., and Edidin, A. (1999). Advances in the processing, sterilization and crosslinking of ultra high molecular weight polye thylene for total joint arthroplasty, Biomaterials 20 1659 1688. Kurtz, S., Villarraga, M., Herr, M., Bergstrom, J., Rimnac, C., Edidin, A. (2002). Thermomechanical behavior of virgin and highly crosslinked ultra high molecular weight polyethylene used in total joint replacements, Biomaterials 23 3681 3697. Maxian, T.A., Brown, T.D., Pederson, D.R., and Callaghan, J.J. (1996). Adaptive finite element modeling of long term polyethylene wear in total hip arthroplasty, Journal of Bone and Joint Surgery 14 668 675. Meding, J., Ritter, M., and Faris, P. (2001). Total knee arthroplasty with 4.4 mm of tibial polyethylene, Clinical Orthopaedics and Related Research 388 112 117. Meyer, D.M., and Tichy J.A. (1999). Lubrication model of an artificial hi p joint: Pressure profile versus inclination angle of the acetabular cup, ASME Journal of Tribology 121 492 498.

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68 Mikosz, R., Andriacchi, T., and Andersson, G. (1988). Model analysis of factors influencing the prediction of muscle forces at the knee. Journal of Orhtopaedic Research 6 205 214. Murataglu, O., Bragdon, C., OConnor, D., Jasty, M., Harris, W., Rizwan, G., and McGarry, F. (1999). Unified wear model for highly crosslinked ultra high molecular weight polyethylene, Journal of Orthpaedic R esearch 13 143 146. Nuo, N. and Ahmed, A.M. (2001). Sagittal profile of the femoral condyles and its application to femorotibial contact analysis, Journal of Biomechanical Engineering 123 18 26. Oonishi, H., Ishimaru, H., and Kato, A. (1996). Effe ct of cross linkage by gamma radiation in heavy doses to low wear polyethylene in total hip prostheses, Journal of Materials science: Materials in Medicine 7 753 763. Pavone, V., Boettner, F., Fichert, S., and Sculco, T. (2001). Total condylar knee ar throplasty, Clinical Orthopaedics and Related Research 388 18 25. Premnath, V., Harris, W.H., Jasty, M., and Merrill, E.W. (1996). Gamma sterilization of UHMWPE articular implants: An analysis of the oxidation problem, Biomaterials 17 1741 1753. Ra nawat, C.S., and Boachieadjei, O. (1988). Survivorship analysis and results of total condylar knee arthroplasty 8 year to 11 year follow up period, Clinical Orthopaedics and Related Research 226 6 13. Schipplein, O.D., and Andriacchi, T.P. (1991). I nteraction between active and passive knee stabilizers during level walking, Journal of Orthopaedic Research 9 113 119. Schmalzried, T.P., and Callaghan, J.J. (1999). Wear in total hip and knee replacements, Journal of Bone and Joint Surgery 81 A 115 136. Schmalzried, T.P., Szuszczewicz, E.S., Northfield, M.R., Akizuki, K.H., Frankel, R.E., Belcher, G., and Amstutz, H.C. (1998). Quantitative assessment of walking activity after total hip or knee replacement, Journal of Bone and Joint Surgery 80 A 54 59. Scuderi, G.R., Insall, J.N., Windsor, R.E., and Moran, M.C. (1989). Survivorship of cemented knee replacements, Journal of Bone and Joint Surgery British Volume 71 798 803. Seireg, A., and Arvikar, R. (1975). The prediction of muscular load sh aring and joint forces in the lower extremities during walking, Journal of Biomechanics 8 89 102.

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69 Sextro, G., Berry, D., and Rand, J. (2001). Total knee arthroplasty using cruciate retaining kinematic condylar prosthsis, Clinical Orthopaedics and Relat ed Research 388 33 40. Taylor, S.J.G. and Walker, P.S. (2001). Force and moments telemetered from two distal femoral replacements during various activities, Journal of Biomechanics 34 839 848. Taylor, S.J.G., Walker, P.S., Perry, J.S., Cannon, S.R., and Woledge, R. (1998). The forces in the distal femur and the knee during walking and other activities measured by telemetry, Journal of Arthroplasty 13 428 437. Van Loon, C., Wisse, M., de Waal Malefijit, M., Jansen, R., and Veth, R. (2000). The ki nematic total knee arthroplasty, Archives of Orthopaedic Trauma and Surgery 120 48 52. Waldman, S.D. and Bryant, J.T. (1997). Dynamic contact stress and rolling resistance model for total knee arthroplasties, Journal of Biomechanical Engineering 119 254 260. Wang, A., Essner, A., Polineni, V.K., Stark, C., and Dumbleton, J.H. (1998). Lubrication and wear of ultra high molecular weight polyethylene in total joint replacements, Tribology International 31 17 33. Wang, A. (2001). A unified theory o f wear of ultra high molecular weight polyethylene in multidirectional sliding, Wear 248 38 47. Williams, J.I., Young, N., and Archinoff, R. (1994). Comparing a disease specific measure and a functional status measurement in assessing the waiting times and outcomes of hip and knee replacements, Quality of Life Reasearch 3 99. Wright, T.M., Rimnac, C.M., Stulberg, S.D., Mintz, L., Tsao, A.K., Klein, R.W., and McCrae, C. (1992). Wear of polyethylene in total joint replacements, Clinical Orthopaedics a nd Related Research 276 126 134. Young, T.H., Cheng, C.K., Lee, Y.M., Chen, L.Y., and Huang, C.H. (1999). Analysis of ultrahigh molecular weight polyethylene failure in artificial knee joints: Thermal effect on long term performance, Journal of Biomed ical Materials Research 48 159 164.

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70 BIOGRAPHICAL SKETCH The author was born in Orlando, Florida, in 1978. Throughout his life he excelled at mathematics and sciences. In high school he earned the Bright Futures scholarship, which helped him to pay for school at the University of Florida. The author majored in mechanical engineering with a minor in computer science as an undergraduate, and re ceived the Society of Tribologists and Lubrication Engineers scholarship award. After graduating in 2001, he decided to continue his work by going to graduate school at the University of Florida.


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Title: Computational modeling: the effects of motion on total knee replacements
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Title: Computational modeling: the effects of motion on total knee replacements
Physical Description: Mixed Material
Creator: Hamilton, Matthew A. ( Author, Primary )
Publication Date: 2003
Copyright Date: 2003

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Source Institution: University of Florida
Holding Location: University of Florida
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COMPUTATIONAL MODELING: THE EFFECTS OF MOTION ON TOTAL KNEE
REPLACEMENTS












By

MATTHEW A. HAMILTON


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


2003










ACKNOWLEDGMENTS

I wish to thank Dr. Greg Sawyer for his guidance and support on not only the present

project, but past and future ones as well. I also appreciate the help of Dr. Ziegert and Dr.

Arakere throughout my time at the University of Florida and for agreeing to review this

work. Special thanks go to all the members of the Tribology Lab (Aaron Ison, Jason

Action, Dan and Pam Dickrell, Darren McGuire, Brian Micklos and Dave Burris) for

help provided in their various fields of expertise.

I would like to thank my parents for their support through all my years in school.

I would like to thank my wife, Melissa. Her encouragement throughout this endeavor has

helped me tremendously. Without her I could not have achieved this goal.















TABLE OF CONTENTS


Page

A C K N O W LED G M E N T S ......................... ................................................................... ii

LIST OF TA BLES ....................................................... ........... .. ............ .. iv

LIST OF FIGURES ............................... .... ...... ... ................. .v

A B S T R A C T ...................................... ..................................................... viii

Chapter

1 IN T R O D U C T IO N ................................................................................ .......... .. .. ... 1

2 BACKGROUND AND LITERATURE REVIEW ......................................................5

3 EN G IN EER IN G A PPR O A CH ........................................................... .....................18

E x p erim en tal D ata ........................................................................... .......... .. .. .. 18
M them atics ..................................................................................................... ..... 23

4 RESULTS AND DISCU SSION .................................. ............... ...............37

W e a r A n a ly sis ....................................................................................................... 3 7
C ro ssin g A n aly sis ................................................................................................. 4 3
Thermal Analysis .................................... .. .......... ............... 47

5 SU M M A R Y A N D CLO SU RE ........................................................... .....................54

A CLASS HEIRARCHY FOR SOFTWARE DEVELOPED.......................................55

B DERIVATION OF ALGORITHM S ........................................ ........................ 58

Temperature Rise Derivation ............. ..... .................................... 58
C crossing M otion D erivation ............................................................ .....................60

L IST O F R E FE R E N C E S ......................................................................... ....................64

B IO G R A PH IC A L SK E TCH ..................................................................... ..................70















LIST OF TABLES


Tablege

2-1 Causes of TKR failure. ...................................... .......................... .9

4-1 Quantitative summary of damage results predicted by the computer simulations for
gait and stair activities with 70-30 and 50-50 load splits. Maximum wear, creep,
and total damage may occur at different locations on the surface .........................40

4-2 Quantitative comparison between retrieval damage and simulation damage predicted
by an activity partition of 70% gait, 30% stair with a 70-30 load split ..................41

4-3 L ist of standard conditions .............................................................. .....................47

4-4 Conditions run off of the 'standard condition' (bold) and corresponding maximum
temperature rise in degrees Celsius for the gait and stair activities accordingly......51















LIST OF FIGURES


Figure page

2-1: Norm al knee vs. prosthetic knee .................................................... .................6

2-2: Percentage of failures in TKRs vs. years implanted.............................................10

3-1: Component design and film strips of the kinematics for the gait and stair rise
activities, the corresponding locus plots of contact pressure centroid is shown to the
far right. .............................................................................18

3-2: In vivo experimental data used as inputs to the dynamic contact model. a)Anterior-
posterior (AP) translation. b) Internal-external (IE) rotation. c) Flexion. d) Axial
force. Kinematic data are from pre-retrieval video fluoroscopy gait and stair
experiments with the femur moving with respect to the tibia. Anterior translation
and external rotation are positive. Axial force data are scaled vertical ground
reaction force data from a patient of similar age, height, weight, and knee flexion
characteristics. ........................................................................2 1

3-3: Schematic of the knee joint showing the partitioning of energy into the femoral and
tibial com ponent. ......................................................................26

3-4: Schematic of the solution approach for temperature rise of element 2A, with
neighboring elements treated as point sources of heat (both frictional heating and
convective cooling). ................................... .. ... ....... ............ 27

3-5: (a) Vector plot of the counterface slip velocities for 17 different instances during
contact. This particular location in the lateral compartment during a stair rise
activity; this location showed significant crossing motion. (b) The motion path of
the femoral component over this location, notice the x-axis is exaggerated by 20
times, a 1-to-i motion path is shown to the right. ...................... ..............30

3-6: (a) Vector plot for tribological intensities shown with the angular coordinate
convention (b) Overlaying plot with angular coordinate c) Scatter plot of
tribological intensity versus angular coordinate compared with uni-directional
motion (delta function) and vari-directional motion (step function)........................32

3-7: A series of simple intensity maps, with corresponding counterface motion and
norm alized crossing severity index. .............................................. ............... 35









3-8: Overlaying plots of velocity vectors, normalized by the largest velocity vector in the
simulation, onto the corresponding elements on the tibial mesh. These vectors are
from a stair rise activity, which shows the greatest degree of crossing motion. ......36

4-1: Damage visualization of the retrieved tibial insert, a) Laser scan showing damage
regions visible to the naked eye. b) Contour map indicating depth of damage zones.
The color bar indicates depth in mm. Stars indicate location of maximum damage
on each side. ....................................................... ................ 3 9

4-2: Damage contour maps predicted by the computer simulations. a) Gait with 70-30
load split. b) Stair with 70-30 load split. c) Gait with 50-50 load split. d) Combined
activity assuming 70% gait, 30% stair with 70-30 load split ...............................40

4-3: Visualization of the static contact pressures predicted by the dynamic model for an
axial load of 3 BW. a) Linear material model. b) Nonlinear material model. Color
bar indicates pressure in MPa. The element grid is 50 x 50, the same as that used in
the dam age predictions. ............................... .... .......... ................ ............. 43

4-4: Contour maps of a) tribological intensity and b) normalized crossing severity index
for elem ents in contact during gait. ............................................... ............... 45

4-5: Contour maps of a) tribological intensity and b) normalized crossing severity index
for elements in contact during stair rise. ...................................... ............... 46

4-6: Contour maps for continuous stair activity for the 'standard condition' of a) the
average frictional heat flux and b) the temperature rise in degrees Celsius. (1Hz,
k, =13 W/(m K), y =0.06, h =30 (W/(m2 -K)) ....................... ....... .........49

4-7: Contour maps for continuous stair activity for the 'standard condition' of a) the
average frictional heat flux and b) the temperature rise in degrees Celsius. (1Hz,
kf =13 W/(m K), y =0.06, h =30 (W/(m2 -K)) ....................... ............. 50

4-8: Plots of maximum temperature for variations in a) activity rate, b) thermal
conductivity of the femoral component, and c) convection heat transfer coefficient. 53

A-1: Class hierarchy of java packages written for this project Part 1.
(ArrayManipulation, DoubleFormat, Encryption, ColorWindow, Library Files)...55

A-2: Class hierarchy forjava packages written for this project Part 2. (Library Files,
Eng M ath, threads). ........................... ........ .... .. .... ............ 56

A-3: Class hierarchy for java packages written for this project -Part 3. (Graphing,
ColorW heel, FileJ_ O ) ...................... .................. ................... .. ...... 57

B-1: Results of crossing intensity analysis with motion paths which are elliptical in
n atu re ............................................................ ................ 6 2









B-2: Results of crossing intensity analysis with motion paths described by a pin rotating
about its central axis while sliding linearly across a surface .................................63











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

COMPUTATIONAL MODELING: THE EFFECTS OF MOTION ON TOTAL KNEE
REPLACEMENTS

By

Matthew A. Hamilton

May 2003

Chair: W. Gregory Sawyer
Department: Mechanical and Aerospace Engineering

Total knee replacements have evolved significantly in the past half-century. A

major development, the use of Ultra-High Molecular-Weight Polyethylene (UHMWPE),

has increased functional life to more than a decade. Although the occurrence of failures

in current TKRs is low, wear and its results still plague the total joint replacement.

Wearing of the UHMWPE tibial bearing changes its geometry allowing the femoral

component to move in ways not intended by manufacturers. This loosening results in

instability and poor functionality in the joint. When the joint no longer functions as

intended, the recourse is revision surgery. Due to the loss of bone surrounding the joint

from osteolysis and the original implant surgery, doctors are left with little to work with

when revising a TKR. This results in success rates of revision surgery is below 70%

causing major concern to patients at higher risk for component failure.

The focus of current research is to increase the life of the components by

developing a tool capable of aiding researchers in the design-stage of TKR production.

By allowing designers to identify possible design flaws prior to prototyping, the testing of

new designs would be expedited. A tool of this nature would help reduce the workload









for knee simulator testing. Testing in a knee simulator is not only time consuming (each

test could take up to four months), but it is also expensive (average cost $40,000). Thus,

new ideas could get from the drawing board to the doctor's office more quickly and at a

lower cost.

The purpose of this project was to develop a tool, which could use fluoroscopic

data from a patient performing a set of exercises, to predict the effect of these motions on

a TKR over a finite period of time. The motions experienced by the surface of the

bearing were evaluated to establish a method of categorizing motions based on their

ability to damage the surface. Wear and creep damage predictions were made based on a

variety of motion inputs. These damage predictions, made using the software, were

compared to actual data from a retrieved tibial bearing component. Finally, the thermal

effects created by motions in the knee were evaluated. Although it is not likely that

thermal effects created by everyday activity could be a factor, those experienced in knee

simulators may be sufficient to affect results.

The tool developed accurately predicted wear geography, which is a first step

toward predicting total damage. Also tools for quantifying multidirectional motion and

temperature rise due to frictional heating were developed. The crossing motion results

give an indication of the severity of crossing in a particular motion. Thermal results from

frictional heating were from 10 to 15 C .














CHAPTER 1
INTRODUCTION

The evolution of the Total Knee Replacement (TKR) began in the early 1940s. The

first generation of TKR consisted of a hinge joint, which connected the femur to the tibia.

Although the mechanism provided short-term pain relief, its motion was limited and had

severe problems with loosening and infection. As a result, the design was abandoned

after a few years. The second-generation prosthesis, the McKeever developed in1957

and the Macintosh developed in 1958, were simply metal plates separating the tibia and

femur. The theory behind the plate was to prevent rubbing of irregular surfaces. These

devices had mild success, but were unpredictable and did not relieve symptoms in many

patients. The next generation of design came from Massachusetts General Hospital. It

was a mold type arthroplasty designed to replace the femoral half of the knee joint, but

like its predecessors it was unpredictable and failed after short periods of use.

A new era of knee replacement came about in the 1960s, when Frank Gunston, an

orthopedist from Sir John Chamley's Hip Center, developed a metal on plastic knee

replacement. The prosthesis was cemented directly to the bone, and was the first metal

and plastic knee (introduced in 1968). In 1972, Dr. John Insall developed a prototype for

the current total knee replacement. To replace all the surfaces in the knee the prosthesis

had three components the femur, tibia and patella. The components were attached to

the bones using bone cement. Since the development of the metal and plastic

replacement researchers have been testing new designs and materials to increase the

versatility and durability of the arthroplasty. A major development was the polymer









Ultra-High Molecular-Weight Polyethylene (UHMWPE). The polymer has been

the choice for bearing material for the past 30 years, replacing the original bearing choice

Polytetrafluoroethylene (Teflon)(Oonishi, Ishimaru, and Kato, 1996). UHMWPE and

Teflon are both biocompatible, however, UHMWPE has a much higher wear resistance.

This plastic has increased the useful-life of TKRs. Current replacements have success

rates above 90% ten-years after implantation, but the need for increased function and life

in the replacements still challenges designers and researchers.

Although the occurrence of failures in current TKRs is 'low,' wear and its results

still plague the total joint replacement. Wear is defined as damage to a solid surface,

generally involving progressive loss of material, due to relative motion between that

surface and a contacting substance or substances. Factors influencing wear are normal

load, sliding-distance, friction-coefficient, nominal temperature and environment. The

wearing of the components in a TKR has a dual effect on the life of the device. Wearing

of the UHMWPE tibial bearing changes its geometry allowing the femoral component to

move in ways not intended by manufacturers. Loosening, as a result of wear, causes

instability and poor functionality in the joint. Another issue troubling designers of these

prostheses is the effect of wear debris generated from the bearing. Although the volume

of material worn away by use has decreased with the advent of UHMWPE, the size of the

particles itself has become an issue. The average size of a UHMWPE wear particle in

knee joint replacements is sub-micron, particles of this size have been labeled the most

"biologically active." The body identifies these particles as foreign bodies and triggers

defense mechanisms to eliminate them. These defense mechanisms are the source of

bone resorption, also called osteolysis, a condition where the bones surrounding the joint









are broken down. This softening of tissue surrounding the TKR causes the components

to become unstable, and eventually failure of the TKR. When the joint no longer

functions as intended the recourse is revision surgery. Due to the loss of bone

surrounding the joint from osteolysis and the original implant surgery, doctors are left

with little to work with when revising a TKR. This results in success rates of revision

surgery below 70%, causing major concern to patients at higher risk for component

failure.

The focus of current research is to increase the life of the components by

developing a tool capable of aiding researchers in the design- stage of TKR production.

By allowing designers to identify possible design flaws prior to prototyping, the testing of

new designs would be expedited. A tool of this nature would help reduce the workload

for knee simulator testing. Testing in a knee simulator is not only time consuming (each

test could take up to four months), but it is also expensive (average cost $40,000). Thus,

new ideas could get from the drawing board to the doctor's office more quickly and at a

lower cost.

The purpose of this project was to develop a tool, which could use fluoroscopic

data from a patient performing a set of exercises, to predict the effect of these motions on

a TKR over a finite period of time. The motions experienced by the surface of the

bearing were evaluated to establish a method of categorizing motions based on their

ability to damage the surface. Wear and creep damage predictions were made based on a

variety of motion inputs. These damage predictions, made using the software, were

compared to actual data from a retrieved tibial bearing component. Finally, the thermal

effects created by motions in the knee were evaluated. Although it is not likely that






4


thermal effects created by everyday activity could be a factor, those experienced in knee

simulators may be sufficient to affect results.














CHAPTER 2
BACKGROUND AND LITERATURE REVIEW

From the beginning, the durability of TKRs has been a major focus for researchers

studying the prostheses. Even the first replacements were studied from the time of

implantation to see if the solution was viable for a long-term use. Although the first

component designs were not sufficient to sustain activity for long periods of time, they

were a valuable first step toward successful replacements. As with any new technology,

fledgling attempts are often failures, but the abilities of designers to recognize problems

and create new solutions leads to success. The evolution from a simple hinge-joint or a

metal plate inserted between two sides of an arthritic joint, to molded metal replacements

of the femur with a plastic bearing component cemented directly to the tibia, and finally

to replacing the ends of the femur and tibia with anatomically correct metal replacements

separated by a molded plastic insert has taken over 50 years. While no standard design

has been set, most have the same basic components with subtle variations

The typical TKR consists of three major parts. The femoral condyle, typically

made of cobalt-chrome, is connected to the femur using rods protruding from the

backside of the component. The tibial plateau is attached to the tibia via a long rod

extending down from the bottom of flat metal plate. Finally, the tibial bearing, a molded

piece of ultra-high molecular-weight polyethylene, is snapped into the tibial plateau

providing a place for the femoral condyle to rest. The way these parts are attached to the

bones, and their shapes vary from one design to the next, but these are generally the parts

that make up the TKR.









The procedure to insert a total knee replacement involves surgically removing

damaged or injured parts of the knee, and replacing them with the components mentioned

above. Initially the muscles and ligaments surrounding the knee are separated from the

bone, leaving the joint exposed. The ends of the femur and tibia are surgically removed

so the pieces of the TKR can be easily attached. Metal rods attached to the backside of

the femoral condyle and tibial plateau are forced into the femur and tibia respectively. In

some cases the two components are cemented to the bones using bone-cement. The

bearing is attached to the tibial plateau, usually it snaps into the surface of the plateau.

Some varieties of TKR require the backside of the patella be removed and replaced with

a UHMWPE piece as well.





Patella Femoral Condyles

Cartilage A

Meniscus _

UHMWPE
Anterior
Cruciate Bearing
Ligament




Tibial
Platform


Figure 2-1: Normal knee vs. prosthetic knee.









In an effort to improve the versatility of the TKR, researchers have set out to

determine what causes the prostheses to fail. The study of knee replacement survivorship

is an active topic demonstrated by Schmalzried and Callaghan (1999) who referenced

over 200 articles. Although many designs for knee replacements exist, the results of the

studies are very similar. Sextro et al. (2001), Pavone et al. (2001), Diduch et al. (1997),

Beuchel et al. (2001) and Meding et al. (2001) all reported less than 10% failure after a

ten-year period. Table 2-1 gives an overview of studies of knee replacement survivorship

published over the last 15 years. Figure 2.2 shows a graph of survivorship rates versus

years in vivo, created from the results of those studies. While the results of these studies

indicated that success rates for the implants are 'high,' a variety of observations were

made in the studies. Bearing thickness was listed as a variable affecting the success of

total knee replacements in multiple studies. Sextro et al. (2001) noted that a bearing

thickness less than 8 mm significantly increased the chance of failure in the component.

Meding et al. (2001) also mentioned bearing thickness, although no failures were shown

as a result of bearing thickness, the study noted a large increase in knee pain scores for

bearings with a thickness of less than 6 mm. Wright et al. (1992) found that 11 of the 12

failures found in their study occurred in bearings having a thickness of 6 mm or less. The

authors concluded the smaller thickness "increased the stresses on and within the

polyethylene."(131) The less-conforming bearing and smaller thickness combined to

produce higher stresses, which are emphasized by the "predominance and the severity of

the delamination and surface deformation ... on the retrieved components."(131) Higher

stresses are associated with an increase in wear debris which "is a contributory factor ...

to long term complications, especially component loosening."(132) Diduch, Insall, Scott,









Scuderi and Font-Rodriguez (1997) commented on the importance of affixing the

components to the bones using cement. The report stated that "cement fixation has also

been shown to be an effective barrier against penetration of polyethylene particulate

debris."(60) This debris is a major cause of aseptic loosening in the TKR components.

Four of the studies (Wright et al., 1992; Sextro et al., 2001; Beuchel et al., 2001; Pavone

et al., 2001) noted osteolysis and wear specifically as causes of failure in TKRs studied.

Osteolysis or bone resorption is the result of the body trying to eliminate wear particles

building up in the tissue and fluid surrounding the knee joint. The failure rates ranged

from 2.2% to 4.4% of knees studied failing due to wear or osteolysis. While these

percentages seem small, over 266,000 total knee replacements are performed in the

United States annually, giving rise to the potential for hundreds of thousands of knees to

be affected by wear and its results within the next decade.

Wear is defined as damage to a solid surface, generally involving progressive loss

of material, due to relative motion between that surface and a contacting substance or

sub-stances. Historically three types have been identified in retrieved total joint

replacements: abrasive wear, adhesive wear and fatigue wear. Abrasive wear is a result

of plowing of UHMWPE surface or scratching of the metallic-femoral-head caused by

loose third-body particles. The third-body particles could be bone chips, bone-cement or

metallic debris trapped between the surfaces of the bearing and the femoral condyle

(Wang, 2001). Adhesive wear occurs when the asperities on the articulating surface

(typically the UHMWPE) are sheared off. In order for this to occur, the interfacial shear

strength must be greater than the bulk shear strength of the material (Bragdon et al.,













1996). Fatigue wear occurs through the initiation and propagation of surface or


subsurface cracks under cyclic loading conditions (Wang, 2001).


Table 2-1. Causes of TKR failure.

Reference Average Follow-up
Period
Ranawat and
Boachie-Adjei 9.5 yrs.
(1988)
Scuderi et al. 15 yrs
(1989)

Kristenson et al.
(1992)yrs

Cadambi et al.
(1994) 4.5 yrs.
(1994)

Diduch et al.
(1997) 8yrs.

Font-Rodriguez et al.
(1997) 7-21 yrs.
(1997)

Duffy et al.
(1998)yrs

Van Loon et al.
(2000) 12yrs


Beuchel et al.
(2001)


Laskin
(2001)

Meding et al.
(2001)

Pavone et al.
(2001)


Sextro et al.
(2001)


Tayot et al.
(2001)


20 yrs.



10 yrs.


10 yrs.



23 yrs.



16 yrs.



11.5 yrs.


Failures
(Percentage)

5
(5.9%)

12
(9.4%)

8
(11%)

30
(11.1%)

5
(4.6%)

57
(2.2%)

2
(2.7%)

11
(11%)



28
(7.5%)


4
(4%)

9
(2.3%)

10
(8.3%)


13
(7.7%)


25
(7.5%)


Failure Causes


1 loose tibial comp., 1 loose patellar comp.,
1 infection

4 loose tibial comp., 2 loose femoral comp.,
3 infections

2 loose components, 5 infections


osteolysis


1 bearing wear, 2 infection


3 comp. instabilities, 10 loose tibial comp.,
15 loose femoral comp., 24 infections


1 bearing wear, 1 osteolysis


4 bearing wear, 2 malposition,
1 persistent pain, 4 infections


1 condylar fracture, 1 loose femoral comp.
2 loose tibial comp., 5 bearing dislocations,
9 bearing wear, 3 ligament instability,
4 deep infection, 3 osteolysis


2 ligament instabilities, 2 infections


8 loose patellar comp., 1 infection


4 osteolysis, 1 perioprosthetic fracture,
2 perisstent pain, 3 infections


1 condylar fracture, 1 instability,
2 loose tibial comp., 1 loose femoral comp.,
4 bearing wear, 4 loose patellar comp.

10 loose components, 15 infections











100,%-hi I I _KIInI wI 0
A A UJta n* l. (1.l> I
F BdA(1i 0 *
a 95 W% M _U n ,M W I









Time !m planted (yre,)
5 20 25ilHAIril X








Figure 2-2: Percentage of failures in TKRs vs. years implanted.




Although wear can occur in many ways, the results of wear are similar, namely,

debris is released from the articulating surface into the surrounding environment. The

effect of the debris in vivo has been studied extensively (Hahn et al., 1996; Howling et
al., 2001). Both of these studies linked UHWPE wear debris found in the fluid and


tissue surrounding the joint to osteolysis. Wear debris in TKRs ranges in size from sub-

micron to tens-of-microns, but Howling et al. characterized particles less than one-

micron in size as the most "biologically active." Wolfarth et al. also made a connection


between sub-micron- sized particles and osteolysis in their 1996 study of UHMWPE

particles in the synovia and tissue surrounding the knee.
Tim. lmnplintndlyru,)

























The search for a polymer capable of withstanding the harsh conditions of the knee

joint reslthoughed inwear can occur in many ways, the discovery of ultra-high molecular-weight polyethylene. UHMWPE is

a polymer characterized by long molecular chains, resulting in an 'ultra-high' molecular

weight. The chains in the polymer are not typically interconnected with extremely strong


bonds allowing the polymer chains to orient themselves in the direction of sliding. By









doing this, the wear resistance of the material increases and makes the polymer a valuable

solid lubricant. This characteristic also creates a problem for the material when exposed

to multidirectional sliding. When the chains orient themselves in the direction of sliding

it makes them more susceptible to breaking from motion perpendicular to the sliding

direction. The issue of multidirectional sliding was addressed by Bragdon et al. (1996)

and Wang (2001). In Wang's 'unified theory', he compared wear rates for UHMWPE

created by linear reciprocating to those from clinically retrieved prostheses. The clinical

retrievals displayed wear rates one to two orders of magnitude greater than those

predicted by linear reciprocating.

An innovation in the processing of UHMWPE which came about specifically to

combat the effects of multidirectional sliding was improved cross-linking. Cross-linking

in UHMWPE is the connection between two adjacent polymer chains via covalent C-C

bonds. Increasing the density of these covalent bonds between chains prevents chain

mobility and orientation as well as increasing the difficulty of breaking one chain from

another. Higher cross-link density has been shown to decrease wear rates in

multidirectional sliding (Wang, 2001). There are several ways to control the cross-link

density, mainly irradiation followed by heat treatment is used. By irradiating the material

carbon bonds are broken leaving free radicals dispersed throughout the polymer. Heat-

treating serves to ensure uniform cross-link density (Wang et al., 1998). After the free

radicals are created, the carbon is available to bond with other molecules. One issue with

cross-linking is preventing other molecules from occupying the open carbon bonds. If

oxygen is present during or after the polymer is processed it can diffuse through the

material and bond to the newly formed free radicals. Exposure to oxygen after









processing dramatically decreases the number of linking reactions (Premmath et al.,

1996) effectively weakening the inter-molecule strength. This oxidation can greatly

reduce the wear resistance of the material. Storage of the material after treatment can

also influence the degree of oxidation. If the UHMWPE is stored in an open-air

environment it allows oxygen time to diffuse throughout the material preventing cross-

linking from occurring. The problems with oxidation lead to processing and storing

UHMWPE in an inert environment leaving many free radicals in the polymer un-bonded.

While oxidation can occur in the body (Premnath et al., 1996), the degree of oxidation is

greatly reduced by preventing oxidation until the component is implanted.

Debris particle generation of UHMWPE remains a clinical issue in total joint

replacements such as knees and hips. Significant advances in modeling and

understanding the origin of wear debris liberation in total hip replacements have been

made over the past decade. One aspect recently discovered is the importance of sliding

direction on wear in UHMWPE (Bragdon et al., 1996; Muratoglu et al., 1999; Burroughs

and Blanchet, 2001; Wang, 2001). Orders of magnitude changes in wear rate with

increasing degrees of crossing motion have been reported.

Linear reciprocation is motion in one direction. Since UHMWPE has the ability to

orient itself in a particular sliding direction, it can accommodate unidirectional motion

very effectively. Conversely, when sliding direction is varied, UHMWPE can exhibit

wear rates two to three orders of magnitude greater than linear tests find (Wang, 2001).

As mentioned above, several groups have researched the effects of multidirectional

sliding. The focus of these papers was motion in hip replacements, because of the hip

replacement's conforming design is conducive to multidirectional motion. In the study









by Bragdon (1996) they found that an increase in the variation of sliding direction

increased the wear rates demonstrated by the UHMWPE. Although Wang's study was

similar, he went a step further and tried to quantify the degree of multidirectional motion

by defining an angle alpha. Wang used this angle to measure degree of

multidirectionality and showed increase in the crossing angle led to an increase in wear.

The conforming and axis-symmetric nature of hip joint replacements has

facilitated the study of their relative motion for fluid mechanics and lubrication (Meyer

and Tichy, 1999) and contact studies (Bragdon et al., 1996; Maxian et al., 1996; Wang,

2001; Kurtz et al., 1999). To date, the complexity of the knee motions has prevented

detailed study of the tibial component. Locus plots, which seek to represent the

kinematics by following the most probable locations of contact on the tibial component,

do not capture the relative motions experienced by a particular location on the surface.

To capture this information it is critical to model the pressure distribution (which spans

multiple surface elements) and the slip velocity over an entire activity cycle tracking the

contact conditions for each surface element.

In order to accurately depict the motions in a knee joint during patient activity Scott

Banks and Andrew Hodge (1996) developed a method for imaging the components in

vivo known as video fluoroscopy. This method is used to determine the kinematics of

TKR components throughout a variety of exercises. To accomplish this, Banks and

Hodge took x-rays of the patients at discrete time intervals during the motion, and

compared the component outlines with predefined libraries of component positions. The

image comparison was made using Fourier transforms of the shapes. The results were

data files containing the relative position of the origin of the femoral condyle with respect









to the tibial bearing's origin, as well as, the orientation of the axes of the femoral

condyle's coordinate system with respect to the axes of the bearing's system. This

technique has since been used in other studies of knee positioning (Asano et al., 2001;

Hoff et al., 1998)

Other groups have tried to accurately reproduce the motions of the knee. Chen et

al. (2001) developed a computational technique to determine preoperatively the

kinematics of a knee replacement design. The program was validated using a theoretical

symmetric knee, so any motion in the medial-lateral direction would be shown as

erroneous. This was not the only attempt to use technology to estimate in vivo motions.

Godest et al. (2002) used computational models to estimate the motions in a knee using

models of muscles and ligaments as well as Finite Element Analysis (FEA) to determine

feasible pressure distributions. The result was pressures ranging from 17 to 22 MPa

depending on the mesh size and angles accurate to +15%.

The ability to accurately predict pressure distributions on the surface of the tibial

component depends on the simulations' capability of reproducing muscle and ligament

forces in the knee accurately. Forces experienced by the knee joint during motions are

not clearly defined to date. While many studies have focused on estimating the forces in

the muscles, tendons, ligaments etc., there is yet to be a perfect solution. The reason for

this is the complexity of the tissue surrounding the knee joint. Not only the muscles

which move the legs, but the tendons and ligaments which hold the joint together must be

accounted for in modeling of the system. Costigan et al (2002) studied the gait and stair

cycles of 35 patients to estimate the kinetics of the knee during these motions. The study

reported maximum contact forces in the distal-proximal direction ranging from 3 to 6









times bodyweight. Because the instrumentation of a knee in vivo is not feasible at this

time many studies have focused on developing models for the various tissues surrounding

the knee (Seireg and Arvikar, 2001; Mikosz et al., 2000; Komistek et al., 2000). Seireg

and Arvikar used a linear springs method to model the muscle and bone forces in the

knee and found maximum force values of six times body weight. This result is supported

by the study by Costigan et al. (2002). Another solution incorporating the springs

method was Mikosz et al. who modeled 13 muscles and various tendons and ligaments to

find a solution for knee forces. Their solution was compared to EMG readings, which

approximate muscle forces, and was validated as a first-order approximation. Komistek

et al incorporated Kane's method of dynamics into a model of the lower extremity. This

study modeled forces from muscles, ligaments and tendons in the ankle, knee and hip

joints. The solution required solving 30 equations with as many unknowns, for several

predetermined positions. The results were compared to available video-fluoroscopy data.

Temperature rise in total hip joint replacements as a result of frictional energy

being dissipated at the contact has been measured in vivo (Bergmann et al., 200 la), in

vitro (Liao et al., 2002), and has been modeled using 2-d and 3-d finite element

techniques (Hu et al., 2001, and Bergmann et al., 2001b, respectively). In the main, the

studies conclude that frictional heating may cause temperature rises on the order of 1-

10 C, and that low friction implant materials are the best way to reduce frictional heating.

Kurtz et al. (2002) reviewed the thermomechanical behavior of UHMWPE. It was

shown that thermal processing, primarily used to increase oxidation resistance,

significantly changed the mechanical behavior of the UHMWPE. Kurtz also determined

that environmental temperatures between 20"C and 60"C do affect mechanical









properties of the polymer, but they can be "accurately predicted using an Arrhenius

model ... ."

Young et al. investigated the effect of heat generated by friction in the knee joint.

It was determined that frictional heating caused temperature rise at the surface, but due to

cooling provided in vivo by synovial fluid the subsurface experienced the highest

temperature rise. Modeling of the bearing yielded a maximum temperature rise below the

surface 1 to 2 mm. This distance corresponds to a maximum subsurface shear stress, and

the combination of the two could result in higher wear for the prosthesis. The authors

conclude that reduction in frictional heating would increase the functional life of a TKR.

Bergmann, Graichen, Rohlmann, Verdonschot and van Lenthe (2001) studied

temperature rise in hip replacements in vivo. Data was recorded from instrumented

components using telemetry. The patients were studied during gait and cycling activities.

Maximum-recorded temperature was 43.1 C after one-hour of walking. The

temperatures for cycling were lower than gait indicating that normal load had a large

influence on the heat produced. This conclusion is supported by the equation for power

produced from frictional heating:

P oc pFV 2.1

The group also published a follow-up paper using FEA to determine the sensitivity

of temperature rises to materials considered for the prostheses. Material properties of the

femoral-head in the replacement were shown to affect the thermal state e.g. a cobalt-

chromium head resulted in temperatures in the synovia of 46 C.

Hu et al. (2001) studied the affect of temperature rise in total hip replacements.

The thermomechanical finite-element simulation showed temperature rise in the hip to be






17


significant in changing the properties of the polymer. The maximum temperature seen in

the simulations was 42.9C (1 10 F), this occurred at a friction coefficient of 0.1. The

results agree with experimental results produced by Bergmann et al. The study noted that

temperature can affect wear rate, creep, fatigue and oxidative degradation. All of these

factors lead to aseptic loosening of the parts and early revision surgery.















CHAPTER 3
ENGINEERING APPROACH

3.1 Experimental Data
Fluoroscopic kinematic data were collected from one total knee arthroplasty

patient: female, age 65 at time of surgery, height 170 cm, mass 70 kg (Harman et al.,

2001). The implanted component was a cemented posterior-cruciate ligament retaining

prosthesis (Series 7000, Stryker Howmedica Osteonics, Inc, Allendale, NJ) with a 6.8

mm thick insert. The Knee Society Clinical Rating System scores for this patient (Insall

et al., 1989) were 97-knee and 80-function after one year and 99-knee and 100-function

after two years. The collected kinematics of this implant and patient are shown in Figure

3-1, with a locus plot describing the path of the contact pressure centroid over the insert

for the various activities.



isometric kinematics locusplos
0% 2% 40% 60% W0%










Figure 3-1: Component design and film strips of the kinematics for the gait and stair rise
activities, the corresponding locus plots of contact pressure centroid is shown to the far
right.









The patient performed treadmill gait and stair rise/descent activities during

fluoroscopic motion analysis (Banks, 1992; Banks and Hodge, 1996; Banks et al., 1997a,

Banks et al., 1997b) 21 months after surgery. Fluoroscopic analysis matches three-

dimensional geometric models of the prosthetic components to the two-dimensional

fluoroscopic images. The technique is accurate to approximately 1 for all rotations and

0.5 mm for translations in the sagittal plane (Banks and Hodge, 1996). Kinematic data

from one representative cycle of each activity was averaged in 50 increments of knee

flexion for stair and 1% increments for gait including stance and swing phases.

A dynamic simulation of the patient's in vivo knee mechanics was created by

incorporating an elastic contact model into a commercial multibody dynamics software

program (Fregly, Sawyer, Banks and Harman, 2002)(Pro/MECHANICA MOTION,

Parametric Technology Corporation, Waltham, MA). This approach was taken to predict

joint kinematics and contact pressures in a fraction of the time possible with current

dynamic finite element methods (Giddings et al., 2001; Godest et al., 2002). The contact

model uses elastic foundation theory (Johnson, 1985; An et al., 1990; Blankevoort et al.,

1991; Li et al., 1997) and treats the tibial insert as an elastic layer contacting a rigid

femoral component. This model formulation accommodates the finite thickness and

dimensions of the tibial insert, conformal or non-conformal contact situations, and linear

or nonlinear polyethylene material properties. The elastic contact model was

implemented as a dynamic link library that can be incorporated into any multibody

dynamics software.

The elastic foundation approach calculates contact pressures on a grid of elements

covering the tibial insert contact surfaces. The elements define a "bed of springs" where









each spring is independent from its neighbors (Johnson, 1985). This approximation

eliminates the integral nature of contact problems, thereby greatly simplifying the

analysis of conformal geometry or nonlinear materials. For any element, given the

interpenetration d between the undeformed tibial and femoral surfaces in the direction of

the local surface normal, the contact pressure p acting on the element can be calculated

from (Johnson, 1985; An et al., 1990; Blankevoort et al., 1991).

(1 -v)E
p= ( d 3.1
((1+v)(1-2v))h

where E is Young's modulus of the elastic layer, v is Poisson's ratio of the elastic

layer, and h is the layer thickness at that location. The interpenetration d for each

element is calculated using the ACIS 3D Toolkit (Spatial Corporation, Westminster, CO).

For a nonlinear material, E can be defined as a nonlinear function ofp (Cripton, 1993), in

which case Eqn 3.1 becomes a nonlinear equation in p that can be solved using standard

root-finding methods (Nufio and Ahmed, 2001). To produce a set of point forces, the

resulting element pressures are multiplied by their corresponding areas. These forces are

replaced with a single equivalent force and torque applied to both bodies for purposes of

dynamic simulation (Kane and Levinson, 1985).

The dynamic contact model used in vivo fluoroscopic measurements (anterior-

posterior translation, internal-external rotation, and flexion; Fig. 3-2 a, b, and c,

respectively) as prescribed kinematic inputs. The model predicted the remaining degrees

of freedom (axial translation, varus-valgus rotation, and medial-lateral translation) via

forward dynamic simulation to ensure compatibility with the applied loads. All

prescribed and predicted motions were for the femur moving with respect to a fixed tibia.









5 10

E -
E GaitGait
0 0
-5 -N.-- 0

-1 0 Stair \ nW -5 \^ Stair




S600 / Git
< 1
-151-----------i -10 l---------

90 2400



Si\ 1O \ \1200 j \
30 / 7 /
LL 60o Gaitr

0 0 I I I I i
0 20 40 60 80 100 0 20 40 60 80 100
Percent of Cycle Percent of Cycle

Figure 3-2: In vivo experimental data used as inputs to the dynamic contact model.
a)Anterior-posterior (AP) translation. b) Internal-external (IE) rotation. c) Flexion. d)
Axial force. Kinematic data are from pre-retrieval video fluoroscopy gait and stair
experiments with the femur moving with respect to the tibia. Anterior translation and
external rotation are positive. Axial force data are scaled vertical ground reaction force
data from a patient of similar age, height, weight, and knee flexion characteristics.



Calculation of accurate slip velocities and contact pressures on individual surface

elements requires the dimensions of the simulated contact patch to match in vivo

conditions. The ability of the elastic foundation contact model to predict contact areas

and pressures was evaluated experimentally using 16 different static loading conditions

(loads of 750, 1500, 2250, and 3000 N and flexion angles of 0, 30, 60, and 90). For each

condition, experimental pressure measurements were made with a Tekscan K- Scan

sensor. Using linear polyethylene material properties, the model was able to predict

experimentally measured average contact pressures to within 0.3 + 0.5 MPa, indicating

that the contact areas were also well predicted. Predictions made using nonlinear material









properties were less accurate. Consequently, a linear rather than nonlinear polyethylene

material model was used in the present study.

Dynamic simulations, representing patient-specific in vivo conditions, were

generated by combining in vivo fluoroscopic measurements with assumed loading

conditions. In the dynamic contact model, the femoral component possessed six degrees

of freedom (DOFs) relative to the tibial insert. Three DOFs (anterior-posterior

translation, internal-external rotation, and flexion) were defined to match fluoroscopically

measured gait and stair kinematics (two separate simulations). The remaining three DOFs

(axial translation, varus-valgus rotation, and medial-lateral translation) were numerically

integrated to predict their motion. An axial force was applied vertically downward to the

femoral component to produce a 70% medial-30% lateral load split at 0 flexion (Johnson

et al., 1981; Schipplein and Andriacchi, 1991; Hurwitz et al., 1998). The axial force

curve for each activity was defined by scaling a vertical ground reaction force curve (Lu

et al., 1997; Taylor et al., 1998, Taylor and Walker, 2001) to be between 0.25 and 3.0

BW (Schipplein and Andriacchi, 1991; Lu et al., 1997; Taylor et al., 1998, Taylor and

Walker, 2001). Ground reaction force data were taken from a patient of similar age,

height, weight, and knee flexion characteristics.

The dynamic contact model generated contact pressures and slip velocities in two

steps. First, a forward dynamics simulation calculated contact forces and kinematics.

Then an inverse dynamics analysis used the results from the forward dynamics simulation

to calculate contact pressures and slip velocities over a 50x50 element grid on each tibial

contact surface.











3.2 Mathematics

A computational wear model was developed to produce element-by-element

damage predictions given the calculated time history of contact pressures and slip

velocities experienced by each element. The model computes total damage depth for each

element as the sum of material removal due to mild wear and surface deformation due to

compressive creep:

6Damage = N" *Wear + Creep Eqn 3.2

where 8Damage is the total damage, 6,ar, is the damage per cycle due to mild wear,

N is the total number of cycles, and 86reep is the damage due to creep. The number of

cycles was based on the number of months implanted assuming 1 million cycles per year

of gait or stair (Schmalzried et al., 1998).

Wear predictions were made in the software using Archard's wear model for mild

wear (Archard and Hirst, 1956). The model predicts the wear depth at a point on the

surface of a sliding contact based on the contact pressure, sliding distance and the wear

rate:


S ear = k p,d, =k p, v, At Eqn 3.3
1=1 1=1

where k is the wear rate (mm3/Nm), p, is the contact pressure (MPa) and the


sliding distance is calculated as the product of the slip velocity magnitude V, (m/s)

multiplied by the time increment At (s). The appropriate value of wear rate was based on

surface roughness values measured from contacting areas of the retrieved femoral

component. The measurements were made using a white-light optical interferometer









(Wyko NT1000, Veeco Instruments, Woodbury, NY). Multiple locations on the surface

yielded results varying from 46 to 275 nm with an average value of 131 nm. Fisher et al.

(1994) published wear rates for UHMWPE, subjected to similar contact conditions, as a

function of R Based on the average R, value, a wear rate of k = 220 x 10"9 mm3/Nm

was chosen.

Ultra-high molecular-weight polyethylene is a viscoelastic and viscoplastic

material, as such it deforms in a time-dependent manner when stressed (Waldman and

Bryant, 1997; Lee and Pienkowski, 1998). This characteristic of the polymer is

responsible for some portion of surface deformation and must be accounted for. Creep

characteristics for medical grade UHMWPE have been published by Lee and Pienkowski

(1998). According to their research, compressive creep can be estimated as a function of

time of compression and contact pressure using the following model:



8cep = 3.491 x103 +7.996 x104 Log NC At -4 --h 3.4

1=1

where all notation is previously defined with the exception of the subscript c

indicating instances in time when the surface pressure p, is non-zero, and h the initial

thickness of the tibial insert. The model assumes pressures are in MPa and time has units

of minutes. The resulting deformation has units matching the units of bearing thickness

e.g. bearing thickness in mm yields deformations in mm.

The thermal modeling begins by assuming a particular partitioning of energy Q (J)

into the femoral component Qf and tibial component Q, respectively. Eqn 3.5 gives an

expression for the energy dissipated in the contact, where p is the friction coefficient, F









is the normal load, V is the slip speed, T is the total activity time, and is an instant in

time during the activity.

T
Q= Q+ Q,= P f, IV 3.5
0

The energy is partitioned into the two bodies as shown in Figure 3-3 and given by

Eqn 3.6. The energy partition rule is based on matching the interface temperature for: (a)

stationary contact area of two half-spaces, and (b) femoral component moving relative to

a stationary tibial component. Both the femoral component and tibial component are

modeled as half spaces of thermal conductivities kf and k, respectively. Partitioning

heat based on a moving heat source (b) is generally preferred when the Peclet number,

Pe, is greater than 10 (Williams et al., 1994; Bhushan, 1999).


Pe<10 Q =Q 1 Q +=Q 1+
Q k, k,
3.6

Pe> 10 Q = 1+ k J ~=OQ 1+ 1
k, 1.6 f JN [ k


Vsl
The Peclet number for the femoral component is defined as Pef = where Vf is
af

a characteristic sliding speed of the femoral component, / is a characteristic half width of

the contact patch, and af is the thermal diffusivity of the femoral component. From Eqn

3.6 it can readily be seen that the moving source partitions greater heat into the femoral

component for Pef > 10. A stationary partitioning of heat is assumed for the remainder

of the modeling and analysis, in part because partitioning greater amounts of heat into the









tibial component should provide an upper bound on the frictional heating likely to be

seen in service.


Figure 3-3: Schematic of the knee joint showing the partitioning of energy into the
femoral and tibial component.



The heat flux q, (W / m2) is found for a tibial element i by partitioning the

average energy dissipated over that element and dividing by the activity time. This is

shown in Eqn 3.7, where P, is the pressure on element i at time .

T

T 1+ "
I k,


The temperature rise 0, at a particular element A is found using the algorithm

shown schematically in Figure 3-4 and given by Eqn 3.8. This algorithm is for square









elements of area A, half width a, and the vector from the origin element A2 to element i

is 4.


ot 1=0
o=1.12 q^"a q, 3.8
k, 21nk, jr ,

The first group in the algorithm is the central temperature rise of the square element

under uniform heat flux q,. The second group in the algorithm is the summation of all of

the contributions of the remaining surface elements to the temperature rise. This

algorithm treats all of the contributing elements as a single point source of heat,

Q, = ,q and uses the classic point source solution discussed in detail in Carslaw and

Jaeger (1959). This algorithm is used for all elements regardless of whether or not they

experience a frictional heat flux.


* frictional heat flux 2a
- convective heat transfer `2a-
Figure 3-4: Schematic of the solution approach for temperature rise of element A2, with
neighboring elements treated as point sources of heat (both frictional heating and
convective cooling).









The elements that do not experience a frictional heat flux during the activity are

exposed to the synovial fluid, and convective cooling of the component occurs through

this surface/fluid interface. The heat flux of these elements not in contact is assumed to

follow Newton's law of cooling, as shown in Eqn 3.9, where 6, is the temperature rise of

the element above ambient temperature 0, = T T1b and h is the convection heat

transfer coefficient with units of W / (m2 K).


= -hO 3.9

The solution for the surface temperature map of the tibial component requires an

iterative solver due to the coupling between the convective cooling heat flux and the

surface temperature. The solution approach is similar to a fixed-point iteration with two

constraints on the solution method: 1) the convective cooling power cannot exceed the

frictional heat, and 2) surface temperatures below ambient are not permitted.

Analysis of crossing motion occurs at the level of the individual surface elements

on the tibial insert and observes the motion of the femoral component relative to each

element. The only motions of interest are those that occur when the element is in contact

with the femoral component, a condition of non-zero contact pressure. Following the

style of presentation previously shown by (Wang, 2001) on crossing motions in hip

contacts, a trimetric view of the tibial surfaces with over-laid velocity vectors for 10

representative elements is shown in Figure 2. This qualitative plot is generated from

stair-rise data, which shows the greatest degree of crossing. It is shown here to illustrate

the very limited degree of crossing motion seen in this design, and to illustrate the need to

develop a formulaic quantitative measure of crossing intensity that can be used to

visualize crossing on all the elements simultaneously.









Figure 3-5 (a) gives a vector plot of the slip velocities for 17 evenly spaced

increments in time, for an element in the lateral compartment of the tibial bearing during

stair rise. As can be seen from the locus plot shown in Figure 3.1 this patient showed

lateral pivoting. The elements in the lateral compartment are in contact for the most

number of time steps; thus, they provide the most interesting vector-plots. In Figure 3-5

(b) the x-axis, which is the medial-lateral axis, of the slip velocity plot is exaggerated by

almost an order of magnitude. In the motion path plot, these slip velocities are multiplied

by the time increment to give incremental displacements of the femoral component over

the element. These incremental motion vectors are strung head to tail to give a motion

path. In this plot the medial lateral axis is exaggerated 20 times, to the right an un-

exaggerated motion path is shown. The arrow heads on the motion path plot are evenly

spaced by time, and some information about sliding speed at various locations along the

path can be gleaned.

Instinctively, one knows that motion in the absence of load is not damaging.

Similarly, motion during lightly loaded contact imparts less energy to the surface than

identical motion during a highly loaded contact. To account for this, a tribological

intensity vector is defined as the product of the contact pressure (P) and the slip vector

(d = V. At), as shown by Eqn 3.10.


tribological intensity = P d


3.10












slip velocity (mrm/s)
-4 -2 0
40
anteril
14
30----- ---- -
30 \
is13




10
ra
-10


17



20


-30

tel
-40
-4 -2 0







motion path (mm)
0








2 -20
-20


S-25


-30 -
1 -20--------


TO 25-------


-30 ---- ^ --


2 4 6


1.5 mm-


medial-lateral axis
Figure 3-5: (a) Vector plot of the counterface slip velocities for 17 different instances
during contact. This particular location in the lateral compartment during a stair rise
activity; this location showed significant crossing motion. (b) The motion path of the
femoral component over this location, notice the x-axis is exaggerated by 20 times, a 1-
to-1 motion path is shown to the right.









Figure 3-6 (a) shows a vector plot oftribological intensity, with an angular

coordinate (0) defined from the positive medial-lateral axis for a vector moving in the

anterior direction (solid) and from the negative medial-lateral axis for a vector moving in

the posterior direction (dashed). This allows all vectors to be placed on a single plot

where the orientation of slip has the same angular coordinate regardless of direction, such

an orientation plot is show in figure 3-6 (b). A scatter plot of tribological intensity versus

angular coordinate is shown in Figure 3-6 (c). It is hypothesized that the most probable

direction of polymer orientation on a particular element coincides with the dominant

orientation oftribological intensity (0) for that element; this is given by Eqn 3.11, where

the subscript (i) denotes a particular time step.



0 = -P1 3.11

1 = i i i
























S uni-directional
I motion
i c"*=0


multi-directional
motion
(*= 0.14


\ K /2 3'/4
Figure 3-6: (a) Vector plot for tribological intensities shown with the angular coordinate
convention (b) Overlaying plot with angular coordinate c) Scatter plot of tribological
intensity versus angular coordinate compared with uni-directional motion (delta function)
and vari-directional motion (step function).


vari-directional
motion
0"*=1


yass









As observed in laboratory tests (Burroughs and Blanchet, 2001) hip joint simulator tests

(Bragdon et al., 1996), and modeled (Wang, 2001; Muratoglu et al., 1999) unidirectional

motions produce significantly less wear than multi-directional motions. In hip and knee

joints multidirectional motion arises from abduction/adduction and internal/external

rotation respectively. A crossing intensity parameter (a) is defined by Eqn 3.12, and

describes the spread oftribological intensity vectors about the dominant orientation

direction (6). This is a statistical formulation for crossing motion intensity as opposed

to an extreme value parameter proposed earlier by Wang, 2001.


: jz(I J At 0- )) 3.12

The crossing intensity of a circular counterface motion gives continuously varying

and evenly distributed (vari-directional) tribological intensity vector orientation and is

defined as co. This is given by Eqn 3.13, where the subscript (o) denotes vari-

directional motion parameters.


go = Po I Ato -,2 3.13


The products of slip distance and contact pressure for this vari-directional motion

are constant and the summation must give the same overall tribological intensity as

shown by Eqn 3.14. The normalized crossing intensity (a *) is given by dividing the

element crossing intensity (Eqn 3.12) by the equivalent vari-directional crossing intensity

(Eqn 3.13), which is shown in Eqn 3.15 in terms of only element parameters.

S314
Po I V-IAto =n- At 3.14-
n I I t









n 2
J(PV At 6-o)
*=-= 3.15

1=1 \ '1 1=1 yJz



Using Eqns 3.11 and 3.15 both the orientation direction and normalized spread of

crossing motion can be found (the crossing motion intensity is defined as a* = 0 for

n =1).

Figure 3-7 shows the algorithm visually overlaying three different motions: (a)

delta function that corresponds to uni-directional motion a* = 0, (b) multi-directional

motion distributed about a* = 0.09, and (c) vari-directional motion distributed uniformly

along the angular coordinate axis a* = 1. Crossing motion is often conceptualized using

a bi-directional diamond pattern, where a square has a crossing angle of 900. Figure 3-8

shows a series of simple counterface motions. The usefulness of this normalized crossing

intensity parameter is immediately obvious; the circular motion gives a* = 1, the

reciprocating motion gives a* = 0, and the narrowing diamond patterns span the range

smoothly.









intensity
map


1R
o 7':


counterface
motion


Figure 3-7: A series of simple intensity maps, with corresponding counterface motion
and normalized crossing severity index.


O /1C 7


1.00





0,87






0.58







0.29







0.10


30P









lO









Velocity Vectors for Selected Elements
E elements with contact
E elements without contact
















Figure 3-8: Overlaying plots of velocity vectors, normalized by the largest velocity
vector in the simulation, onto the corresponding elements on the tibial mesh. These
vectors are from a stair rise activity, which shows the greatest degree of crossing motion.














CHAPTER 4
RESULTS AND DISCUSSION

4.1 Wear Analysis
Five computational wear predictions (two activities with two load splits, and one

partition of activities; 70% gait and 30% stair) were compared to the actual wear depths

and patterns measured on the tibial insert retrieved from the patientpost-mortem. The

total time of implantation at retrieval was 51 months. For both the predictions and the

retrieval, visualizations of the wear contours were generated using commercial automatic

inspection software (Geomagic Qualify, Raindrop Geomagic, Research Triangle Park,

NC). Using the wear predictions, the center of each contact element on the tibial insert

surface was displaced by the calculated damage depth 8r6,, in the direction of the local

surface normal. A "worn" polygonal surface model was created from these points, and

the software generated a contour plot of the deviations between the original and worn

surfaces.

The retrieval showed scratching, burnishing, and tractive striations on the articular

surfaces (Harman et al., 2001). Pitting and delamination were not observed. A three-

dimensional scan was obtained of the worn insert (Fig. 4.la) and a matched unworn insert

using a laser scanner (Vivid 900, Minolta Corporation, Ramsey, NJ) possessing a

manufacturer-reported accuracy of +0.04mm. Once the point clouds generated by the

laser scans were converted to polygonal surface models and aligned by the software, a

retrieval wear contour plot was also generated (Fig. 4. b). To determine a threshold for









reporting retrieval wear, the unworn insert was aligned with the insert CAD model and

the maximum deviation between contact surfaces (0.25 mm) determined.

Qualitatively, the damage regions predicted by the computer simulations were in

good agreement with the clinical wear regions (compare Fig. 4.2 with Fig. 4.1). The

medial wear scars for the 70-30 gait case (Fig. 4.2a) extended along a focalized track to

the anterior medial comer of the insert, similar to the retrieval. In contrast, the medial

wear scars for the 70-30 (Fig. 4.2b) stair case extended broadly to the posterior rim of the

insert, enlarging the region predicted by the gait cases. The lateral wear scars for the 70-

30 gait case extended more anteriorly than in the retrieval, but the anterior border on the

lateral side in the 70-30 stair case corresponded well with the retrieval. Altering the load

split to 50-50 decreased medial damage while increasing lateral damage for both gait

(Fig. 4.2c) and stair (not shown). For a 70% gait, 30% stair partitioning of activities

based on linear rules of mixture, the damage area for a 70-30 load split (Fig. 4.2d) was a

combination of the gait (Fig. 4.2a) and stair (Fig. 4.2b) damage areas. For both gait and

stair, the lateral wear regions were more central in the anterior-posterior direction than

were the medial regions, similar to the retrieval, and possessed a posterior border of

extremely similar shape and location to the retrieval.

The predicted locations of maximum damage were in good agreement with the

retrieval (stars in Fig. 4.1b and Fig 4.2). On the lateral side, the location of maximum

damage was the same in all four simulations and was consistent with the retrieval. On the

medial side, the maximum damage location was shifted toward the tibial eminence for the

gait simulations (Fig. 4a and c) and posteriorly for the stair simulations (Fig. 4b).

However, when a 70% gait, 30% stair partitioning of activities was considered (Fig. 4d),








the predicted maximum damage location on the medial side also became consistent with

the retrieval.

Quantitatively, the simulations predicted maximum total damage depths on the
same order of magnitude as those measured from the retrieved insert (Table 4-1). The

predicted maximum damage depths ranged from 0.9 mm to 1.8 mm. The predicted creep

deformation was approximately one-third to one-half of the total damage. The 70-30 load

split for gait and stair activities exhibited approximately equal medial and lateral

maximum damage depths, whereas the 50-50 load cases produced about 1.5 times deeper

damage on the lateral side. Total damage area was greater for gait than for stair, while

total damage volume was approximately 50% larger for stair than for gait. Smaller medial

loads (50-50 split) decreased the damage volume in the medial compartment and

increased damage volume in the lateral compartment such that the total damage volume

was unaffected by load split.





a b -
a b 0.80




(,__ /j __

S. 0.25
Medial Lateral
Figure 4-1: Damage visualization of the retrieved tibial insert, a) Laser scan showing
damage regions visible to the naked eye. b) Contour map indicating depth of damage
zones. The color bar indicates depth in mm. Stars indicate location of maximum damage
on each side.





























Medial Lateral


11.10






0.01


d





Pl'l


Figure 4-2: Damage contour maps predicted by the computer simulations. a) Gait with
70-30 load split. b) Stair with 70-30 load split. c) Gait with 50-50 load split. d) Combined
activity assuming 70% gait, 30% stair with 70-30 load split.


Table 4-1: Quantitative summary of damage results predicted by the computer
simulations for gait and stair activities with 70-30 and 50-50 load splits. Maximum wear,
creep, and total damage may occur at different locations on the surface.

Load Gait Stair
Split Damage Medial Lateral Total Medial Lateral Total
70-30 Wear depth (mm) 0.6 0.7 0.7 0.8 -
Creep depth (mm) 0.6 0.4 -0.8 0.6
Total depth (mm) 1.1 1.0 1.5 1.4 -
Area (mm2) 360 304- 663 342 216 558
Volume (mm3) 126 57 183 20-1 77 282
50-50 Weai depth (mm) 0.5 0.9 0.6 1.1 -
Creep depth (mm) 0.5 0.5 0.7 0.8 -
Total depth (mm) 0.9 1.4 1.3 1.8 -
Area (mm2) 328 349 677 327 236 563
Volume im W'i 94 89 184 152 121 273


I-









Combining damage predictions from the two activities (70% gait, 30% stair)

resulted in damage similar to the retrieved implant (Table 4-2). The predicted locations

of maximum damage depth were the same as on the retrieved insert (Fig. 4-2 (d)).

Maximum damage depths for the retrieval were 0.7 mm medial and 0.8 mm lateral versus

1.0 mm and 1.1 mm for the simulation. The combined case predicted 87% of the total

damage area on the retrieval, 84% medially and 91% laterally. The medial-lateral ratio

for damage depth was 0.88 for the retrieval and 0.89 for the simulation while for damage

area it was 1.38 for the retrieval and 1.28 for the simulation.




Table 4-2: Quantitative comparison between retrieval damage and simulation damage
predicted by an activity partition of 70% gait, 30% stair with a 70-30 load split.


Retrievala Simulation"
Damage Medial Lateral Total Medial Lateral Total
Total depth (mm) 0.7 0.8 1.0 1.1 -
Area (mm2) 422 305 727 354 277 632
"See Harman et al. 42001) for measurement details.
bActivity partition: 70% gait, 30% stair with 70-30 load split.



Damage results predicted by the software were made using a mixture of stair and

gait kinematics. Although the average person does not spend 30% of their time on stairs,

the flexed-knee under high load motion performed when climbing stairs can be likened to

a variety of everyday activities e.g. sitting in or rising from a chair, kneeling to tie a shoe,

getting into or out of bed. While these activities are less frequent the summation of all

the motions is accounted for in the 30% estimate.









Despite its computational advantages, the current contact model formulation has

limitations. It does not account for viscoelastic material properties (Waldman and Bryant,

1994; Waldman and Bryant, 1997), friction (Sathasivam and Walker, 1997), or how

pressure applied at one location affects the displacement of other locations (Johnson,

1985). However, the most significant issue is the use a linear material model, which

explains the more focalized damage regions seen in the predictions compared to the

retrieval. This material model was chosen over a nonlinear model for two reasons. First, a

linear model is more in line with the guiding concept of using models with previously

published, well-established parameter values. Second, in recent simulations of a different

knee implant using the same dynamic contact model, a linear model matched static

contact pressure measurements better than did a nonlinear material model (Cripton, 1993)

for 16 different loading conditions (loads of 750, 1500, 2250, and 3000 N and flexion

angles of 0, 30, 60, and 900; Fregly et al., 2002). The value of Young's modulus that

reproduced the experimental data (400 MPa) was the same as the value reported by Kurtz

et al. (2002). Use of a nonlinear material model (Cripton, 1993) in the simulations

produces the same "dog-bone" contact patch (due to end effects) as the linear material

model but with more uniform contact pressures across the patch (Fig. 4-3). Thus, a

nonlinear material model with well-established parameter values would produce more

uniform damage predictions in the medial-lateral direction, similar to the retrieval, but

would not likely cause dramatic changes in the depth or distribution of predicted damage.









E elements with 32
no contact












0

Figure 4-3: Visualization of the static contact pressures predicted by the dynamic model
for an axial load of 3 BW. a) Linear material model. b) Nonlinear material model. Color
bar indicates pressure in MPa. The element grid is 50 x 50, the same as that used in the
damage predictions.








4.2 Crossing Analysis
The algorithms discussed in the mathematics section of Chapter 3 were applied to

the patient kinematic data for two activities: gait and stair rise, and contour plots of

tribological intensity and normalized crossing motion intensity were generated (figures 4-

4 and 4-5). Both gait and stair activities showed limited crossing intensity, with

maximum a* = 0.04 and a* = 0.09 respectively. The coincidence of highest crossing

intensity with greatest tribological intensity occurs on the lateral compartments for both

activities. This is believed to be due to the pivoting nature of this patient's kinematics.

The crossing motions observed for this patient suggest that uniform bi-directional

patterns with 100 of included angle are a reasonable screening motion for pin-on-disk









testing. It is unclear how significant this degree of crossing is to the tribological behavior

of the implanted UHMWPE. A model previously proposed by Wang for multidirectional

sliding of UHMWPE offered a dependence of wear rate on maximum included angle

(2a) as shown in Eqn 4.1.

k sin 2a .
kxc 1-- 4.1
2a

In the study by Wang (2001) experiments with pin-on-disk multidirectional sliding

produced one order of magnitude increases in wear rate with as little as 150 of crossing

motion.

Conforming knee prostheses are currently available, with designed lateral or

medial pivoting. The challenges in evaluating the potential tribological impact of such

designs require evaluation of both tribological intensity and crossing intensity. Perhaps,

overall tribological severity is the product of the two.








Gait


elements with
no contact


0 100 200


500 600


Tribological Severity P V At (MPa-mm)


n elements with
no contact


0 0.01 0,02


0.03 O.C


8I


Crossing Motion Severity o*

Figure 4-4: Contour maps of a) tribological intensity and b) normalized crossing severity
index for elements in contact during gait.


(









Stair Rise


Elements with
no contact


Tribological Severity P At


(MPa-mm)


n elements with
no contact


0.03


0.06


0,10


Crossing Motion Severity o*

Figure 4-5: Contour maps of a) tribological intensity and b) normalized crossing severity
index for elements in contact during stair rise.


I I I II I 'I~iiiI
















4.3 Thermal Analysis
The values of the various variables needed in this code that defines the 'standard

condition' for both gait and stair rise activities are given in Table 4-3. The standard

condition assumes a 1 Hz activity with a Cobalt-Chrome femoral component, a low

convective heat transfer coefficient, and a low Peclet number partitioning of heat.




Table 4-3: List of standard conditions


elastic modulus E

Poisson's ratio v

thermal conductivity

tibial component kt

femoral component kf

convection coefficient h

cycle time (stair & gait) T

friction coefficient u


value

460.

0.46


0.4

13.

30.

1.0

0.06


units

MPa





W/(rmK)

W/(rmK)

W/(m2*K)

sec


references)

Kurtzet al, 2002

Bartelet al, 1995









Kakac and Yener, 1994

Hall and Unsworth, 1997


The heat flux contour plot as described by Eqn 3.7 is shown in figure 4-6 (a) and

4-7 (a) for the gait and stair activities respectively. The resulting temperature rise

contour plots are shown in figure 4-6 (b) and 4-7 (b) for the gait and stair activities

respectively. These temperature results converge quickly, less than 100 iterations. For

both gait and stair rise activity maximum temperature occurs on the medial compartment.









The code was run varying three different parameters off of the standard condition;

these were activity frequency, heat transfer coefficient, and thermal conductivity of the

femoral component. Table 4-4 summarizes these data.

The choice of femoral component thermal conductivity for the standard condition

corresponds to a cobalt-chrome component, which is widely used. The maximum contact

temperature is extremely sensitive to the choice of femoral component thermal

conductivity. At kf = 0.4 half of the frictional heating is carried away by the femoral

component and half goes in the tibial component. At kf = 40.0 only 1% of the frictional

heating goes into the tibial component.

The stationary analysis partitions heat to the respective bodies as a function of

their thermal conductivities only. This is considered reasonable for low values of Peclet

number. However, for the standard condition (Cobalt-Chrome femoral component)

characteristic sliding speed is Vf = 0.100 m / s, representative contact half-length is

/ = 0.002 m, and the thermal diffusivity is a, = 3.4x106, thus Pef z 60. The energy

partition and corresponding maximum temperature rise for this standard condition is

Qt = 3% Ql & Omax = 10.50C and Q, = 0.6% Qo,, & Omax = 2.50C for stationary and

moving analysis respectively. In figure 6b the upper horizontal axis, x2, gives the ratio

of the heat going into the tibial component to the total frictional heat; thus the graph can

be used to calculate temperature rise for any partitioning of energy over the range

Qt = 1% Qtotz to Q = 50% Qtota .









Gait (1 Hz)


0 10 20 30 40

Average Heat Flux (mW/mm2)




















0 1 2 3 4 5 6 7 8 9

Temperature Rise (Celcius)
Figure 4-6: Contour maps for continuous stair activity for the 'standard condition' of a)
the average frictional heat flux and b) the temperature rise in degrees Celsius. (1Hz,
kf =13 W/(m.K), = 0.06, h = 30 (W/(m2 K))









Stair Rise(1Hz)

medial lateral
Q=4.8 W Q=1.9 W




Elements with
no contact








0 10 20 30 40 50

Average Heat Flux (mW/mm2)


0 2 4 6 8 10 12 14

Temperature Rise (Celcius)
Figure 4-7: Contour maps for continuous stair activity for the 'standard condition' of a)
the average frictional heat flux and b) the temperature rise in degrees Celsius. (1Hz,
kf =13 W/(m. K), p = 0.06, h = 30 (W/(m2 -K))










Table 4-4: Conditions run off of the 'standard condition' (bold) and corresponding
maximum temperature rise in degrees Celsius for the gait and stair activities accordingly.

femoral
component convection maximum temperature
cyde thermal heat transfer rise
frequency conductvity cofficient gait stair

(Hz) A,(W/m. K) h (W/m 2 K) SF., (K) m(K)

0.50 13,00 30.0 4.5 7.2
I 0.75 13-00 30.0 6.B 11.
1 i 13X1 3U0 9.1 14.
125 13,00 30.0 11, 1 8
S 1 5 13.00 30.0 14. 22.
1,75 13,00 41 1 16, 25.
2,00 13 00 30,0 18, 29


1 X, '- 40 Su '3 150. I'O.]
C 1.00 1.30 30,0 71, 110.
S 1.00 4'-u 30.0 28, 44-
S 1.00 11.0 3 9.1 14
1.00 410 iJ 30.0 3.0 48


1 3X'i I 1.,0 10, 16,1

S 1.00 13.00 3.0 10. 16-
1 '1 13,00 Idll 97 15,

1| 1J 1U X 9,1 14.
S 1,00 13,00 100,0 6.5 11.




The activity frequencies were varied between 0.5Hz and 2Hz. The nearly linear

trend of frictional heating with activity frequency (Fig. 4-8 (a)) is expected. This can be

most easily seen in Eqn 3.7, where the frictional heat flux is directly proportional to the

sliding speed. The activity period is proportional to the kinematic path, which is

essentially constant, divided by the speed. Activity frequency is the reciprocal of the









activity period; thus, activity frequency is directly proportional to speed. Similarly,

though not shown, linear dependencies with friction coefficient were expected and found.

The very weak dependence on convection coefficient (Fig. 4-8 (c)) is somewhat

surprising, but perhaps fortuitous. Because the frictional heat flux was discretized and

time averaged for the various elements, the tibial surface has islands of frictional heating

surrounded by regions of convective cooling. Any element that experiences contact is

precluded from providing cooling. During these activities, elements are continuously

exposed to both fictional heating and convective cooling. This mixture of heating and

cooling would yield a lower average heat flux than that used by this solution procedure.

In vivo the knee does not go for thousands of cycles without a break. As

discussed above this steady-state analysis takes average heat fluxes to predict the contact

temperature. The time required for this system to reach steady-state was not explored.

Thus, many interrupted activities may not reach the temperatures predicted here. The

analysis is more similar to joint simulator machines that can run for hours or days without

stopping.

The feasibility of these results are interesting given that many studies on the

temperature rises for hip replacements find 1-10 C and the present results for the knee

replacement are generally in line with those findings. Overall the analysis here aims to

provide a simple first model for thermal analysis during continuous activities such as gait

and stair for total knee replacements with UHMWPE tibial bearings. The findings from

the standard conditions are most likely overestimates of actual steady-state temperature

rises, primarily because of the heat partitioning and time averaging of the heat-flux,







53



which precludes frictional heating and convective cooling to operate on the same


element.


activity period (s)
2.0 1.33 1.00 0.80


a.67 0.57 0.5


30

25

'5 : ------------------_1---
20

15



^ --


0.50 0.75


1.00 1.25 1.50
activity frequency (Hz)


1.75 2.00


aCD


3T

E
Q.


percent of frictional energy going into tibial compo nent
50% 29% 9% 4% 191
300__


100

50
30



--
10

5

0.4 1.0 4.0 10. 40.
oernvo common ent thermal conductivity kf (W/(m-K)


20


1 5


5--"--------1-----






0
5 --- -- __----- ------_____




1 3 10 30 10
convection heat transfer coefficient h ((W/mtK))


Figure 4-8: Plots of maximum temperature for variations in a) activity rate, b) thermal
conductivity of the femoral component, and c) convection heat transfer coefficient.









CHAPTER 5

SUMMARY AND CLOSURE

Patients specific kinematic data derived from fluoroscopy has been a source of inputs for

making wear predictions. Unfortunately, these measurements are based on 2-D x-ray

images, and have uncertainties based on lack of depth perception and inaccuracies of the

3-D image matching. In the sagittal plane the errors can be as much as 0.5 mm, which is

significant when wear depths are on the order of 1 mm. Fortunately, another group on

campus has been working with multibody dynamics software and was able to import the

data and optimize the motion of the two components based on elastic foundation. This

group was capable of resolving unknowns which could not be determined through video

fluoroscopy. The refined kinematics files were used with a custom software package for

producing wear predictions, creep predictions, crossing motion analysis and thermal

effects of gait and stair-rise exercises. The predictions were made using gait motion and

stair-rise motion separately, but to predict damage for a retrieval a combination of the

two motions was considered. Assuming the gait motion made up 70% of the patients

total motion and the remainder was stair-rise motion, the area where damage occurred

and the depth of the damage coincided with that found from the laser-scanned retrieval.

Although this analysis has only been made for one implant and one patient, preliminary

results are promising. Future work available for this project is refining the contact model

used in the multibody dynamics software and implementing creep recovery models into

the damage prediction software. Also validation of the thermal analysis could be

performed using a motion which is easily reproduced experimentally.




















APPENDIX A

CLASS HEIRARCHY FOR SOFTWARE DEVELOPED


I---------------------------,

c-mpnent canvas -----------
'mC I ColorWi ndow

Container Window IlI ColorPickWindow


P-1
java.awt -Panel




Java.lang

I Object I


ActlonLocll

Constants
I -" i
Element

ElementlnmeStpte
.[I
Facet

Mdel I

Model Rolate

Moderrime Gep

S rtAlgarthm I

WearAnalyale--
L_ _______ ___ ___i


I. . .-


ArrayManipulation

ArrayReslze

L
*_____---------_ --___-

DoubleFormnat

-I DoIbleFormat

L-------^^fl

Encryption

Encryption

L____ ____-_____


Figure A-1: Class hierarchy of java packages written for this project Part 1.
(ArrayManipulation, DoubleFormat, Encryption, ColorWindow, Library Files).


CLASS

Sextendi

*i n impllmmn..i


I








56


-- -- - -- - ----------------,
javalang Java.awt
|QIJet ~- cmpenarnt cnva|s

I-- -I I--
LibrmryFils


Fme -- MainPanel






I 0
--- "-- -- ------

Eng_Math -ds MyrAnoi
--- w---int-r Myrn it.r
RoVectaathio
-| y~nlmaLrk
S I l MAnim. ni. r














Figure A-2: Class hierarchy for java packages written for this project Part 2.
(Library Files, Eng Math, threads).
--- --- -- -- -- I -- -













(Library Files, Eng_Math, threads).

























r --------------I
Java.lang I java.awt


J-.t -a-- CWai- --n







-------------------


I nI


Graphing
I Graph ColorWheel

I I I lem

--- -- 6r.pP.. I -- --- -- --
SGiaphFrama I clelI












SBankser
JpeEncoVader
I - -
iPSJlerO






BMPFI" |
I 1 sUPFIII )


Figure A-3: Class hierarchy for java packages written for this project Part 3.
(Graphing, ColorWheel, File IO).


n











APPENDIX B
DERIVATION OF ALGORITHMS

B.1 Temperature Rise Derivation
The temperature rise analysis begins with the differential heat conduction equation:

a20 a20 a20 1 dO
-+- +--- B-l
x2 +y2 8Z2 a dt

This equation assumes a homogeneous solid, and is satisfied by the following:

{(x x')2(y y_ 2 Z,)2}
S= 4at B-2
3
8pC, ()at)2


where Q is the heat released (J), p is the density C is the specific heat



Sk land a is the thermal diffusivity and t is time (s). Knowing that
kg -K) s

r2 =(x x)2 +(y-y')2 +(z-z')2 B-3

equation (2) simplifies to

-2
0 = e4at B-4
3
8pC, (nrat)

The above temperature rise is for a point source instantaneously releasing a packet of heat

at a distance r from the point in question. The time that has passed since the heat was

released is t. This solution can be integrated from the initial time corresponding to the

first packet being released to the final time to give a temperature rise from a constant heat

source at a constant distance producing a fixed flux for a known time period. The

integral is described below:









T 2
0 =e"4dt B-5
o 8pCP (nat) 2

where q is the energy released per unit time (W) and T is the total time the heat is

released from the point source. By substituting t = T -t' the solution for temperature rise

becomes


0 f e4a(T t')dt B-6
3= B-6
"8pC[ra(T- t')7

further substituting r = (T t')1/2 the equation can be rewritten as


o= 3 e 4 dc B-7
S4pC, [na

This equation can be evaluated and simplified to


6 = erfc r B-8
41rkr r 4fat )

where k is the thermal conductivity of the substance. If the equation is considered at very

long times (t -> o), the temperature rise becomes only a function of material properties

and distance from the heat source:


0=-- B-9
47rkr

As mentioned earlier this solution is for a infinite solid or 'whole'-space, however,

the solution required is for a 'half -space indicating that the temperature rise would be

doubled. Therefore,


0-- = B-10
21kr









This solution is considered for surface elements which are squares. Finding the

solution for the maximum temperature rise on the surface requires integrating over the

entire surface.


B.2 Crossing Motion Derivation
The starting point for determining the crossing intensity of a motion path is to

define a vector whose direction will be used to measure all the directions against. This

vector provides a reference for the other direction vectors. In this case, a vector from

medial to lateral was chosen. Tribological intensity is a variable defined in this analysis

as

T=P-d B-11

where P is the pressure at any point and d is the sliding distance over which the pressure

is applied. In the software, each element has a time history of pressure, slip velocity and

time increment. The equation for T can be rewritten as follows:

T, =P I At, B-12

Tribological intensity for each timestep is calculated from the pressure (P) at the

timestep, the magnitude of the slip velocity ( V, ) at the timestep and the period of time

which the two act (At,).

An average direction of sliding is then calculated by using the angle created

between the sliding vector and the arbitrary reference vector. The angle was measured,

as shown in Figure 3-6, so that a unidirectional motion would yield a delta function. The

angular measurement was weighted using tribological intensity as follows:











0- = -1 B-13
P, 1-1 At,


The angle 0 represents the most probable sliding direction, or the direction receiving the

most energy input. This value is used to determine how close to linearity the motion is

by using a standard deviation calculation:


1 A ) B-14


This equation's result holds little value alone, but by normalizing it with respect to a

motion which is varidirectional (a circular motion) whose tribological intensity is the

same as the motion above the normalized value gives a result which is useful. The

equivalent circular path can be described as follows:


P, P = 1 PV At 2
=i ( 2 B-15

n 1
The normalized crossing motion intensity is defined as




oo o V1) Ato (


Various cases of motion and their corresponding crossing motion intensity are

described in Figures 3-7, B-l and B-2.






62


intensity intensity counterface
vectors map motion




1:1


0R .082

0.75:1



111 IC 0,59






SR 0.12
t^I l-^lr 0l ^ 30







0.10:1









in nature.

















intensity
map












O /
0 ^t


counterface
motion


1.00


1.0 1/2 7 0.69


1/27 0.61


=300






a=1001


0 2C n


0.50 1/2i 0.36






0.17 1/27 0.13


Figure B-2: Results of crossing intensity analysis with motion paths described by a pin
rotating about its central axis while sliding linearly across a surface.


intensity
vectors


S= 900
'-ft.


0.87


- 1/2













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BIOGRAPHICAL SKETCH

The author was born in Orlando, Florida, in 1978. Throughout his life he excelled at

mathematics and sciences. In high school he earned the Bright Futures scholarship,

which helped him to pay for school at the University of Florida. The author majored in

mechanical engineering with a minor in computer science as an undergraduate, and

received the Society of Tribologists and Lubrication Engineers scholarship award. After

graduating in 2001, he decided to continue his work by going to graduate school at the

University of Florida.