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1.
The objective of this study was to validate the MRI-based joint contact modeling methodology in the radiocarpal joints by comparison of model results with invasive specimen-specific radiocarpal contact measurements from four cadaver experiments. We used a single validation criterion for multiple outcome measures to characterize the utility and overall validity of the modeling approach. For each experiment, a Pressurex film and a Tekscan sensor were sequentially placed into the radiocarpal joints during simulated grasp. Computer models were constructed based on MRI visualization of the cadaver specimens without load. Images were also acquired during the loaded configuration used with the direct experimental measurements. Geometric surface models of the radius, scaphoid and lunate (including cartilage) were constructed from the images acquired without the load. The carpal bone motions from the unloaded state to the loaded state were determined using a series of 3D image registrations. Cartilage thickness was assumed uniform at 1.0 mm with an effective compressive modulus of 4 MPa. Validation was based on experimental versus model contact area, contact force, average contact pressure and peak contact pressure for the radioscaphoid and radiolunate articulations. Contact area was also measured directly from images acquired under load and compared to the experimental and model data. Qualitatively, there was good correspondence between the MRI-based model data and experimental data, with consistent relative size, shape and location of radioscaphoid and radiolunate contact regions. Quantitative data from the model generally compared well with the experimental data for all specimens. Contact area from the MRI-based model was very similar to the contact area measured directly from the images. For all outcome measures except average and peak pressures, at least two specimen models met the validation criteria with respect to experimental measurements for both articulations. Only the model for one specimen met the validation criteria for average and peak pressure of both articulations; however the experimental measures for peak pressure also exhibited high variability. MRI-based modeling can reliably be used for evaluating the contact area and contact force with similar confidence as in currently available experimental techniques. Average contact pressure, and peak contact pressure were more variable from all measurement techniques, and these measures from MRI-based modeling should be used with some caution.  相似文献   

2.
This study was undertaken to assess magnetic resonance imaging (MRI)-based radiocarpal surface contact models of functional loading in a clinical MRI scanner for future in vivo studies, by comparison with experimental measures from three cadaver forearm specimens. Experimental data were acquired using a Tekscan sensor during simulated light grasp. Magnetic resonance (MR) images were used to obtain model geometry and kinematics (image registration). Peak contact pressures (PPs) and average contact pressures (APs), contact forces and contact areas were determined in the radiolunate and radioscaphoid joints. Contact area was also measured directly from MR images acquired with load and compared with model data. Based on the validation criteria (within 25% of experimental data), out of the six articulations (three specimens with two articulations each), two met the criterion for AP (0%, 14%); one for peak pressure (20%); one for contact force (5%); four for contact area with respect to experiment (8%, 13%, 19% and 23%), and three contact areas met the criterion with respect to direct measurements (14%, 21% and 21%). Absolute differences between model and experimental PPs were reasonably low (within 2.5 MPa). Overall, the results indicate that MRI-based models generated from 3T clinical MR scanner appear sufficient to obtain clinically relevant data.  相似文献   

3.
Measurement of static alignment of articulating joints is of clinical benefit and can be determined using image-based registration. We propose a method that could potentially improve the outcome of image-based registration by using initial manual registration. Magnetic resonance images of two wrist specimens were acquired in the relaxed position and during simulated grasp. Transformations were determined from voxel-based image registration between the two volumes. The volumes were manually aligned to match as closely as possible before auto-registration, from which standard transformations were obtained. Then, translation/rotation perturbations were applied to the manual registration to obtain altered initial positions, from which altered auto-registration transformations were obtained. Models of the radiolunate joint were also constructed from the images to simulate joint contact mechanics. We compared the sensitivity of transformations (translations and rotations) and contact mechanics to altering the initial registration condition from the defined standard. We observed that with increasing perturbation, transformation errors appeared to increase and values for contact force and contact area appeared to decrease. Based on these preliminary findings, it appears that the final registration outcome is sensitive to the initial registration.  相似文献   

4.
OBJECTIVE: Ex vivo studies have suggested that cartilage contact areas and pressure are of high clinical relevance in the etiology of osteoarthritis in patients with patellar subluxation. The aims of this study were therefore to validate in vivo measurements of contact areas with 3D open magnetic resonance imaging (MRI), and to study knee joint contact areas in patients with patellar subluxation at different angles of knee flexion in comparison with healthy subjects. METHODS: 3D-MRI data sets of 12 healthy volunteers and eight patients with patellar subluxation were acquired using a standard clinical (1.5 T) and an open (0.2 T) MRI scanner. We compared femoro-patellar and femoro-tibial contact areas obtained with two different sequences from open MRI [dual-echo-steady-state (DESS) and fast-low-angle-shot (FLASH) sequences] with those derived from standard clinical 1.5 T MRI. We then analyzed differences in joint contact areas between healthy subjects and patients with patellar subluxation at 0 degree, 30 degrees, and 90 degrees of knee flexion using open MRI. RESULTS: The correlation of the size of contact areas from open MRI with standard clinical MRI data ranged from r = 0.52 to 0.92. Open-MRI DESS displayed a smaller overestimation of joint contact areas (+21% in the femoro-patellar, +12% in the medial femoro-tibial, and +19% in the lateral femoro-tibial compartment) than FLASH (+40%, +37%, +30%, respectively). The femoro-patellar contact areas in patients were significantly reduced in comparison with healthy subjects (-47% at 0 degree, -56% at 30 degrees, and -42% at 90 degrees of flexion; all p < 0.01), whereas no significant difference was observed in femoro-tibial contact areas. CONCLUSIONS: Open MRI allows one to quantify joint contact areas of the knee with reasonable accuracy, if an adequate pulse sequence is applied. The technique permits one to clearly identify differences between patients with patellar subluxation and healthy subjects at different flexion angles, demonstrating a significant reduction and lateralization of contact areas in patients. In the future, application of this in vivo technique is of particular interest for monitoring the efficacy of different types of surgical and conservative treatment options for patellar subluxation.  相似文献   

5.
The purposes of this study were to determine the in situ functional and material properties of articular cartilage in an experimental model of joint injury, and to quantify the corresponding in situ joint contact mechanics. Experiments were performed in the anterior cruciate ligament (ACL) transected knee of the cat and the corresponding, intact contralateral knee, 16 weeks following intervention. Cartilage thickness, stiffness, effective Young’s modulus, and permeability were measured and derived from six locations of the knee. The total contact area and peak pressures in the patellofemoral joint were obtained in situ using Fuji Pressensor film, and comparisons between experimental and contralateral joint were made for corresponding loading conditions. Total joint contact area and peak pressure were increased and decreased significantly (=0.01), respectively, in the experimental compared to the contralateral joint. Articular cartilage thickness and stiffness were increased and decreased significantly (=0.01), respectively, in the experimental compared to the contralateral joint in the four femoral and patellar test locations. Articular cartilage material properties (effective Young’s modulus and permeability) were the same in the ACL-transected and intact joints. These results demonstrate for the first time the effect of changes in articular cartilage properties on the load transmission across a joint. They further demonstrate a substantial change in the joint contact mechanics within 16 weeks of ACL transection. The results were corroborated by theoretical analysis of the contact mechanics in the intact and ACL-transected knee using biphasic contact analysis and direct input of cartilage properties and joint surface geometry from the experimental animals. We conclude that the joint contact mechanics in the ACL-transected cat change within 16 weeks of experimental intervention.  相似文献   

6.
Two intrinsic (scapholunate and lunotriquetral) and two extrinsic (radiolunate and radiocapitate) wrist ligaments were studied at high and low elongation rates (1 and 100 mm/min). Statistically significant differences among all four ligaments were noted for the viscoelastic and elastic components of stress versus strain for the fully recoverable strain and early permanent deformation stress for all ligaments. Intrinsic ligaments became permanently deformed at statistically significantly higher strain levels than the extrinsic ligaments and accept larger permanent deformation at strain levels below evident fiber failure. Ultimate strength data demonstrated significant rate dependency for stress and strain for all ligaments. Intrinsic ligaments failed statistically greater stress and strain levels than the extrinsic group. Some clinical implications of these findings are discussed.  相似文献   

7.
Abstract

Finite element modeling (FEM) can predict hip cartilage contact mechanics. This study investigated how subject-specific boundary conditions and joint geometry affect acetabular cartilage contact mechanics using a multi-scale workflow. For two healthy subjects, musculoskeletal models calculated subject-specific hip kinematics and loading, which were used as boundary conditions for FEM. Cartilage contact mechanics were predicted using either generic or subject-specific FEM and boundary conditions. A subject-specific mesh resulted in a more lateral contact. Effects of subject-specific boundary conditions varied between both subjects. Results highlight the complex interplay between loading and kinematics and their effect on cartilage contact mechanics.  相似文献   

8.
Patellofemoral osteoarthritis and its potential precursor patellofemoral pain syndrome (PFPS) are common, costly, and debilitating diseases. PFPS has been shown to be associated with altered patellofemoral joint mechanics; however, an actual variation in joint contact stresses has not been established due to challenges in accurately quantifying in vivo contact kinematics (area and location). This study developed and validated a method for tracking dynamic, in vivo cartilage contact kinematics by combining three magnetic resonance imaging (MRI) techniques, cine-phase contrast (CPC), multi-plane cine (MPC), and 3D high-resolution static imaging. CPC and MPC data were acquired from 12 healthy volunteers while they actively extended/flexed their knee within the MRI scanner. Since no gold standard exists for the quantification of in vivo dynamic cartilage contact kinematics, the accuracy of tracking a single point (patellar origin relative to the femur) represented the accuracy of tracking the kinematics of an entire surface. The accuracy was determined by the average absolute error between the PF kinematics derived through registration of MPC images to a static model and those derived through integration of the CPC velocity data. The accuracy ranged from 0.47 mm to 0.77 mm for the patella and femur and from 0.68 mm to 0.86 mm for the patellofemoral joint. For purely quantifying joint kinematics, CPC remains an analytically simpler and more accurate (accuracy <0.33 mm) technique. However, for application requiring the tracking of an entire surface, such as quantifying cartilage contact kinematics, this combined imaging approach produces accurate results with minimal operator intervention.  相似文献   

9.
Understanding in vivo joint mechanics during dynamic activity is crucial for revealing mechanisms of injury and disease development. To this end, laboratories have utilized computed tomography (CT) to create 3-dimensional (3D) models of bone, which are then registered to high-speed biplanar radiographic data captured during movement in order to measure in vivo joint kinematics. In the present study, we describe a system for measuring dynamic joint mechanics using 3D surface models of the joint created from magnetic resonance imaging (MRI) registered to high-speed biplanar radiographs using a novel automatic registration algorithm. The use of MRI allows for modeling of both bony and soft tissue structures. Specifically, the attachment site footprints of the anterior cruciate ligament (ACL) on the femur and tibia can be modeled, allowing for measurement of dynamic ACL deformation. In the present study, we demonstrate the precision of this system by tracking the motion of a cadaveric porcine knee joint. We then utilize this system to quantify in vivo ACL deformation during gait in four healthy volunteers.  相似文献   

10.
Characterisation of hip joint contact forces is essential for the definition of hip joint prosthesis design requirements. In vivo hip joint contact force measurements have been made using instrumented hip joint prostheses. However, to allow determination of the range of values of joint contact force and their directions relative to anatomical structures in a range of subject groups sufficient to form an agreed data base it is necessary to adopt a different approach without the use of an implanted transducer. The use of mathematical models of the lower limb to examine the forces in soft tissues and at the joints has provided valuable insight into internal loading conditions. Several authors have proposed mathematical musculo-skeletal models. However, there have been only limited attempts at validation of these models. It is possible to use the results of in vivo force measurements from instrumented prostheses to validate the results calculated using the mathematical models. In this study two subjects with instrumented hip joint prostheses were studied. Forces at the hip joints were calculated using a three-dimensional model of the leg. Walking at slow, normal and fast speeds (0.97-2.01m/s), weight transfer from two to one leg and back again, and sit to stand were studied. Direct comparisons were made between the 'gold standard' measured hip joint contact forces and the calculated forces. There was general agreement between the calculated and measured forces in both pattern and magnitude. There were, however, discrepancies. Reasons for these differences in results are discussed and possible model developments suggested.  相似文献   

11.
Abnormal, excessive stresses acting on articular joint surfaces are speculated to be one of the causes for joint degeneration. However, articular surface stresses have not been studied systematically, since it is technically difficult to measure in vivo contact areas and pressures in dynamic situations. Therefore, we implemented a numerical model of articular surface contact using accurate surface geometries. The model was developed for the cat patellofemoral joint. We demonstrated that small misalignments of the patella relative to the femur change the joint contact mechanics substantially for a given external load. These results suggest that misalignment might be studied as one of the factors causing articular cartilage disorder and joint degeneration.  相似文献   

12.
Fuji film has been widely used in studies aimed at obtaining the contact mechanics of articular joints. Once sealed for practical use in biological joints, Fuji Pressensor film has a total effective thickness of 0.30 mm, which is comparable to the cartilage thickness in the joints of many small animals. The average effective elastic modulus of Fuji film is approximately 100 MPa in compression, which is larger by a factor of 100-300 compared to that of normal articular cartilage. Therefore, inserting a Pressensor film into an articular joint will change the contact mechanics of the joint. The measurement precision of the Pressensor film has been determined systematically; however, the changes in contact mechanics associated with inserting the film into joints have not been investigated. This study was aimed at quantifying the changes in the contact mechanics associated with inserting sealed Fuji Pressensor film into joints. Spherical and cylindrical articular joint contact mechanics with and without Pressensor film and for varying degrees of surface congruency were analyzed and compared by using finite element models. The Pressensor film was taken as linearly elastic and the cartilage was assumed to be biphasic, composed of a linear elastic solid phase and an inviscid fluid phase. The present analyses showed that measurements of the joint contact pressures with Fuji Pressensor film will change the maximum true contact pressures by 10-26 percent depending on the loading, geometry of the joints, and the mechanical properties of cartilage. Considering this effect plus the measurement precision of the film (approximately 10 percent), the measured joint contact pressures in a joint may contain errors as large as 14-28 percent.  相似文献   

13.
A validation study was conducted to determine the extent to which computational ankle contact finite element (FE) results agreed with experimentally measured tibio-talar contact stress. Two cadaver ankles were loaded in separate test sessions, during which ankle contact stresses were measured with a high-resolution (Tekscan) pressure sensor. Corresponding contact FE analyses were subsequently performed for comparison. The agreement was good between FE-computed and experimentally measured mean (3.2% discrepancy for one ankle, 19.3% for the other) and maximum (1.5% and 6.2%) contact stress, as well as for contact area (1.7% and 14.9%). There was also excellent agreement between histograms of fractional areas of cartilage experiencing specific ranges of contact stress. Finally, point-by-point comparisons between the computed and measured contact stress distributions over the articular surface showed substantial agreement, with correlation coefficients of 90% for one ankle and 86% for the other. In the past, general qualitative, but little direct quantitative agreement has been demonstrated with articular joint contact FE models. The methods used for this validation enable formal comparison of computational and experimental results, and open the way for objective statistical measures of regional correlation between FE-computed contact stress distributions from comparison articular joint surfaces (e.g., those from an intact versus those with residual intra-articular fracture incongruity).  相似文献   

14.

Background

Transection of the canine cranial cruciate ligament (CCL) is a well-established osteoarthritis (OA) model. The effect of CCL loss on contact pressure and joint alignment has not been quantified for stifle loading in standing. The purposes of the study were to measure femorotibial contact areas and stresses and joint alignment following transection of the CCL in an ex vivo model. We hypothesized that transection of the CCL would lead to abnormal kinematics, as well as alterations in contact mechanics of the femorotibial joint.

Methodology/Principal Findings

Eight canine hindlimbs were tested in a servo-hydraulic materials testing machine using a custom made femoral jig. Contact area and pressure measurements, and femorotibial rotations and translations were measured in the normal and the CCL–deficient stifle in both standing and deep flexion angles.We found that at standing angle, transection of the CCL caused cranial translation and internal rotation of the tibia with a concurrent caudal shift of the contact area, an increase in peak pressure and a decrease in contact area. These changes were not noted in deep flexion. At standing, loss of CCL caused a redistribution of the joint pressure, with the caudal region of the compartment being overloaded and the rest of the joint being underloaded.

Conclusion

In the Pond-Nuki model alterations in joint alignment are correlated with shifting of the contact points to infrequently loaded areas of the tibial plateau. The results of this study suggest that this cadaveric Pond-Nuki model simulates the biomechanical changes previously reported in the in-vivo Pond-Nuki model.  相似文献   

15.
Anatomical differences among squirrels are usually most evident in the comparison of flying squirrels and nongliding squirrels. This is true of wrist anatomy, probably reflecting the specializations of flying squirrels for the extension of the wing tip and control of it during gliding. In the proximal row of carpals of most squirrels, the pisiform articulates only with the triquetrum, but in flying squirrels there is also a prominent articulation between the pisiform and the scapholunate, providing a more stable base for the styliform cartilage, which supports the wing tip. In the proximal wrist joint, between these carpals and the radius and ulna, differences in curvature of articular surfaces and in the location of ligaments also correlate with differences in degree and kind of movement occurring at this joint, principally reflecting the extreme dorsal flexion and radial deviation of the wrist in flying squirrels when gliding. The distal wrist joint, between the proximal and distal rows of carpals, also shows most variation among flying squirrels, principally in the articulations of the centrale with the other carpal bones, probably causing the distal row of carpal bones to function more like a single unit in some animals. There is little variation in wrist musculature, suggesting only minor evolutionary modification since the tribal radiation of squirrels, probably in the early Oligocene. Variation in the carpal bones, particularly the articulation of the pisiform with the triquetrum and the scapholunate, suggests a different suprageneric grouping of flying squirrels than previously proposed by McKenna (1962) and Mein (1970). J. Morphol. 246:85-102, 2000. Published 2000 Wiley-Liss, Inc.  相似文献   

16.
Finite element methods have been applied to evaluate in vivo joint behavior, new devices, and surgical techniques but have typically been applied to a small or single subject cohort. Anatomic variability necessitates the use of many subject-specific models or probabilistic methods in order to adequately evaluate a device or procedure for a population. However, a fully deformable finite element model can be computationally expensive, prohibiting large multisubject or probabilistic analyses. The aim of this study was to develop a group of subject-specific models of the patellofemoral joint and evaluate trade-offs in analysis time and accuracy with fully deformable and rigid body articular cartilage representations. Finite element models of eight subjects were used to tune a pressure-overclosure relationship during a simulated deep flexion cycle. Patellofemoral kinematics and contact mechanics were evaluated and compared between a fully deformable and a rigid body analysis. Additional eight subjects were used to determine the validity of the rigid body pressure-overclosure relationship as a subject-independent parameter. There was good agreement in predicted kinematics and contact mechanics between deformable and rigid analyses for both the tuned and test groups. Root mean square differences in kinematics were less than 0.5 deg and 0.2 mm for both groups throughout flexion. Differences in contact area and peak and average contact pressures averaged 5.4%, 9.6%, and 3.8%, respectively, for the tuned group and 6.9%, 13.1%, and 6.4%, respectively, for the test group, with no significant differences between the two groups. There was a 95% reduction in computational time with the rigid body analysis as compared with the deformable analysis. The tuned pressure-overclosure relationship derived from the patellofemoral analysis was also applied to tibiofemoral (TF) articular cartilage in a group of eight subjects. Differences in contact area and peak and average contact pressures averaged 8.3%, 11.2%, and 5.7% between rigid and deformable analyses in the tibiofemoral joint. As statistical, probabilistic, and optimization techniques can require hundreds to thousands of analyses, a viable platform is crucial to component evaluation or clinical applications. The computationally efficient rigid body platform described in this study may be integrated with statistical and probabilistic methods and has potential clinical application in understanding in vivo joint mechanics on a subject-specific or population basis.  相似文献   

17.
Mathematical models of small animals that predict in vivo forces acting on the lower extremities are critical for studies of musculoskeletal biomechanics and diseases. Rabbits are advantageous in this regard because they remodel their cortical bone similar to humans. Here, we enhance a recent mathematical model of the rabbit knee joint to include the loading behavior of individual muscles, ligaments, and joint contact at the knee and ankle during the stance phase of hopping. Geometric data from the hindlimbs of three adult New Zealand white rabbits, combined with previously reported intersegmental forces and moments, were used as inputs to the model. Muscle, ligament, and joint contact forces were computed using optimization techniques assuming that muscle endurance is maximized and ligament strain energy resists tibial shear force along an inclined plateau. Peak forces developed by the quadriceps and gastrocnemius muscle groups and by compressive knee contact were within the range of theoretical and in vivo predictions. Although a minimal force was carried by the anterior cruciate and medial collateral ligaments, force patterns in the posterior cruciate ligament were consistent with in vivo tibial displacement patterns during hopping in rabbits. Overall, our predictions compare favorably with theoretical estimates and in vivo measurements in rabbits, and enhance previous models by providing individual muscle, ligament, and joint contact information to predict in vivo forces acting on the lower extremities in rabbits.  相似文献   

18.
Quantifying the complex loads at the patellofemoral joint (PFJ) is vital to understanding the development of PFJ pain and osteoarthritis. Discrete element analysis (DEA) is a computationally efficient method to estimate cartilage contact stresses with potential application at the PFJ to better understand PFJ mechanics. The current study validated a DEA modeling framework driven by PFJ kinematics to predict experimentally-measured PFJ contact stress distributions. Two cadaveric knee specimens underwent quadriceps muscle [215 N] and joint compression [350 N] forces at ten discrete knee positions representing PFJ positions during early gait while measured PFJ kinematics were used to drive specimen-specific DEA models. DEA-computed contact stress and area were compared to experimentally-measured data. There was good agreement between computed and measured mean and peak stress across the specimens and positions (r = 0.63–0.85). DEA-computed mean stress was within an average of 12% (range: 1–47%) of the experimentally-measured mean stress while DEA-computed peak stress was within an average of 22% (range: 1–40%). Stress magnitudes were within the ranges measured (0.17–1.26 MPa computationally vs 0.12–1.13 MPa experimentally). DEA-computed areas overestimated measured areas (average error = 60%; range: 4–117%) with magnitudes ranging from 139 to 307 mm2 computationally vs 74–194 mm2 experimentally. DEA estimates of the ratio of lateral to medial patellofemoral stress distribution predicted the experimental data well (mean error = 15%) with minimal measurement bias. These results indicate that kinematically-driven DEA models can provide good estimates of relative changes in PFJ contact stress.  相似文献   

19.
The long-term goal of our research is to understand the mechanism of osteoarthritis (OA) initiation and progress through experimental and theoretical approaches. In previous theoretical models, joint contact mechanics was implemented without consideration of the fluid boundary conditions and with constant permeability. The primary purpose of this study was to investigate the effect of fluid boundary conditions at the articular surfaces on the contact mechanics, in terms of load sharing and fluid flow properties using variable permeability. The tested conditions included totally sealed surfaces, open surfaces, and open surfaces with variable permeability. While the sealed surface model failed to predict relaxation times and load sharing properly, the class of open surface models (open surfaces with constant permeability, and surfaces with variable permeability) gave good agreement with experiments, in terms of relaxation time and load sharing between the solid and the fluid phase. In particular, the variable permeability model was judged to be the most realistic of the three models, from a biological and physical point of view. This model was then used to simulate joint contact in the early and late stages of OA. In the early stages of OA, the model predicted a decrease in peak contact pressure and an increase in contact area, while in the late stages of OA, peak pressures were increased and contact areas were decreased compared to normal. These findings agree well with experimental observations.  相似文献   

20.
Knee osteoarthritis (OA) detrimentally impacts the lives of millions of older Americans through pain and decreased functional ability. Unfortunately, the pathomechanics and associated deviations from joint homeostasis that OA patients experience are not well understood. Alterations in mechanical stress in the knee joint may play an essential role in OA; however, existing literature in this area is limited. The purpose of this study was to evaluate the ability of an existing magnetic resonance imaging (MRI)-based modeling method to estimate articular cartilage contact area in vivo. Imaging data of both knees were collected on a single subject with no history of knee pathology at three knee flexion angles. Intra-observer reliability and sensitivity studies were also performed to determine the role of operator-influenced elements of the data processing on the results. The method's articular cartilage contact area estimates were compared with existing contact area estimates in the literature. The method demonstrated an intra-observer reliability of 0.95 when assessed using Pearson's correlation coefficient and was found to be most sensitive to changes in the cartilage tracings on the peripheries of the compartment. The articular cartilage contact area estimates at full extension were similar to those reported in the literature. The relationships between tibiofemoral articular cartilage contact area and knee flexion were also qualitatively and quantitatively similar to those previously reported. The MRI-based knee modeling method was found to have high intra-observer reliability, sensitivity to peripheral articular cartilage tracings, and agreeability with previous investigations when using data from a single healthy adult. Future studies will implement this modeling method to investigate the role that mechanical stress may play in progression of knee OA through estimation of articular cartilage contact area.  相似文献   

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