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Bearing Movement After Oxford Unicompartmental Knee Arthroplasty: A Mathematical Model

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Abstract

Movements of the meniscal bearing on the tibial plateau after Oxford knee arthroplasty (Biomet Orthopedics, Inc, Warsaw, Ind) reflect tibiofemoral movement. They have been shown to be load dependent. This article describes the use of a mathematical model of the knee in the sagittal plane to show that movement differences under load are caused by differences in ligament strains induced by different combinations of external loads and muscle forces. During passive flexion, roll-back of the femur on the tibia is required to minimize the forces transmitted by the ligaments.

Backward movement of the meniscal bearing on the tibial plateau during passive flexion and forward movement during passive extension are seen before wound closure during Oxford knee arthroplasty (Biomet Orthopedics, Inc, Warsaw, Ind).¹ Bearing movements of 12.7 mm in the medial compartment and 15 mm in the lateral compartment have been measured during passive flexion to 90° in vitro after bicompartmental arthroplasty.² Bearing movements measured in vivo after medial and lateral unicompartmental arthroplasty by comparing static radiographs taken in extension and at 90° flexion revealed significantly smaller values.³ The differences between in vitro and in vivo measurements were attributed to the presence of passive muscle forces.

Recently, fluoroscopic methods were used to track bearing movement continuously over the range of motion (ROM) during passive and active extension/flexion and during a steep step-up exercise (Price et al, unpublished data, 2003). Active extension exhibited significantly less forward-bearing movement on the tibial plateau than did passive extension, whereas step-up exhibited backward-bearing movement. In this article, the investigators use a mathematical model of the replaced knee in the sagittal plane to show that activity-dependent differences in bearing movements arise because different activities require different patterns of external loads and muscle forces leading to different patterns of ligament strain. The development of the model is built on earlier work by Zavatsky and O’Connor,^4,5 Gill and O’Connor,⁶ and Lu and O’Connor.^7,8

Materials and Methods

Figure 1 shows a model of the replaced knee in the sagittal plane drawn by the computer in three positions: in extension and at 60° and 120° of flexion. The prosthesis comprises a circular femoral component, a flat tibial component, and a fully conforming meniscal-bearing situated between these two components, but not connected to either component. The geometry of the model bones and the positions on them of the model ligament attachment areas were based on anatomic studies by Feikes.⁹ The model ligament architecture, i.e. the relationship between the position on the femur of the point of origin of a fascicle and the position on the tibia of its insertion, were based on reports by Friederich et al¹⁰ and Mommersteeg et al¹¹ and confirmed by Feikes.⁹

The anterior cruciate ligament (ACL) was modeled as a single array of fascicles, originating along a straight line on the femur and inserting along a straight line on the tibia, the distance of the point of insertion of a fascicle along the tibial attachment line being proportional to the distance of its origin along the femoral insertion line, also called proportional mapping.⁴ The posterior cruciate ligament (PCL) was modeled as a two-bundle array, with the fascicles originating along two separate intersecting straight lines on the femur and inserting along a single straight line on the tibia, again with proportional mapping. The calculations included similar models of the medial collateral ligament and lateral collateral ligament, but for clarity these models are not shown in the diagrams.

The flat articular surface of the model tibial component lies below the ACL attachment line and above the PCL attachment line, giving a posterior slope 7° relative to the axis of the bone. The model femoral component was positioned on the femur so that its articular surface was a least square fit to the surface of the natural joint. The distance of the center of curvature of the circular femoral component above the articular surface of the tibial component was equal to sum of the radius of the femoral component and the thickness of the chosen meniscal bearing. The anteroposterior (AP) position of the femoral component along the tibial component was more difficult to determine and was determined through trial and error.

Figure 2 illustrates the trial-and-error process in three drawings. At each chosen flexion angle, the AP position of the femur on the tibia was estimated. If the estimate placed the femur too far posterior, as in the first drawing in Figure 2, the ACL and medial collateral ligament were stretched and loaded in tension while the PCL and lateral collateral ligament were completely slack. If the estimate placed the femur too far anterior, as in the third drawing Figure 2, the PCL and lateral collateral ligament were stretched and loaded in tension while the ACL and medial collateral ligament were slack. At each flexion angle, the AP position of the femur on the tibia was therefore chosen to be that at which both sets of ligaments were just slack, a zero ligament tension criterion. The model in Figure 1, therefore, represents passive motion with no external loads, with the ligaments just slack and unloaded and with the articular surfaces just in contact but transmitting zero compressive force. Figure 2 can be viewed as a model of passive stability where ligament forces are recruited progressively to resist AP movement of the tibia on the femur.

In contrast, we also modeled active extension under gravity with the femur horizontal and the leg raised or lowered by the tension in the quadriceps mechanism transmitted into the tibia via the patellar tendon (Figure 3). Again, the AP position of the femur on the tibia was determined by trial and error, but in this case, the calculated forces in the ACL and medial collateral ligament had to balance the components of the weight and patellar tendon forces parallel to the tibial plateau. At each estimated AP position of the femur on the tibia, the moment of the tendon force about the center of the knee balanced that of the weight of the leg. Ligament forces were calculated from ligament strains by using stress/strain relations for ligament tissue based on measurements by Woo et al.^5,12

Results

Calculations were performed for a number of bearing thicknesses, varying in increments of 1 mm (Figure 4). If the chosen bearings were too thin by even 1 mm, the ligaments remained slack over the range of flexion and there was a region of instability at each flexion angle in which the tibia could be moved backward and forward without resistance in the form of ligament force. If the chosen bearing were too thick by even 1 mm, both ACL and PCL remained stretched and loaded over the flexion range. With the correct choice of bearing thickness, 5 mm in this illustration, it was possible to find an AP position of the bearing on the tibia at which both ligaments were just unloaded, and this process gave the model of passive motion shown in Figure 1.

Figure 4 plots the calculated AP position of the model meniscal bearing against flexion angle during passive flexion/extension (Figure 1) and active flexion/extension (Figure 3). In passive flexion to 9°, the bearing moves marginally forward approximately 0.1 mm on the tibial plateau and then moves backward 7.5 mm during further flexion to 90°. These movements can be seen in Figure 1, which shows flexion to 120° with additional backward bearing movement.

In contrast, backward bearing movement from extension to 66° during active flexion is reduced from 4.1 mm for passive motion to 1.7 mm. During further flexion, bearing movement is approximately the same during passive and active flexion.

Figure 5 plots the length of the anterior fascicle of the model ACL against flexion angle for both passive and active flexion. In passive flexion, the anterior fascicle of the ACL changes its length during 90° flexion by less than ±0.4% and may be regarded as effectively isometric. Similarly, the most anterior fascicle of the posterior bundle of the model PCL remained isometric to within ±0.6% during passive flexion. During active flexion, the anterior fascicle of the ACL is strained during flexion to 66° and is slack thereafter. Maximum ACL strain occurs at approximately 22° flexion. The most anterior fascicle of the posterior PCL bundle is slack up to 66° and experiences small strains during further flexion.

Discussion

The differences in bearing movements between active and passive flexion are readily explained by the differences in ligaments strains.

During passive extension and flexion, some ligament fascicles remain just sufficiently tight to hold the articular surfaces together and all others are slack. This requires roll-back of the femur on the tibia during flexion accommodated by backward translation of the meniscal bearing on the tibial plateau. The most anterior fascicle of the ACL and the most anterior fascicle of the posterior bundle of the PCL remain isometric.

Activity has a major effect on bearing position, because the AP movement of the tibia on the femur and associated bearing movements required to strain only half the fascicles of the cruciate ligaments at a given flexion angle (Figure 2) are of the same order as the movements that occur passively during flexion/extension (Figure 1).

During active flexion and extension with the femur horizontal, the patellar tendon force is largest in extension where the weight of the leg has its longest lever arm about the center of the knee. At extension, all ACL fascicles are just tight in the unloaded joint, but are all stretched under load to balance the muscle and weight forces (Figure 3). As a result, the tibia is moved anteriorly relative to its position in the unloaded joint and the meniscal bearing moves posteriorly on the plateau (Figure 4). As the joint flexes, the posterior fascicles of the ACL go slack; only a proportion of the more anterior fascicles are loaded. As a result, the strain of the most anterior fascicle actually increases although the total force in the ligament decreases. The fascicle strain is greatest at 22° flexion, (Figure 5), and the differences in bearing position between passive and active flexion are greatest at this position (Figure 4). The strains in the PCL above 66° are small in comparison, and the differences between active and passive bearing positions are also small (Figure 4).

Alternate loading of the ACL and PCL is related to the direction of the patellar tendon relative to the tibia. The tendon rotates about the tibial tubercle as the joint flexes, (Figure 1). At extension, the tendon is directed superiorly and anteriorly so that its force loads the ACL and unloads the PCL. At 120° flexion, the tendon is directed superiorly and posteriorly, and its force then loads the PCL and unloads the ACL. The magnitudes of the ligament forces and the transition from ACL to PCL loading depend on how the tendon force varies over the flexion range.

Force in the hamstrings in flexion pulls the tibia backward relative to the femur and strains the PCL. Activities involving hamstrings would move the meniscal bearing anteriorly relative to its position in the unloaded joint. Tibiofemoral movement is dependent on the independent variation of tendon direction and tendon force.

The differences in bearing position predicted by the model (Figure 4) correspond well with measurements made fluoroscopically in 12 patients after Oxford medial arthroplasty by Price et al (unpublished data, 2003). Goodfellow et al¹³ show measurements from one of the patients, with differences between passive and active flexion similar to those predicted by the model, but more profound differences between passive extension and step up, an activity in which patellar tendon forces are largest in flexion. During step-up, the bearing moved backwards on the tibial plateau as the joint was extended.

Similar differences between active and passive tibiofemoral movements of the normal intact knee have been reported³ and most recently confirmed by Freeman’s group.¹⁴ These differences can also be explained by differences in ligament loading, but Huss et al^15,16 have shown that indentation of the cartilage layers under load also contributes to activity-dependent differences in tibiofemoral movement.

References

1. Goodfellow J, O’Connor J. The mechanics of the knee and prosthesis design. J Bone Joint Surg Br. 1978; 60:358-369.
2. Goodfellow JW, O’Connor J. Clinical results of the Oxford knee. Surface arthroplasty of the tibiofemoral joint with a meniscal bearing prosthesis. Clin Orthop Relat Res. 1986; 205:21-42.
3. Bradley J, Goodfellow JW, O’Connor JJ. A radiographic study of bearing movement in unicompartmental Oxford knee replacements. J Bone Joint Surg Br. 1987; 69:598-601.
4. Zavatsky AB, O’Connor JJ. A model of human knee ligaments in the sagittal plane. Response to passive flexion. Proc Inst Mech Eng [H]. 1992; 206:125-134.
5. Zavatsky AB, O’Connor JJ. A model of human knee ligaments in the sagittal plane. 2. Fibre recruitment under load. Proc Inst Mech Eng [H]. 1992; 206:135-145.
6. Gill HS, O’Connor JJ. Biarticulating two-dimensional computer model of the human patellofemoral joint. Clin Biomech (Bristol, Avon). 1996; 11:81-89.
7. Lu TW, O’Connor JJ. Fibre recruitment and shape changes of knee ligaments during motion: as revealed by a computer graphics-based model. Proc Inst Mech Eng [H]. 1996; 210:71-79.
8. Lu TW, O’Connor JJ. Lines of action and moment arms of the major force-bearing structures crossing the human knee joint: comparison between theory and experiment. J Anat. 1996; 189 (Pt 3):575-585.
9. Feikes J. The Mobility and Stability of the Human Knee Joint [PhD thesis]. Oxford, England: University of Oxford; 1999.
10. Friedrich N, Muller W, O’Brien W. Klinische anwendung biomechanisched und funktionell anatomischer am kniegelenk. Orthopäde. 1992; 21:41-50.
11. Mommersteeg TJ, Kooloos JG, Blankevoort L, Kauer JM, Huiskes R, Roeling FQ. The fibre bundle anatomy of human cruciate ligaments. J Anat. 1995; 187 (pt 2):461-71.
12. Woo SL-Y, Adams DJ. The tensile properties of human anterior cruciate ligamenet (ACL) and ACL graft tissues. In: Daniel DM, Akeson WH, O’Connor JJ, eds. Knee Ligaments, Structure, Function Injury and Repair. New York, NY: Raven Press; 1990.
13. Goodfellow J, O’Connor J, Dodd C, Murray D. Unicompartmental Arthroplasty with the Oxford Knee. Oxford, England: Oxford University Press; 2006.
14. Hill PF, Vedi V, Williams A, Iwaki H, Pinskerova V, Freeman MA. Tibiofemoral movement 2: the loaded and unloaded living knee studied by MRI. J Bone Joint Surg Br. 2000; 82:1196-1198.
15. Huss RA, Holstein H, O’Connor JJ. The effect of cartilage deformation on the laxity of the knee joint. Proc Inst Mech Eng [H]. 1999; 213:19-32.
16. Huss RA, Holstein H, O’Connor JJ. A mathematical model of forces in the knee under isometric quadriceps contractions. Clin Biomech (Bristol, Avon). 2000; 15:112-122.

Authors

Dr O’Connor is from the University of Oxford, Oxford, England, and Dr Imran is from Ajman University, Ajman, United Arab Emirates.

Dr O’Connor is a consultant for the sponsor and has received financial contributions from them in the past 12 months. Dr Imran has no financial interests in the materials mentioned herein.