Figure 3.14 Sagittal plane model of the knee.
Figure 3.14 shows a model of the knee in the sagittal plane, at extension, 60° and 120° flexion (see website, Animation 2). The joint is represented as a four-bar linkage with the two bones and isometric fibres of the two cruciate ligaments acting as the four links. It is a model of passive flexion/extension. The articular surface of the tibia is taken to be flat, a compromise between the convex lateral tibial plateau and the concave medial plateau of the human knee. The fibres of the ACL and PCL are assumed to remain isometric during passive flexion. The instantaneous centre of the joint, through which the flexion axis passes, lies at the intersection of the two isometric fibres. Figure 3.14 shows that it moves backwards on the tibia during flexion and forwards during extension as the two ligament fibres rotate relative to each other.
The shape of the femoral condyle is calculated from the condition that, in each position, the condyle must touch the tibial plateau at the point where the perpendicular to that plateau passes through the flexion axis (the line C in each diagram) (the ‘common normal theorem’) (O’Connor et al., 1989). When a similar calculation is performed assuming the tibial plateau to be a concave circle (the medial tibial plateau) or a convex circle (the lateral tibial plateau), the calculated shape of the corresponding condyle fits well with parasagittal sections taken through the medial or lateral condyles respectively of a human knee (O’Connor et al., 1989).
For these cruciate ligament fibres to remain isometric and for the surfaces to remain in contact during flexion and extension, the model femur has to roll backwards while sliding forwards on the tibia during passive flexion and vice versa during extension. Any other movement would require the application of load, the stretching or slackening of the fibres, the separation or indentation of the surfaces. Because the model tibial plateau has been taken to be flat, the backward excursion of the centre of curvature of the model femoral condyle during flexion is the same as that of the contact point. As Figure 3.9 above demonstrates, this is not true for concave or convex tibial plateaux.
During flexion to 140°, the isometric fibre of the model ACL rotates through 40° towards the tibia. The isometric fibre of the PCL rotates through 40° away from the tibia, as shown in Zuppinger’s original sketch of the four-bar linkage model (Pinskerova et al., 2000). The rotation of the model PCL about its tibial insertion is similar in magnitude and direction to that shown by Nakagawa et al. (2004).
The sulcus of the trochlear groove in Figure 3.14(a) is modelled as a circle anterior to the tibial facet of the femur; it appears not to be continuous with the femoral condyle because images of the trochlear flanges have been omitted (but see Figs. 3.17 and 3.18 below). The patella is modelled as a rectangle with two articular surfaces, the more posterior making contact with the trochlea in extension and at 60° (Gill & O’Connor, 1996). By 120°, the more anterior articular surface of the patella, modelling its medial and lateral facets, makes contact with the model femoral condyle.
Apart from the assumptions about the geometry of the trochlea and the patellar surfaces, the model also assumes that the patellar tendon remains of constant length and that the line of action of the patellofemoral contact force passes through the intersection of the patellar and quadriceps tendons. (This latter condition has been used as the basis of an experimental method to measure the patellofemoral force in cadaver specimens under load. (Miller et al., 1998)) As a result, during trochlear contact, the patella rolls proximally on the trochlea during passive flexion and vice versa during passive extension. Such rolling movements have been observed by Goodfellow et al. (1976) (see Fig. 5.1) and discussed in Chapter 5 below in the context of patello-femoral arthritis. These authors also observed the transition to condylar contact in the flexed knee.
The magnitude of the backwards rotation of the patellar tendon during flexion agrees well with measurements on intact specimens by Buff et al. (1988) and has been used as a measure of tibiofemoral movement after arthroplasty (see Fig. 3.27). Gill and O’Connor (1996) showed that about two thirds of the rotation of the patellar tendon is due to the cam-like shape of the distal femur and one third due to the rolling of the femur on the tibia.
The model of Figure 3.14 also shows how straight lines representing the lines of action of the hamstrings and gastrocnemius tendons change their directions relative to the bones at the knee during flexion. The model has been used to discuss how antagonistic extensor/flexor action may be used to protect ligaments after injury or repair (O’Connor, 1993).
An animation of this model will be found as Animation 2 on the website. It shows roll-back of the femur on the tibia, the rotation of the isometric cruciate fibres about their insertions on the tibia and that the transition from trochlear patellar contact to condylar contact occurs at about 99° of flexion. The total roll-back of the model femur on the tibia over 120° is 11.5 mm, in reasonable agreement with the average movement (11.3 mm) of the contact points medially (7.9 mm) and laterally (14.7 mm) calculated by Feikes from her experiments (Table 3.1 above).