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A mathematical model of forces in the knee under isometric quadriceps contractions
Clinical Biomechanics 15 (2000) 112±122
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A mathematical model of forces in the knee under isometric quadriceps contractions Richard A. Huss a,*, Horst Holstein a, John J. OÕConnor b b
a Department of Computer Science, University of Wales Penglais, Aberystwyth, Ceredigion, Wales SY23 3DB, UK Department of Engineering Science and Orthopaedic Engineering Centre, University of Oxford Nueld Orthopaedic Centre, Headington, Oxford OX3 7LD, UK
Received 14 April 1999; accepted 20 July 1999
Abstract Objective. To predict the kneeÕs response to isometric quadriceps contractions against a ®xed tibial restraint. Design. Mathematical modelling of the human knee joint. Background. Isometric quadriceps contraction is commonly used for leg muscle strengthening following ligament injury or reconstruction. It is desirable to know the ligament forces induced but direct measurement is dicult. Methods. The model, previously applied to the Lachmann or ÔdrawerÕ tests, combines an extensible ®bre-array representation of the cruciate ligaments with a compressible Ôthin-layerÕ representation of the cartilage. The model allows the knee con®guration and force system to be calculated, given ¯exion angle, restraint position and loading. Results. Inclusion of cartilage deformation increases relative tibio±femoral translation and decreases the ligament forces generated. For each restraint position, a range of ¯exion angles is found in which no ligament force is required, as opposed to a single ¯exion angle in the case of incompressible cartilage layers. Conclusions. Knee geometry and ligament elasticity are found to be the most important factors governing the jointÕs response
to isometric quadriceps contractions, but cartilage deformation is found to be more important than in the Lachmann test. Relevance Estimation of knee ligament forces is important when devising exercise regimes following ligament injury or reconstruction. The ®nding of a Ôneutral zoneÕ of zero ligament force may have implications for rehabilitation of the ligament-injured knee. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Knee; Articular cartilage; Quadriceps; Ligaments
~ Flp Flpx ; Flpy
1. Notation (x, y) dr Fq ~ Fl Flx ; Fly ~ Fla Flax ; Flay
coordinate system ®xed in tibia distance of restraining force below tibial plateau quadriceps force ligament force exerted on tibia, acting at point ~ rl rlx ; rly , of magnitude Fl anterior cruciate ligament force exerted on tibia, acting at point ~ rla rlax ; rlay , of magnitude Fla
~ Fp Fpx ; Fpy ~ Fc Fcx ; Fcy ~ Fr Frx ; Fry ~ d dx ; dy
*
Corresponding author. E-mail address: [email protected] (R.A. Huss).
/
0268-0033/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 8 - 0 0 3 3 ( 9 9 ) 0 0 0 5 9 - 5
posterior cruciate ligament force exerted on tibia, acting at point ~ rlp rlpx ; rlpy , of magnitude Flp patellar tendon force exerted on tibia, acting at point ~ rlp rpx ; rpy , of magnitude Fp contact force exerted on tibia, acting at point ~ rc rcx ; rcy , of magnitude Fc restraining force exerted on tibia, acting at point ~ rr rrx ; rry , of magnitude Fr displacement of femur from reference position knee ¯exion angle
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2. Introduction This paper describes the response of the knee to isometric quadriceps contraction with the aid of a twodimensional quasi-static model of the joint, previously applied to the study of anterior/posterior (A/P) loading [1]. Isometric quadriceps contraction is one of a number of exercises (along with dynamic quadriceps exercise with the knee ¯exed, hamstrings exercise, and cycling without full extension) commonly used in rehabilitation following ligament injury or reconstruction. During isometric quadriceps contraction, the knee is maintained at a constant ¯exion angle by a restraining force perpendicular to the tibia when the quadriceps are under load. In rehabilitation, it is desirable to know the ligament forces induced by the applied muscle forces, with the aim of minimising the risk of further damage to the ligaments whilst allowing muscle rehabilitation. Since direct measurement of ligament forces is dicult, a mathematical model is used to provide estimates of their values. Experimental measurements of relative tibio±femoral displacements have, however, been made, for example in vivo by Howell [2] and in vitro by Hirokawa et al. [3]. Many mathematical models of the knee joint have been devised; early theoretical models of isometric quadriceps contractions considered the net shear force at the joint, using experimental data relating to the joint position and the magnitude of the restraining force. A recent three-dimensional, quasi-static knee model is described by Blankevoort and Huiskes [4] and Mommersteeg et al. [5,6]; three-dimensional measurements of the tibial and femoral articular surface geometry are used, with deformable articular cartilage surfaces, and each ligament is represented by a bundle of individual ®bre bundles. The contact geometry, and ligament insertion sites and force±length relationships are measured from individual cadaver specimens [5,6], or obtained by an optimisation procedure [4]. The model uses a large system of non-linear equations to predict the ¯exing and extending motion of the knee, ligament forces, and A/P and varus±valgus laxity, but muscle forces have not yet been considered. Shelburne and Pandy [7±9] describe a sagittal-plane model which incorporates 11 elastic elements representing the ligaments; the patella is treated as a rectangle. The model has been applied to studying the relationships between muscle forces, external loads, and cruciate ligament forces at the knee during isometric exercises. All articular cartilage layers are rigid, in contrast to the present investigation. OÕConnor et al. [10,11], using a model with cruciate ligaments represented by inextensible lines determined by a four-bar kinematic linkage [12], calculated the varying geometry of the quadriceps, hamstring and gastrocnemius muscle tendons as the knee ¯exed and
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extended, and also the ligament and contact forces resulting from given muscle loads. The patellofemoral joint model assumed that the patellar tendon and quadriceps force action lines intersect at a point which moves on a circular path about the centre of the femurÕs trochlear facet. Zavatsky and OÕConnor [13] extended this model by incorporating a representation of the cruciate ligaments used previously to study the kneeÕs response to A/P loading (the Lachmann or ÔdrawerÕ tests) [14,15]. In this model, the cruciate ligaments were represented by continuous arrays of elastic ®bres joining line-segment attachments on the tibia and femur. The femur and tibia were thereby able to undergo small relative translations parallel to the tibial plateau under quadriceps loading. An anatomical model of the patellofemoral joint was described by Gill and OÕConnor [16]. This represented the patella as a rectangular block rolling and sliding on a circular facet on the femur; at high ¯exion angles, contact transferred from the trochlea to the femoral condyles. The orientation of the patella was determined using both geometric and force equilibrium constraints. This patella model has been incorporated into knee models used to calculate ligament and contact forces under a variety of external loading situations, using the approach of Zavatsky and OÕConnor [13]. Many of the predictions of this model were found to be in good agreement with experiment [17]. Huss et al. [1,18] extended the model of A/P laxity described by Zavatsky and OÕConnor [14,15] by additionally allowing the joint surfaces to deform under load. The contact forces were calculated using a ÔthinlayerÕ constitutive equation for articular cartilage, and a simpli®ed geometrical representation of the contact area. The addition of cartilage deformation was found to have a signi®cant eect on the force/displacement relationship, but was less important than ligamentous deformation. In this paper, the ligament model of Zavatsky and OÕConnor [14,15], the patella model of Gill and OÕConnor [16], and the contact force calculation of Huss et al. [1,18] are combined to produce a model of the response of the knee to isometric quadriceps contractions in which both ligaments and cartilage layers are able to deform. The contribution of cartilage deformation is determined by comparing the results to those of a model in which the surfaces are rigid. Consequences for rehabilitation are discussed. 3. Methods 3.1. Loading regime and geometry The system of forces acting on the tibia is shown in Fig. 1. A restraining force ~ Fr resists extension of the leg
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of the ligaments where the ®bres are taut to give the ligament force. The line of action is found by considering moments about an arbitrarily-chosen point. Full details of the ligament calculations are given by Zavatsky and OÕConnor [14,15]. 3.3. Contact model The contact force is calculated as described in Huss et al. [1,18] using a thin-layer approximation. The femoral condyles are treated as rigid spheres indenting uniform planar elastic layers (representing the articular cartilage) bonded to a rigid substrate. The calculations take into account the varying pro®le of the femoral condyles with ¯exion angle, described by OÕConnor et al. [19]. 3.4. Patella model
Fig. 1. Forces applied to the tibia in response to a quadriceps force Fq . Dotted lines represent the unloaded state.
when the subject applies a quadriceps force ~ Fq . The restraining force is applied at a chosen distance dr below the tibial plateau. The quadriceps force is transmitted to the tibia by the patellar tendon, which exerts a force ~ Fp . A tibio±femoral contact force ~ Fc is generated, and there may also be forces ~ Fla and ~ Flp in the anterior and posterior cruciate ligaments. The collateral ligaments are omitted from the current analysis. The coordinate system is ®xed in the tibia with its origin at O, at the front of the tibial insertion of the anterior cruciate ligament (ACL). The x-axis is directed posteriorly parallel to the tibial plateau and the y-axis is directed proximally. As in the model of A/P laxity [1,18], the joint con®guration is described by a ¯exion angle / giving the position of the joint in the absence of any external forces, and a displacement ~ d dx ; dy of the femur relative to the tibia from this neutral position. 3.2. Ligament model The cruciate ligaments are represented by continuous arrays of elastic ®bres joining line-segment attachments on the tibia and femur. The natural lengths of the ®bres are de®ned by specifying a ¯exion angle for each ligament or part of a ligament, at which all ®bres are just taut with ~ d ~ 0. Under load, the strain can be found for each ®bre, given its natural length and the joint con®guration (/ and ~ d). A non-linear stress±strain relationship is used to determine the stress, which is integrated over those parts
The model incorporates the patella mechanism of Gill and OÕConnor [16]. The patellar tendon is represented as an inextensible string connecting the tibial tubercule and the inferior pole of the patella, with the quadriceps tendon lying parallel to the femur (unless it wraps around the femur). Additional geometric and forcebalance considerations allow the orientation of the patellar tendon to be found, given / (and, in the present model, ~ d) together with the force ratios for the patellar tendon, quadriceps tendon and patellofemoral contact. The present analysis is restricted to trochlear contact. 3.5. Equilibrium equations Static equilibrium requires the total force acting on the tibia and the total ¯exing moment to be zero. In a 2D model, static equilibrium requires Fcx Fpx Frx Flax Flpx 0;
1
Fcy Fpy Fry Flay Flpy 0;
2
rcx Fcy ÿ rcy Fcx rpx Fpy ÿ rpy Fpx rrx Fry ÿ rry Frx rlax Flay ÿ rlay Flax rlpx Flpy ÿ rlpy Flpx 0:
3
For a choice of model parameters including dr and input variables /, Fr , all measurable externally, we seek the values of Fq , dx , dy , Frx , Fry , rrx , rry , Flax , Flay , rlax , rlay , Flpx , Flpy , rlpx , rlpy , Fcx , Fcy , rcx , rcy , Fpx , Fpy , rpx , and rpy and have three equilibrium equations, Eqs. (1)±(3) on these 23 unknowns. We use the solution of the contact problem to give Fcx , Fcy , rcx , and rcy as functions of /, dx and dy , giving four additional equations. The method of Zavatsky and OÕConnor gives Flax , Flay , rlax , rlay , Flpx , Flpy , rlpx , and rlpy as functions of /, dx and dy (eight additional equations). The patella model of Gill and OÕConnor gives Fpx , Fpy , rpx , and rpy as functions of Fq , /, dx and dy (four equations). Given dr and Fr , it is
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possible to calculate Frx , Fry , rrx , rry (four equations). In total, 23 equations are therefore available. 3.6. Solution technique In fact, Fq rather than Fr was treated as chosen input for the purpose of presenting results. The chosen inputs
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were /, Fq , and dr ; quantities to be solved for were dx , dy , and Fr . An initial approximate solution was obtained as follows: 1. With the ligaments assumed to be inextensible and the articular cartilage incompressible, the lines of action of the patellar tendon, contact and restraining forces and the magnitude of the patellar tendon force
Fig. 2. Horizontal component of displacement dx versus quadriceps force Fq for ¯exion angles of 0°, 20°, 40°, 60°, 80° and 100°. dr 200 mm: (a) Compressible cartilage. (b) Incompressible cartilage.
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were calculated at the speci®ed ¯exion angle / with ~ d ~ 0. 2. The intersection of the patellar tendon and contact force lines of action was found and was used as described by Zavatsky and OÕConnor [13] to ascertain whether an ACL or a PCL force was required for equilibrium, in the absence of deformable tissues. 3. Eqs. (1)±(3) were solved for Fr , Fl and Fc .
4. Using the compressible cartilage model, the value of dy required to produce the value of Fc calculated in step 3 was found by solving the equation Fc dy Fc step 3:
4
5. Using the extensible ligament model, the value of dx required to produce the value of Fl calculated in step 3 was found by solving the equation Fl fdx ; dy step 4; /g Fl step 3:
5
Fig. 3. Ligament force Fl versus quadriceps force Fq for ¯exion angles of 0°, 20°, 40°, 60°, 80° and 100°. dr 200 mm: (a) Compressible cartilage. (b) Incompressible cartilage.
R.A. Huss et al. / Clinical Biomechanics 15 (2000) 112±122
With the value of Fr thereby calculated, and the lines of action of the other forces rede®ned by the values of dx and dy , the equilibrium equations were resolved. This process was repeated iteratively to satisfy a convergence criterion. For comparison, the calculations were repeated with the articular cartilage incompressible. Fc was then independent of dy , which was ®xed at 0 mm. Given /, Fq and dr , the variables to be solved for were dx , Fc and Fr . The solution technique otherwise was analogous to that already described.
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4. Results Fig. 2(a) and (b) compare the compressible and incompressible models, respectively, and show that the absolute value of horizontal displacement increases with increasing quadriceps force, tending towards an asymptotic value as described by Zavatsky and OÕConnor [13]. Inclusion of compressible cartilage layers increases the displacement by between 36% at 40° ¯exion and 105% at 0° ¯exion. Anterior displacement occurs at lower ¯exion angles and posterior displacement
Fig. 4. Ligament force Fl versus quadriceps force Fq and ¯exion angle /. dr 200 mm. The thick solid line in (a) indicates the boundary of the region within which Fl 0; thick dotted lines indicate the line in (b) along which Fl 0: (a) Compressible cartilage. (b) Incompressible cartilage.
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at higher ¯exion angles, as described by Mandtl et al. [20] Figs. 3 and 4 show that cartilage deformation substantially reduces the ligament forces induced by quadriceps loading, the reduction being more than 50% in some cases. For both models, the ACL is loaded at lower ¯exion angles and the PCL at higher ¯exion
angles, consistent with the directions of tibial displacements in Fig. 2. With compressible cartilage layers, the ligament forces also tend more rapidly towards asymptotic values. Fig. 4(a) also shows that cartilage deformation introduces a region of zero ligament force at around / 62.5° with the restraining load positioned 200 mm below the tibial plateau, whereas in Fig. 4(b) the ligament force is zero only at a single ¯exion angle of 62.5°, as described by Zavatsky and OÕConnor [13]. It can be seen from Figs. 2±4 that the results broadly con®rm those of Zavatsky and OÕConnor [13], despite the slightly dierent patella models used. The region of zero ligament force moves to higher ¯exion angles as the restraint position is moved distally (increasing dr ). Larger ligament forces are required for equilibrium at high ¯exion angles with the restraint near the knee, and at low ¯exion angles with the restraint located more distally. Fig. 5 shows qualitative agreement between the theory and HirokawaÕs experiment [3]. The variation of the horizontal displacement with ¯exion angle and quadriceps force is similar in form, but the calculation underestimates the measurement. Fig. 6 shows better agreement between the theory and HowellÕs experiment [2] but the calculation if anything overestimates the measurement.
5. Discussion Inclusion of cartilage deformation in the present model is found to increase the dynamic laxity of the joint signi®cantly, more so than for passive laxity. The
Fig. 5. Horizontal component of displacement dy versus ¯exion angle for six values of quadriceps load. dr 430 mm: (a) Experimental results of Hirokawa et al. [13], redrawn from Table 1 of their paper. (b) Results of incompressible cartilage model. (c) Results of compressible cartilage model.
Fig. 6. Horizontal component of displacement dx versus ¯exion angle, showing a histogram of the experimental results of Howell [14] under maximal quadriceps loading, and the model results under 2500 N quadriceps forces with compressible and incompressible articular cartilage.
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additional freedom causes a region of zero ligament force in which the patellar tendon, contact and restraining forces intersect. The model results are in broad agreement with the experiments of Hirokawa et al. [3] and Howell [2], and suggest how isometric quadriceps exercises following cruciate ligament injury may be planned to minimise danger to the ligaments; the region of zero ligament force may broaden slightly the range of possible exercises. The greater eect on dynamic laxity, compared to passive laxity [1], suggests that reproducing normal passive laxity in a replacement with more rigid surfaces is not a guarantee of normal behaviour under load.
and OÕConnor, although the critical value of dr is slightly higher than their value due to the dierent patella models used. Once the restraining force is proximal to the intersection of the patellar tendon and contact force lines of action at full extension, it ceases to be possible to ®nd a ¯exion angle at which Fl 0 N with the quadriceps loaded. The range of / with zero Fl is signi®cantly wider at dr 200 mm than it is for higher values of dr . The condition for zero ligament force is illustrated in Fig. 7 for a quadriceps force of 1000 N. For comparison, the line of no ligament force is also shown for the incompressible cartilage model.
5.1. Antero-posterior laxity
5.4. Correlation with experimental results
Cartilage deformation has been found to make a more signi®cant contribution to the dynamic anteroposterior laxity of the joint (compare Fig. 2(a) and (b)) than was found in the study of the passive Lachmann test [1]. In the latter the contact force has to balance only the vertical component of the ligament force, whereas in the Isometric Quadriceps test it also has to balance the relatively larger component of the patellar tendon force.
Fig. 5 illustrates the A/P tibial displacements induced by a variety of quadriceps loads; the experimental results of Hirokawa et al. [3] are shown along with those of the incompressible and compressible cartilage models. Hirokawa et al. used a radiographic technique to measure the relative tibio±femoral displacement of cadaveric specimens, with the femur ®xed and the tibia held at a constant ¯exion angle via a ®ve degree-of-freedom universal joint. Quadriceps forces were simulated using a variety of weights attached to the quadriceps tendon. Tibial rotation was also measured. The value of dr used in the model calculations was 430 mm. The general features are similar to the experimental results: displacement increases non-linearly with load; anterior tibial displacement is observed from extension to a transitional angle around 80±90°, with a peak at approximately 30°, and posterior displacements at higher angles.
5.2. Region of zero ligament force Deformation of articular cartilage allows the ligaments to slacken, leading to a general reduction of ligament forces, and possibly a region of zero ligament force. In this region, the tightening eect of the external forces is exactly balanced by the slackening eect of cartilage deformation. The thick continuous line in Fig. 4(a) indicates the boundary of the region of no ligament force. The region contains the dotted line de®ning the ÔcriticalÕ ¯exion angle (62.5°) of zero ligament force according to the incompressible cartilage model [13]. In that model, a ligament force is not required when the restraining force, patellar tendon force and contact force pass through a single point. The additional freedom imparted by cartilage deformation enables concurrency of the forces over a range of ¯exion. 5.3. Eect of varying restraint position Zavatsky and OÕConnor [13] found that the position of zero ligament force moved to higher ¯exion angles as the restraining force moved more distally. The same eect is found when the cartilage layers are allowed to deform, except that a range of angles is now found for zero ligament force. Below dr 134 mm, the region of no ligament force vanishes; again, this agrees with the results of Zavatsky
Fig. 7. Graph of restraint position versus ¯exion angle, for a quadriceps force of 1000 N. The solid line shows where no ligament force is required for equilibrium with incompressible cartilage (after Zavatsky and OÕConnor [5], Fig. 4(a)), and the dotted lines show the corresponding boundaries of no ligament force with compressible cartilage.
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The experimentally-measured magnitude of the tibial displacement is however considerably larger than that calculated in either model, reaching a maximum of 8 mm as opposed to 2 mm in the incompressible cartilage model, or 2.5 mm when the cartilage layers are allowed to deform. Hirowaka et al. [3] note that rotation of the tibia may introduce an error into their results, and provide estimates of the resulting errors. These reduce the maximum to around 6.3 mm. This is still considerably more than that predicted by the compressible cartilage model. To approach the experimental values of the tibial displacement, very considerable and unphysical changes to the elasticity of the model articular cartilage and ligament ®bres are required. The discrepancy may be due to the greater laxity shown by cadaver than by normal knees, or to rotation of the tibia, which is ignored in this 2D model. Omission from the model of the constraint oered by the collateral ligaments is likely to result in predicted displacements being higher than in real life, and so cannot be the cause of this discrepancy. Howell [2] used an arthrometer to measure the anterior tibial translation during maximum isometric quadriceps contraction in vivo in 22 normal patients at / 15°, 30°, 45°, 60°, and 75°. The restraining force ~ Fr was applied 290 mm below the joint line. Fig. 6 shows the resulting tibial displacements. The model calculations using a quadriceps force of 2500 N now underestimate the measurements. (No actual force values were measured by Howell.) 5.5. Clinical relevance As the main features of the results are broadly similar to those reported by Zavatsky and OÕConnor [13], similar clinical lessons may be drawn. The analysis suggests that, with deformable cartilage, ligament forces do not increase steadily with increasing quadriceps force but reach relatively small asymptotic values (Fig. 3(a)). This could explain why relatively weak and slender ligaments can survive even in the presence of large muscle forces. The model predictions allow rehabilitation regimes following cruciate ligament injury or repair to be planned in such a way as to minimise ligament forces. The discovery of a region of zero ligament force in the compressible cartilage model, as opposed to a single ¯exion angle, may permit a wider range of ¯exion angles to be used in such treatments, or make slight inaccuracies in setting up exercises less critical. Very substantial quadriceps forces may be borne within this region without risking damage to the ligaments. When the cartilage is made incompressible (e.g. following total knee replacement), this eect is removed. The previous study of the Lachmann test [1] suggested that cartilage deformation made relatively little
contribution to the passive laxity of the knee. When the surfaces are replaced by the more rigid components of a prosthesis, the passive laxity could appear to be quite normal. However the signi®cant contribution of cartilage deformation to dynamic laxity, in the presence of muscle forces, implies that a replaced knee may exhibit signi®cantly less dynamic laxity than the normal. 5.6. Limitations of the present approach The present approach has a number of limitations. Only isometric quadriceps contractions are considered, whereas in reality it is common for more than one muscle group to act simultaneously, and loading is likely to be dynamic. Development of the present model to cover more complex loading regimes is possible. Various important structures within the knee are omitted from the model, particularly the collateral ligaments and the menisci. Omission of the collateral ligaments simpli®es the analysis; they are also found to be relatively sensitive to the choice of attachment areas and to contribute relatively little to the total ligament forces under A/P loading [15]. They do however have a large impact on tibial rotation (which occurs during isometric quadriceps exercise) and abduction/adduction [21]. Under isometric quadriceps loading, Harfe et al. [21] found that the medical collateral ligament (MCL) slackened in the range 15° < / < 45° and tightened in the range 90° < / < 120°, whereas the lateral collateral ligament (LCL) slackened at all ¯exion angles. This suggests that including the MCL in the model would provide an additional restraint at higher ¯exion angles, resulting in less tibial translation than is currently predicted, while the inclusion of the LCL would have little eect. The present calculations underestimate HirokawaÕs measurements of tibial displacement [3], suggesting that constraints oered by the menisci to antero-posterior movement are small. However, because the menisci spread the compressive force over larger areas of cartilage and therefore reduce the contact pressures, their omission is likely to lead to an overestimate of the extent of cartilage deformation. A purely 2D model naturally has limitations. A 2D ligament model cannot fully represent the physical cruciate ligaments, and the patterns of ®bre recruitment and extension following relative tibio±femoral displacement. The four-bar linkage omits important aspects of the kneeÕs kinematics, particularly the ÔscrewhomeÕ mechanism (the external rotation of the tibia during the ®nal stage of extension). The transverse forces within the joint are also ignored; these will be more important in the present model than when purely A/P forces were being discussed, as the quadriceps has a marked inclination in the frontal plane. A 3D kinematic
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mechanism [22,23], and 3D modelling of the cruciate ligaments [24], helps to address these diculties. The present contact force calculation could also be improved. The true 3D shape of the tibial and femoral surfaces is not taken fully into account, though the effective local spherical radius of curvature of the femoral condyles is used to modify the force calculation [1]. Articular cartilage is actually non-linear, viscoelastic and non-isotropic; it has been modelled in detail as a biphasic system comprising a solid matrix and a liquid phase by Mow et al. [25]. Over time scales of a few seconds, however, its response to load is nearly instantaneous and it is therefore appropriate to model it as an elastic solid in the current circumstances.
6. Conclusions Isometric quadriceps contractions were studied using a 2D quasistatic model of the knee incorporating deformable ligaments and articular cartilage. The model calculations presented have outcomes that are broadly comparable to those of equivalent models with incompressible cartilage and inextensible ligament ®bres. Compressible cartilage therefore is of secondary importance in knee modelling. Whereas an inextensible ligament model can give only the ratios of forces, the extensible ligament models, both with and without cartilage deformation, show displacements and ligament forces which vary non-linearly with the applied quadriceps force and tend to asymptotic values. Incorporation of compressible cartilage layers reduces the values of the ligament forces and increases the relative A/P translation between the tibia and femur. In the absence of compressible cartilage layers, consideration of the lines of action of the patellar tendon, contact, restraining and cruciate ligament forces shows whether an ACL or a PCL force is required for equilibrium at a given ¯exion angle; it is found that for each ¯exion angle there is a single restraint position at which no ligament forces are required for equilibrium. When the cartilage is made compressible, the increased ¯exibility aorded to the joint results in a range rather than a single value of restraint position leading to zero ligament force. Comparing the results to those of Zavatsky and OÕConnor [13], it is clear that knee geometry alone is sucient for most purposes for predicting whether ligament forces will be required in a given loading regime, and that adding ligament elasticity gives a reasonable approximation to the relative tibio±femoral displacements, and hence ligament forces. Cartilage deformation is a less important, but still signi®cant, modi®er of this behaviour.
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Acknowledgements RAH was supported during the course of this work by a University of Wales, Aberystwyth Postgraduate Studentship.
References [1] Huss RA, Holstein H, OÕConnor JJ. The eect of cartilage deformation on laxity of the knee joint. (Proc Instn Mech Engrs Part H) J Eng Med 1999;213(H1):19±32. [2] Howell SM. Anterior tibial translation during a maximum quadriceps contraction: is it clinically signi®cant? Am J Sports Med 1990;18(6):573±78. [3] Hirokawa S, Solomonow M, Lu Y, Lou Z-P, DÕAmbrosia R. Anterior-posterior and rotational displacement of the tibia elicited by quadriceps contraction. Am J Sports Med 1992;20(3):299±306. [4] Blankevoort L, Huiskes R. Technical note: validation of a threedimensional model of the knee. J Biomech 1996;29(7):955±61. [5] Mommersteeg TJA, Huiskes R, Blankevoort L, Kooloos JGM, Kauer JMG, Maathuis PGM. Technical note: a global veri®cation study of a quasi-static knee model with multi-bundle ligaments. J Biomech 1996;29(12):1659±64. [6] Mommersteeg TJA, Huiskes R, Blankevoort L, Kooloos JGM, Kauer JMG. An inverse dynamics modelling approach to determine the restraining function of human knee ligament bundles. J Biomech 1997;30(2):139±46. [7] Shelburne KB, Pandy MG. A musculoskeletal model of the knee for evaluating ligament forces during isometric contractions. J Biomech 1997;30(2):163±76. [8] Pandy MG, Shelburn KB. Dependence of cruciate-ligament loading on muscle forces and external load. J Biomech 1997;30(10):1015±24. [9] Shelburne KB, Pandy MG. Determinants of cruciate-ligament loading during rehabilitation exercise. Clin Biomech 1998;13(6):403±13. [10] OÕConnor JJ, Shercli TL, Fitzpatrick D, Bradley J, Daniel DM, Biden E, Goodfellow JW. Geometry of the knee. In: Daniel DM, Akeson WH, OÕConnor JJ, editors. Knee ligaments: structure, function, injury, and repair. New York: Raven Press, 1990: 163±200. [11] OÕConnor JJ, Shercli TL, Fitzpatrick D, Biden E, Goodfellow JW. Mechanics of the knee. In: Daniel DM, Akeson WH, OÕConnor JJ, editors. Knee ligaments: structure, function, injury, and repair. New York: Raven Press, 1990:201±38. [12] Huson A. The functional anatomy of the knee joint: The closed kinematic chain as a model of the knee joint. In: The knee joint. Recent advances in basic research and clinical aspects. Proceedings of the International Congress, Rotterdam. Amsterdam: Excerpta Medica, International Congress Series 1993;324:163±68. [13] Zavatsky AB, OÕConnor JJ. Ligament forces at the knee during isometric quadriceps contractions. (Proc Instn Mech Engrs Part H) J Eng Med 1993;207(H1):7±18. [14] Zavatsky AB, OÕConnor JJ. A model of human knee ligaments in the sagittal plane: Part 1. Response to passive ¯exion. (Proc Instn Mech Engrs Part H) J Eng Med 1992;206(H3):125±34. [15] Zavatsky AB, OÕConnor JJ. A model of human knee ligaments in the sagittal plane: Part 2. Fibre recruitment under load. (Proc Instn Mech Engrs Part H) J Eng Med 1992;206(H3):135±45. [16] Gill HS, OÕConnor JJ. Biarticulating two-dimensional computer model of the human patellofemoral joint. Clin Biomech 1996;11(2):81±9.
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[17] Lu T-W, OÕConnor 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 (3):575±85. [18] Huss RA, Holstein H, OÕConnor JJ. A two-dimensional computer model of the eect of cartilage deformation on laxity of the knee joint. Aberystwyth: University of Wales, Aberystwyth, Department of Computer Science; 1997 Tech. Report No.: UWA-DCS97-026. [19] OÕConnor JJ, Shercli TL, Biden E, Goodfellow JW. The geometry of the knee in the sagittal plane. (Proc Instn Mech Engrs Part H) J Eng Med 1989;203:223±33. [20] Mandtl PR, Daniel DM, Biden E, Stone ML. Tibial translation with quadriceps: an in vitro study of the eect of load placement, ¯exion angle, and ACL sectioning. Trans Orthop Res Soc 1987;33:243.
[21] Harfe DT, Chuinard CR, Espinoza LM, Thomas KA, Solomovow M. Elongation patterns of the collateral ligaments of the human knee. Clin Biomech 1998;13(3):163±75. [22] Wilson DR, OÕConnor JJ. A three-dimensional geometric model of the knee for the study of joint forces in gait. Gait and Posture 1997;5(2):108±15. [23] Wilson DR, Feikes JD, OÕConnor JJ. Ligaments and articular contact guide passive knee ¯exion. J Biomech 1998;31(12):1127± 36. [24] Zavatsky AB, OÕConnor JJ. Three-dimensional geometrical models of human knee ligaments. (Proc Instn Mech Engrs Part H) J Eng Med 1994;208(H4):229±40. [25] Mow VC, Hou JS, Owens JM, Ratclie A. Biphasic and quasilinear viscoelastic theories for hydrated soft tissues. In: Mow VC, Ratclie A, Woo SL-Y, editors. Biomechanics of diarthroidal joints. New York: Springer, 1990:215±60.
www.elsevier.com/locate/clinbiomech
A mathematical model of forces in the knee under isometric quadriceps contractions Richard A. Huss a,*, Horst Holstein a, John J. OÕConnor b b
a Department of Computer Science, University of Wales Penglais, Aberystwyth, Ceredigion, Wales SY23 3DB, UK Department of Engineering Science and Orthopaedic Engineering Centre, University of Oxford Nueld Orthopaedic Centre, Headington, Oxford OX3 7LD, UK
Received 14 April 1999; accepted 20 July 1999
Abstract Objective. To predict the kneeÕs response to isometric quadriceps contractions against a ®xed tibial restraint. Design. Mathematical modelling of the human knee joint. Background. Isometric quadriceps contraction is commonly used for leg muscle strengthening following ligament injury or reconstruction. It is desirable to know the ligament forces induced but direct measurement is dicult. Methods. The model, previously applied to the Lachmann or ÔdrawerÕ tests, combines an extensible ®bre-array representation of the cruciate ligaments with a compressible Ôthin-layerÕ representation of the cartilage. The model allows the knee con®guration and force system to be calculated, given ¯exion angle, restraint position and loading. Results. Inclusion of cartilage deformation increases relative tibio±femoral translation and decreases the ligament forces generated. For each restraint position, a range of ¯exion angles is found in which no ligament force is required, as opposed to a single ¯exion angle in the case of incompressible cartilage layers. Conclusions. Knee geometry and ligament elasticity are found to be the most important factors governing the jointÕs response
to isometric quadriceps contractions, but cartilage deformation is found to be more important than in the Lachmann test. Relevance Estimation of knee ligament forces is important when devising exercise regimes following ligament injury or reconstruction. The ®nding of a Ôneutral zoneÕ of zero ligament force may have implications for rehabilitation of the ligament-injured knee. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Knee; Articular cartilage; Quadriceps; Ligaments
~ Flp Flpx ; Flpy
1. Notation (x, y) dr Fq ~ Fl Flx ; Fly ~ Fla Flax ; Flay
coordinate system ®xed in tibia distance of restraining force below tibial plateau quadriceps force ligament force exerted on tibia, acting at point ~ rl rlx ; rly , of magnitude Fl anterior cruciate ligament force exerted on tibia, acting at point ~ rla rlax ; rlay , of magnitude Fla
~ Fp Fpx ; Fpy ~ Fc Fcx ; Fcy ~ Fr Frx ; Fry ~ d dx ; dy
*
Corresponding author. E-mail address: [email protected] (R.A. Huss).
/
0268-0033/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 8 - 0 0 3 3 ( 9 9 ) 0 0 0 5 9 - 5
posterior cruciate ligament force exerted on tibia, acting at point ~ rlp rlpx ; rlpy , of magnitude Flp patellar tendon force exerted on tibia, acting at point ~ rlp rpx ; rpy , of magnitude Fp contact force exerted on tibia, acting at point ~ rc rcx ; rcy , of magnitude Fc restraining force exerted on tibia, acting at point ~ rr rrx ; rry , of magnitude Fr displacement of femur from reference position knee ¯exion angle
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2. Introduction This paper describes the response of the knee to isometric quadriceps contraction with the aid of a twodimensional quasi-static model of the joint, previously applied to the study of anterior/posterior (A/P) loading [1]. Isometric quadriceps contraction is one of a number of exercises (along with dynamic quadriceps exercise with the knee ¯exed, hamstrings exercise, and cycling without full extension) commonly used in rehabilitation following ligament injury or reconstruction. During isometric quadriceps contraction, the knee is maintained at a constant ¯exion angle by a restraining force perpendicular to the tibia when the quadriceps are under load. In rehabilitation, it is desirable to know the ligament forces induced by the applied muscle forces, with the aim of minimising the risk of further damage to the ligaments whilst allowing muscle rehabilitation. Since direct measurement of ligament forces is dicult, a mathematical model is used to provide estimates of their values. Experimental measurements of relative tibio±femoral displacements have, however, been made, for example in vivo by Howell [2] and in vitro by Hirokawa et al. [3]. Many mathematical models of the knee joint have been devised; early theoretical models of isometric quadriceps contractions considered the net shear force at the joint, using experimental data relating to the joint position and the magnitude of the restraining force. A recent three-dimensional, quasi-static knee model is described by Blankevoort and Huiskes [4] and Mommersteeg et al. [5,6]; three-dimensional measurements of the tibial and femoral articular surface geometry are used, with deformable articular cartilage surfaces, and each ligament is represented by a bundle of individual ®bre bundles. The contact geometry, and ligament insertion sites and force±length relationships are measured from individual cadaver specimens [5,6], or obtained by an optimisation procedure [4]. The model uses a large system of non-linear equations to predict the ¯exing and extending motion of the knee, ligament forces, and A/P and varus±valgus laxity, but muscle forces have not yet been considered. Shelburne and Pandy [7±9] describe a sagittal-plane model which incorporates 11 elastic elements representing the ligaments; the patella is treated as a rectangle. The model has been applied to studying the relationships between muscle forces, external loads, and cruciate ligament forces at the knee during isometric exercises. All articular cartilage layers are rigid, in contrast to the present investigation. OÕConnor et al. [10,11], using a model with cruciate ligaments represented by inextensible lines determined by a four-bar kinematic linkage [12], calculated the varying geometry of the quadriceps, hamstring and gastrocnemius muscle tendons as the knee ¯exed and
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extended, and also the ligament and contact forces resulting from given muscle loads. The patellofemoral joint model assumed that the patellar tendon and quadriceps force action lines intersect at a point which moves on a circular path about the centre of the femurÕs trochlear facet. Zavatsky and OÕConnor [13] extended this model by incorporating a representation of the cruciate ligaments used previously to study the kneeÕs response to A/P loading (the Lachmann or ÔdrawerÕ tests) [14,15]. In this model, the cruciate ligaments were represented by continuous arrays of elastic ®bres joining line-segment attachments on the tibia and femur. The femur and tibia were thereby able to undergo small relative translations parallel to the tibial plateau under quadriceps loading. An anatomical model of the patellofemoral joint was described by Gill and OÕConnor [16]. This represented the patella as a rectangular block rolling and sliding on a circular facet on the femur; at high ¯exion angles, contact transferred from the trochlea to the femoral condyles. The orientation of the patella was determined using both geometric and force equilibrium constraints. This patella model has been incorporated into knee models used to calculate ligament and contact forces under a variety of external loading situations, using the approach of Zavatsky and OÕConnor [13]. Many of the predictions of this model were found to be in good agreement with experiment [17]. Huss et al. [1,18] extended the model of A/P laxity described by Zavatsky and OÕConnor [14,15] by additionally allowing the joint surfaces to deform under load. The contact forces were calculated using a ÔthinlayerÕ constitutive equation for articular cartilage, and a simpli®ed geometrical representation of the contact area. The addition of cartilage deformation was found to have a signi®cant eect on the force/displacement relationship, but was less important than ligamentous deformation. In this paper, the ligament model of Zavatsky and OÕConnor [14,15], the patella model of Gill and OÕConnor [16], and the contact force calculation of Huss et al. [1,18] are combined to produce a model of the response of the knee to isometric quadriceps contractions in which both ligaments and cartilage layers are able to deform. The contribution of cartilage deformation is determined by comparing the results to those of a model in which the surfaces are rigid. Consequences for rehabilitation are discussed. 3. Methods 3.1. Loading regime and geometry The system of forces acting on the tibia is shown in Fig. 1. A restraining force ~ Fr resists extension of the leg
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of the ligaments where the ®bres are taut to give the ligament force. The line of action is found by considering moments about an arbitrarily-chosen point. Full details of the ligament calculations are given by Zavatsky and OÕConnor [14,15]. 3.3. Contact model The contact force is calculated as described in Huss et al. [1,18] using a thin-layer approximation. The femoral condyles are treated as rigid spheres indenting uniform planar elastic layers (representing the articular cartilage) bonded to a rigid substrate. The calculations take into account the varying pro®le of the femoral condyles with ¯exion angle, described by OÕConnor et al. [19]. 3.4. Patella model
Fig. 1. Forces applied to the tibia in response to a quadriceps force Fq . Dotted lines represent the unloaded state.
when the subject applies a quadriceps force ~ Fq . The restraining force is applied at a chosen distance dr below the tibial plateau. The quadriceps force is transmitted to the tibia by the patellar tendon, which exerts a force ~ Fp . A tibio±femoral contact force ~ Fc is generated, and there may also be forces ~ Fla and ~ Flp in the anterior and posterior cruciate ligaments. The collateral ligaments are omitted from the current analysis. The coordinate system is ®xed in the tibia with its origin at O, at the front of the tibial insertion of the anterior cruciate ligament (ACL). The x-axis is directed posteriorly parallel to the tibial plateau and the y-axis is directed proximally. As in the model of A/P laxity [1,18], the joint con®guration is described by a ¯exion angle / giving the position of the joint in the absence of any external forces, and a displacement ~ d dx ; dy of the femur relative to the tibia from this neutral position. 3.2. Ligament model The cruciate ligaments are represented by continuous arrays of elastic ®bres joining line-segment attachments on the tibia and femur. The natural lengths of the ®bres are de®ned by specifying a ¯exion angle for each ligament or part of a ligament, at which all ®bres are just taut with ~ d ~ 0. Under load, the strain can be found for each ®bre, given its natural length and the joint con®guration (/ and ~ d). A non-linear stress±strain relationship is used to determine the stress, which is integrated over those parts
The model incorporates the patella mechanism of Gill and OÕConnor [16]. The patellar tendon is represented as an inextensible string connecting the tibial tubercule and the inferior pole of the patella, with the quadriceps tendon lying parallel to the femur (unless it wraps around the femur). Additional geometric and forcebalance considerations allow the orientation of the patellar tendon to be found, given / (and, in the present model, ~ d) together with the force ratios for the patellar tendon, quadriceps tendon and patellofemoral contact. The present analysis is restricted to trochlear contact. 3.5. Equilibrium equations Static equilibrium requires the total force acting on the tibia and the total ¯exing moment to be zero. In a 2D model, static equilibrium requires Fcx Fpx Frx Flax Flpx 0;
1
Fcy Fpy Fry Flay Flpy 0;
2
rcx Fcy ÿ rcy Fcx rpx Fpy ÿ rpy Fpx rrx Fry ÿ rry Frx rlax Flay ÿ rlay Flax rlpx Flpy ÿ rlpy Flpx 0:
3
For a choice of model parameters including dr and input variables /, Fr , all measurable externally, we seek the values of Fq , dx , dy , Frx , Fry , rrx , rry , Flax , Flay , rlax , rlay , Flpx , Flpy , rlpx , rlpy , Fcx , Fcy , rcx , rcy , Fpx , Fpy , rpx , and rpy and have three equilibrium equations, Eqs. (1)±(3) on these 23 unknowns. We use the solution of the contact problem to give Fcx , Fcy , rcx , and rcy as functions of /, dx and dy , giving four additional equations. The method of Zavatsky and OÕConnor gives Flax , Flay , rlax , rlay , Flpx , Flpy , rlpx , and rlpy as functions of /, dx and dy (eight additional equations). The patella model of Gill and OÕConnor gives Fpx , Fpy , rpx , and rpy as functions of Fq , /, dx and dy (four equations). Given dr and Fr , it is
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possible to calculate Frx , Fry , rrx , rry (four equations). In total, 23 equations are therefore available. 3.6. Solution technique In fact, Fq rather than Fr was treated as chosen input for the purpose of presenting results. The chosen inputs
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were /, Fq , and dr ; quantities to be solved for were dx , dy , and Fr . An initial approximate solution was obtained as follows: 1. With the ligaments assumed to be inextensible and the articular cartilage incompressible, the lines of action of the patellar tendon, contact and restraining forces and the magnitude of the patellar tendon force
Fig. 2. Horizontal component of displacement dx versus quadriceps force Fq for ¯exion angles of 0°, 20°, 40°, 60°, 80° and 100°. dr 200 mm: (a) Compressible cartilage. (b) Incompressible cartilage.
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were calculated at the speci®ed ¯exion angle / with ~ d ~ 0. 2. The intersection of the patellar tendon and contact force lines of action was found and was used as described by Zavatsky and OÕConnor [13] to ascertain whether an ACL or a PCL force was required for equilibrium, in the absence of deformable tissues. 3. Eqs. (1)±(3) were solved for Fr , Fl and Fc .
4. Using the compressible cartilage model, the value of dy required to produce the value of Fc calculated in step 3 was found by solving the equation Fc dy Fc step 3:
4
5. Using the extensible ligament model, the value of dx required to produce the value of Fl calculated in step 3 was found by solving the equation Fl fdx ; dy step 4; /g Fl step 3:
5
Fig. 3. Ligament force Fl versus quadriceps force Fq for ¯exion angles of 0°, 20°, 40°, 60°, 80° and 100°. dr 200 mm: (a) Compressible cartilage. (b) Incompressible cartilage.
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With the value of Fr thereby calculated, and the lines of action of the other forces rede®ned by the values of dx and dy , the equilibrium equations were resolved. This process was repeated iteratively to satisfy a convergence criterion. For comparison, the calculations were repeated with the articular cartilage incompressible. Fc was then independent of dy , which was ®xed at 0 mm. Given /, Fq and dr , the variables to be solved for were dx , Fc and Fr . The solution technique otherwise was analogous to that already described.
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4. Results Fig. 2(a) and (b) compare the compressible and incompressible models, respectively, and show that the absolute value of horizontal displacement increases with increasing quadriceps force, tending towards an asymptotic value as described by Zavatsky and OÕConnor [13]. Inclusion of compressible cartilage layers increases the displacement by between 36% at 40° ¯exion and 105% at 0° ¯exion. Anterior displacement occurs at lower ¯exion angles and posterior displacement
Fig. 4. Ligament force Fl versus quadriceps force Fq and ¯exion angle /. dr 200 mm. The thick solid line in (a) indicates the boundary of the region within which Fl 0; thick dotted lines indicate the line in (b) along which Fl 0: (a) Compressible cartilage. (b) Incompressible cartilage.
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at higher ¯exion angles, as described by Mandtl et al. [20] Figs. 3 and 4 show that cartilage deformation substantially reduces the ligament forces induced by quadriceps loading, the reduction being more than 50% in some cases. For both models, the ACL is loaded at lower ¯exion angles and the PCL at higher ¯exion
angles, consistent with the directions of tibial displacements in Fig. 2. With compressible cartilage layers, the ligament forces also tend more rapidly towards asymptotic values. Fig. 4(a) also shows that cartilage deformation introduces a region of zero ligament force at around / 62.5° with the restraining load positioned 200 mm below the tibial plateau, whereas in Fig. 4(b) the ligament force is zero only at a single ¯exion angle of 62.5°, as described by Zavatsky and OÕConnor [13]. It can be seen from Figs. 2±4 that the results broadly con®rm those of Zavatsky and OÕConnor [13], despite the slightly dierent patella models used. The region of zero ligament force moves to higher ¯exion angles as the restraint position is moved distally (increasing dr ). Larger ligament forces are required for equilibrium at high ¯exion angles with the restraint near the knee, and at low ¯exion angles with the restraint located more distally. Fig. 5 shows qualitative agreement between the theory and HirokawaÕs experiment [3]. The variation of the horizontal displacement with ¯exion angle and quadriceps force is similar in form, but the calculation underestimates the measurement. Fig. 6 shows better agreement between the theory and HowellÕs experiment [2] but the calculation if anything overestimates the measurement.
5. Discussion Inclusion of cartilage deformation in the present model is found to increase the dynamic laxity of the joint signi®cantly, more so than for passive laxity. The
Fig. 5. Horizontal component of displacement dy versus ¯exion angle for six values of quadriceps load. dr 430 mm: (a) Experimental results of Hirokawa et al. [13], redrawn from Table 1 of their paper. (b) Results of incompressible cartilage model. (c) Results of compressible cartilage model.
Fig. 6. Horizontal component of displacement dx versus ¯exion angle, showing a histogram of the experimental results of Howell [14] under maximal quadriceps loading, and the model results under 2500 N quadriceps forces with compressible and incompressible articular cartilage.
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additional freedom causes a region of zero ligament force in which the patellar tendon, contact and restraining forces intersect. The model results are in broad agreement with the experiments of Hirokawa et al. [3] and Howell [2], and suggest how isometric quadriceps exercises following cruciate ligament injury may be planned to minimise danger to the ligaments; the region of zero ligament force may broaden slightly the range of possible exercises. The greater eect on dynamic laxity, compared to passive laxity [1], suggests that reproducing normal passive laxity in a replacement with more rigid surfaces is not a guarantee of normal behaviour under load.
and OÕConnor, although the critical value of dr is slightly higher than their value due to the dierent patella models used. Once the restraining force is proximal to the intersection of the patellar tendon and contact force lines of action at full extension, it ceases to be possible to ®nd a ¯exion angle at which Fl 0 N with the quadriceps loaded. The range of / with zero Fl is signi®cantly wider at dr 200 mm than it is for higher values of dr . The condition for zero ligament force is illustrated in Fig. 7 for a quadriceps force of 1000 N. For comparison, the line of no ligament force is also shown for the incompressible cartilage model.
5.1. Antero-posterior laxity
5.4. Correlation with experimental results
Cartilage deformation has been found to make a more signi®cant contribution to the dynamic anteroposterior laxity of the joint (compare Fig. 2(a) and (b)) than was found in the study of the passive Lachmann test [1]. In the latter the contact force has to balance only the vertical component of the ligament force, whereas in the Isometric Quadriceps test it also has to balance the relatively larger component of the patellar tendon force.
Fig. 5 illustrates the A/P tibial displacements induced by a variety of quadriceps loads; the experimental results of Hirokawa et al. [3] are shown along with those of the incompressible and compressible cartilage models. Hirokawa et al. used a radiographic technique to measure the relative tibio±femoral displacement of cadaveric specimens, with the femur ®xed and the tibia held at a constant ¯exion angle via a ®ve degree-of-freedom universal joint. Quadriceps forces were simulated using a variety of weights attached to the quadriceps tendon. Tibial rotation was also measured. The value of dr used in the model calculations was 430 mm. The general features are similar to the experimental results: displacement increases non-linearly with load; anterior tibial displacement is observed from extension to a transitional angle around 80±90°, with a peak at approximately 30°, and posterior displacements at higher angles.
5.2. Region of zero ligament force Deformation of articular cartilage allows the ligaments to slacken, leading to a general reduction of ligament forces, and possibly a region of zero ligament force. In this region, the tightening eect of the external forces is exactly balanced by the slackening eect of cartilage deformation. The thick continuous line in Fig. 4(a) indicates the boundary of the region of no ligament force. The region contains the dotted line de®ning the ÔcriticalÕ ¯exion angle (62.5°) of zero ligament force according to the incompressible cartilage model [13]. In that model, a ligament force is not required when the restraining force, patellar tendon force and contact force pass through a single point. The additional freedom imparted by cartilage deformation enables concurrency of the forces over a range of ¯exion. 5.3. Eect of varying restraint position Zavatsky and OÕConnor [13] found that the position of zero ligament force moved to higher ¯exion angles as the restraining force moved more distally. The same eect is found when the cartilage layers are allowed to deform, except that a range of angles is now found for zero ligament force. Below dr 134 mm, the region of no ligament force vanishes; again, this agrees with the results of Zavatsky
Fig. 7. Graph of restraint position versus ¯exion angle, for a quadriceps force of 1000 N. The solid line shows where no ligament force is required for equilibrium with incompressible cartilage (after Zavatsky and OÕConnor [5], Fig. 4(a)), and the dotted lines show the corresponding boundaries of no ligament force with compressible cartilage.
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The experimentally-measured magnitude of the tibial displacement is however considerably larger than that calculated in either model, reaching a maximum of 8 mm as opposed to 2 mm in the incompressible cartilage model, or 2.5 mm when the cartilage layers are allowed to deform. Hirowaka et al. [3] note that rotation of the tibia may introduce an error into their results, and provide estimates of the resulting errors. These reduce the maximum to around 6.3 mm. This is still considerably more than that predicted by the compressible cartilage model. To approach the experimental values of the tibial displacement, very considerable and unphysical changes to the elasticity of the model articular cartilage and ligament ®bres are required. The discrepancy may be due to the greater laxity shown by cadaver than by normal knees, or to rotation of the tibia, which is ignored in this 2D model. Omission from the model of the constraint oered by the collateral ligaments is likely to result in predicted displacements being higher than in real life, and so cannot be the cause of this discrepancy. Howell [2] used an arthrometer to measure the anterior tibial translation during maximum isometric quadriceps contraction in vivo in 22 normal patients at / 15°, 30°, 45°, 60°, and 75°. The restraining force ~ Fr was applied 290 mm below the joint line. Fig. 6 shows the resulting tibial displacements. The model calculations using a quadriceps force of 2500 N now underestimate the measurements. (No actual force values were measured by Howell.) 5.5. Clinical relevance As the main features of the results are broadly similar to those reported by Zavatsky and OÕConnor [13], similar clinical lessons may be drawn. The analysis suggests that, with deformable cartilage, ligament forces do not increase steadily with increasing quadriceps force but reach relatively small asymptotic values (Fig. 3(a)). This could explain why relatively weak and slender ligaments can survive even in the presence of large muscle forces. The model predictions allow rehabilitation regimes following cruciate ligament injury or repair to be planned in such a way as to minimise ligament forces. The discovery of a region of zero ligament force in the compressible cartilage model, as opposed to a single ¯exion angle, may permit a wider range of ¯exion angles to be used in such treatments, or make slight inaccuracies in setting up exercises less critical. Very substantial quadriceps forces may be borne within this region without risking damage to the ligaments. When the cartilage is made incompressible (e.g. following total knee replacement), this eect is removed. The previous study of the Lachmann test [1] suggested that cartilage deformation made relatively little
contribution to the passive laxity of the knee. When the surfaces are replaced by the more rigid components of a prosthesis, the passive laxity could appear to be quite normal. However the signi®cant contribution of cartilage deformation to dynamic laxity, in the presence of muscle forces, implies that a replaced knee may exhibit signi®cantly less dynamic laxity than the normal. 5.6. Limitations of the present approach The present approach has a number of limitations. Only isometric quadriceps contractions are considered, whereas in reality it is common for more than one muscle group to act simultaneously, and loading is likely to be dynamic. Development of the present model to cover more complex loading regimes is possible. Various important structures within the knee are omitted from the model, particularly the collateral ligaments and the menisci. Omission of the collateral ligaments simpli®es the analysis; they are also found to be relatively sensitive to the choice of attachment areas and to contribute relatively little to the total ligament forces under A/P loading [15]. They do however have a large impact on tibial rotation (which occurs during isometric quadriceps exercise) and abduction/adduction [21]. Under isometric quadriceps loading, Harfe et al. [21] found that the medical collateral ligament (MCL) slackened in the range 15° < / < 45° and tightened in the range 90° < / < 120°, whereas the lateral collateral ligament (LCL) slackened at all ¯exion angles. This suggests that including the MCL in the model would provide an additional restraint at higher ¯exion angles, resulting in less tibial translation than is currently predicted, while the inclusion of the LCL would have little eect. The present calculations underestimate HirokawaÕs measurements of tibial displacement [3], suggesting that constraints oered by the menisci to antero-posterior movement are small. However, because the menisci spread the compressive force over larger areas of cartilage and therefore reduce the contact pressures, their omission is likely to lead to an overestimate of the extent of cartilage deformation. A purely 2D model naturally has limitations. A 2D ligament model cannot fully represent the physical cruciate ligaments, and the patterns of ®bre recruitment and extension following relative tibio±femoral displacement. The four-bar linkage omits important aspects of the kneeÕs kinematics, particularly the ÔscrewhomeÕ mechanism (the external rotation of the tibia during the ®nal stage of extension). The transverse forces within the joint are also ignored; these will be more important in the present model than when purely A/P forces were being discussed, as the quadriceps has a marked inclination in the frontal plane. A 3D kinematic
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mechanism [22,23], and 3D modelling of the cruciate ligaments [24], helps to address these diculties. The present contact force calculation could also be improved. The true 3D shape of the tibial and femoral surfaces is not taken fully into account, though the effective local spherical radius of curvature of the femoral condyles is used to modify the force calculation [1]. Articular cartilage is actually non-linear, viscoelastic and non-isotropic; it has been modelled in detail as a biphasic system comprising a solid matrix and a liquid phase by Mow et al. [25]. Over time scales of a few seconds, however, its response to load is nearly instantaneous and it is therefore appropriate to model it as an elastic solid in the current circumstances.
6. Conclusions Isometric quadriceps contractions were studied using a 2D quasistatic model of the knee incorporating deformable ligaments and articular cartilage. The model calculations presented have outcomes that are broadly comparable to those of equivalent models with incompressible cartilage and inextensible ligament ®bres. Compressible cartilage therefore is of secondary importance in knee modelling. Whereas an inextensible ligament model can give only the ratios of forces, the extensible ligament models, both with and without cartilage deformation, show displacements and ligament forces which vary non-linearly with the applied quadriceps force and tend to asymptotic values. Incorporation of compressible cartilage layers reduces the values of the ligament forces and increases the relative A/P translation between the tibia and femur. In the absence of compressible cartilage layers, consideration of the lines of action of the patellar tendon, contact, restraining and cruciate ligament forces shows whether an ACL or a PCL force is required for equilibrium at a given ¯exion angle; it is found that for each ¯exion angle there is a single restraint position at which no ligament forces are required for equilibrium. When the cartilage is made compressible, the increased ¯exibility aorded to the joint results in a range rather than a single value of restraint position leading to zero ligament force. Comparing the results to those of Zavatsky and OÕConnor [13], it is clear that knee geometry alone is sucient for most purposes for predicting whether ligament forces will be required in a given loading regime, and that adding ligament elasticity gives a reasonable approximation to the relative tibio±femoral displacements, and hence ligament forces. Cartilage deformation is a less important, but still signi®cant, modi®er of this behaviour.
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Acknowledgements RAH was supported during the course of this work by a University of Wales, Aberystwyth Postgraduate Studentship.
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