Introduction

Patellar malalignment and maltracking can increase the compressive stress between the patellar facets and trochlear ridge, resulting in anterior knee pain and ultimately cartilage injury.10 Previous studies have investigated the interactions between the internal and external forces acting at the tibiofemoral (TF) joint, for example, the relationship between the knee adduction moment and medial compartmental contact force during gait.48 Less is known about the forces transmitted by the patellofemoral (PF) joint during daily activity, especially how the knee muscles act to distribute load across the medial and lateral facets of the patella. This is an important knowledge gap given the frequency of PF pain in the general population, particularly among those who are active,35 and the causative link between joint loading and the progression of knee osteoarthritis.6

Higher loads borne by the lateral patellar facet may be expected from an examination of bone anatomy alone. Using the Wiberg patella-classification system, Tecklenburg et al.40 reported that 90% of all patellae have a larger lateral facet. In an anatomical study of 92 patellae, Baldwin and House5 similarly found that the lateral patellar facet is 25% wider than the medial facet. A larger lateral patellar facet may reflect higher loads transmitted by the lateral PF compartment during daily activities like walking in the same way that a larger medial femoral condyle and medial tibial plateau19 are indicative of a higher medial TF contact force in gait.30,37,39

Load distribution at the PF joint may be more directly explained by the interactions between the muscles and bones which meet at the knee. The angle between the lines of action of the quadriceps muscles and patellar tendon in the frontal plane, often referred to as the Q-angle, normally measures 10°-15° with the knee fully extended.38 The Q-angle induces a laterally-directed force on the patella, which is resisted primarily by the vastus medialis, the medial retinacular structures, and the lateral ridge of the trochlea.23 Thus, a normal anatomical arrangement may predispose the PF joint to higher contact forces in the lateral compartment.

Experiments on intact cadaver knees have examined how the orientation of the quadriceps muscles in the frontal plane22,34 (Q-angle), force distribution within the sub-regions of the quadriceps,1,15 and application of a hamstring force11,27 affect medial-lateral load distribution at the PF joint during simulated activity. Elias et al.11 found that at least two-thirds of the contact force acting at the PF joint was transmitted by the lateral patellar facet at all flexion angles during simulated knee extension. Ramappa et al.34 found that mean and peak contact pressures were higher on the lateral patellar facet than the medial facet during simulated squatting in the normal knee, and that the pressure on the lateral facet increased further (by up to 20%) when the Q-angle was increased to simulate PF malalignment. Goudakos et al.15 reported higher forces transmitted by the lateral PF compartment in stair climbing but not in level walking. One limitation of cadaver experiments is that the loading experienced by the knee during dynamic activities like walking is difficult to reproduce in vitro.1,15,22,27

Computational models have also been used to calculate the force acting at the PF joint in daily activities such as walking,7,18,21 but no study to our knowledge has estimated the forces transmitted by the medial and lateral compartments of the PF joint during gait. Using a musculoskeletal model of the whole body in conjunction with finite element analysis, Akbarshahi et al.2 found the force on the lateral patellar facet to be higher than that on the medial facet in stair climbing. Others have also found that the force transmitted by the lateral patellar facet is higher when only a quadriceps force is applied to extend the knee.12,20 A better understanding of load distribution at the PF joint during daily activities like walking may lead to improved surgical and physiotherapy methods to treat PF disorders, particularly if the forces transmitted to the patella can be explained in terms of the actions of the individual knee muscles and adjacent structures.

The primary goal of the present study was to describe and explain load distribution across the medial and lateral facets of the patella over one cycle of normal walking. Our specific aims were firstly, to calculate the forces acting in the medial and lateral compartments of the PF joint over one gait cycle; and secondly, to explain load distribution at the PF joint in terms of the balance of forces applied by the quadriceps muscles, patellar tendon and trochlear groove. Based on the size difference between the medial and lateral patellar facets, the orientation of the quadriceps relative to the patellar tendon in the frontal plane (Q-angle), and the results of previous in vitro experiments and computational modelling studies, we expected the force acting on the lateral patellar facet to be higher over the course of an entire gait cycle.

Materials and Methods

Participants

Six healthy young people participated in this study (3 males, 3 females; age, 26.9 ± 5.2 years; height, 171 ± 10 cm; mass, 67.7 ± 7.6 kg; BMI, 23.1 ± 1.3 kg/m2) (Table 1). Each individual gave informed consent prior to data collection. Exclusion criteria were no history of knee pain, no previous knee surgery, and no contraindications to receiving a magnetic resonance (MR) or computed-tomography (CT) scan. The Human Research Ethics Committee at the University of Melbourne granted ethics approval for this study.

Table 1 Demographic data for each of the six individuals who participated in this study
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Gait Experiments and Imaging

Each participant walked on level ground at their self-selected speed. Full-body motion, ground reaction forces and 6-degree-of-freedom (6-DOF) kinematics of the TF and PF joints were measured simultaneously for one complete gait cycle. Full-body motion was measured using nine motion capture cameras (VICON, Oxford, UK) sampling at 200 Hz while ground reaction forces were recorded from three force plates sampling at 1000 Hz. A mobile biplane X-ray imaging system (MoBiX) measured 6-DOF movements of the femur, tibia and patella of the right knee. Each participant also received a high-resolution CT scan (Siemens, Munich, Germany) of their right knee, which was subsequently used for kinematic processing of the biplane X-ray images. For each participant, the maximum radiation dosage received from both biplane X-ray imaging and the CT scan was limited to 0.18 mSv.16 An anatomical reference frame was fit to each bone as described by Gray et al.16 Full details of this experimental procedure have been reported previously.16

A fat-suppressed proton-density-weighted turbo spin-echo (0.625 × 0.625 × 3 mm) MR scan of each participant’s right knee was acquired using a 3T scanner (Siemens, Munich, Germany). Articular cartilage lining the medial and lateral femoral condyles, tibial plateaux, and patellar facets were manually segmented from the MR scan using 3D Slicer33 and subsequently processed in Geomagic Studio (3D Systems, USA) to create participant-specific geometric models.17

Measurement of Patellar Facet Size and Cartilage Thickness

For each participant, the width and area of the posterior surface of the patella were calculated directly from the participant-specific geometric models using Geomagic Studio. The medial and lateral patellar facets were divided using a plane through the peak of the patellar ridge. The width of the overall patella was found by selecting the most medial and lateral points on the patella’s posterior surface and measuring the distance between these points along the medial-lateral axis. The widths of the medial and lateral patellar facets were then determined by measuring the distances between those same points and the plane defining the patellar ridge along the medial-lateral axis. The medial, lateral and total surface areas of the patella were found by manually identifying each region on the posterior surface of the bone. Cartilage thickness was measured directly on the MR images by calculating the shortest distance between the bone-cartilage interfaces in each region of the knee.

Model Development

A three-dimensional (3D) musculoskeletal model of the body was created in OpenSim 3.38 and scaled to each participant. The model consisted of 13 rigid segments with 32 DOFs. The ipsilateral knee was comprised of a 6-DOF TF joint and a 6-DOF PF joint whereas the contralateral knee was represented as a 1-DOF translating hinge joint as described by Delp et al.8 A 6-DOF joint was used to define the 3D position and orientation of the pelvis relative to the ground. The hip, ankle and back joints were represented as 3-DOF, 2-DOF, and 3-DOF joints, respectively. Eighty muscle-tendon units actuated the skeleton, with each unit represented as a three-element muscle in series with tendon.32 A first-order model was used to relate neural excitation to muscle activation.46 Twelve Hunt-Crossley spheres (six spheres under each foot) were used to simulate foot-ground contact, each sphere applying normal, frictional, and damping forces.29

Contact forces transmitted by the medial and lateral compartments of the TF and PF joints were simulated using four elastic foundation models.21 The geometries of the contacting surfaces were based on participant-specific articular cartilage models segmented from MRI. The values of the elastic modulus, Poisson’s ratio and dissipation constant were assumed to be 5.0 MPa,21 0.4525 and 8.04 for all cartilage models. Uniform cartilage thickness was assumed within each elastic foundation model. Cartilage thickness was calculated separately for each participant and for each of the four elastic foundation models used to simulate contact at the medial and lateral compartments of the TF and PF joints.

The geometries of the knee ligaments and patellar tendon were modelled using thirteen nonlinear elastic bundles.36 The anterior cruciate ligament (ACL) and posterior cruciate ligament (PCL) were each represented by an anterior and posterior bundle. The medial collateral ligament (MCL) was represented by a superficial layer comprised of three bundles and a deep layer comprised of two bundles. The lateral collateral ligament (LCL) was represented as a single bundle. The patellar tendon (PT) was represented by medial, lateral and central bundles. Ligament stiffness and reference strain values were adapted from Shelburne et al.36 Attachment sites were manually identified on the geometric bone models segmented from each participant’s CT scan.

Validation of Model Response Against Cadaver Experiments

Model validation was accomplished by comparing model-predicted TF and PF joint biomechanics against the results of in vitro experiments reported in the literature.14,47 The TF and PF models were validated separately to eliminate the influence of mechanical coupling between these two joints.16 For example, anterior-posterior translation of the tibia relative to the femur affects the line-of-action of the patellar tendon and can therefore alter the mechanics of the PF joint. The geometry and mechanical properties of the model knee ligaments were validated by simulating the response of the TF joint model to anterior-posterior drawer and internal-external rotation. Specifically, anterior-posterior shear forces of 100 N and internal-external rotation torques of 5 Nm were applied to the model tibia with the femur held fixed from full extension to 90° of knee flexion, and the simulation results then compared against corresponding experimental data.14 Similarly, model predictions of PF joint mechanics for a simulated leg raise were validated against the results of cadaver experiments reported by Zavatsky et al.47 In these simulations, a prescribed force was applied to the rectus femoris muscle to extend the knee slowly from 90° to 5° over 1 second. Only passive forces were applied by the vasti muscles in these simulations.

Walking Simulation

A dynamic optimization problem for walking was solved by modifying the data-tracking direct collocation framework described by Lin et al.29 to include a PF joint. In this approach the states (32 generalized coordinates, 32 generalized velocities, 80 muscle-fiber lengths, 80 muscle-fiber velocities, and 80 muscle activations) and controls (80 muscle excitations) were each discretized into 70 nodes over the duration of the gait cycle. Each collocation problem was solved using the ‘fsolve’ nonlinear system solver in MATLAB (Mathworks, Natick, MA, USA) to minimize the sum of three terms: defect errors, a data-tracking constraint, and a periodic constraint29:

$$ J = J_{\text{Defects}} + J_{\text{Tracking}} + J_{\text{Periodicity}} . $$
(1)

The first term in Eq. (1) was introduced to minimize the sum of the squared errors resulting from trapezoidal approximations of the dynamic equations of motion29:

$$ J_{\text{Defects}} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{W} \cdot \left( {\frac{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}_{n + 1} - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}_{n} }}{{t_{n + 1} - t_{n} }} - \frac{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}_{n}^{'} + \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}_{n + 1}^{'} }}{2}} \right)^{2} , $$
(2)

where x is the vector of states, x′ is the vector of state derivatives, and W is the vector of weighting factors. The weighting factors assigned to the generalized velocities at the PF joint were set to zero, which allowed the quadriceps forces to change without producing large increases in the defect errors associated with the position of the patella.

The second term in Eq. (1) was introduced to minimize the squared differences between 13 model-predicted quantities and corresponding experimental data:

$$ J_{\text{Tracking}} = \mathop \sum \limits_{n = 1}^{70} \left( {\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{q}^{\text{model}} \left( n \right) - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{q}^{ \exp } \left( n \right)} \right)^{2} + \left( {\underline{GRF}^{\text{model}} \left( n \right) - \underline{GRF}^{ \exp } \left( n \right)} \right)^{2} } \right), $$
(3)

where the superscripts model and exp denote simulated and experimental data, respectively; qexp is the vector of generalized coordinates tracked during a simulation, and includes the ipsilateral knee flexion angle and the 6-DOF position and orientation of the pelvis; GRFexp is the vector of ground reaction forces and is comprised of the three components of the ground reaction force applied to the ipsilateral leg and the contralateral leg.

The third term in Eq. (1) was introduced to enforce symmetry in the states and controls that were not tracked (377 in total) by requiring these variables to be identical at the beginning and end of the walking cycle:

$$ J_{\text{Periodicity}} = \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}_{{t_{0} }} - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}_{{t_{f} }} } \right)^{2} + \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u}_{{t_{0} }} - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u}_{{t_{f} }} } \right)^{2} $$
(4)

where \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}_{{t_{0} }} \) and \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}_{{t_{f} }} \) are vectors of states at the beginning and end of the gait cycle, respectively; and \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u}_{{t_{0} }} \) and \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u}_{{t_{f} }} \) are vectors of controls at the beginning and end of the gait cycle, respectively.

An initial guess for the dynamic optimization simulation was developed through a two-step procedure. In the first step an inverse kinematics problem was solved to minimize differences between the positions of virtual markers in the model and reflective markers used in the motion capture experiments. With all ipsilateral knee DOFs except TF flexion locked, a computed muscle control (CMC) algorithm was used to determine the muscle excitations needed to produce the computed kinematic trajectories.41 Ligament, joint contact and ground reaction forces were not simulated in this step of the procedure. In the second step, with the ipsilateral TF flexion angle and hip angles prescribed and the PF joint free to move, a forward dynamics simulation was performed to calculate the position of the patella consistent with the muscle excitations determined from CMC. The resulting PF kinematics were then combined with the states and controls obtained from CMC and resampled into 70 nodes to create an initial guess for the dynamic optimization simulation.

Validation of Model Response Against Gait Experiments

For each participant the dynamic optimization solution was validated by comparing the full-body motion, ground reaction forces, muscle activation patterns, and 6-DOF TF and PF joint kinematics measured over one gait cycle against the corresponding model predictions. For each participant, the centroids of cartilage contact at the TF and PF joints were also calculated at each time point during the gait cycle using both the model-predicted and measured kinematics. Separate medial and lateral contact centroids were computed for the TF joint while only a single contact centroid was found for the PF joint. Each participant’s cartilage models were converted into point clouds with a fine density (average point spacing of 0.1 mm) and transformed into the positions dictated by the model-predicted or measured kinematics. Overlapping points between the two point-cloud sets at each joint (e.g., the patellar cartilage point cloud and the femoral cartilage point cloud for the PF joint) defined the contact volume. The centroid of the contact volume was mapped to the closest point on the bone models for visualization.

Muscle Contributions to Patellofemoral Contact Force

For each participant, the dynamic optimization solution was analyzed to determine the contributions of the individual muscles to the net contact force applied to the patella. The contributions of the quadriceps muscles (vastus lateralis, vastus medialis, vastus intermedius, and rectus femoris), patellar tendon, and patellar inertia to the PF contact force were found by calculating the compressive (anterior-posterior) and shear (superior-inferior and medial-lateral) components of each of these forces at each time point during the gait cycle (Fig. 1). The inertial force due to patellar motion was found by multiplying the mass and acceleration of the patella. All forces were expressed in the patellar reference frame. The sum of the forces applied by the quadriceps muscles, patellar tendon and patellar inertia in each direction must equal the PF contact force acting in that direction for dynamic equilibrium to hold at each time point of the gait cycle.

Figure 1
figure 1

Free-body diagram illustrating the forces acting on the model patella. The right knee of a representative participant is shown at 20% of the gait cycle with the knee flexion angle at 14°. The patellar inertia force acts at the center of mass of the patella and is not shown here. VM, vastus medialis, VI, vastus intermedius, VL, vastus lateralis, RF, rectus femoris, PT, patellar tendon, PF, patellofemoral contact force. The gray lines represent the patellar reference frame with the positive directions indicated.

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Data Analysis

For each participant, the accuracy of the model calculations relative to experimental data was assessed by computing Root Mean Square Errors (RMSEs) and Pearson’s correlation coefficients (R). RMSEs were calculated between the model-predicted and measured kinematics, ground reaction forces, and centroids of cartilage contact while R values were computed between the model-predicted and measured TF and PF kinematics. For both the stance and swing phases of the gait cycle, peak contact forces acting in the medial and lateral PF compartments were also compared using a two-tailed paired Student’s t-test. A significance level of p < 0.05 was used throughout.

Results

Patellar Facet Size and Cartilage Thickness

The average width and area of the posterior surface of the patella were 41.7 ± 4.4 mm and 1130.1 ± 184.4 mm2, respectively. Consistent with measurements reported by Baldwin and House,5 the lateral patellar facet was wider than the medial facet (26.1 ± 1.3 mm and 15.6 ± 2.5 mm, respectively), and its surface area was accordingly larger (672.1 ± 111.3 mm2 and 458.0 ± 80.2 mm2 for the lateral and medial patellar facets, respectively). Mean cartilage thicknesses measured at the medial and lateral TF compartments were 4.50 ± 0.34 mm and 4.91 ± 0.53 mm, respectively, compared to 5.11 ± 0.49 mm measured at the PF compartment (see Table S1 in Supplementary Material for cartilage thicknesses measured for all six participants).

Comparison of Model Response with Cadaver Experiments

Model predictions for the 100 N anterior-posterior drawer and 5 Nm axial torque tests were within one standard deviation of the in vitro measurements at all flexion angles except near full extension (Fig. 2). Model-predicted PF joint motion and PF contact force for the simulated leg raise were also within one standard deviation of results obtained from cadaver experiments, except for lateral patellar tilt between 0° and 30° of tibiofemoral flexion (Fig. 3a). The calculated ratio of PF contact force to quadriceps muscle force was in good agreement with experiment at all flexion angles (Fig. 3b, compare blue line and red dots).

Figure 2
figure 2

Passive behavior of the model knee (blue lines) when a 100 N anterior-posterior force (top) and a 5 Nm internal-external rotation torque (bottom) were applied from 0° (full extension) to 90° of tibiofemoral flexion. The red lines are measurements reported by Gollehon et al.14 for intact cadaver knees (n = 16). The shaded regions represent ± 1 standard deviation from the mean for both model and experiment. Anterior tibial translation and internal tibial rotation are positive.

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Figure 3
figure 3

(a) Model-predicted PF contact force and PF joint motion (blue lines) plotted as a function of the tibiofemoral flexion angle when a prescribed quadriceps (rectus femoris) force was applied to extend the knee. Results obtained for a single representative model are compared against measurements obtained from intact cadaver knees (n = 12) reported by Zavatsky et al.47 The shaded regions represent ± 1 standard deviation for the experimental data. Patellar flexion, lateral patellar rotation and lateral patellar tilt are positive. (b) Model-predicted PF contact force normalized by the applied quadriceps force and plotted as a function of the tibiofemoral flexion angle. The blue line is the ratio of PF contact force to quadriceps force corresponding to the simulated knee extension movement shown in (a). The green line is a third-order polynomial fit to all model-predicted data points obtained for the walking simulations generated for all six participants. The green shaded region shows the standard error in the polynomial line fit obtained using MATLAB’s ‘polyfit’ function. The red dots are measurements obtained from intact cadaver knees reported by Mason et al.31 for a knee extension movement.

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Comparison of Model Response with Gait Experiments

The sequence and timing of model-predicted active muscle forces were consistent with EMG measurements obtained for young healthy people walking at their self-selected speeds (Fig. 4). In both model and experiment the vastus medialis, vastus lateralis and hamstrings co-contracted during early stance while the ankle plantarflexors (soleus and gastrocnemius) were activated in late stance. Differences between model and experiment were also evident. Vastus medialis and lateralis were turned off during late stance and early swing in the model whereas moderate levels of EMG activity are observed in vivo. In addition, muscle EMG measurements indicate that the rectus femoris is activated throughout the gait cycle, however this muscle was turned off during late swing in the model.

Figure 4
figure 4

Model-predicted active muscle forces (blue lines) compared against measured EMG activity (red shaded regions) for one gait cycle. The EMG data were obtained from 8 healthy young individuals walking at their preferred speeds.9 No EMG data were available for the vastus intermedius. The blue shaded areas represent ± 1 standard deviation for the model calculations.

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Model-predicted full-body kinematics and ground reaction forces were in good agreement with measured values (Table 2). Model-predicted knee kinematics for walking were also within one standard deviation of experiment, except for external tibial rotation and joint distraction (Fig. 5). The tibia was more externally rotated and distracted in the model, with increased external tibial rotation corresponding with greater medial patellar tilt. Model-predicted tibiofemoral flexion, external tibial rotation, and lateral tibial shift showed strong correlations with measured kinematics (R > 0.7), indicating good agreement in the shapes of these curves (Table 3). RMS errors quantifying the absolute differences between model-predicted and measured TF kinematics were < 4.0 mm for all three translations, < 6.0° for flexion and < 2.0° for abduction. Internal-external tibial rotation was associated with the largest RMS error (9.5°) consistent with an excessive amount of external tibial rotation predicted in the model. Model-predicted patellar flexion, anterior-posterior patellar translation and superior-inferior patellar translation were each nearly perfectly correlated with the corresponding kinematic measurements (R ≈ 1), while RMS errors were < 6.0° for all three rotations and < 5.0 mm for medial-lateral patellar shift and anterior-posterior patellar translation.

Table 2 Root-mean square errors (RMSE) between the model-predicted and measured full-body motion and ground reaction forces calculated for one cycle of walking.
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Figure 5
figure 5

Comparison of model-predicted (blue) and measured (red) joint kinematics and cartilage contact locations obtained for one gait cycle. The first and second columns show comparisons of tibiofemoral (TF) (a) and patellofemoral (PF) (b) joint kinematics. For both model and experiment, the solid and dotted lines represent the stance and swing phases, respectively, while the shaded areas represent ±1 standard deviation from the mean. The third column compares the calculated and measured paths of cartilage contact for the TF joint (a) and PF joint (b). Locations of the cartilage contact centroids are given in mm. Tibiofemoral flexion, tibial abduction, external tibial rotation, lateral tibial shift, anterior tibial drawer and distraction are positive for the TF joint, while patellar flexion, lateral patellar rotation, lateral patellar tilt, lateral patellar shift, anterior patellar translation and superior patellar translation are positive for the PF joint.

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Table 3 Root-mean square errors (RMSE) and Pearson’s correlation coefficients (R) quantifying differences between model-predicted and measured 6-DOF tibiofemoral joint (TFJ) and patellofemoral joint (PFJ) kinematics and centroids of cartilage contact.
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The centroids of cartilage contact predicted by the model at the TF and PF joints were consistent with corresponding measurements obtained from biplane X-ray imaging and MRI (RMS errors ≤ 6.0 mm) (Fig. 5; Table 3). In the model and experiment, the centroid of PF joint contact was located on the lateral patellar facet close to the central vertical ridge throughout the gait cycle.

Tibiofemoral Joint Loading

The mean total TF contact force peaked twice during stance, the magnitudes of the first and second peaks being 3.1 times and 2.9 times body weight (BW), respectively (Fig. 6a). Mean peak forces in the medial compartment were greater than those in the lateral compartment. In early stance the mean peak contact force in the medial TF compartment was 1.6 BW compared to 1.4 BW in the lateral compartment, whereas in late stance the peak medial and lateral TF contact forces were 1.7 BW to 1.2 BW, respectively.

Figure 6
figure 6

Total contact force (blue lines), medial contact force (green lines), and lateral contact force (red lines) calculated for the tibiofemoral (TF) joint (a) and the patellofemoral (PF) joint (b) over one gait cycle. The solid lines represent mean values for all six participants and each shaded area represents ± 1 standard deviation.

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Force Distribution at the Patellofemoral Joint

The mean total PF contact force also displayed two peaks: the first occurred in early stance and its magnitude was 0.56 BW while the second peak was slightly higher (0.58 BW) and occurred during swing (Fig. 6b). The standard deviation in the magnitude of the first peak was twice as large as that associated with the second peak. The force acting on the lateral patellar facet was much higher than that on the medial facet. Lateral PF contact force peaked first during early stance (0.54 BW) and then again in early swing (0.51 BW). Both peaks in the lateral PF contact force were significantly higher than the corresponding peaks in the medial PF contact force (p = 0.012 and p = 0.001 for stance and swing, respectively). The medial PF contact force was close to zero for much of the stance phase and peaked at 0.15 BW during early swing.

The patellar tendon contributed most to the first peak in the anterior (compressive) PF contact force during early stance whereas the quadriceps contributed most to the second peak in early swing (Fig. 7 Top row, red and blue lines in column 1). The vastus medialis and lateralis contributed substantially to the first peak in the PF compressive force while the vastus intermedius and rectus femoris made relatively minor contributions (Top row, columns 2 and 3). Although rectus femoris contributed substantially to the second peak in the PF compressive force, the summed contribution of the three vasti was still greater (Top row, columns 2 and 3).

Figure 7
figure 7

Anterior-posterior (top row), superior-inferior (middle row), and medial-lateral (bottom) components of the total patellofemoral (PF) contact force calculated for normal walking. Shown are the mean values of the forces acting on the patella for all 6 participants. All forces are expressed in the patellar reference frame. The force due to the quadriceps muscles (blue line) was obtained by summing the separate forces from the vastus lateralis (purple line), vastus medialis (green line), vastus intermedius (orange line), and rectus femoris (pink line) in each direction. The heavy black line (Total Muscle) represents the sum of the forces contributed by the quadriceps muscles and patellar tendon (red line). The gray stippled area (Net) represents the net force acting on the patella and was obtained by adding the forces contributed by the quadriceps muscles (all four portions), patellar tendon, and patellar inertia (thin black line). The gray shaded area (PF Contact Force) represents the total contact force acting on the patella in each direction. The gray stippled area is equal and opposite to the gray shaded area as dynamic equilibrium is satisfied in each direction. Anterior, superior and lateral forces are positive.

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The quadriceps and patellar tendon applied nearly equal and opposite forces to the patella in the superior-inferior direction, hence the shear component of the PF contact force in this direction was relatively small (Fig. 7 Middle row, gray shaded region). Rectus femoris, vastus medialis and vastus lateralis contributed most of the PF superior shear force during stance (Middle row, columns 2 and 3).

The quadriceps and patellar tendon each applied a lateral force to the patella, and their combined effect was a laterally-directed shear force that peaked during early stance, late stance and early swing (Fig. 7 Bottom row, black line). Peak PF lateral shear force reached 0.2 BW during early stance compared to ~ 0.1 BW in late stance and early swing. The quadriceps dominated the lateral shear force applied to the patella throughout the gait cycle (Bottom row, cf. blue and black lines in column 1). The vastus lateralis and patellar tendon were the major contributors to the first peak in early stance, whereas rectus femoris contributed substantially to the second peak in late stance, and vastus lateralis and intermedius accounted for a large fraction of the third peak in early swing.

Muscle action (all four portions of the quadriceps plus the patellar tendon) dominated the PF contact force in each direction (Fig. 7, cf. heavy black lines with stippled area in column 4). The contribution of inertia due to patellar motion was relatively small, except near toe-off when patellar accelerations increased (thin black lines in column 4).

Discussion

We used a 12-DOF model of the knee-joint complex (6-DOF for each of the TF and PF joints) embedded in a full-body musculoskeletal model to calculate load distribution at the PF joint for one cycle of normal walking. The lateral patellar facet was on average larger than the medial patellar facet, and the force transmitted by the lateral facet was found to be correspondingly higher during both stance and swing. The vastus medialis, vastus lateralis and patellar tendon contributed substantially to the first peak in the PF contact force during stance, whereas all three portions of the vasti and rectus femoris were responsible for the second peak during swing. A higher force transmitted by the lateral patellar facet was explained mainly by the laterally-directed shear force applied by the quadriceps muscles, especially the vastus lateralis, vastus intermedius and rectus femoris.

Model-predicted TF and PF joint motion were validated against in vivo measurements of 6-DOF TF and PF kinematics measured for healthy gait. The time histories of tibial abduction-adduction, medial-lateral tibial shift, patellar flexion-extension, medial-lateral patellar rotation, anterior-posterior patellar translation, and superior-inferior patellar translation calculated in the model were each within one standard deviation of corresponding biplane X-ray imaging measurements reported by Gray et al.16 (Fig. 5). The model also reproduced rapid superior translation of the patella just prior to heel-strike, which Gray et al.16 noted as a characteristic feature of PF kinematics for normal walking. However, internal-external tibial rotation, anterior-posterior tibial drawer, tibial compression-distraction and medial-lateral patellar shift did not compare as favorably, possibly due to inaccuracies in the stiffness characteristics assumed for the model knee ligaments and/or other soft-tissue structures present in the native knee that were not included in the model. For example, lateral shift of the patella during early stance (Fig. 5) may have been exaggerated in the model because the actions of the medial patellofemoral ligament and medial patellar retinaculum were not considered. Lenhart et al.25 found that internal-external tibial rotation and anterior-posterior tibial drawer were sensitive to the ligament properties assumed in their model, which may also explain some of the differences between model and experiment observed in the present study. Despite these discrepancies the predicted centroids of cartilage contact were reasonably consistent with measurements obtained from biplane X-ray imaging and MRI (Fig. 5).

Overall, there was general agreement between the calculated values of TF and PF contact forces and results reported by others. The shape and magnitude of the total TF contact force were consistent with the findings of previous modelling and experimental studies (Fig. 8). Sritharan et al.39 found that the vasti and gastrocnemius muscles generated most of the TF contact force in early and late stance, respectively, in agreement with the muscle timing calculated in the present study (Fig. 4). Consistent with previous PF models, our calculations show a peak in total PF contact force during early stance (Fig. 8) when the vasti are active. However, not all previous model predictions show a peak in the total PF contact force during swing, as we found here (Fig. 8). The latter result was due to the ratio of PF contact force to quadriceps force being greater when the knee is flexed42 (Fig. 3b), as is the case during swing. Thus, even a small force developed by the quadriceps muscles during swing can be amplified and transmitted as a much greater contact force to the patella.

Figure 8
figure 8

Total tibiofemoral (TF) and patellofemoral (PF) joint contact forces (thick black lines) calculated in the present study compared to results reported in the literature.

Full size image

The magnitude of our predicted total PF joint contact force was generally lower than that found in previous studies (Fig. 8). The medial and lateral PF retinacula were not included in the model, and their combined actions may serve to increase the contact force between the patella and trochlear groove. The method used to calculate the PF contact force, which is a specific strength of the present study, likely also contributed to our predicted PF joint contact force being lower than that found previously. PF contact force is dependent on quadriceps muscle force, which also depends on the moment arm of the quadriceps and the relative movements of the femur, tibia and patella. The dynamic optimization solution derived in the present study permitted all of these parameters to be determined simultaneously, allowing for a more accurate simulation of knee extensor mechanics, as opposed to a two-step modelling approach, where quadriceps muscle force is found using a simplified knee model that does not allow for changes in PF joint kinematics.2,18 Finally, we modelled the patellar tendon as a nonlinear elastic element, as opposed to an inextensible structure,21,37 allowing for a more realistic simulation of the knee-extensor mechanism.

Our calculations indicate that most of the compressive load acting at the PF joint during walking is transmitted by the lateral patellar facet. Van Kampen and Huiskes44 found that external rotation of the tibia increases lateral patellar shift, and our model simulations show a larger amount of external rotation than that observed experimentally (Fig. 5). However, this difference between model and experiment is not likely to have had a significant effect on our estimates of load distribution at the PF joint for two reasons: firstly, most of the PF contact force was transmitted by the lateral patellar facet in all six of our participants, including those individuals who exhibited the same amount of external tibial rotation as that measured from biplane X-ray imaging (see Supplementary Material, Fig. S1); and secondly, the predicted and measured centroids of cartilage contact were both located on the lateral patellar facet close to the central vertical ridge in all six individuals (Fig. 5). Instead, a higher contact force in the lateral PF compartment is most likely explained by the relative orientations of the quadriceps muscles and patellar tendon in the frontal plane as described by the Q-angle.12 We found that the net shear force acting in the frontal plane was directed laterally throughout the gait cycle and that the quadriceps muscles, especially the vastus lateralis, vastus intermedius and rectus femoris, contributed a large fraction of this laterally-directed force (Fig. 7). These findings are consistent with those reported by others: Chen et al.7 found the net shear force in the frontal plane to be in the lateral direction for an entire cycle of walking, while Lenhart et al.26 showed that strengthening the vastus lateralis shifted the centroid of PF joint pressure laterally and resulted in a higher medially-directed patellar contact force in running. Interestingly, in vivo kinematic measurements obtained for a static forward lunge indicate that the PF contact point may move medially when the knee is flexed beyond 90°.24 Future studies should investigate force distribution at the PF joint during functional activities that require greater knee flexion than level walking, such as stair climbing and squatting.

Our analysis indicates that the vastus medialis, vastus lateralis and patellar tendon are the major contributors to the first peak in PF contact force, and that all four portions of the quadriceps contribute to the second peak (Fig. 7). However, the potential for the hamstring muscles to contribute to the PF contact force should not be underestimated. In a series of experiments performed on intact cadaver knees, Li et al.27 found that the application of hamstring force induced a posterior translation of the tibia which resulted in increased pressure between the patella and the trochlear groove. We found that the PF compressive force and posterior (shear) component of the resultant hamstring force both peaked in synchrony, first during early stance and then again in late stance and early swing. Furthermore, the PF compressive force was positively correlated with the posterior hamstring shear force (R = 0.61) (see Supplementary Material, Fig. S2). Together, these results suggest that the PF compressive force may also be modulated by hamstring muscle action during walking.

There are limitations of our study which ought to be considered when interpreting the results. The medial patellofemoral ligament, which is a stabilizer against lateral patellar shift when the knee is extended,3 was not included in the model. Nonetheless, the model-predicted centroid of PF joint contact compared favorably with experiment (Fig. 5). A second potential limitation was the way in which the quadriceps muscles were assumed to insert onto the patella. The quadriceps have complicated patellar insertion dynamics, with the relative anterior-posterior alignment of the muscles inserting on the quadriceps tendon and the volumes of the oblique portions of the vastus lateralis and vastus medialis varying considerably between individuals.45 These variations were not accounted for here. Third, the menisci, which play an important role in both the stability and load transmission of the TF joint,21 were not represented in the model, which may affect load-sharing between the muscles, ligaments and articular surfaces of the bones. Fourth, the upper body was represented by a single trunk segment and arm swing was not modelled, though the impact of arm swing on sagittal plane gait kinematics and kinetics appears to be small.43 Finally, a simple 1-DOF translating hinge was used to model the contralateral knee to minimize computational time during a simulated gait cycle.

In summary, this study used a 3-body, 12-DOF knee model to simulate both TF and PF joint biomechanics for one cycle of normal walking, and the model predictions were validated through a direct comparison with accurate knee kinematic measurements obtained from mobile biplane X-ray imaging. A large fraction of the PF compressive force was transmitted by the lateral patellar facet and is explained by the lateral pull of the quadriceps muscles, specifically, the vastus lateralis, vastus intermedius and rectus femoris.