Computational biomechanics of human knee joint in maximum voluntary isometric extension with focus on the role of joint center positioning

Abstract

Maximum voluntary isometric contraction (MVIC) extension assesses knee quadriceps function, strength, neuromuscular recovery and rehabilitation programs. We use a musculoskeletal (MS) model of the lower limb that incorporates a finite element (FE) of a cadaver knee. Muscle/ligament/contact forces, tissue stresses/strains and passive reaction moments are computed while simulating MVIC extension in seated posture at three knee flexion angles (KFA) (30°, 60°, 90°). Three input parameters of MVIC extension moment, lever arm of the posteriorly-directed external force, and coactivation in knee flexors were each varied at four levels using a Taguchi orthogonal design. Sensitivity of output parameters to these input variables were estimated. The location of the joint center where moment equations are verified was also varied. Results demonstrated a significant increase in quadriceps, patellar tendon, and patellofemoral contact forces with KFA (p < 0.001) and MVIC moment (p < 0.001). Greater lever arm of the external force increased ACL forces. In contrast to OpenSim simulation of the same MVIC, changes in the joint center location affected only the passive reaction moments that varied to maintain equilibrium while the computed muscle forces and internal ligament/contact loads remained unchanged. Findings highlight inherent limitations in MS models that idealize the entire knee as a joint located at its center of rotation.

Introduction

As the most frequently injured joint in the human body¹, the knee is subject to large mechanical loads and relative motions resulting in complex internal interactions among its components. Activation in muscles crossing the knee controls and stabilizes the joint in normal and injured conditions during daily living^2,3. The quadriceps femoris group plays a central role in many knee pathologies, such as osteoarthritis, anterior cruciate ligament (ACL) injury and knee anterior pain, and as such is a major focus in diagnosis and rehabilitation aimed at the evaluation and restoration of the knee extensor mechanism⁴. While recent injury prevention strategies focus toward a more holistic approach and training paradigms⁵, maximum voluntary contraction extension is an open kinetic chain manoeuvre commonly carried out to assess quads muscle strength, guide therapy, and evaluate recovery outcome.

Previous studies on the maximum isometric knee contraction (MVIC) in extension show that the torque peaks at ~ 45°–60° knee flexion angle (KFA); healthy adults typically generate ~ 2.9–3.5 Nm/kg at ~ 60° that drops to ~ 1.3–2.2 Nm/kg at near full extension⁶. This trend is attributed mainly to the changes in muscle lever-arms⁶. Knee strength is dependent on sex, age, physical fitness and body mass; males and younger individuals exhibit higher torques⁶. Most MVIC extension tests are performed in a seated posture (hip at ~ 90°), using dynamometers with the lower leg resisting a posterior force applied near the ankle joint⁷. Under these conditions, quadriceps, especially vastus lateralis and vastus medialis, are highly activated reaching 80–90% of their maximum EMG⁸. Simultaneously, coactivation in hamstrings increases with joint angle and contraction intensity^9,10. The MVIC extension remains, however, contentious and of limited value in patients following surgery or with knee pain, ACL injury and advanced osteoarthritis that commonly demonstrate reduced quadriceps and elevated hamstring activities which together limit net extension torque^11,12.

The MVIC testing is simple, reproducible, and sensitive to changes in muscle strength, making it thus a standard assessment tool in clinical, sports and diagnostic environments. Comprehensive analysis of the knee joint in MVIC to estimate muscle forces and internal load distribution requires, however, realistic musculoskeletal (MS) model studies^13–15. Coupled MS and finite element (MS-FE) models allow estimation of muscle forces, joint loads, and detailed tissue stresses-strains. MVIC simulations serve also as an effective tool to verify the fidelity of MS models^15,16. While earlier experimental studies have extensively characterized knee extension MVIC strength and muscle EMG activation, related computational modeling of knee joint mechanics remains scarce. Mesfar and Shirazi-Adl (2008) simulated open-kinetic-chain knee extensions under low resistant force of 30 N, and reported that quadriceps force, patellar tendon (PT) tension, and tibiofemoral (TF)-patellofemoral (PF) contact forces-pressures peaked at 90° knee flexion, while the ACL force peaked near full extension¹⁴. Resistant forces of much larger magnitude (~ 450 N) are however reported in extension MVIC efforts. In another study, Hume et al. (2018) simulated isometric knee extensions at various joint angles using two MS-FE models without menisci: one deformable with ligamentous laxity and articular contact, and another with a simplified kinematically prescribed joint non-deformable¹³. The joint maximum moment was then estimated while assuming simultaneous peak activation in all primary muscles and 5% coactivation in all antagonist muscles.

While global inverse dynamics can be carried out about any point, an important concern in all MS model studies with simplified joint simulations is the likely effects on results when positioning the reference point (or joint center) about which the internal moment equations of motion are maintained. Earlier studies carried out sensitivity analyses on the anterior-posterior^15,17,18 and cranial-caudal¹⁹ positioning of spherical joint centers and clearly demonstrated substantial effects of joint center location on estimated muscle forces and internal loads. To compute reliable results in MS models when incorporating an idealized joint, the reference point should coincide with the joint ‘center-of-reaction’, i.e., the point about which the sum of moments of external loads and muscle forces approaches zero suggesting thus no residual reaction (passive) moments. The location of this point, however, is neither known a priori nor fixed under applied loads and motions. This point is also distinct from the joint center of rotation at which the reaction moment is commonly and erroneously neglected^16,18.

The objective of this study is hence to parametrically investigate open-kinetic chain MVIC knee extension at 30°, 60°, and 90° of flexion using a coupled MS-FE model of a female knee joint^15,20. We aim to (1) compute muscle forces and internal load distribution during a seated extension MVIC as the torque, lever arm of the resistant force and coactivation in flexor muscles vary each at 4 different levels, and (2) explore the effect of changes in the joint centre location on foregoing predictions. In doing so, we will also compare our predictions with those of the scaled OpenSim (Gait 2392) model. It is hypothesized that (1) muscle forces and internal stresses peak at 60-90^o knee flexion angles and (2) in accordance with our earlier works²¹ and while maintaining equilibrium, muscle forces and internal loads but not the reaction moments remain identical in our MS-FE model (where the knee joint is simulated as a deformable body) though not in the OpenSim model (where an idealized joint is fixed at a joint center), irrespective of the changes in the location of the joint reference point.”

Methods

Coupled MS-FE model

The knee extension MVIC was simulated using an existing, subject-specific MS-FE model of the lower limb reconstructed from a female cadaver specimen with no visual abnormalities in cartilage layers, menisci, ligaments and bony structures (27 years of age)²² and validated through numerous investigations^{15,20,23–27} (Fig. 1). The bony structures (femur, tibia, and patella) were modeled as distinct rigid bodies. Twelve muscles crossing the knee were incorporated with origins, insertions, and wrapping path geometries based on the OpenSim Gait2392 model^28,29, scaled to match our female subject (61.9 kg body mass, 169 cm body height). Muscle paths included bone wrapping and via-points for realistic anatomical trajectories at different knee flexion angles of 30^o, 60^o, and 90^o. The muscle PCSAs were taken for our female subject from regression equations²⁹.

Fig. 1.

Schematic of the simulated seated maximum voluntary isometric contraction (MVIC) knee extension exercise using: (A) OpenSim Gait 2392 model, and (B) MS-FE model used in this study.

Ligaments were represented as nonlinear tension-only springs with initial pre-strains³⁰. Articular cartilage was modeled either as a depth-dependent fibril-reinforced composite (refined model, ~ 1 mm mesh size) or as a homogeneous compressible isotropic material (coarse model, ~ 2 mm mesh size). In a recent study using both models in gait, negligible differences between the predictions of these models were found³¹; Pearson’s correlation r > 0.98 for ACL and muscle forces as well as normalized root mean square of differences at ~ 5% for contact forces^27,31,32. Menisci were modeled as fiber-reinforced composites with nonlinear collagen fiber orientation in both circumferential and radial directions. All parametric (Taguchi method) sensitivity analyses employed the coarse model to reduce computational demands and facilitate convergence. In our earlier mesh verifications of the same knee model, global output measures (muscle, ligament, and contact forces) were highly consistent between refined and coarse models (r > 0.98 with ~ 5% nRMS differences)³². A limited number of cases were also analyzed with the refined model to more accurately compute contact pressure distributions.

To simulate the seated testing posture of subjects during MVIC, the entire lower extremity in the anatomical upright position was initially rotated by 20° anteriorly at the pelvis followed by the flexion of 70° at the femur to achieve an upright trunk-hip flexion of 90°. The pelvis and femur were subsequently fixed to replicate seated subjects. The lower leg and foot were then rotated about the femoral mid-epicondylar axis to target knee flexion angles (KFA) of 30°, 60°, or 90°, with the remaining five degrees of freedom of the tibia and 6 degrees of freedom of the patella left free for an unconstrained passive flexion. Following the application of ligament pre-strains, all 3 tibial rotations were locked, while translational degrees of freedom remained free. A resistant force was applied at the distal tibia normal to its already rotated position, generating the experimentally reported net MVIC extension moment about the femoral epicondylar axis (Fig. 1). Gravitational forces of the foot (7.98 N) and shank (29.78 N) were also considered at their respective mass centers (Fig. 1).

Muscle force estimation

To resolve the muscle redundancy, a static optimization framework was employed while maintaining the equilibrium equations. The cost function minimized the sum of cubed muscle stresses (force divided by physiological cross-sectional area, PCSA)³³:

1

subject to knee joint flex-ext moment Eq. 

2

and inequality equations on muscle forces:

3

where is the muscle force, F_pi is the passive force, PCSA_i is the physiological cross-sectional area, σ_max is the maximum muscle stress, and is the muscle moment arm in the ith muscle. Given the minimal length changes in muscles (< 10%), passive force contributions were overlooked. Nevertheless, and in general, the unknown muscle forces F_i could represent the total passive and active force components. The coupled FE and optimization analyses were implemented using ABAQUS v6.12 (Dassault Systems, Providence, RI, USA) and MATLAB R2013a (The MathWorks, Natick, MA) with the Optimization Toolbox (Fmincon).

Parametric analysis

To investigate the sensitivity of output measures of interest (muscle forces, ligament forces, joint contact stresses/forces/areas, and passive joint moments) to key biomechanical factors, a parametric analysis was performed. Three primary independent input variables were considered; magnitude of the MVIC extension moment, lever arm of the resistant force applied onto the distal tibia near the ankle, and coactivation flexion moment of all hamstrings and gastrocnemii muscles (muscle j from 1 to n):

Knee coactivation flexion moment in hamstrings and gastrocnemii (Eq. 4):

4

These 3 parameters were taken each at 4 levels within respective physiologically relevant ranges reported in the literature (Table 1). This was repeated at three distinct KFAs of 30°, 60°, and 90°.

Table 1.

Factors and their corresponding levels used in the parametric analysis (mean moments were 85.4 Nm at 30^o knee angle, 136.8 Nm at 60^o and 142.4 Nm at 90^o).

Parameter/level	1	2	3	4
Coactivity moment (% MVIC)	0%	3%	6%	9%
Lever arm (mm)	260	310	360	410
MVIC moment (Nm)	Mean -	Mean -	Mean +	Mean +

For the MVIC extension moment, reported values vary with sex, age and testing positions^12,34–37. The data collected in supine or prone postures or from mixed-sex cohorts are less suitable for our female-specific seated model. Silva et al. (2003), however, reported MVIC knee extension torques for a predominantly female cohort in the seated posture. To account for variations across individuals, the MVIC extension moments of healthy subjects reported in Silva et al. (2003)¹⁰ at different knee angles were normalized to their mean body mass (79.8 kg) and applied in our model study. Consequently, the mean MVIC extension moments at 30°, 60°, and 90° of knee flexion were selected as 1.38, 2.21, and 2.30 Nm/kg, respectively¹⁰. These values were varied at four levels (mean ± 0.5 SD and mean ± SD/6) to encompass some physiological variability while avoiding convergence difficulties in our nonlinear MS-FE model studies when subject to larger moments.

The location of the resistant force on the tibia was also varied across four levels to examine the influence of the load location on the computed output measures. Specifically, lever arms of 260 mm, 310 mm, 360 mm, and 410 mm (measured from the mid-epicondylar axis of femur) were prescribed as fixed absolute distances in the subject-specific model to represent differences in external loading. The same locations were replicated in OpenSim so that the moment arm about the epicondylar axis matched.

The muscle coactivation of hamstrings and gastrocnemii muscles as knee flexors were also varied as another input parameter. Some levels of cocontraction in knee flexors have been recorded during MVIC knee extension efforts^12,38,39. Based on these findings, four levels (0%, 3%, 6%, and 9%) of coactivation moment (Mc in Eq. 4) were considered as a proportion of the MVIC extension moment. This coactivation equation was imposed as an additional constraint equation when solving for unknown muscle forces¹⁵.

To efficiently explore the interaction between these three independent input factors while reducing computational cost, a Taguchi L16 orthogonal array design was employed⁴⁰. This design yielded 16 unique simulation scenarios at each of the three KFAs, resulting in a total of 48 nonlinear coupled MS-FE simulations. Key biomechanical outputs such as muscle forces, ligament forces, TF and PF contact forces/stresses/areas, and passive reaction moments were extracted and analyzed across these scenarios. Additional analyses were carried out in few cases under pure MVIC moment instead of the resistant force to examine the effect of an idealized loading simulation on results.

In addition to the above parametric study, separate analyses were performed to estimate the passive reaction moments of the knee joint that likely vary^15,17,18 with the location of the joint center (reference point). In MS modeling, the location of the joint center is often assumed rather arbitrarily close to the joint center of rotation, particularly when idealized frictionless joints are employed. However, altering this reference point could affect calculated reaction moments when larger joint reaction (compression) force exists. To evaluate this effect, reaction joint moments were recomputed at different reference points along the joint posterior–anterior and medial-lateral axes placed at 10 mm intervals along the joint posterior-anterior and medial-lateral axes with the origin at the mid-epicondylar point (MEP), defined as the midpoint of the trans-epicondylar axis connecting the medial and lateral femoral epicondyles on the subject-specific geometry. We investigate if and how the changes in the joint center position influence estimated muscle forces, internal loads and passive joint moments.

OpenSim analyses

For comparisons, some analyses were repeated using the OpenSim Gait2392 MS model²⁸. The model was scaled with the OpenSim’s standard Scale Tool to reflect the anthropometric dimensions (169 cm, 61.9 kg) of the subject used in our MS-FE simulations (geometric segment scaling). Muscle-tendon parameters (e.g., optimal fiber length and tendon slack length) were updated using OpenSim’s default scaling rules, and no additional manual tuning was performed beyond matching muscle maximum isometric forces across platforms (MS-FE and OpenSim models). Simulations were performed at KFAs of 30°, 60°, and 90° with external forces applied at the identical anatomical location near the ankle using OpenSim’s External Loads tool. Static optimization was used to resolve muscle redundancy, minimizing the sum of cubed muscle activations. The optimization upper bound was taken as × PCSA, consistent with our MS-FE model; Thus, both platforms used the same objective function minimizing the sum of cubed muscle activations. To evaluate the influence of the knee joint reference point (center) location on predicted muscle forces, the tibia was first rotated about the epicondylar axis to the target KFA followed by shifting the tibial coordinate frame in the anteroposterior or mediolateral direction. The origin of this offset frame, aligned with the mid-epicondylar axis of the femur (MEP), was displaced by ± 10 mm, ± 20 mm, and ± 30 mm along the axis, resulting in additional static optimization simulations per each KFA.

Statistical analysis

An analysis of variance (ANOVA) was carried out to evaluate the main effects and interactions of the four independent variables, KFA, MVIC extension moment, lever arm of the resistant force, and coactivation moment level, on dependent biomechanical outcomes (muscle forces, ligament loading, contact mechanics, passive joint moments). Post hoc comparisons were conducted using Tukey’s HSD test where significance was detected (p < 0.05).

Results

Quadriceps generated very large forces that markedly increased with KFA (Fig. 2A); the resultant force jumped from 2227.9 ± 267.3 N at 30°, to 4764.4 ± 611.3 N at 60°, and peaked at 6324 ± 1034.3 N at 90° (SD values represent variability across the Taguchi parametric cases). The hamstrings carried much smaller forces of 39.3 ± 31 N at 30°, 55.7 ± 43.5 N at 60°, and 74.3 ± 63.4 N at 90° (p > 0.01) with the medial hamstrings (ST and SM) most solicited (mean: 62%, range: 0-77%). The gastrocnemii (GL and GM combined) forces increased with KFA from 70.9 ± 56.3 N at 30° to 118.3 ± 92.9 N at 60° and 194.5 ± 161.7 N at 90° (p > 0.01). Sensitivity analysis revealed that forces in quadriceps significantly increased with MVIC moment (p < 0.001) and coactivity level (p < 0.001) but not with the lever arm (p = 0.543) (Fig. 2B). Forces in flexors also increased with the coactivation moment (p < 0.01).

Fig. 2.

(A) Estimated muscle forces (mean ± SD) at 30°, 60°, and 90° KFAs during MVIC knee extension. (B) Sensitivity of estimated muscle forces to variations in coactivation level (left), external load’s lever arm (middle), and MVIC extension moment (right). Muscle abbreviations: BF_lh – biceps femoris long head; GA – gracilis; RF – rectus femoris; SR – sartorius; SM – semimembranosus; ST – semitendinosus; BF_sh – biceps femoris short head; VIM – vastus intermedius; VL – vastus lateralis; VMO – vastus medialis; GL – lateral gastrocnemius; GM – medial gastrocnemius.

The posterior cruciate ligament (PCL) was slightly loaded only at 90° of knee flexion (17.7 ± 32.4 N; range: 0–108.9 N) with the major portion (~ 82%) carried by its anterolateral bundle (Fig. 3A). In contrast, ACL was almost slack at 90° (< 1 N) but tense at lower flexion angles (260.4 ± 172 N (range: 0–468.1 N) at 60°, and 292.7 ± 82.7 N (range: 127.5–412.7 N) at 30° (both anteromedial and posterolateral bundles were at < 1 N at 90°), with the posterolateral bundle resisting a larger portion at 30° and 60°. Sensitivity analysis demonstrated significant increases in ACL forces with both MVIC moment (p < 0.01) and lever arm (p < 0.01), while variations in flexors coactivation had no significant effect on ACL loading (p = 0.295) (Fig. 3B). The patellar tendon (PT) forces increased with the knee flexion angle from 2161.4 ± 246.1 N (range: 1732.9–2533.8 N) at 30° to 4456.5 ± 801.4 N (range: 3277–6023.2 N) at 90° (p < 0.001). The sensitivity analysis demonstrated that PT forces increased with the MVIC extension moment and coactivity level (p < 0.001) but not lever arm (p = 0.187) (Fig. 3C). The ratio of PT force over quadriceps force dropped with KFA; from 0.958 ± 0.002 at 30^o to 0.703 ± 0.021 (p < 0.001) at 90^o (Fig. 3D) while the sensitivity analysis showed that the PT/Q ratio did not significantly change with MVIC extension moment, lever arm and coactivity (p > 0.07).

Fig. 3.

(A) Forces (mean ± SD) in PCL and ACL (N) during MVIC extension exercise at different KFAs. PCL_al and PCL_pm are, respectively, anterolateral and posteromedial bundles of PCL. ACL_am and ACL_pl are, respectively, anteromedial and posterolateral bundles of ACL. (B) Sensitivity of the ACL force. (C) Sensitivity of the PT force to variations in coactivation level (left), external load’s lever arm (middle), and MVIC extension moment (right). (D) The ratio of computed PT over quadriceps forces during MVIC extension exercise at different KFAs compared to those reported in the literature.

The TF contact force initially increased with KFA; from 1532.9 ± 197.7 N (2.5 ± 0.3 BW) at 30°, to 2377.1 ± 297.7 N (3.9 ± 0.5 BW) at 60° (p < 0.001), but then dropped to 2121.7 ± 408.6 N (3.5 ± 0.7 BW) at 90^o (p > 0.01) (Fig. 4A). The medial compartment carried a larger share of TF contact load only at 60° KFA (~ 56%). Total PF contact force significantly increased with KFA, from 1991.2 ± 246.7 N at 30°, to 3419.6 ± 981.9 N at 60°, and 5608.4 ± 981.9 N at 90° (p < 0.01) (Fig. 4A). Sensitivity analysis demonstrated that PF contact force significantly increased with the MVIC moment (p < 0.001) but remained largely unaffected by variations in coactivation or lever arm (p > 0.9) (Fig. 4B). The PF contact area increased with KFA from 336 ± 28 at 30° to 568 ± 39 at 60° and 641 ± 56 at 90°. The PF contact stresses followed the same trends; peak values increased from ~ 11 MPa at 30° to 13 MPa at 60° and 21 MPa at 90° while contact areas shifted in the proximal direction (Fig. 5). The ratio of the PF contact force to the quadriceps force did not follow a linear trend and decreased from 0.89 ± 0.001 at 30° to 0.72 ± 0.03 at 60° but then increased back to 0.89 ± 0.03 at 90°.

Fig. 4.

(A) Tibiofemoral (TF) and patellofemoral (PF) contact forces (mean ± SD) during MVIC extension at different KFAs. TF forces are also partitioned between the medial and lateral plateaus. (B) Sensitivity of PF contact forces to variations in coactivation level (left), external load’s lever arm (middle), and MVIC extension moment (right).

Fig. 5.

Contact pressure (MPa) distribution on the PF articular cartilage at different KFAs angles for the reference case of MVIC knee extension (mean extension moment without co-activation constraints and with the resistance force applied 390 mm distal to the knee) (D: distal, P: proximal, M: medial and L: lateral).

The passive reaction flexion/extension (Flex-Ext) moment of the knee joint (calculated as the differential between the moments of all external forces and all muscle forces about a changing reference point located at MEP level) varied linearly at all 3 KFAs with the anterior-posterior location of the joint reference point (Fig. 6A). The passive moment disappeared with ~ 23 mm, 27 mm and 39 mm anterior shifts relative to MEP at 30°, 60° and 90° of KFA and changed direction thereafter to become positive (extension) (Fig. 6A). The sensitivity analysis (Fig. 6B) showed a significant increase in the passive Flex-Ext moment with increasing MVIC moment (p < 0.001) while the lever arm and coactivity did not have significant impacts (p > 0.07).

Fig. 6.

(A) The internal passive (reaction) flex-ext moment across different KFAs, calculated at varying anterior–posterior positions of the joint center for reference case of MVIC knee extension (mean extension moment without co-activation constraints and with the resistance force applied 390 mm distal to the knee). (B) Sensitivity of the passive FE moment of the knee joint calculated about the mid epicondylar point to variations in coactivation level (left), external load’s lever arm (middle), and MVIC extension moment (right).

Comparison of predicted muscle forces in our MS-FE model with those in OpenSim revealed large discrepancies in general that varied with the joint center location (Fig. 7). At 60° KFA and the default mid-epicondylar point (MEP) as the joint center, the total quadriceps force in OpenSim reached ~ 3577 N, much smaller that 4717 N in our MS-FE model. However, as the joint reference point shifted ~ 20 mm anteriorly, a sharp increase in quadriceps force (up to ~ 4723 N) was found in OpenSim with almost no changes in our MS-FE model (~ 1.5%). Conversely, a 30 mm posterior shift in the joint center decreased quadriceps force in OpenSim by ~ 28%. Estimated muscle forces and as a result internal loads in OpenSim are hence very sensitive to the assumed location of the joint center while the MS-FE model results remain independent of such changes (Fig. 7).

Fig. 7.

Comparison of estimated quadriceps forces for reference case of MVIC knee extension (mean extension moment without co-activation constraints and with the resistance force applied 390 mm distal to the knee) from the OpenSim Gait 2392 model and MS-FE model at varying anterior–posterior positions of the joint center. Results of MS-FE model do not change with the joint center position.

Discussion

In this study we used a coupled MS-FE model of the lower extremity and simulated MVIC extension tests in seated postures while varying the resistant force (or equivalently the torque), coactivation in knee flexor muscles and lever arm of resistant force on the distal shin close to the ankle joint. The location of the knee joint center was also altered and results compared to those of a scaled OpenSim model. Our predictions confirmed the hypotheses that (1) with the exception of ACL, muscle forces and internal loads increased with KFA and that (2) in clear contrast to OpenSim results and as expected, all estimations except the knee passive (reaction) moments remained unchanged as the joint center altered. In OpenSim, the sensitivity of predicted muscle forces to joint center positioning is a direct consequence of the idealized rigid-body joint in which the joint center is fixed with no passive reaction moment to maintain moment equilibrium.

Quadriceps force increased from ~ 2.2 kN at 30° to 6.3 kN at 90° (Fig. 2), a trend consistent with earlier predictions of increasing quadriceps and patellar tendon forces with KFA under a resistant force⁶. This peak force (~ 6.3 kN) remains within the range reported for maximal effort in a subject weighing ~ 62 kg (~ 10 BW)⁴¹. In all simulated knee positions, the vastus lateralis (VL) was the most activated of the quadriceps, followed by the vastus medialis (VMO), rectus femoris (RF) and vastus intermedius (VIM). The estimated activations at 90° flexion (~ 94% in VL, 70% in VM and 62% in RF) aligns with previous measurements showing KFA-dependent increases in quadriceps activation⁴² and dominant contribution in VL compared to VMO and RF^36,43. The coactivity in flexor muscles also followed similar trend with KFA; lower coactivation at 30° and higher at 90° (hamstrings ~ 74 N; gastrocnemius ~ 195 N). The increasing trends in coactivity with MVIC moment and KFA are corroborated by others⁴⁴.

The patellar tendon (PT) force increased with the MVIC extension moment and quadriceps force but at different proportions depending on KFA; the PT/quadriceps force ratio decreased with KFA (Fig. 3D), dropping from around unity at 30° to ~ 0.7 at 90°. This implies the crucial role of the PF articulation and contact force that grows with KFA as reported elsewhere^45–48. The substantial increase in PT, quadriceps and resulting PF contact forces point to a higher risk of anterior knee pain.

The total TF contact force in this open kinetic chain strength test was lowest at 30° (2.5 ± 0.3 BW), peaked at 60° (3.9 ± 0.5 BW) and dropped back at 90^o (3.5 ± 0.7 BW). This non-monotonic pattern, in agreement with earlier measurements⁴⁹, is due mainly to the changes in the orientations of quadriceps and PT relative to the tibial plateau during flexion. The medial versus lateral tibial compartmental loading does not follow the patterns estimated in the stance phases of level walking and stair ascent^20,23 where the medial side carries larger contact forces. In contrast, the PF contact force monotonically rose with KFA, roughly three-fold from ~ 2.0 kN at 30° to ~ 5.6 kN at 90° (~ 9 BW). Results align with prior estimations under open-kinetic chain extension with much smaller load⁵⁰ as well as running and drop landing⁵¹. The PF contact area also increased with KFA (≈ 336 mm² at 30° to 641 mm² at 90°), in general agreement with earlier simulations¹⁴ and measurements^51–54. The PF Contact stresses rose sharply (peak of ~ 11 MPa at 30° to 21 MPa at 90°, Fig. 5), far exceeding our previously reported maximums of 4.83 MPa at 20% stance in stair ascent or 2.46 MPa at 25% of level walking²⁰. Earlier studies reported peak PF contact stresses of ~ 7–13 MPa (contact area ~ 600 mm²) under a PF force of ~ 5 BW⁵⁵ and ~ 13 MPa (contact area of ~ 550 mm²) under PF forces of ~ 10–11 BW⁵¹.

A key finding of this study is the relative sensitivity of our coupled MS-FE model results versus OpenSim estimations when shifting the joint center in the anterior-posterior and medial-lateral directions. In our MS-FE model, repositioning the reference point (joint center), when maintaining moment equilibrium equations, altered only the passive (reaction) moment evaluated as the differential between moments of all external forces and those of all muscle forces crossing the knee (Fig. 8). Muscle, contact and ligament forces remained almost unchanged irrespective of the joint center considered. This is quite expected as in any structure under external loads, the dynamic equilibrium of loads should be maintained irrespective of the location about which they are computed.

Fig. 8.

Schematic of the variation of the internal passive (reaction) moment at different joint centers calculated as the differential between the moments of muscle forces and external loads. Results at MEP position (mid epicondylar axis) and its anterior and posterior shifts are shown.

In clear contrast, all OpenSim results substantially altered when changing the joint default center set at the mid-epicondylar (MEP) point, especially in the anterior-posterior directions (Figs. 7). For example, at 60° KFA, the OpenSim model predicted lower total quadriceps force (~ 3577 N vs. 4717 N in the MS-FE model). The difference, however, diminished as the joint center shifted anteriorly so that at 20 mm anterior shift in the default joint center, OpenSim’s quadriceps force jumped to ~ 4723 N (an increase of ~ 32%) matching the MS-FE value. Conversely, a 30 mm posterior shift in the default joint center substantially reduced OpenSim quadriceps forces by ~ 28%, while the MS-FE result remained virtually unchanged during these shifts (changes < ± 2%) (Fig. 7). In view of the little changes in the flex-ext moment of external forces as the joint center shifts in the anterior-posterior direction, the reason for foregoing alterations lies mainly in the consideration of no passive contribution in OpenSim irrespective of where the joint center (default or not) is located. It is to be emphasized that even at the joint instantaneous center of rotation, such assumption yields erroneous estimations as the knee joint is not a hinge joint especially in presence of large compression forces. The differences between predictions of these two models (Fig. 7) are expected to directly depend on the magnitude of joint compression and are hence expected to increase at heavier physical activities. These results corroborate earlier findings on the effect of joint center positioning^{15,17–19,21} and underlines the significance of the realistic simulation of the knee joint itself. Small differences in the definition of the joint center in conventional MS models (often taken as the joint center of rotation) was also reported to cause large errors in predicted muscle forces and internal loads^18,56.

In addition, another important but related observation is that the internal knee joint passive (reaction) moment depends directly on the location of the joint center and alters as it shifts (especially in the anterior-posterior direction in our MVIC extension simulations) (Fig. 6). In other words, to obtain identical muscle, contact and ligament forces when the joint center is shifted, the internal knee joint reaction moment should also be adjusted accordingly (see Fig. 6) as clearly observed at our earlier trunk biomechanical model studies as well²¹. In other words, the internal passive (reaction) moment-rotation properties of the knee joint should be modified as its position alters from a location to another. Essentially, a detailed finite element joint model automatically takes care of this issue irrespective of the location of the reference point. Had the joint center been placed on the joint center of reaction, the MS models with idealized joints (as OpenSim here) would yield similar results as our coupled MS-FE models. The location of the reaction center, however, is not known a priori and alters under loading. Thus, the MS-FE simulation can realistically capture the “active-passive” interplay; the passive reaction moment automatically adjusts to changes in the location of the joint center where moment equilibrium equations are maintained.

The posteriorly directed resistant force (applied at distal shin) plays a critical role in the joint mechanics in general and in ACL-PCL forces, in particular. This was investigated by applying an idealized pure MVIC moment with no resistant force in 3 cases under the mean extension moment, 390 mm lever arm and without co-activation. Comparison of results demonstrated that ACL forces markedly increased in the absence of the posteriorly-directed shear force, rising from 117.7 N, 43.5 N, and 0 N at 30°, 60°, and 90° flexion to 439.2 N, 431.1 N, and 65.4 N, respectively. Foregoing substantial increases are due to the absence of the posteriorly-directed external forces in pure moment simulations that tend to unload ACL forces by applying tibial posterior drawer forces. In contrast, in the absence of the resistant force, PCL forces totally disappeared. Joint contact forces were also affected by this idealization of joint loading in MVIC. This corroborates earlier studies on the effect of joint restraint in flexion and extension exercises⁵⁰ and reaffirms the importance of realistic simulation of loading in MVIC activity.

The current complex MS-FE simulations, have limitations that must be considered when interpreting the findings. First; all results were based on a single female cadaveric knee model. While this subject-specific model has extensively been validated in earlier works^25,50, it does not capture the inherent variability in the knee anatomy and musculature of the general population. Therefore, absolute predictions should be interpreted with some caution while the trends and the MS-FE vs. OpenSim comparisons are expected to hold and remain valid. Second and despite the inclusion of co-activation moment parameters; muscle forces were estimated using static optimization that may not fully reflect the co-contraction patterns observed in pathological knee conditions (e.g., ACL deficient or OA). Structural and material properties of the joint tissues (cartilage, menisci, ligaments) were adopted from the literature and previous validations. Additionally, the musculature in our model was developed after scaling the OpenSim Gait2392 model and Rajagopal et al. reference values for muscle architecture²⁸. Finally, the optimization criterion of sum of cubed muscle stresses along with antagonistic activities in flexor muscles defined as a prescribed global flexor moment that parametrically varied at 4 levels (Table 1) were employed.

In conclusion, quadriceps, patellar tendon, and patellofemoral contact forces as well as internal passive moment substantially increased with KFA and applied MVIC moment, while the ACL force was primarily loaded at lower KFAs. More distal location of the external resistant force reduced the magnitude of the posterior shear on the knee resulting in a substantial increase in the ACL force and drop in the PCL force. In clear contrast to OpenSim estimations of MVIC activity, changes in the joint center location affected only the internal passive moments with all other results (muscle forces, internal load distribution) remaining almost unchanged. Results also highlight the fundamental shortcomings in MS models that regularly, for the verification of moment equilibrium equations, simulate the knee (or its individual compartments) as an idealized joint located at an assumed center of rotation (or pressure)^56–58. This would further deteriorate under larger joint compression forces expected in heavier daily activities.

Author contributions

P.S. and A.S.A. jointly contributed to the study design, model development, data analysis, and manuscript preparation. P.S. was responsible for developing the musculoskeletal models, performing simulations, and preparing the figures and tables. A.S.A. conceived the study, supervised the work, provided ongoing guidance, and drafted the manuscript.

Funding

This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), grant number RGPIN5595 to A. Shirazi-Adl.

Data availability

The datasets generated during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.