Accurate joint kinematics plays an important role in estimating joint kinetics in musculoskeletal simulations. Biplanar fluoroscopic (BPF) systems have been introduced to measure skeletal kinematics with six degrees-of-freedom. The purpose of this study was to model knee kinematic coupling using knee kinematics during walking, as measured by the BPF system. Seven healthy individuals (mean age, 23 ± 2 yr) performed treadmill walking trials at 1.2 m/s. Knee kinematics was regressed separately for the swing and stance phases using a generalized mixed effects model. Tibial anterior translation function was $y=0.20x−3.09$ for the swing phase and $y=0.31x−0.54$ for the stance phase, where $x$ was the flexion angle and $y$ was the tibial anterior translation. Tibial lateral and inferior translation were also regressed separately for the stance phase and the swing phase. Tibial external rotation was $y=−0.002x2+0.19x−0.64$ for the swing phase and $y=−0.19x−1.22$ for the stance phase. The tibial adduction rotation function was also calculated separately for the stance and swing phase. The study presented three-dimensional coupled motion in the knee during the stance and swing phases of walking, and demonstrated the lateral pivoting motion found in previous studies. This expanded understanding of secondary knee motion functions will benefit musculoskeletal simulation and help improve the accuracy of calculated kinetics.

## Introduction

Kinematics and kinetics of the knee joint have been studied to understand the pathomechanics and progression of knee joint diseases, such as osteoarthritis and anterior cruciate ligament injury [1,2]. Indeed, knee joint kinetics such as joint contact force and joint moments are associated with the pathomechanics of joint disease [3]. Also, excessive and repetitive joint contact force and joint moments could damage the knee cartilage and meniscus [4]. However, it is difficult to quantify knee laxity and joint contact loads unless a force sensor is implanted in the human joint. Prior studies estimated joint kinetics from external forces and joint kinematics through musculoskeletal simulation [5,6], wherein accurate joint kinematics play important roles. A knee model with a one degree-of-freedom hinge joint that ignores secondary knee kinematics decreases the accuracy of estimated ligament and cartilage loading patterns in musculoskeletal simulation [5,7]. Also, the moment arms of leg muscles, which determine joint kinetics results in musculoskeletal simulations, are sensitive to secondary knee kinematics for a musculoskeletal model simulation [8]. Therefore, accurate three-dimensional joint kinematics helps improve the accuracy of kinetics and joint load estimation in musculoskeletal simulations.

The knee joint primarily has a flexion–extension degree-of-freedom accompanying important secondary motions such as anterior–posterior (AP), medial–lateral (ML), and superior–inferior (SI) translation, as well as internal–external (IE) and varus–valgus (VV) rotation. The secondary motions are determined by bony contact geometry, ligaments, menisci, muscle coordination, and external loading conditions such as the ground reaction forces for different activities [9]. Frequently, the knee is modeled as a hinge joint with a single degree-of-freedom. Simplified knee joint models neglect the secondary motions of the knee joint, thereby reducing the accuracy of the kinetics calculations for the joint. Other studies have used predefined secondary motions as functions of the knee flexion angle [1012], reflecting rollback in the knee [13,14]. Also, the standard full-body models in opensim include a planar knee joint model that has predefined secondary motions [11]. However, the two-dimensional planar model deals with only AP and SI translations in the knee, ignoring IE and VV rotations that are also important kinematics that affect joint contact locations and medial–lateral force balance in the knee. Moreover, the function for secondary knee motion was determined from experiments with cadaveric knees and nonambulatory data [15]. The secondary motions can be different between weight-bearing and nonweight-bearing conditions [16]. The secondary motions in the knee can change depending on whether the knee is engaged in walking, lunging, passive flexion, or another activity. Thus, the secondary motion during in vivo walking cannot be extrapolated from nonambulatory motion of cadaveric knees [8,17,18].

Measuring the secondary motions in the knee with skin markers involves skin artifact [19,20]. Recently, a biplanar fluoroscopic (BPF) system was introduced to measure in vivo skeletal kinematics [2125]. The BPF system captures in vivo skeletal images and calculates three-dimensional intact kinematics of joints with state-of-the-art accuracy. Previously, the translational and rotational accuracies of the BPF system were studied using cadaveric knees mounted on linear and rotary stages [26] and in human subjects [15,2732]. The BPF system has been used to reveal different knee kinematic patterns during daily activities such as ground walking [30,31], treadmill walking [31,32], stair walking [30], flexion–extension [29], and during the treadmill walking in ACL reconstructed knees [27,32]. Furthermore, Hume et al. [15] created knee kinematics splines as functions of knee flexion angle from in vivo knee kinematics during a nonweight bearing knee extension and applied them to musculoskeletal simulation. The purpose of this study was to measure the three-dimensional skeletal kinematics of the femur and tibia in the normal knee during walking and to model knee kinematic coupling. Secondary motions of the knee kinematics were modeled as functions of knee flexion angle for the stance phase and the swing phase during in vivo walking.

## Methods

### Data Acquisition.

This study was approved by the Institutional Review Board at Chung-Ang University. Seven healthy individuals (all males, age 23 ± 2 yr, weight 63 ± 6 kg, and height 170 ± 3 cm) without history of knee injury participated in this study after providing informed consent. A BPF system (KMC-1400ST, Comed Medical, Sungnam, Korea) was setup around a treadmill (FDM-TDSL, Zebris, Isny im Allgäu, Germany). The BPF system was approved by the Korean Food and Drug Administration. X-ray tubes were operated at 55 kVp and 10 mA. Charge coupled device cameras attached to image intensifiers acquired knee joint X-ray images at 27 frames per second with image resolution of 1024 × 1024 pixels and exposure time of 4 ms. Participants performed a walking trial on the treadmill at 1.2 m/s [33], and anterior–posterior and lateral fluoroscopic images of the right knee were captured with a BPF system. We also acquired biplanar X-ray images of the knee during a static standing trial to determine the knee reference pose. Participants also underwent computed tomography (CT) imaging of the knee.

### Biplanar Fluoroscopic Data Processing.

Three-dimensional triangular mesh models of the distal femur and proximal tibia were obtained from the CT data. Mesh models of the femur and tibia were created from CT data using the seg3d open-source software package [34]. The triangular mesh models of the femur and tibia and the fluoroscopic images from the BPF system were loaded into a three-dimensional graphical environment in our custom software, which was developed with matlab (MathWorks, Natick, MA) and vtk (Kitware, Clifton Park, NY) [22]. The femur and tibia models were manually registered to the fluoroscopic images for the midswing phase to the midstance phase during walking due to the overlap of the bones in the images (Fig. 1). The standing trial was processed in the same manner. The precision of the rotation and translation was 0.6 deg and 0.6 mm, respectively, when the registration was repeated five times by the same individual.

### Knee Kinematic Coupling.

The anatomical coordinate systems on the femur and tibia were configured (Fig. 2) and their relative movements were quantified according to the method suggested by the International Society of Biomechanics [35,36]. The kinematics for standing posture were used as a reference knee pose.

Tables for secondary knee motions versus knee flexion angle were prepared separately for swing and stance phases. The tables were used to formulate knee kinematic functions using a generalized mixed effects model. The knee flexion angle and the individual were set as fixed and random effects in the statistical model, respectively, and regression analyses were performed on AP, ML, and SI translations and IE and VV rotations using IBM SPSS Statistics for Windows (version 23.0, released 2015, IBM Corp., Armonk, NY).

The mixed effect model was defined as
$y=Xβ+Zu+ε$

Here, $y$ is kinematic coupling or knee secondary motions, $X$ is a designed fixed effect, $Z$ is a designed random effect, $β$ is a fixed effect coefficient with an intercept and slope, $u$ is a random effect variance with an intercept and slope, which allows for individual difference, and ε is the observation error. The Akaike information criterion, which measures the quality of a statistical model for a given data set, was calculated to determine the order of the polynomial regression using SPSS. The order of the random effect polynomial equation with intercept was the same as that of a fixed effect polynomial equation. Fixed effect coefficients with a 95% confidence interval were calculated. The variance by intercept and flexion angle as the random effect was calculated.

To compare the knee kinematics results of this study to previous results measured by a BPF system during normal walking, the anatomical coordinate systems of the femur and tibia were set according to the method of Kefala et al. [30] and the kinematics were recalculated. Also, the contact paths on the articular surfaces in the medial and lateral condyles were calculated to determine whether the distinct lateral pivot pattern during walking appears as a combination of the knee kinematic functions [18]. The closest point pairs were calculated using a weighted average of the distance between contact surfaces [37].

The effect of the kinematic coupling function on musculoskeletal simulation results was tested in the OpenSim simulation platform. Three human musculoskeletal models with different knee kinematic constraints were prepared. The first model was the original gait 2354 model of opensim, which had AP and SI translation constraints [11]. The gait 2354 model was modified to have knee kinematics functions calculated from swing phase results for the second model and from stance phase results for the third model. The moment arms of four muscles (biceps femoris long head, biceps femoris short head, rectus femoris, and vastus intermedius) were estimated during walking (gait 2354 sample data) in the three models (Fig. 7). Root-mean-squared-deviations (RMSD) and Pearson correlations between moment arms estimated by the modified gait 2354 models and the original 2354 model were calculated to quantify the difference of moment arm between modified gait models and original gait model.

## Results

The average (±standard deviation) duration for the swing phase—between midswing and heel-strike—was 200.5 ± 48.7 ms, and that for the stance phase—between heel-strike and midstance—was 280.5 ± 45.3 ms for the treadmill walking experiment. The processed swing and stance phases together represented 40.5 ± 4.3% of a normal walking gait cycle.

The orders of the polynomial functions of secondary knee motions were determined by the Akaike information criterion. The AP translation for the swing phase was regressed by a first-order polynomial function, and the AP translation for the stance phase was regressed by a first-order polynomial. The AP translation functions were $y=0.20x−3.09$ and $y=0.31x−0.54$ for the swing phase and stance phase, respectively, where $y$ was the tibial anterior translation, and $x$ was the flexion angle. The variances by intercept of knee AP translation function were 12.2 mm2 and 9.1 mm2 for the swing phase and stance phase, respectively. The variances by flexion angle of knee AP translation function were 0.001 mm2 and 0.02 mm2 for the swing phase and stance phase, respectively.

The results of the kinematic functions for AP, ML, and SI are shown in Fig. 3 and summarized in Table 1 along with the variance by intercept and flexion angle.

The IE rotation for the swing phase was regressed by a second-order polynomial function, and the IE rotation for the stance phase was regressed by a first-order polynomial. The IE rotation functions were $y=−0.002x2+0.19x−0.64$ and $y=−0.19x−1.22$ for the swing phase and stance phase, respectively, where $y$ was the tibial external rotation, and $x$ was knee flexion angle. The results of the kinematic functions for IE and VV are shown in Fig. 4 and summarized in Table 1 along with the variance by intercept and flexion angle.

The knee kinematics results of seven healthy subjects during treadmill walking, which was used to formulate the knee kinematic coupling function, were recalculated using another joint angle representation method [30]. The AP translation and IE and VV rotation conformed to those reported in a previous study that measured knee kinematics during normal walking using a BPF system, as shown in Fig. 5.

The movements of the closest point pairs between the femur and tibia were visualized on the distal femoral surface and proximal tibial surface during the stance phase for knee flexion from 0 to 25 deg (Fig. 6). The lines connecting the closest points on the femoral surface moved parallel during the stance phase. Meanwhile, the closest points on the medial plateau had a larger range of motion than those on the lateral plateau.

The largest RMSD was observed in the rectus femoris which were 5.4 mm and 6.0 mm for stance and swing phases, respectively. The smallest RMSD was observed in the biceps femoris-long head which were 2.3 mm and 5.6 mm for stance and swing phases, respectively. Pearson correlations of all four muscles were larger than 0.9. The Pearson correlation of vastus intermedius was the largest with 0.96 and 0.99 for stance phase and swing phase, respectively, while that of vastus intermedius was the smallest with 0.93 and 0.94 for stance phase and swing phase, respectively.

## Discussion

We formulated knee kinematic coupling functions for the secondary motions as functions of knee flexion angle by regressing the in vivo knee kinematics measured by a BPF system using a generalized mixed effects model. Previous studies on two-dimensional planar knee models did not include IE rotation of the knee [11,12], though the range of IE rotation is approximately 10 deg during the stance phase of walking. The knee kinematic coupling functions including all three-dimensional secondary knee motions will help improve accuracy for calculating kinetics in musculoskeletal simulations.

We showed the in vivo secondary knee kinematics during the weight-bearing, normal walking condition. in vivo knee kinematics during walking is different from those of passive knee flexion in cadaveric knees [16,38]. The secondary knee motions are also affected by loading conditions, such as weight-bearing [9,16,39]. However, previous studies calculated the secondary knee motions from experiments on cadaveric knees [1012]. Accordingly, the secondary motion functions derived from the passive knee flexion of cadaveric knees are inadequate for musculoskeletal simulation of the knee during walking. Previously, Hume et al. [17] measured knee kinematics during daily activities (knee extension, gait, step down and walking pivot turn) using a BPF system, but they calculated the spline functions of knee kinematics only from knee kinematics measured during nonweight bearing active extension. They reported that the results of musculoskeletal simulation using the knee kinematics spline from in vivo kinematics varied from the results of cadaveric knee kinematics. Therefore, the previous gait model using a kinematics function based on cadaveric specimen kinematics during passive flexion could not reproduce in vivo knee kinematics during walking. We focused on developing a knee kinematics function that can be used in normal gait models based on in vivo knee kinematics measured by the BPF system. In our study, the knee kinematic coupling functions were formulated for the stance and the swing phases separately, because secondary knee motions would be altered by the knee joint loading condition [16].

We used a generalized mixed effects model to normalize the effect by individual when considering the relationships between the flexion angle and other motion degrees-of-freedom whose couplings were all statistically significant at p < 0.05. Meanwhile, the variance by intercept was as high as 12.2 mm2 and 28.0 deg2 for AP translation and IE rotation during the swing phase, respectively, and the values were 9.1 mm2 and 10.7 deg2 during the stance phase. This implies that the individual differences in motion and anatomy can cause offset in the coupled motion, but there is still a significant functional relationship relative to the flexion angle.

The visualization of the closest points on the medial and lateral regions of the knee showed distinct lateral pivot patterns that have been observed in previous studies of knee kinematics during walking using biplanar fluoroscopic systems [18,23,40]. Our lateral pivot results of the in vivo knee during walking differed from the medial pivot motions during passive knee flexion extension of the cadaveric knee [13,41] and nonambulatory activities [42].

A previous study generated secondary knee motions using a participant-specific musculoskeletal knee model with six degrees-of-freedom during passive and active knee flexion and walking [7]. The secondary knee motions were quite different between activities [7,9]. Secondary knee motions also affected muscle activation patterns during walking in a simulation study [7]. Slight variations in secondary knee motions changed the moment arms of muscles and ligaments when estimating muscle and joint force [8]. Thus, it is important to implement 3D secondary knee kinematics during a musculoskeletal simulation of walking. The estimated moment arms of the modified gait 2354 models using knee kinematic coupling functions varied from those of the original gait 2354 model (Fig. 7). Also, our data indicated that the estimated moment arms from models using kinematic coupling from the swing phase and stance phase were different. The knee kinematic coupling function and the resulting moment arms of muscles may improve the accuracy of musculoskeletal simulation of the stance and swing phase of gait especially for the population group tested in this study.

A limitation of this study is that the knee kinematics were calculated from treadmill walking data between midswing and midstance, which represents 40.5% of the full gait cycle. Thus, our regressed functions represent part of the stance and swing phases. Our biplanar fluoroscopic system was a fixed system; thus, overlap between the limbs was inevitable. The target knee was out of the BPF imaging space from midstance to midswing. We tested only male subjects in this study, and knee kinematics may vary by sex [43,44]. The treadmill speed was set at 1.2 m/s, which is the average walking speed of the Korean population [33]. However, the advantage of our study is that the skeletal kinematics were directly measured with state-of-the-art accuracy. Implementation of knee kinematic coupling could increase the accuracy of musculoskeletal simulations by providing more physiological movement of the joint and more accurate moment arms for the muscles and ligaments.

## Acknowledgment

This work was supported by the Basic Science Research Program through the NRF (NRF-2017R1A2B2010763) funded by the Ministry of Science and ICT of the Republic of Korea. This research was supported by the Chung-Ang University Graduate Research Scholarship in 2016.

## Funding Data

• Ministry of Science and ICT of the Republic of Korea (Funder ID: 10.13039/501100004083).

• Chung-Ang University (Funder ID: 10.13039/501100002460).

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