## 4 Discussion  
A new MbMBT scheme considering the kinematic constraints provided by the personalized model of the tibofemoral joint was developed in the current study. In general, the MbMBT approach could more precisely determine the 3D tibofemoral kinematics than the conventional SbMBT, especially in the translation components (Tables 1-3). This improvement can be attributed to the proposed tibofemoral joint models, as they provide physical constraints to prevent the nonphysiological motion of the tibia relative to the femur. Simultaneous image registration of the two bones also helps ensure that the originally accurate kinematic components from the SbMBT would not be compromised by joint model imperfection and motion-related kinematic variability. However, the personalized kinematic model was shown to not effectively diminish the 6-DOF errors of individual bones (Figure 6), similar to the strategies of the anti-collision maneuver at the joint [30; 31]. To better reflect the physiological deformation of the ligament in the joint model, the present study first estimated the ligament length in 3D space (i.e., \(L_{3d}\)) from its sagittal plane projection length (i.e., \(L_{2d}\)) using the trained RF model with the prior tibofemoral kinematics obtained from single-body 3D/2D image registration (step-1). The approach was developed to account for the fact that the ligament length changes along with the tibofemoral motion [43] and presents individual differences. Since the fluoroscopic detector is often configured to capture ML view images of the knee joint, the out-of-plane translation error of the bone should have a minor influence on the estimation of the \(L_{2d}\). Thus, the relatively accurate \(L_{2d}\) was taken to serve as the baseline length of the ligament, and its scaling factors to \(L_{3d}\) were predicted with the trained model. The results indicated that the predicted scaling could yield significantly more accurate PCL lengths than those obtained with uniform scaling (Figure 5D). The length of the ACL predicted with RF regression showed the least accurate result among the three ligaments, in which the RMS error was near 0.6 mm close to that obtained with uniform scaling (Figure 5D). This may indicate that the scaling factors between the sagittal and 3D lengths of the ACL were not sufficiently explained by tibofemoral motions. Further exploration is required to better predict the ACL length variation. The presented three-step MbMBT scheme originated from the observations that the SbMBT using single-plane images could normally determine accurate in-plane translations and rotations, moderate accuracies in out-of-plane rotations and poor out-of-plane translations [21; 25; 28], which often propagated to result in artifactual tibofemoral dislocation. Thus, the MIR with two-level optimization in step-two was designed to solely amend the z-axis translation of the tibia by the tibofemoral kinematic model, while the other DOFs were merely fine-tuned to ensure image matching between the DRR and the fluoroscopic image. With the strategy, the outer-level optimization left the control of the least accurate z-axis translation to the inner-level subproblem, where the tibofemoral model dominates the relative z-axis translation of two bones. After completion of the step-two analysis, the model parameters, including variations in ligament length and joint compression, were updated according to the tibofemoral kinematics to better represent the \"subject-specific\" and \"motion-related\" soft tissue deformation during a particular task. Polynomial fitting can also ensure the smoothness of the tissue deformation against knee flexion, which enables improvement of the smoothness of tibofemoral motion across multiple image frames under the guidance of the joint model in step-three analysis. The kinematic constraints provided by the personalized tibofemoral model were demonstrated to effectively ameliorate three-axis translational precision of the joint during all the tested functional activities (Tables 1-3).