Dual fluoroscopic imaging and CT-based finite element modelling to estimate forces and stresses of grafts in anatomical single-bundle ACL reconstruction with different femoral tunnels

This study was approved by our institutional review board, and informed consent was obtained by the subject. One healthy volunteer (male; 26 years; body height 175 cm; body weight 65 kg) was recruited for this study. The subject had a right dominant limb and no history of knee injuries, surgeries or systemic diseases. Physical examinations (Lachman test and pivot shift test) were performed to rule out knee pathologies. The motion of the right knee was analysed (Fig. 1a).

Computed tomography (CT) and dual fluoroscopic imaging techniques have been described in detail previously [10, 11]. Similar methods to investigate in vivo knee kinematics have been used in our previous work [12, 13]. CT scans (SOMATOM Definition AS + ; Siemens) of the right knee joint ranging from approximately 15 cm proximal and 15 cm distal to the joint line (thickness, 0.6 mm; resolution, 512 × 512 pixels) were obtained. The images were then imported into Mimics (v19.0, Materialise NV, Leuven, BE). CT Bone Segmentation (threshold:1250–3680; fill holes) and Smoothing (iterations:7; smooth factor:0.7) were used to calculate 3D models of the femur and tibia. The CT-based models were exported in STL format (Fig. 1b). The knee of the subject was simultaneously imaged using two fluoroscopes (BV Pulsera; Philips) as the subject performed a lunge motion (full extension to ~ 100° of flexion). The cumulative radiation dose was 4.95 mGy during the whole test, which was less than that of a conventional abdominal CT scan. Considering that the target motion took only 5 s in the overall 32 s, it was possible that the radiation dose would reduce markedly when the procedure was maturely used. Next, the fluoroscopic images were imported into MATLAB (R2013a; MathWorks) and positioned in the imaging planes based on the projection geometry of the fluoroscopes when the subject was scanned. Finally, the STL models were imported into the software, viewed from the directions corresponding to the fluoroscopic X-ray source used to acquire the images, and manually manipulated in six degrees of freedom (6DOF) with the software until the projections of the model matched the outlines of the in vivo fluoroscopic images taken when the subject performed the knee motion (Fig. 1c).

Local femoral and tibial coordinate systems were established before the imaging procedure. The femoral origin was located at the midpoint of the geometric centre axis, a line connecting the centres of the spheres fit to the lateral and medial posterior femoral condyles, representing the mediolateral axis. The anteroposterior axis was perpendicular to the plane defined by the geometric centre axis and the femoral shaft. The proximodistal axis was set to be perpendicular to the two other axes. The tibial origin was the projection point of the femoral origin on the tibial plateau. In order to make the calculated kinematic data easier to be applied in the finite element analysis subsequently, the three axes of the tibial coordinate system were parallel to the femoral coordinate system (Fig. 2a). The 3D translation was quantified as the relative displacement between the origins of the femur in the tibial coordinate system. The angular rotations were determined by the femoral coordinate system with respect to the tibial coordinate system using Euler angles in the following sequence: mediolateral axis (Z axis), anteroposterior axis (X axis) and proximodistal axis (Y axis) [14]. The tibial coordinate system was defined as fixed coordinate system and the femoral coordinate system as floating coordinate system. After matching the projections of the models to their corresponding outlines, the rotation matrix of the floating coordinate system relative to the fixed coordinate system was calculated. Through the rotation matrix, the relative translation and rotation of the two coordinate systems can be calculated according to the sequence described above. The 6DOF in each frame were connected in series to form the 6DOF changes in knee flexion (Fig. 3).

The femoral bony landmarks (lateral intercondylar ridge and the edge of femoral articular cartilage) were identified and used to determine the femoral footprint of the ACL [2, 3] (Fig. 4a). Twenty-one femoral attachment points were mapped to cover the footprint completely. Each point was 2 mm apart from the others. In addition, we categorized these points based on their location (anterior, middle and posterior; high, middle and low) (Fig. 4b, c). The tibial attachment point was identified as the midpoint of the line that connected the centres of the double bundle of the ACL on the tibial footprint [15] (Fig. 4d). Femoral tunnels were simulated using the anteromedial portal for each attachment point. The tibial tunnel was simulated at 15° in the anterior view and 55° in the lateral view (Fig. 5). The grafts measured 8 mm in diameter and 10 cm in depth into the femoral and tibial tunnels and were constructed as single, soft cylindrical solids. All the modelling procedures were performed using SolidWorks (v2018, Dassault Systemes, Massachusetts, USA).

The bones were considered shells and defined as rigid bodies. The grafts were considered solids and were defined as nearly incompressible, transversely isotropic hyperelastic neo-Hookean materials with the following strain-energy function: \(\Psi = \frac{1}{{2D}}In(J^{2} ) + C_{1} (\overline{{I_{1} }} ^{2} - 3) + F_{2} (\lambda ) \), where C₁ is the neo-Hookean constant and D is the inverse of the bulk modulus k = 1/D (C1 = 1.95, D = 0.00687) [16]. A free meshing technique was used for the grafts using eight-node linear hexahedral elements with hourglass control (e.g. element type C3D8RH). In order to achieve balance between the accuracy of stress and the computational time, a mesh sensitivity analysis was performed by stepwise upsizing the mesh size, according to a previous study [17]. The convergence tolerance was set as a variation of stress within 5% from the previous model with higher mesh density. As a result, the optimum mesh size of grafts was between 0.9 and 1.1 mm. The models of graft included an average of 6000-plus nodes with 5000-plus elements. Bonded contact between the graft and internal surface of the femoral tunnel and frictionless contact between the graft and tibial tunnel were defined. The femoral and tibial coordinate systems were established as described above. The reference points of the femur and tibia were their origins in the coordinate systems.

To replicate the ACL reconstruction completely, a model at 20° of knee flexion and an initial graft tension of 80 N were used [18, 19]. The boundary and loading conditions were implemented in two different steps to imitate the actual state of the knee after ACL reconstruction and during the knee lunge motion as the researchers did previously [15]. First, the femur and tibia were fixed, and an initial graft tension of 80 N was applied on the end surface of the graft along the tibial tunnel direction. Second, the tibia and the portion of the graft in the tibial tunnel were fixed, and the femur was translated three-dimensionally and rotated angularly according to the 6DOF calculation (Fig. 2b). Knee motion was simulated at each of the flexion angles and quasi-static simulation was performed. We simulated the motions by assuming a state. At a certain flexion angle, 6DOF was applied to the femoral model relative to the tibial model, causing the deformation of the graft model relative to the initial position. The femur position was simulated in 10° increments as outer loadings for the graft. The reaction force of the femur and maximal principal stress of the graft were subsequently calculated (Fig. 6). Finite element analysis was performed in Abaqus (v2018; Dassault Systèmes SE, Vélizy-Villacoublay, FR), executed on a 64-bit Windows operating system with Quad-core and eight-thread Inter Core platforms configured with 24 GB of RAM. Simulation required approximately 10 min at each knee flexion angle. Total required time for performing this analysis was about 35 h.

The graft force first increased and then decreased after the maximum force was reached, which occurred at approximately 30° of flexion (Fig. 7a, b). The graft force was significantly affected by the graft placement locations in the higher–lower direction (P < 0.001). At high degrees of flexion, the high location led to a consistently larger force; the middle location led to anatomical graft behaviour similar to that of a normal knee, while the low location led the graft force to decrease early to nearly zero. No significant differences in the graft force were found when the graft was placed in the anterior–posterior direction (P = 0.971) (Table. 1A). At each tested flexion angle, significant differences were found among the high, middle and low locations (Table 2). The statistical results between pairs of the three locations are shown in Table 3a.

During the overall process of lunge simulation, the maximal principal stress of the graft appeared near the entrance to the femoral tunnel. The graft stress did not change significantly before 30° of flexion was reached. Afterwards, it first increased and then decreased after the maximum stress was reached, which occurred at 90° of flexion (Figs. 7c, d). Significant differences in the graft stress were found in the higher–lower direction (P < 0.001) but not in the anterior–posterior direction (P = 0.104) (Table 1A). However, significant differences in the graft stress were found between the anterior and posterior locations (P = 0.037) (Table 1B). The low and posterior locations showed lower levels of stress than the other locations. From 40° to 100° of flexion, significant differences were found among the high, middle and low locations. At 100° of flexion, significant differences were found among the anterior, middle and posterior locations (Table 2). The statistical results between the pairs of the three locations are shown in Table 3A, 3B.