The patella geometry in this study was obtained from a computer tomography (CT) image of the left knee of a healthy 26-year-old male volunteer with a height of 182 centimeters and a weight of 75 kg. The CT scanning parameters and post-processing process could be found in Supplementary Material 1. The images were then recorded onto an optical disc in the international standard digital imaging and communications in medicine (DICOM) format for archival purposes.

The DICOM-formatted CT slice images of the knee were imported into MIMICS® 21.0 (Materialize’s Interactive Medical Image Control System/Materialize NV, Belgium). Following noise reduction, the determination of optimal bone tissue boundaries within the threshold range of 226–1691 Hounsfield units, elimination of soft tissue surrounding skeletal images, selective segmentation based on anatomical structures, region-growing, surface gap filling, and smoothing of various patellar components, the geometric model of the patella was successfully constructed (Fig. 1b).

The model was then exported in STL format files, which underwent repair and optimization using the Geomagic Studio (3D Systems Inc., NC, USA). Subsequently, the model surface’s triangular facets were fitted with a smooth and continuous surface, ultimately generating a cohesive surface model. The patellar solid model was imported into the Creo Parametric 10.0 (PTC Inc., USA) in IGES format [27]. By applying a 1 mm inward offset, the cortical and trabecular bone portions of the patellar solid model were separately established [28].

In the Creo Parametric program, we first establish a fracture line on the central horizontal line of the patella to create an AO/OTA 34-C1 type fracture. Subsequently, we created models of the patellar ligament and the quadriceps tendon to simulate the structure of a knee extension mechanism. Then, we constructed models for the K-wire and the tension band, setting the diameter of the K-wire at 2.0 mm and the tension band at 1.8 mm. The proximal end of the K-wires of ARTBW was bent 180°. The simplified model of traditional TBW did not do any bending. Finally, we assemble the models of the patellar fracture, the K-wire, and the tension band. Two K-wire was inserted into the patella in the coronal plane, with the main body of the K-wire placed at a depth of one-half. In the sagittal view, the K-wire’s depth or sagittal placement was centered. In ARTBW models, the bent end of the K-wire was inserted into the patella body, and the insertion depth of the bent proximal K-wire accounted for one-fourth of the total length of the patella. The tension band was wrapped around the K-wires in a figure-eight configuration. Thus, we obtained two models fixed with K-wire and tension bands for the patellar fractures.

The resulting models were then imported into the Abaqus/CAE 2020 program (Dassault System Inc., Waltham, MA, USA) for finite element analysis. In this study, the patella, patellar ligament, and quadriceps tendon were all set to be isotropic, homogeneous linear elastic materials. Then, constructing mesh for all solid models in Abaqus (Fig. 1c). The material properties were determined according to previous studies [29,30,31]. The type of mesh elements, the number of elements for all solid models, and the corresponding material parameters are displayed in Table 1.

Cortical bone was set in bonded contact with cancellous bone, while the internal fixation device was placed in contact with the patella with a friction coefficient of 0.2. The fracture fragments were also set in contact with each other with a friction coefficient of 0.45. The friction coefficient between K-wires and tension was set to 0.1 [27]. Full constraints were applied to the proximal end of the quadriceps tendon and the distal end of the patellar tendon, with the force applied to the patellar surface (Fig. 1d). When the knee joint was flexed at angles of 20°, 45°, and 90°, the force on the lower, middle, and upper thirds of the patellofemoral joint surface was applied at 2.0, 3.5, and 4.4 MPa, respectively [32, 33] (Fig. 1e).

We measured the maximum von Mises stress of the internal fixation (K-wires and tension band) and the articular surface during knee flexion at 20°, 45°, and 90°. Additionally, we assessed the fracture’s maximum displacement gap (\(\:d\)), the displacement angle (\(\:\theta\:\)) after loading, and the rotational angle of the two K-wires under two surgical approaches and loading conditions (Fig. 1f).

The mechanical texting used synthetic bone (SYNBONE 1600, Switzerland) with two nylon straps threaded through the fracture line to simulate the quadriceps and patellar tendon (Fig. 2a). The titanium K-wires and steel wire were used to fix the patella fracture. Twelve patellar models were cut transversally at the center of the patella with a line saw to simulate the patellar transverse fracture. Then they were randomly assigned to two groups. One group was fixed with the traditional TBW, whereas the other group employed the modified ARTBW.

a-b. The extensor mechanism model and mechanical testing setting. c. The K-wire pull-out experiment model (n = 3 for each group) d. The K-wire pull-out testing device and setting. e. The fracture displacement distance of anti-rotation tension band wiring (ARTBW) and tension band wiring (TBW) after dynamic tensile testing. f. The loading value on the testing model until fracture displacement reaches to 2 mm. g. The failure pull-out load of ARTBW and traditional TBW (p < 0.05)

Bionix® Tabletop Test Systems (MTS Systems, USA) was used to carry out the biomechanical test. The distal end of the extensor mechanism (patellar tendon) was fixed, and the other end was attached to the steel cable. The pulley on the test machine was adjusted so that the cable was positioned at a 45-degree from the floor (Fig. 2b).

To compare the mechanical performance between the two groups, we conducted static tensile tests and dynamic fatigue experiments. In these tests, the testing machine initially applied a preload of 50 N to the extensor mechanism, and the fracture gap at both ends under 50 N was measured. After machine zero adjusting, an axial load was applied. In the static tensile test, the fracture gap was measured using a vernier caliper at each 20 N increment until the change in the fracture gap reached 2 mm compared to the initial gap. In the dynamic tensile tests, a cyclic load of 100–250 N at a rate of 200 mm/min was then applied to the testing models for 50 cycles. We collected the fracture displacement after 50 cycles under 100 N [34].

Moreover, to test the pull-out strength of different K-wire insertion methods, a K-wire pull-out experiment was conducted. In this experiment, two types of K-wires were implanted into the cancellous bone block with 20 PCF (Sawbones 1522 − 315, USA) (Fig. 2c). The synthetic bone material properties were consistent with previous studies [35, 36]. For the traditional TBW group, the K-wires were directly drilled in, while for the ARTBW group, the wires were bent at the proximal end after drilling and then hammered into the cancellous bone model. A strong rope was hooked onto the bent part of the K-wire’s proximal end, and an upward load was applied at a rate of 10 mm/min, with data collected at a rate of 100 Hz (Fig. 2d). The force at the onset of K-wire extraction from the cancellous bone block was recorded.

The biomechanical data were presented as mean ± standard deviation (SD). For the statistical analysis, the Man Whitney-U test was conducted with *P < 0.05 considered statistically significant. Statistical analyses were carried out utilizing SPSS 27 (IBM, USA).