## 4 Discussion  
Previous studies investigating the biomechanical effects of CA design variations on extraction space closure have provided important insights into CA treatment optimization. [PERSON] et al. demonstrated that the vertical height of the clear aligner extraction space significantly influenced the patterns of anterior tooth retraction and canine tipping when clear aligner extraction space height parameters were systematically varied [17,18]. Another finite element study [16] examined effects of different extraction space designs (non-pontic, full-pontic, half-pontic, and beam design configurations) and initially observed similar crown displacements, distal inclinations, and distal rotations across all designs. However, as tooth movement progressed, canine movement patterns diverged significantly among the various designs, with the beam design configuration demonstrating superior effectiveness in controlling bodily tooth movement. These previous research findings indicate that extraction space configuration is a critical factor influencing the biomechanical performance of CA treatments. Building on this foundation, the present study systematically analyzed effects of buccolingual width variations in the extraction space on canine tooth movement patterns, providing novel insights into the optimal beam design parameters for enhanced treatment outcomes. The results of this study clearly demonstrate that both the presence and width of the beam significantly influence the biomechanical response during canine retraction. While distal movement of the canine was observed across all experimental groups, the movement patterns varied considerably depending on the width of the buccolingual beam. Group 0 (0 mm beam width) exhibited the greatest crown displacement and the most pronounced tipping behavior. This indicates that without structural support in the extraction space, tooth movement control is compromised, resulting in an increase in unwanted tipping components. These findings are consistent with those of [PERSON] et al., who demonstrated that lower heights were effective for bodily movement, suggesting that the tipping effect was enhanced because the aligner completely bypassed the extraction space and transmitted forces only to the adjacent teeth, thereby displacing the line of force away from the center of resistance of the crown [17,18]. Conversely, as the beam width increases, both the final distal displacement and final tipping angles decrease. These results indicate that a greater beam width not only reduces the linear displacement, but also effectively controls unwanted tipping movements, thereby promoting more favorable bodily tooth movement [21,22, 23]. Previous studies have demonstrated that the utility of design elements, such as attachments, significantly depends on their ability to accurately transmit forces and moments to enhance the efficiency and predictability of tooth movement [11,24,25,26,27]. Similarly, the results of this study revealed that the buccolingual width of the extraction space is a critical design variable that influences force and moment transmission, thereby controlling tooth movement patterns. The analysis of forces and moments applied to the canine demonstrated a clear correlation with the beam width parameters. As the beam width increased, the initial force transmitted to the canine increased proportionally, whereas the initial moment decreased inversely. Specifically, when beam width increased from 0 mm to 3 mm, the initial force applied to the canine rose from approximately 9.2 N to 9.4 N, while the initial moment decreased from 24.0 Nmm to 20.8 Nmm. As tooth movement progressed, force patterns remained consistent with the initial observations; however, moment characteristics exhibited a reversal, with higher values maintained in wider beam configurations during the later treatment stages.