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This paper focuses on a singular system with a sign-changing potential in Γ, a bounded domain with a Lipschitz boundary in ℝ d . By imposing appropriate conditions on the weight potential, which is allowed to change sign, we establish the existence of multiple solutions using the shape optimization approach. This study represents one of the earliest endeavors to explore and analyze the occurrence of multiple solutions in fractional singular systems involving sign-changing potentials. By explicitly addressing this particular aspect, our paper contributes significantly to the limited body of literature that exists in this specific field. [ABSTRACT FROM AUTHOR]

Establishing good screw-bone structural stability is conducive to reducing the risk of postoperative screw loosening. Screw insertion torque is an objective index for evaluating screw-bone structural stability. Therefore, accurate prediction of screw insertion torque can improve the preoperative evaluation of patients, optimize the surgical plan, and improve the surgical effect. At present, the correlation between different bone assessment methods and screw insertion torque is unclear. The aim of this study was to evaluate the correlation between different bone assessment methods and screw insertion torque and to optimize the predictive performance of screw insertion torque through mathematical modeling combined with different radiology methods. Prospective cross-sectional study. Seventy-seven patients with preoperatively available DXA, CT and MRI data who underwent spinal fixation surgeries between October 2022 and September 2023 and 357 sets of screw data were included in this analysis. Spinal, vertebrae-specific and screw trajectory's BMD were measured preoperatively by different imaging modalities. Intraoperative screw insertion torque was measured using an electronic torque wrench. Pearson linear correlation, scatter plots and univariate linear regression were used to evaluate the correlation between different bone evaluation methods and screw insertion torque. Different bone evaluation methods were fitted into the prediction model of screw torque and the related equations were obtained. Screw insertion torque had the strongest positive correlation with the volumetric bone mineral density (vBMD) of the screw trajectory (Pedicle screw insertion torque (PSIT): R = 0.618, p<.001; Terminal screw insertion torque (TSIT): R = 0.735, p<.001). A weak negative correlation was found between the screw insertion torque and level specific vertebral bone quality (VBQ) (PSIT: R = -0.178, p=.001; TSIT: R = -0.147, p=.006). We also found that the PSIT was strongly correlated with the TSIT (R = 0.812, p<.001). Compared to other bone quality assessment methods, screw trajectory vBMD may be better predict the magnitude of screw insertion torque. In addition, we further optimized preoperative assessments by constructing a mathematical model to better predict screw insertion torque. In conclusion, clinicians should select appropriate preoperative bone quality assessment methods, identify potential low-torque patients, optimize surgical plans, and ultimately improve screw insertion accuracy and reduce postoperative screw loosening rate. [ABSTRACT FROM AUTHOR]

Background and Objective: The deepening of the nasolabial fold (NLF) is a common occurrence during the aging process, necessitating the development of an objective method to evaluate changes in NLFs. The objective of our study was to introduce a method of depth deviation based on three‐dimensional (3D) image for evaluating the efficacy of hyaluronic acid (HA) injection in correcting NLFs. Methods: Between May 2019 and November 2019, we recruited 56 Chinese volunteers with moderate to severe NLFs for this study. The participants received HA injections and were followed up for a period of 12 months. Pre and posttreatment, we collected the Wrinkle Severity Rating Scale (WSRS) scores and 3D images. To assess the efficacy of NLF correction, we used the maximum value of depth deviation (MVD) based on 3D images and the Global Aesthetic Improvement Scale (GAIS) score. Additionally, we compared the relationship between MVD and other clinical parameters. Results: Fifty‐four female participants completed the study and were included in the analysis. The HA injection resulted in a significant decrease in NLF severity (p < 0.05). One month after the injection, the average GAIS score was 4.72 ± 0.45, and the average MVD was 2.10 ± 0.56 mm. Both the GAIS score and the average MVD exhibited a decline over time during the follow‐up period, with the average MVD showing a significant correlation with the GAIS score at all time points (p < 0.05). Conclusions: Using 3D technology, we demonstrated that the quantitative data provided by MVD can serve as an objective method for evaluating the correction of NLFs. [ABSTRACT FROM AUTHOR]