Computed Tomography(CT) is one of the successful methods, in which mathematical theories are combined with engineering technology. Volume CT will replace the cross-sectional CT that has been widely used. In this paper two new cone vertex orbits, which are circle-and-arc and regular triangular pyramid lines, have been proposed and their complete conditions for exact reconstruction have been obtained. After having studied circle-and-n-line cone vertex orbit which is one of the important orbits in volume CT, we have given a better proof about the complete condition in circle-and-line and circle-and-2-line cone vertex orbit and have obtained some proper conclusions. Finally, we extend the above results to circle-and-n-line vertex orbit. These results will be useful for the design of direct volume imaging.
Algorithms ; Artifacts ; Computer Simulation ; Tomography, X-Ray Computed ; methods

Objective:To explore the application value of multiple indexes of high-frequency ultrasound in the diagnosis of shoulder stiffness, and to provide data support and theoretical basis for the diagnosis and treatment of shoulder stiffness.Methods:Clinical data of 261 patients with unilateral shoulder pain with limited mobility admitted to the Orthopedic Trauma, Hebei Medical University Third Hospital from January to December 2023 were retrospectively analyzed. With the results of shoulder arthroscopy as the gold standard, the patients were divided into 127 patients with shoulder stiffness (observation group) and 134 patients with non-shoulder stiffness and shoulder pain (control group). The fluid accumulation of long head of the biceps tendon (LHBT), echo of rotator interval (RI) and blood flow signal, thickness of coracohumeral ligament (CHL), thickness of subacromial-subdeltoid bursa (SASD), fluid accumulation and blood flow signal, fluid accumulation of posterior glenohumeral joint capsule (PJ), the inferior capsule thickness (ICT) and the thickness of inferior glenohumeral ligament (IGHL) were observed and measured in the two groups in addition, the CHL difference, ICT difference and IGHL difference between the affected side and the contralateral side were calculated, and the difference between the two groups of ultrasound indicators was compared and the multivariate logistic stepwise regression analysis was performed, and the receiver operating characteristic (ROC) curve was drawn to evaluate the diagnostic value of each indicator for shoulder stiffness.Results:RI hypoecho with increased blood flow signal, SASD thickening, SASD effusion, SASD blood flow signal detection rate, IGHL thickness, IGHL difference and ICT difference in observation group were significantly higher than those in the control group (all P<0.05). Multivariate logistic stepwise regression analysis showed that SASD thickening, SASD effusion and IGHL difference were independent predictors of shoulder stiffness (all P<0.05). The area under the curve of SASD thickening, SASD effusion and IGHL difference in the diagnosis of shoulder stiffness were 0.550, 0.540 and 0.636, respectively. The diagnostic sensitivity was 89.8%, 10.2% and 60.6%, and the specificity was 20.1%, 97.8% and 67.2%, respectively. The area under the curve was 0.676, the diagnostic sensitivity was 59.8%, and the specificity was 75.4%. Conclusions:High-frequency ultrasound is an effective imaging method to assist clinical diagnosis of shoulder stiffness. SASD thickening, SASD effusion and IGHL difference may be independent predictors of shoulder stiffness. The combined application of these three indicators has higher clinical application value and provides favorable ultrasound indicators for clinical differential diagnosis of shoulder stiffness.