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Background: Scoliosis causes abnormal spinal curvature and torsional rotation of the vertebrae and has implications for human suffering and societal cost. In differential geometry, Writhe describes three-dimensional curvature. Differential geometric quantities can inform better diagnostic metrics of scoliotic deformity. This evaluation could help physicians and researchers study scoliosis and determine treatments., Methods: Eight adult lumbar spine CT scans were analyzed in custom MATLAB programs to estimate Writhe and Cobb angle. Five patients exhibited scoliotic curvature, and three controls were asymptomatic. Vertebral centroids in three-dimensional space were determined, and Writhe was approximated. A T-test determined whether the affected spines had greater Writhe than the controls. Cohen's D test was used to determine effect size., Results: Writhe of scoliotic spines (5.4E-4 ± 2.7E-4) was significantly higher than non-scoliotic spines (8.2E-5 ± 1.1E-4; p = 0.008)., Conclusion: Writhe, a measure of curvature derived from 3D imaging, is significantly greater in scoliotic than in non-scoliotic spines. Future directions must include more subjects and examine writhe as a marker of scoliosis severity, progression, and response to treatment., (© 2023. The Author(s), under exclusive licence to Scoliosis Research Society.)

Several classification systems of the spinal curves in adolescent idiopathic scoliosis (AIS) have been developed to guide surgical decision-making. The current classification systems are based on the spinal deformity patterns or deformity magnitudes in one or two anatomical planes. Considering the 3D nature of the spinal deformity in AIS, these classifications fail to capture the spine's curve in its entirety. We proposed a classification based on the axial plane and showed that mathematical analysis of the 3D spinal curve, using differential geometry, supports the differences between the subtypes in this classification system. We calculated the writhe and twist of the entire spinal centerline, elements of the Călugăreanu-White-Fuller theorem, in a cohort of 30 right thoracic AIS patients. We also classified this cohort manually based on the vertebral level at which the direction of vertebral rotation caudal to the thoracic curve changes: Lumbar in Group I (V-shaped axial projection) or thoracolumbar in Group II (S-shaped axial projection). The writhe and twist of the spinal curve were significantly different between these manual classification subgroups. Our manual classification distinguished the axial subgroups of right thoracic AIS supported by mathematical specifications of the entire curve in three dimensions. Graphical abstract.

The rate of decrease of gallstone diameter appeared to be linear with oral bile acid treatment time, as estimated by inspection of graphic data of individual patient serial oral cholecystograms. A theoretical basis for this model was derived. The hypothesis of diameter decrease proportional to treatment time was tested with data from 223 patients with radiolucent gallbladder stones up to 20 mm in diameter treated with 7-8 or 14-15 mg.kg-1.day-1 of either ursodiol or chenodiol for 1 year, followed up after 3, 6, and 12 months of treatment. Linearity of individual sets of data points was tested by mathematical and statistical methods, including polynomial curve fitting, linear regression analysis, quadratic vs. linear analyses of covariance, and prediction of 12-month stone size by linear extrapolation from diameters at 0 and 6 months. Most patients (greater than 70%) had rates of gallstone diameter decrease that were almost linear with treatment time, independently of different dissolution rates on a given regimen. Additional treatment time needed may be estimated from cholecystograms or ultrasonograms performed before treatment and after several months.