Shape Approximation and Size Difference of the Upper Part of the Talus: Implication for Implant Design of the Talar Component for Total Ankle Replacement

The implant design of the talar component for total ankle replacement (TAR) should match the surface morphology of the talus so that the replaced ankle can restore the natural motion of the tibiotalar joint and may reduce postoperative complications. The purpose of this study was to introduce a new 3D fitting method (the two-sphere fitting method of the talar trochlea with three fitting resection planes) to approximate the shape of the upper part of the talus for the Chinese population. 90 models of the tali from CT images of healthy volunteers were used in this study. Geometrical fitting and morphological measurements were performed for the surface morphology of the upper part of the talus. The accuracy of the two-sphere fitting method of the talar trochlea was assessed by a comparison of previously reported data. Parameters of the fitting geometries with different sizes were recorded and compared. Results showed that compared with previously reported one-sphere, cylinder, and bitruncated cone fitting methods, the two-sphere fitting method presented the smallest maximum distance difference, indicating that talar trochlea can be approximated well as two spheres. The radius of the medial fitting sphere R M was 20.69 ± 2.19 mm which was significantly smaller than the radius of the lateral fitting sphere R L of 21.32 ± 1.88 mm. After grouping all data by the average radius of fitting spheres, the result showed that different sizes of the upper part of the talus presented significantly different parameters except the orientation of the lateral cutting plane, indicating that the orientation of the lateral cutting plane may keep consistent for all upper part of the talus and have no relationship with the size. The linear regression analyses demonstrated a weak correlation ( R 2 < 0.5 ) between the majority of parameters and the average radius of the fitting spheres. Therefore, different sizes of the upper part of the talus presented unique morphological features, and the design of different sizes of talar components for TAR should consider the size-specific characteristics of the talus. The parameters measured in this study provided a further understanding of the talus and can guide the design of different sizes of the talar components of the TAR implant.

Capturing Surface Complementarity in Proteins using Unsupervised Learning and Robust Curvature Measure

The structure of a protein plays a pivotal role in determining its function. Often, the protein surface’s shape and curvature dictate its nature of interaction with other proteins and biomolecules. However, marked by corrugations and roughness, a protein’s surface representation poses significant challenges for its curvature-based characterization. In the present study, we employ unsupervised machine learning to segment the protein surface into patches. To measure the surface curvature of a patch, we present an algebraic sphere fitting method that is fast, accurate, and robust. Moreover, we use local curvatures to show the existence of “shape complementarity” in protein-protein, antigen-antibody, and protein-ligand interfaces. We believe that the current approach could help understand the relationship between protein structure and its biological function and can be used to find binding partners of a given protein.

Relation of Computed Tomography-Based Static Axial Radioulnar Incongruence Measurements under General Anaesthesia and Dynamic, In Vivo RUI during the Walk in Canine Elbow Joints with and without Medial Coronoid Process Disease

Objective This study aimed to evaluate the correlation of static axial radioulnar incongruence (sRUI) measured under general anaesthesia with the real in vivo dynamic RUI (dRUI) during walking. Study design This was a prospective clinical study that included 6 sound elbows (5 dogs) and 7 medial coronoid process disease (MCPD) affected elbows (6 dogs). Materials and Methods Static axial radioulnar incongruence was measured using the sphere fitting technique on computed tomography-based three-dimensional (3D) models of radius and ulna. The in vivo pose of radius and ulna was derived from radiostereometric analysis during the walk and transferred onto previously calculated 3D models. Dynamic RUI was measured on those adjusted models using the sphere fitting technique, providing a measurement of RUI over time during walk. Results Mean sRUI was 0.2 mm (standard deviation [SD]: 0.30) in control and 1.4 mm (SD: 0.73) in elbow joints with MCPD; being significantly different (p = 0.0035; confidence interval [CI]: 0.4772–1.8824). Mean dRUI in controls (−0.4 mm; SD: 0.47) was significantly different (p = 0.0004; CI: 0.9918–2.5225) from dRUI in the affected elbows (1.4 mm; SD: 0.73). Comparison of sRUI and dRUI within each group showed difference in the control group (0.2 vs. −0.4 mm; p = 0.0138; CI: 0.1820–1.0014). In affected elbows, no difference between sRUI and dRUI was found (1.4 vs. 1.4 mm; p = 0.8963). Conclusion In normal elbow joints, sRUI does not represent the in vivo condition during weight bearing. Dynamic and slightly negative RUI occurs during loading (0.2 mm positive to −0.4 mm negative RUI). In MCPD affected elbows with sRUI, no dynamic change of RUI occurs during the walk.

Holographic characterization and tracking of colloidal dimers in the effective-sphere approximation

A colloidal dimer scatters laser light to form an in-line hologram that is clearly distinguishable from the hologram of a single sphere. Fitting to an effective-sphere model rapidly measures the dimer's three-dimensional position and orientation.

Sphere Fitting with Applications to Machine Tracking

We suggest a provable and practical approximation algorithm for fitting a set P of n points in R d to a sphere. Here, a sphere is represented by its center x ∈ R d and radius r > 0 . The goal is to minimize the sum ∑ p ∈ P ∣ p − x − r ∣ of distances to the points up to a multiplicative factor of 1 ± ε , for a given constant ε > 0 , over every such r and x. Our main technical result is a data summarization of the input set, called coreset, that approximates the above sum of distances on the original (big) set P for every sphere. Then, an accurate sphere can be extracted quickly via an inefficient exhaustive search from the small coreset. Most articles focus mainly on sphere identification (e.g., circles in 2 D image) rather than finding the exact match (in the sense of extent measures), and do not provide approximation guarantees. We implement our algorithm and provide extensive experimental results on both synthetic and real-world data. We then combine our algorithm in a mechanical pressure control system whose main bottleneck is tracking a falling ball. Full open source is also provided.

A sphere fitting approach to determine the hip joint centre of the horse

Accurate identification of the hip joint centre (HJC) is crucial for the correct estimation of knee and hip joint loads and kinematics, which is particularly relevant in orthopaedic surgery and musculoskeletal modelling. Several methods have been described for calculation of the HJC in humans, however, no studies have used these methods in the horse despite a similar need for improved evaluation of hip joint biomechanics in rehabilitation and musculoskeletal modelling. This preliminary study uses the commonly used functional method (least-squares sphere fit) to determine the HJC in three equid cadavers. Bone pins with reflective markers attached were drilled into the tuber coxae (TC), tuber ischium (TI), tuber sacrale (TS), greater trochanter (GT), third trochanter (TT) and lateral femoral condyle (FC) of the uppermost limb of the cadavers positioned in lateral recumbency. Three repetitions of passive movements consisting of pro-and retraction, ab- and adduction and circumduction were performed. The HJC was calculated using a least-squares sphere fitting method and presented as a distance from the TC based on a percentage of the TC to TI vector magnitude. Mean (± standard deviation) of the HJC is located 52.4% (± 3.9) caudally, 0.2% (± 6.5) dorsally, and 19.8% (± 4.2) medially from the TC. This study is the first to quantify the HJC in horses ex vivo using a functional method. Further work (in vivo and imaging) is required to validate the findings of the present study.

New Objective Refraction Metric Based on Sphere Fitting to the Wavefront

Purpose. To develop an objective refraction formula based on the ocular wavefront error (WFE) expressed in terms of Zernike coefficients and pupil radius, which would be an accurate predictor of subjective spherical equivalent (SE) for different pupil sizes.Methods. A sphere is fitted to the ocular wavefront at the center and at a variable distance,t. The optimal fitting distance,topt, is obtained empirically from a dataset of 308 eyes as a function of objective refraction pupil radius,r0, and used to define the formula of a new wavefront refraction metric (MTR). The metric is tested in another, independent dataset of 200 eyes.Results. For pupil radiir0≤2 mm, the new metric predicts the equivalent sphere with similar accuracy (<0.1D), however, forr0>2 mm, the mean error of traditional metrics can increase beyond 0.25D, and the MTR remains accurate. The proposed metric allows clinicians to obtain an accurate clinical spherical equivalent value without rescaling/refitting of the wavefront coefficients. It has the potential to be developed into a metric which will be able to predict full spherocylindrical refraction for the desired illumination conditions and corresponding pupil size.