A new fluoroscopic view for evaluation of anteromedial cortex reduction quality during cephalomedullary nailing for intertrochanteric femur fractures: the 30° oblique tangential projection

Background Anteromedial cortex-to-cortex reduction is a key parameter for stable reconstruction of the fracture fragments during the intertrochanteric fracture fixation. This paper introduces the oblique fluoroscopic projection as a novel method to evaluate the quality of anteromedial cortical apposition. Methods Three proximal femur specimens were marked with steel wires along five anatomic landmarks: Greater trochanter, Lesser trochanter, Intertrochanteric line, Anterolateral tubercle and the Anteromedial cortical line. After obtaining the standard femoral neck AP and lateral fluoroscopic images, the C-arm was rotated by every 5°increments until a clear tangential view of the antero-medial-inferior corner cortex was observed. 98 cases of intertrochanteric hip fractures were enrolled from April 2018 to October 2019. After fixation with the nails, the intra-operative anteromedial cortex reduction quality was evaluated from the AP, the true lateral, and the new anteromedial oblique fluoroscopic images. The fluoroscopic results were compared with the post-operative 3D-CT reconstruction images. Results The specimen study showed that internal rotation of the C-arm to approximately 30 ° can remove all the obscure shadows and clearly display the antero-medial-inferior cortical tangent line. Clinically,the positive, neutral and negative apposition of different cortices via intra-operative fluoroscopic images showed79, 19 and 0 cases of medial cortical apposition in AP views; 2, 68 and 28 cases of anterior cortices in lateral views;and 22, 51 and 25cases of anteromedial cortical apposition in oblique views respectively. The post-operative 3D-CT reconstruction images revealed that the final anteromedial cortical contact was noted in 62 cases (63.3%), and lost in 36 cases (36.7%). The overall coincidence rate between intra-operative fluoroscopy and post-operative 3D-CT was 63.3% (62/98) in AP view,79.6% (78/98) in lateral view, and 86.7% (85/98) in oblique view(p < 0.001). Negative cortical apposition in oblique view was highly predictive of a final loss of cortical support on 3D CT (24/25 cases, 96%).And non-negative cortical apposition in oblique view was highly associated with true cortical support on 3D CT images (61/73 cases, 83.6%) (p < 0.001). Conclusions Besides the AP and lateral projections, an anteromedial oblique view of 30° certifies to be a very useful means for evaluation of the fracture reduction quality of anteromedial cortical apposition.

MINIMAL DECOMPOSITION OF BINARY FORMS WITH RESPECT TO TANGENTIAL PROJECTIONS

Let C ⊂ ℙn+1 be a rational normal curve and let X ⊂ ℙn be one of its tangential projection. We describe the X-rank of a point P ∈ ℙn in terms of the schemes evincing the C-rank or the border C-rank of the preimage of P.

On Sha's Secondary Chern–Euler Class

For a manifold with boundary, the restriction of Chern's transgression form of the Euler curvature form over the boundary is closed. Its cohomology class is called the secondary Chern– Euler class and was used by Sha to formulate a relative Poincaré–Hopf theorem under the condition that the metric on the manifold is locally product near the boundary. We show that the secondary Chern–Euler form is exact away from the outward and inward unit normal vectors of the boundary by explicitly constructing a transgression form. Using Stokes’ theorem, this evaluates the boundary term in Sha's relative Poincaré–Hopf theorem in terms of more classical indices of the tangential projection of a vector field. This evaluation in particular shows that Sha's relative Poincaré–Hopf theorem is equivalent to the more classical law of vector fields.

A Projection Method Approach to Constrained Dynamic Analysis

The paper presents a unified approach to the dynamic analysis of mechanical systems subject to (ideal) holonomic and/or nonholonomic constraints. The approach is based on the projection of the initial (constraint reaction-containing) dynamical equations into the orthogonal and tangent subspaces; the orthogonal subspace which is spanned by the constraint vectors, and the tangent subspace which complements the orthogonal subspace in the system’s configuration space. The tangential projection gives the reaction-free (or purely kinetic) equations of motion, whereas the orthogonal projection determines the constraint reactions. Simplifications due to the use of independent variables are indicated, and examples illustrating the concepts are included.

The random tangential projection of a surface

Suppose a smooth surface S is exposed to a penetrating beam of parallel line rays and that, wherever a ray has a tangency with S, there is a corresponding point registration in an image plane H placed perpendicular to the beam. The registering rays form a cylindrical surface which registers in H as a smooth curve C (except possibly for a number of cusps). Properties of C when the beam is taken isotropic random in space are investigated, and stereological applications are noted. The phenomenon of screening of projecting rays, which occurs for example with opaque surfaces exposed to light beams, is considered. Limiting processes permit extensions of the formulae to the cases in which S contains curved edges and in which any boundary curves of S also register. Finally, various related types of random tangent to a surface are considered.

The random tangential projection of a surface

Suppose a smooth surface S is exposed to a penetrating beam of parallel line rays and that, wherever a ray has a tangency with S, there is a corresponding point registration in an image plane H placed perpendicular to the beam. The registering rays form a cylindrical surface which registers in H as a smooth curve C (except possibly for a number of cusps). Properties of C when the beam is taken isotropic random in space are investigated, and stereological applications are noted. The phenomenon of screening of projecting rays, which occurs for example with opaque surfaces exposed to light beams, is considered. Limiting processes permit extensions of the formulae to the cases in which S contains curved edges and in which any boundary curves of S also register. Finally, various related types of random tangent to a surface are considered.