A Mathematical Method for Calculating the Anterior and Superior Axial Coverage Angle of Acetabulum to Femoral 5 Head based on 3D Coordinate System

Abstract

Background: Direct imaging measurement of the anterior and superior axial coverage angle of acetabulum to femoral head is difficult. This study aimed to build the mathematical formulas to calculate them using the available parameters, e.g., acetabular abduction angle. Results: A 3D coordinate system was established by setting the center of the hip joint rotation basing as coordinate center O. In the drawn diagrams of anterior and superior axial coverage angle of acetabulum to femoral head, femoral neck anteversion was set as a, acetabular abduction angle as b, acetabular anteversion as A, and femoral neck-shaft angle as J. In the absence of the anteversion of acetabular and femoral neck, the superior axis coverage angle of acetabulum to the femoral head equals to b + 180-J. The anterior axial coverage angle of acetabulum to femoral head equals to 90 degrees. In the presence of the acetabular and femoral neck anteversion, the superior axis coverage angle of acetabulum to the femoral head equals to arctg {tgb / [cosacos (A + a)]} + 180-J; the anterior axis coverage angle of acetabulum to the femoral head equals to 90- arctg [tg (A + a) / cos {arctg [tgb / (cosa + a)] - (J-90)}]. The close relationship between the acetabulum and femoral head spatial orientation angle was observed in the absence of acetabular and femoral neck anteversion. The superior axis coverage angle of acetabulum to the femoral head was positively correlated with acetabular abduction angle and negatively with femoral neck-shaft angle. While the anterior axial coverage angle of acetabulum to femoral head was not dependent on acetabular abduction angle and femoral neck-shaft angle. In the presence of acetabular and femoral neck anteversion, both anterior and superior axial coverage angle of acetabulum to femoral head were relevant to acetabular abduction angle and femoral neck-shaft angle. The superior axial coverage angle of acetabulum to femoral head positively correlated with acetabular abduction angle, negatively with femoral neck-shaft angle, and positively with acetabular and femoral neck anteversion. The anterior axial coverage angle of acetabulum to femoral head was negatively correlated with acetabular abduction angle, positively with femoral neck-shaft angle and negatively with acetabular and femoral neck anteversion. Conclusions: The formulas were built based on a 3D coordinate system established in the current study. The formula reflects the inner relationship of spatial orientation angle of acetabulum to femoral head, which will offer a simple, novel and efficient mathematical method to calculate the anterior and superior axial coverage angle.