A mathematical simulation of the tip-apex distance and the calcar-referenced tip-apex distance for intertrochanteric fractures reduced with lag screws

BackgroundAs a predictor of the risk of lag screw cutout, it was recommended that keeping tip-apex distance (TAD) < 25 mm and placing the screw centrally or inferiorly, but positioning the lag screw too inferiorly in the head would produce TAD > 25 mm. We aim to simulate various positions of the lag screw in the femoral head and identify whether 25 mm is a suitable cut-off value that favours all sizes of femoral heads with intertrochanteric fractures of the hip.MethodsUsing a general mathematical software, the positions of the screw tip points were simulated. The virtual anterior–posterior and lateral views were then visualised, and the locus of the screw tips was projected into a Cartesian coordinate system according to the TAD and calcar-referenced tip-apex distance (CalTAD) formulas. Each original virtual anterior–posterior and lateral image was zoomed and compiled to match a calculated average image. The screw tip points were recorded, traced and compiled into volumes which could be used to visualise the screw's movements and positioning within the femoral head. The extracted volumes were calculated when 10 mm < TAD < 25 mm and 10 mm < CalTAD < 25 mm, and the region where these two volumes overlapped was also calculated. Suitable positions for the screw tip were then assessed.ResultsFor the TAD calculation, the shape of the traced screw tip points had a pancake-like appearance, while the CalTAD plot produced a teardrop-shaped region. The volume ratios of TAD, CalTAD and overlapping region relative to the femoral head volume were respectively 3.51 ± 1.30%, 5.19 ± 1.62% and 2.64 ± 1.32%. The volumes of the traced TAD, CalTAD and overlapping regions increased slower than the volume of an idealised sphere.ConclusionPositioning the lag screw should address geometrical effects of both tip-apex distance and femoral head size, with an emphasis on measuring the position of the screw tip for the suitable zone by volume ratio. The previous 25 mm TAD cut-off value should be adjusted according to the individual femoral head size.