Determination of the point position for brachytherapy from the shift-projection method using the least squares method and geometrical solutions

We have derived formulas by the least squares method and six sets of geometrical solutions for calculating the position for brachytherapy from shift-projection images on one or two films. There are four types of expression for each coordinate in both methods, the shift-film technique with double exposures on a film and the stereo-shift technique with two films. Some geometrical solutions for the shift-film technique are equal to well-known conventional formulas. In the case of the stereo-shift technique, each conventional formula for the two coordinates is equal to the mean of the two geometrical solutions of each coordinate, and the one for the other coordinate is equal to one of the geometrical solutions. Formulas for the shift-film technique can be easily reduced to those for the stereo-shift technique. The error of the position calculated by the least squares method was the smallest among all the formulas for the shift-film technique in simulation.