Study of ill-conditioning of Linac stereotactic irradiation subspaces using singular values decomposition analysis

A Linac stereotactic irradiation space is characterized by different angular separations of beams because of the geometry of the stereotactic irradiation. The regions of the stereotactic space characterized by low angular separations are one of the causes of ill-conditioning of the stereotactic irradiation inverse problem. The singular value decomposition (SVD) is a powerful mathematical analysis that permits the measurement of the ill-conditioning of the stereotactic irradiation problem. This study examines the ill-conditioning of the stereotactic irradiation space, provoked by the different angular separations of beams, using the SVD analysis. We subdivided the maximum irradiation space (MIS: (AA)AP x (AA)RL = 180 degrees x 180 degrees) into irradiation subspaces (ISSs), each characterized by its own angular separation. We studied the influence of ISSs on the SVD analysis and the evolution of the reconstruction quality of well defined three-dimensional dose matrices in each configuration. The more the ISS is characterized by low angular separation the more the condition number and the reconstruction inaccuracy are increased. Based on the above results we created two reduced irradiation spaces (RIS: (AA)AP x (AA)RL = 180 degrees x 140 degrees and (AA)AP x (AA)RL = 180 degrees x 120 degrees) and compared the reconstruction quality of the RISs with respect to the MIS. The more an irradiation space is free of low angular separations the more the irradiation space contains useful singular components.