Reverse mapping of normal tissue complication probabilities onto dose volume histogram space: the problem of randomness of the dose volume histogram sampling

A very important issue in contemporary inverse treatment radiotherapy planning is the specification of proper dose-volume constraints limiting the treatment planning algorithm from delivering high doses to the normal tissue surrounding the tumor. Recently we have proposed a method called reverse mapping of normal tissue complication probabilities (NTCP) onto dose-volume histogram (DVH) space, which allows the calculation of appropriate biologically based dose-volume constraints to be used in the inverse treatment planning. The method of reverse mapping requires random sampling from the functional space of all monotonically decreasing functions in the unit square. We develop, in this paper, a random function generator for the purpose of the reverse mapping. Since the proposed generator is based on the theory of random walk, it is therefore designated in this work, as a random walk DVH generator. It is theoretically determined that the distribution of the number of monotonically decreasing functions passing through a point in the dose volume histogram space follows the hypergeometric distribution. The proposed random walk DVH generator thus simulates, in a random fashion, trajectories of monotonically decreasing functions (finite series) that are situated in the unit square [0, 1] X [1,0] using the hypergeometric distribution. The DVH generator is an important tool in the study of reverse NTCP mapping for the calculation of biologically based dose-volume constraints for inverse treatment planning.