Generalization of the normalized dose-response gradient to non-uniform dose delivery

A generalization of the standard dose-response gradient to arbitrarily hete rogeneous dose distributions has been developed. The generalized dose-respo nse gradient is the scalar product of the vector representing the dose dist ribution and the gradient of the dose-response relation with respect to tha t dose vector. It is shown that, for a tumor, the individual gamma -values for each portion of the tumor divided by the corresponding local tumor cont rol probability should be added to get the total value for the heterogeneou sly irradiated tumor. This corresponds to summing up the contributions of a ll tumor volumes so that the total Value of the gradient is related to the logarithm of the total tumor clonogen number. General expressions are also derived for the change in the dose-response relation as a function of a cha nge in the delivered dose distribution.