Generalization of the Normalized Dose-response Gradient to Non-uniform Dose Delivery

A generalization of the standard dose-response gradient to arbitrarily heterogeneous dose distributions has been developed. The generalized dose-response gradient is the scalar product of the vector representing the dose distribution and the gradient of the dose-response relation with respect to that dose vector. It is shown that, for a tumor, the individual &#110 -values for each portion of the tumor divided by the corresponding local tumor control probability should be added to get the total value for the heterogeneously irradiated tumor. This corresponds to summing up the contributions of all 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 change in the delivered dose distribution.