Self-consistent tumor control probability and normal tissue complication probability models based on generalized EUD

Traditional methods to compute the tumor control probability (TCP) or normal tissue complication probability (NTCP) typically require a heterogeneous radiation dose distribution to be converted into a simple uniform dose distribution with an equivalent biological effect. Several power-law type dose-volume-histogram reduction schemes, particularly Niemierko's generalized equivalent uniform dose model [Med. Phys. 26, 1000 (1999)], have been proposed to achieve this goal. In this study, we carefully examine the mathematical outcome of these schemes. We demonstrate that (1) for tumors, with each tumor cell independently responding to local radiation dose, a closed-form analytical solution for tumor survival fraction and TCP can be obtained; (2) for serial structured normal tissues, an exponential power-law form relating survival to functional sub-unit (FSU) radiation is required, and a closed-form analytical solution for the related NTCP is provided; (3) in the case of a parallel structured normal tissue, when NTCP is determined solely by the number of the surviving FSUs, a mathematical solution is available only when there is a non-zero threshold dose and/or a finite critical dose defining the radiotherapy response. Some discussion is offered for the partial irradiation effect on normal tissues in this category; (4) for normal tissues with alternative architectures, where the radiation response of FSU is inhomogeneous, there is no exact global mathematical solution for SF or NTCP within the available schemes. Finally, numerical fits of our models to some experimental data are also presented.