Modeling dose response in the presence of spatial variations in dose rate

Nonuniform dose rates are an inevitability in treatments involving internal sources, arising from electronic disequilibrium effects as well as nonuniformity in activity distribution. These dose-rate nonuniformities are of consequence for protracted treatments (when dose-delivery times are of the order of cell-repair times). The influence of nonuniform dose rates on tumor control probability (TCP) has thus been considered. A model for TCP has been developed by merging established (linear-quadratic based) TCP models for dose nonuniformity, with dose-rate effects as influenced by cell repair and proliferation capacities. This model has been examined by considering treatment of spherical tumors of varying sizes filled with uniform distributions of several beta-emitting isotopes. Dose (or dose-rate) volume histograms (DVHs) were calculated for the combinations of tumor size and isotope, and applied to the developed TCP model. Comparison of the results identified several characteristics of the effect of nonuniform dose rate, including the balance between minimum dose and cell number as they vary with tumor size, the dominance of minimum dose (dose rate) on TCP, and the influence of cell-proliferation effects on effective delivered dose (and the effective DVH). The model was also used to determine TCPs for simulated 90Y-labeled microsphere treatments of liver metastases using both uniform and clustered-microsphere models for activity distributions, and for varying tumor size. Despite significantly higher doses being achieved via clustered (nonuniform) activity distributions, the minimum dose for clustered distributions is consistently lower than that of the corresponding uniform distributions, and TCP is always higher for the uniform distributions.