Optimizing normal tissue sparing via spatiotemporal optimization under equivalent tumor-radical efficacy.

Objective: Spatiotemporal optimization in radiation therapy involves determining the optimal number of dose delivery fractions (temporal) and the optimal dose per fraction (spatial). Traditional approaches focus on maximizing the biologically effective dose (BED) to the target while constraining BED to organs-at-risk (OAR), which may lead to insufficient BED for complete tumor cell kill. This work proposes a formulation that ensures adequate BED delivery to the target while minimizing BED to the OAR. Approach: A spatiotemporal optimization model is developed that incorporates an inequality constraint to guarantee sufficient BED for tumor cell kill while minimizing BED to the OAR. The model accounts for tumor proliferation dynamics, including lag time (delay before proliferation begins) and doubling time (time for tumor volume to double), to optimize dose fractionation. Main results: The proposed formulation is implemented for proton modality. The performance of our method is evaluated for varying lag times and doubling times. The results show that the mean BED to the target consistently meets the minimum requirement for tumor cell kill. Additionally, the mean BED to the OAR varies based on tumor proliferation dynamics. In the prostate case with lag time of 7 days and doubling time of 2 days, it is observed that the mean BED delivered to femoral head is lowest at approximately 20 fractions, making this an optimal choice. While in the head-and-neck case, the mean BED to the OAR decreases as the number of fractions increases, suggesting that a higher number of fractions is optimal. Thus, the proposed model effectively determines the optimal fractionation strategy under different tumor proliferation conditions. Significance: A spatiotemporal optimization model is presented that minimizes BED to the OAR while ensuring sufficient BED for tumor cell kill. By incorporating tumor lag time and doubling time, the approach identifies optimal number of fractions. This model can be extended to support hyperfractionation or accelerated fractionation strategies, offering a versatile tool for clinical treatment planning.