Mathematical Proof that Simultaneous Irradiation is Not Optimal for Two Tumors

When fractionated external beam radiotherapy is used for more than one target (in the following, it shall be two) near a critical organ at risk (OAR), a radiation plan that simultaneously irradiates two targets is often employed. However, we prove mathematically that this plan is not optimal in the sense that the biological effect (BE) on the OAR is minimized. Based on the assumption that the OAR receives a fraction of the dose intended for the tumor, the minimization problem for the BE on the OAR was treated under the constraint that the radiation effects on the tumors are fixed. This problem is the maximum determination of constrained optimization problems. We use the bordered Hessian matrix for this discussion. Simple linear quadratic model was employed for the radiation effects. Simultaneous irradiation, the concurrent delivery of radiation to two tumors per session (e.g., 30 fractions of 2 Gy each) is mathematically shown to be suboptimal. Alternative irradiation strategies yielded superior outcomes in the simulation using real-world data. When two tumors near/in one OAR are targeted, there are non-simultaneous irradiation schedule which can reduce radiation-induced damage to the OAR more efficiently than simultaneous irradiation keeping the same biological effect to the tumor.