In most textbooks on nuclear physics1–3 the set of radioactive decay chain equations is solved analytically for a maximum of three nuclides. The general Bateman solution4 is given as a final result or only with a brief mention of the elaborate recursive procedure needed to obtain it.3 Here, a short method for obtaining the general solution is demonstrated.

Adaptive stereotactic body radiation therapy (SBRT) for prostate cancer (PC) by the 1.5 T MR-linac currently requires online planning by an expert user. A fully automated and user-independent solution to adaptive planning (mCycle) of PC-SBRT was compared with user's plans for the 1.5 T MR-linac. Fifty adapted plans on daily magnetic resonance imaging scans for 10 patients with PC treated by 35 Gy (prescription dose [D_p]) in 5 fractions were reoptimized offline from scratch, both by an expert planner (manual) and by mCycle. Manual plans consisted of multicriterial optimization (MCO) of the fluence map plus manual tweaking in segmentation, whereas in mCycle plans, the objectives were sequentially optimized by MCO according to an a-priori assigned priority list. The main criteria for planning approval were a dose ≥95% of the D_p to at least 95% of the planning target volume (PTV), V_33.2 (PTV) ≥ 95%, a dose less than the D_p to the hottest cubic centimeter (V₃₅ ≤ 1 cm³) of rectum, bladder, penile bulb, and urethral planning risk volume (ie, urethra plus 3 mm isotropically), and V₃₂ ≤ 5%, V₂₈ ≤ 10%, and V₁₈ ≤ 35% to the rectum. Such dose-volume metrics, plus some efficiency and deliverability metrics, were used for the comparison of mCycle versus manual plans. mCycle plans improved target dose coverage, with V_33.2 (PTV) passing on average (±1 SD) from 95.7% (±1.0%) for manual plans to 97.5% (±1.3%) for mCycle plans (P < .001), and rectal dose sparing, with significantly reduced V₃₂, V₂₈, and V₁₈ (P ≤ .004). Although at an equivalent number of segments, mCycle plans consumed moderately more monitor units (+17%) and delivery time (+9%) (P < .001), whereas they were generally faster (–19%) in terms of optimization times (P < .019). No significant differences were found for the passing rates of locally normalized γ (3 mm, 3%) (P = .059) and γ (2 mm, 2%) (P = .432) deliverability metrics. In the offline setting, mCycle proved to be a trustable solution for automated planning of PC-SBRT on the 1.5 T MR-linac. mCycle integration in the online workflow will free the user from the challenging online-optimization task.