Abstract
Intensity-modulated radiation therapy (IMRT) is a type of external beam radiation therapy used in cancer treatment. In IMRT, the prescribed radiation dose can be administered such that it is maximized on the cancerous tumor while sparing the surrounding healthy tissues. The total dose is divided into fractions across time intervals, called a fractionation scheme. To find the best fractionation scheme and beamlet intensities for total dose, optimization models are used. In this paper, a non-convex mixed-integer nonlinear programming model has been proposed wherein the spatiotemporal changes of the biological properties of the tumor due to tumor cell re-oxygenation, redistribution, and re-population that occur as the treatment progresses have been considered. Also, the dose constraints over both cumulative limits and per-fraction limits have been considered in the model. The output of this model is called the fractionation scheme and beamlet intensities considering biological changes in tumor cells (FBBTs). When the FBBTs are compared with conventional fractionation scheme and beamlet intensities (CFB) which do not include the biological properties of the tumor, it is observed that the FBBTs are more efficacious than the CFBs. To get FBBTs for datasets that resemble realistic tumors, an algorithm based on simulated annealing has been developed and used.
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Appendices
Appendix 1
Analysis of the scaling factor s
The instances of small datasets that could be solved to optimality by the BARON solver are used to find the effect of changing values of scaling factor s on the objective function values of the optimal FBBTs (Table 9).
The values of the objective functions do not change for any of the instances with the changes in the s values. This suggests that the optimal FBBTs can be obtained using any value of s.
Appendix 2
Detailed result of instance number 2 in Table 2 of small datasets
The first attempt was to find the FBBT by the BARON solver and get an exact solution. The dose deposition values and the number of tumor cells left over after every treatment session are given in Table 10. It can be seen that as tumor point T3 is heterogeneous with a higher number of tumor cells at the start, the total dose deposited on the same in the entire treatment is higher than that on the other tumor points.
After the 6th treatment session, there are still tumor cells left that are larger in number than the acceptable limit b. Hence, N is updated to \(N + 5\). But the solver does not give any solution in the computational time limit of 2 h. So the data of the instance are passed to the model based on SA with \(N = 11\). With the parameters mentioned in “FBBT of small datasets” section, a solution is obtained in 1.53 h but with number of tumor cells left over after treatment greater than b. Again the SA model is solved with \(N = 16\). Table 11 gives the dose depositions and the number of tumor cells left over after every treatment session. Although the obtained FBBT is near optimal, the solution time is 1.49 h and number of tumor cells left over after the 14th treatment session is lesser than b which is the actual target of the treatment planning. Hence, only 14 treatment sessions are conducted in all. Here again the total dose deposited on the tumor point T3 is higher than that on the other tumor points.
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Ghatkar, S. Optimization of fractionation schemes and beamlet intensities in intensity-modulated radiation therapy with changing cancer tumor properties. Decision 46, 385–407 (2019). https://doi.org/10.1007/s40622-019-00229-2
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DOI: https://doi.org/10.1007/s40622-019-00229-2