The standard linear-quadratic (LQ) survival model for external beam radiotherapy is reviewed with particular emphasis on studying how different schedules of radiation treatment planning may be affected by different tumour repopulation kinetics. The LQ model is further examined in the context of tumour control probability (TCP) models. The application of the Zaider and Minerbo non-Poissonian TCP model incorporating the effect of cellular repopulation is reviewed. In particular the recent development of a cell cycle model within the original Zaider and Minerbo TCP formalism is highlighted. Application of this TCP cell-cycle model in clinical treatment plans is explored and analysed.
The standard linear-quadratic survival model for radiotherapy is used to investigate different schedules of radiation treatment planning to study how these may be affected by different tumour repopulation kinetics between treatments. The laws for tumour cell repopulation include the logistic and Gompertz models and this extends the work of Wheldon et al (1977 Br. J. Radiol. 50 681), which was concerned with the case of exponential re-growth between treatments. Here we also consider the restricted exponential model. This has been successfully used by Panetta and Adam (1995 Math. Comput. Modelling 22 67) in the case of chemotherapy treatment planning. Treatment schedules investigated include standard fractionation of daily treatments, weekday treatments, accelerated fractionation, optimized uniform schedules and variation of the dosage and alpha/beta ratio, where alpha and beta are radiobiological parameters for the tumour tissue concerned. Parameters for these treatment strategies are extracted from the literature on advanced head and neck cancer, prostate cancer, as well as radiosensitive parameters. Standardized treatment protocols are also considered. Calculations based on the present analysis indicate that even with growth laws scaled to mimic initial growth, such that growth mechanisms are comparable, variation in survival fraction to orders of magnitude emerged. Calculations show that the logistic and exponential models yield similar results in tumour eradication. By comparison the Gompertz model calculations indicate that tumours described by this law result in a significantly poorer prognosis for tumour eradication than either the exponential or logistic models. The present study also shows that the faster the tumour growth rate and the higher the repair capacity of the cell line, the greater the variation in outcome of the survival fraction. Gaps in treatment, planned or unplanned, also accentuate the differences of the survival fraction given alternative growth dynamics.
We argue that the results published by Ai [Phys. Rev. E 67, 022903 (2003)] on "correlated noise in logistic growth" are not correct. Their conclusion that, for larger values of the correlation parameter lambda , the cell population is peaked at x=0, which denotes a high extinction rate, is also incorrect. We find the reverse behavior to their results, that increasing lambda promotes the stable growth of tumor cells. In particular, their results for the steady-state probability, as a function of cell number, at different correlation strengths, presented in Figs. 1 and 2 of their paper show different behavior than one would expect from the simple mathematical expression for the steady-state probability. Additionally, their interpretation that at small values of cell number the steady-state probability increases as the correlation parameter is increased is also questionable. Another striking feature in their Figs. 1 and 3 is that, for the same values of the parameters lambda and alpha, their simulation produces two different curves, both qualitatively and quantitatively.
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