We would like to comment on the recent paper published by . These authors have suggested a modification to the linear-quadratic (LQ) model in order to better describe the cellular dose response at large dose per fraction. Because the LQ model predicts a dose response whose slope increases with dose, cell lines that exhibit a constant final slope tend to have a poor fit to the LQ model at higher doses. The modification of the LQ model that Guerrero and Li have proposed is a welcome and useful progression of the LQ formalism. The manner in which Guerrero and Li introduced this modification, however, was without a mechanistic justification, and could appear to be arbitrary. The purpose of this letter is to show that Guerrero and Li's modification to the LQ model does have a mechanistic justification, and to show that the mechanistic basis is where sublethal lesion interaction is not considered insignificant as compared to sublethal lesion repair, and further that sublethal lesion interaction is modelled as a linear process.We have previously shown (Carlone 2004, Carlone et al 2003 that the LQ model can be derived using the compartmental model shown in figure 1. In this figure, the quantity A represents the number of potential target sites in a cell; B is the number of sites within the cell where sub-lethal damage has been created; and C is the number of sites where irreparable damage exists. When a cell is irradiated, the production of B lesions occurs at a rate of 2p multiplied by R, the dose rate. The constant p is defined as the yield per unit dose of sub-lethal lesions 1 (Dale 1985). In figure 1, the depletion of sublethal lesions by interaction is considered to be small as compared to repair, and so lesion formation can be considered to be two separate components as shown in figure 1. Assuming exponential repair of sublethal lesions, as shown on the left of figure 1, the steady state value of B is determined by solving:The solution for B(t) is:Next formation of irreparable lesions is considered. This is accomplished by either of the two pathways shown on the right of figure 1. Non-repairable lesions are directly produced at 1 The multiplier 2 is included because there are two potential target pairs required for sub-lethal lesions to interact. This factor was originally introduced when single strand lesions were thought to interact to form an non-repairable double strand lesion. Thus there were 2 × A potential sites for a sublethal lesion. It could be omitted from the current derivation; however it is left here to preserve the derivation of .
Radiobiological parameter estimates for prostate cancer are obtained from both a homogeneous (individual) and heterogeneous (population) tumor control model based on Poisson statistics and the linear quadratic model of cell survival. Parameter estimates for both models are highly correlated: statistically equivalent fits are achievable using either (1) linear quadratic (LQ) parameters with low numbers of radioresistant tumor stem cells, or (2) LQ parameters with corresponding larger number of radiosensitive tumor stem cells. A theoretical framework is developed to explain this correlation. A Monte Carlo error analysis based on binomial statistics is used to estimate confidence intervals for all parameter estimates. It was found that both the homogeneous and heterogeneous models produce approximately equivalent estimates of radiobiological parameters, including the alpha/beta ratio. However, the 95% confidence interval for the alpha/beta ratio derived from the heterogeneous model are considerably larger than those derived from the homogeneous model, which indicate the homogeneous model overestimates the statistical significance of the alpha/beta estimate.
A formula for the alpha/beta ratio is derived using the heterogeneous (population averaged) tumor control model. This formula is nearly identical to the formula obtained using the homogeneous (individual) tumor control model, but the new formula includes extra terms showing that the alpha/beta ratio, the ratio of the mean value of a divided by the mean value of beta that would be observed in a patient population, explicitly depends on the survival level and heterogeneity. The magnitude of this correction is estimated for prostate cancer, and this appears to raise the mean value of the ratio estimate by about 20%. The method also allows investigation of confidence limits for alpha/beta based on a population distribution of radiosensitivity. For a widely heterogeneous population, the upper 95% confidence interval for the alpha/beta ratio can be as high as 7.3 Gy, even though the population mean is between 2.3 and 2.6 Gy.
Irradiation of longitudinally adjacent PTVs with Helical TomoTherapy (HT) may be clinically necessary, for example in treating a recurrent PTV adjacent to a previously‐treated volume. In this work, the parameters which influence the cumulative dose distribution resulting from treating longitudinally adjacent PTVs are examined, including field width, pitch, and PTV location. In‐phantom dose distributions were calculated for various on‐ and off‐axis cylindrical PTVs and were verified by ion chamber and film measurement. Dose distributions were calculated to cover 95% of the PTV by the prescribed dose (DP) using 25 and 50 mm long HT fields with pitches of either 0.3 or 0.45. These dose distributions where then used to calculate the 3D dose distribution in the junction region between two PTVs. The best junction uniformity was obtained for fields of equal width, with larger fields providing better intra‐PTV dose homogeneity than smaller fields. Junctioning fields of different widths resulted in a much larger dose inhomogeneity, but this could be improved significantly by dividing the junction end of the PTV treated with the smaller field into multiple (up to 4) sub‐PTVs, with the prescribed dose in each sub‐PTV decreasing with proximity to the junction region. This provided a PTV matching with dose homogeneity similar to that achieved when junctioning two PTVs, both irradiated by the 50 mm field, and provided a distribution where 95% of the PTV received at least the prescribed dose, with maximum excursions from prescribed dose varying from −19% to +13%. We conclude that junctioning adjacent PTVs is possible. Treating longitudinally adjacent PTVs with different widths is a challenge, but dose uniformity is improved by breaking PTVs into multiple contiguous sub‐PTVs modified to feather (broaden) the effective junctioning region.PACS number: 87.55.D
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