Applying the equivalent uniform dose formulation based on the linear-quadratic model to inhomogeneous tumor dose distributions: Caution for analyzing and reporting

Abstract

We apply the concept of equivalent uniform dose (EUD) to our data set of model distributions and intensity modulated radiotherapy (IMRT) treatment plans as a method for analyzing large dose inhomogeneities within the tumor volume. For large dose nonuniformities, we find that the linerar-quadratic based EUD model is sensitive to the linear-quadratic model parameters, alpha and beta, making it necessary to consider EUD as a function of these parameters. This complicates the analysis for inhomogeneous dose distributions. EUD provides a biological estimate that requires interpretation and cannot be used as a single parameter for judging an inhomogeneous plan. We present heuristic examples to demonstrate the dose volume effect associated with EUD and the correlation to statistical parameters used for describing dose distributions. From these examples and patient plans, we discuss the risk of incorrectly applying EUD to IMRT patient plans.

Figure 1

Cumulative dose volume histogram (CDVH) examples of clinically delivered Peacock plans. The abscissa is defined as percentage of maximum dose to the tumor. The four dose distributions (CDVH#) are summarized by the averaged equivalent uniform dose (EUD), mean dose (Gy), standard deviation (SD), the prescription dose at per cent of maximum dose, the dose per fraction (Gy/fx), and the minimum dose (shown in Table I).

Figure 2

Generated CDVHs to model Peacock patient plans and characterized by the standard deviation (SD). The maximum dose is equal to 66.7 Gy.

Figure 3

Results from example CDVHs indicating variability in equivalent uniform dose (EUD) with respect to the linear‐quadratic (LQ) parameters, α and β. The ratio of EUD to mean dose is plotted as a function of standard deviation for $\alpha =0.2$, 0.3, and 0.4 with $\alpha /\beta =10$ and 2 Gy per fraction. The mean and minimum dose in Gy are plotted as a function of standard deviation.

Figure 4

Variations in EUD with respect to ratios of $\alpha /\beta =7$, 10, and 13 for $\alpha =0.3$.

Figure 5

Variation in EUD with respect to fractionation dose equal to 1, 2, and 4 Gy per fraction. EUD is calculated with $\alpha /\beta =10$ and $\alpha =0.3$.

Figure 6

Step function examples demonstrating the differences in EUD due to dose‐volume effects. Even though standard deviation is small, EUD may be very small relative to the mean dose. In addition, EUD can have very different values for the same standard deviation. The volume effects on EUD are seen in #S2 and #S4 where the different volumes are associated with the minimum dose. EUD is calculated with $\alpha /\beta =10$ and $\alpha =0.3$.

Figure 7

EUD for sample of Peacock patient plans, calculated for $\alpha =0.3$ and $\alpha /\beta =10$, plotted with standard deviation, mean dose, and minimum dose. EUD correlates better with minimum dose than the other statistical parameters.