The choice of statistical methods for comparisons of dosimetric data in radiotherapy

Abstract

Purpose: Novel irradiation techniques are continuously introduced in radiotherapy to optimize the accuracy, the security and the clinical outcome of treatments. These changes could raise the question of discontinuity in dosimetric presentation and the subsequent need for practice adjustments in case of significant modifications. This study proposes a comprehensive approach to compare different techniques and tests whether their respective dose calculation algorithms give rise to statistically significant differences in the treatment doses for the patient.

Methods: Statistical investigation principles are presented in the framework of a clinical example based on 62 fields of radiotherapy for lung cancer. The delivered doses in monitor units were calculated using three different dose calculation methods: the reference method accounts the dose without tissues density corrections using Pencil Beam Convolution (PBC) algorithm, whereas new methods calculate the dose with tissues density correction for 1D and 3D using Modified Batho (MB) method and Equivalent Tissue air ratio (ETAR) method, respectively. The normality of the data and the homogeneity of variance between groups were tested using Shapiro-Wilks and Levene test, respectively, then non-parametric statistical tests were performed. Specifically, the dose means estimated by the different calculation methods were compared using Friedman's test and Wilcoxon signed-rank test. In addition, the correlation between the doses calculated by the three methods was assessed using Spearman's rank and Kendall's rank tests.

Results: The Friedman's test showed a significant effect on the calculation method for the delivered dose of lung cancer patients (p <0.001). The density correction methods yielded to lower doses as compared to PBC by on average (-5 ± 4.4 SD) for MB and (-4.7 ± 5 SD) for ETAR. Post-hoc Wilcoxon signed-rank test of paired comparisons indicated that the delivered dose was significantly reduced using density-corrected methods as compared to the reference method. Spearman's and Kendall's rank tests indicated a positive correlation between the doses calculated with the different methods.

Conclusion: This paper illustrates and justifies the use of statistical tests and graphical representations for dosimetric comparisons in radiotherapy. The statistical analysis shows the significance of dose differences resulting from two or more techniques in radiotherapy.

Figure 1

Density plots showing the deviation of normality and the positive skewness (i.e., clustering at the lower end (left) of the scale). Negative skewness would have shown clustering at the higher end of the scale. A normal curve using the mean and standard deviation of the data as parameters has been drawn on top of the histograms to show the deviation of normality.

Figure 2

Flowchart of statistical methods for parametric data.

Figure 3

Flowchart of statistical methods for non-parametric data.

Figure 4

Boxplot indicating the minimum and maximum values, the 25th percentile, 50th percentile (median) and 75th percentile.

Figure 5

Frequency distribution of dose differences between PBC with MB (left) and ETAR (right). For the majority of beams, the calculated dose is lower with MB and ETAR than with PBC.

Figure 6

The bars show the mean dose that would be delivered using the different methods. Error bars indicate the 95% confidence interval adjusted for repeated-measures.

Figure 7

p-values estimated by bootstrap procedure, indicating the average p-value for each sample-sizes going from 5 to 62. The left panel corresponds to the comparison between MB and PBC, and right panel to the comparison between ETAR and PBC. The red dashed line corresponds to a significance threshold of 0.05 and the blue dashed line to an adjusted significance threshold of 0.025 (0.05/2 to correct for two comparisons).

Figure 8

Statistical power as a function of sample size using Wilcoxon test.

Figure 9

Correlation between PBC with MB (left) and ETAR (right). The line is computed using a least square regression method, and is defined by its slope b₁ and intercept b₀ as: Y_i = b₀ + b₁ X _i, where Y_i is the outcome, X _i is the i ^th participant’s score of the predictor variable.