Evaluating the Propagation of Uncertainties in Biologically Based Treatment Planning Parameters

Abstract

Biologically based treatment planning is a broad term used to cover any instance in radiotherapy treatment planning where some form of biological input has been used. This is wide ranging, and the simpler forms (e.g., fractionation modification/optimization) have been in use for many years. However, there is a reluctance to use more sophisticated methods that incorporate biological models either for plan evaluation purposes or for driving plan optimizations. This is due to limited data available regarding the uncertainties in these model parameters and what impact these have clinically. This work aims to address some of these issues and to explore the role that uncertainties in individual model parameters have on the overall tissue control probability (TCP)/normal tissue control probability (NTCP) calculated, those parameters that have the largest influence and situations where extra care must be taken. In order to achieve this, a software tool was developed, which can import individual clinical DVH's for analysis using a range of different TCP/NTCP models. On inputting individual model parameters, an uncertainty can be applied. Using a normally distributed random number generator, distributions of parameters can be generated, from which TCP/NTCP values can be calculated for each parameter set for the DVH in question. These represent the spread in TCP/NTCP parameters that would be observed for a simulated population of patients all being treated with that particular dose distribution. A selection of clinical DVHs was assessed using published parameters and their associated uncertainties. A range of studies was carried out to determine the impact of individual parameter uncertainties including reduction of uncertainties and assessment of what impact fractionation and dose have on these probabilities.

Keywords: biological optimization; biologically based treatment planning; normal tissue complication probability (NTCP); tumor control probability (TCP); uncertainty.

Figure 1

Data collected for Analysis 1, normal tissue complication probability (NTCP) calculations using the Lyman Kutcher Burman (LKB) model for the rectum. (A–C) The impact on the overall NTCP uncertainties as a result of increasing uncertainty in the individual parameters. Values used for α/β, n, D₅₀, and m were 300 cGy, 0.1, 7,500 cGy, and 0.1, respectively, and the uncertainties applied are expressed as a fraction of each parameter. (D) shows the relationship between D_eff (as a fraction of the D₅₀) and the uncertainty in the final NTCP calculated for different levels of uncertainty in the D₅₀ parameter; lines are for guiding the eye only.

Figure 2

The relationship between the uncertainty in n (A) and m (B) with the overall uncertainty in the NTCP calculated for a selection of patients with ranging values of rectum Deff. Values used for α/β, n, D₅₀, and m were 230 ± 60 cGy, 0.12, 7,600 ± 190 cGy, and 0.15, respectively (18). The uncertainties applied are expressed as a fraction in question and uncertainties are applied to either n or m individually with an uncertainty of zero used for the parameter not being assessed.

Figure 3

(A) The NTCP vs. D_eff curve for grade 2 late toxicity of the rectum (18). Values used for α/β, n, D₅₀, and m were 230 cGy, 0.12, 7,600 cGy, and 0.15, respectively. Simulated rectum NTCP values for a selection of patients with uncertainties of 60 and 190 cGy applied to α/β and D₅₀ parameters, respectively, have been plotted onto the curve for a selection of patients with different D_eff. Each point represents one simulation. (B,C) The simulated results when an additional uncertainty is applied to both m and n of 0.1 (B) and 0.3 (C).

Figure 4

Data collected for Analysis 1, tumor control probability (TCP) calculations using the Lind model for the prostate and prostate PTV. PTV and prostate data displayed in Panel (A, B) respectively for Pt A. Patient B PTV data displayed in Panel (C). Panels show the impact on the overall TCP value as a result of increasing individual parameters. Values used for α/β, D₅₀ and γ were 180 cGy, 4518 cGy and 1.16 respectively and the uncertainties applied are expressed as a fraction of each parameter. Panel (D) shows the relationship between D_eff (as a fraction of the D₅₀) and the uncertainty in the final TCP calculated for different levels of uncertainty in the D₅₀ parameter.

Figure 5

The solid line shows the TCP vs. D_eff curve for the prostate PTV using values of 180 cGy, 4,518 cGy, and 1.16 for α/β, D₅₀, and γ respectively (18). Simulated prostate TCP values for a selection of patients with uncertainties of 47 cGy (26%) and 113 cGy (2.5%) applied to α/β and D₅₀ parameters, respectively, have been plotted onto the curve for a selection of patients with different D_eff. (A) Shows no uncertainty applied to m for the simulated patients and (B,C) show uncertainties in γ of 0.1 and 0.3 respectively. D_eff for each simulation was calculated using the generalized equivalent uniform dose (EUD) formula, with parameter a set to –10.