^{1,a)}

### Abstract

**Purpose:**

To characterize a class of optimization formulations used to handle systematic and random errors in radiation therapy, and to study the differences between the methods within this class.

**Methods:**

The class of robust methods that can be formulated as minimax stochastic programs is studied. This class generalizes many previously used methods, ranging between optimization of the expected and the worst case objective value. The robust methods are used to plan intensity-modulated proton therapy (IMPT) treatments for a case subject to systematic setup and range errors, random setup errors with and without uncertain probability distribution, and combinations thereof. As reference, plans resulting from a conventional method that uses a margin to account for errors are shown.

**Results:**

For all types of errors, target coverage robustness increased with the conservativeness of the method. For systematic errors, best case organ at risk (OAR) doses increased and worst case doses decreased with the conservativeness. Accounting for random errors of fixed probability distribution resulted in heterogeneous dose. The heterogeneities were reduced when uncertainty in the probability distribution was accounted for. Doing so, the OAR doses decreased with the conservativeness. All robust methods studied resulted in more robust target coverage and lower OAR doses than the conventional method.

**Conclusions:**

Accounting for uncertainties is essential to ensure plan quality in complex radiation therapy such as IMPT. The utilization of more information than conventional in the optimization can lead to robust target coverage and low OAR doses. Increased target coverage robustness can be achieved by more conservative methods.

The author thanks Anders Forsgren and Björn Hårdemark for valuable comments on the paper. The research was supported by the Swedish Research Council (VR).

I. INTRODUCTION

II. METHODS

II.A. Uncertainties

II.B. Notation

II.C. Optimization functions

II.D. Accounting for systematic errors

II.D.1. Expected value optimization

II.D.2. Worst case optimization

II.D.3. Conditional value at risk optimization

II.D.4. Minimax stochastic programming

II.E. Accounting for random errors

II.F. Combining systematic and random errors

II.G. Patient geometry

II.H. Computational study

III. RESULTS

III.A. Systematic errors

III.B. Random errors with fixed probability distribution

III.C. Random errors with uncertain standard deviation

III.D. Systematic errors and random errors with fixed probability distribution

III.E. Systematic errors and random errors with uncertain standard deviation

IV. DISCUSSION

V. CONCLUSION

### Key Topics

- Probability theory
- 30.0
- Optimization
- 29.0
- Dosimetry
- 14.0
- Radiation therapy
- 7.0
- Radiation treatment
- 6.0

##### A61N5/00

## Figures

The C-shaped geometry. The solid line indicates where the line doses are taken and the dashed lines indicate the beam directions. The radius of the inner arc of the target is 1.5 cm and that of the outer arc is 5 cm. The OAR has radius 1cm and the external ROI has radius 8 cm.

The C-shaped geometry. The solid line indicates where the line doses are taken and the dashed lines indicate the beam directions. The radius of the inner arc of the target is 1.5 cm and that of the outer arc is 5 cm. The OAR has radius 1cm and the external ROI has radius 8 cm.

Color table indicating the dose levels for the dose distributions. The tick labels denote percentage of the reference dose level, which is 1 for total doses and 0.5 for beam doses.

Color table indicating the dose levels for the dose distributions. The tick labels denote percentage of the reference dose level, which is 1 for total doses and 0.5 for beam doses.

Total doses for the robust methods for systematic errors. (a)–(c) Nominal scenario; and (d)–(f) isocenters shifted 0.5 cm to the right and density two standard deviations lower than measured.

Total doses for the robust methods for systematic errors. (a)–(c) Nominal scenario; and (d)–(f) isocenters shifted 0.5 cm to the right and density two standard deviations lower than measured.

Beam doses for the robust methods for systematic errors. (a)–(f) Nominal scenario; and (g)–(l) isocenters shifted 0.5 cm to the right and density two standard deviations lower than measured.

Beam doses for the robust methods for systematic errors. (a)–(f) Nominal scenario; and (g)–(l) isocenters shifted 0.5 cm to the right and density two standard deviations lower than measured.

DVH families for the robust methods for systematic errors over the 89 systematic error scenarios. The dashed lines correspond to the nominal scenario DVHs.

DVH families for the robust methods for systematic errors over the 89 systematic error scenarios. The dashed lines correspond to the nominal scenario DVHs.

(a) Worst (lowest target , highest OAR ), mean, and best (highest target , lowest OAR ) case trade-off curves for the target importance weight in [100, 1000]; and (b) DPHs. Both figures are resulting from the robust methods for systematic errors evaluated over the 89 systematic error scenarios.

(a) Worst (lowest target , highest OAR ), mean, and best (highest target , lowest OAR ) case trade-off curves for the target importance weight in [100, 1000]; and (b) DPHs. Both figures are resulting from the robust methods for systematic errors evaluated over the 89 systematic error scenarios.

Total nominal scenario doses for the expected value optimization for random errors with fixed standard deviation for the number of fractions (a) *n* = 30 and (b) *n* = ∞; and (c) line doses for *n* in {1, 5, 30, ∞} taken along the line shown in Fig. 1.

Total nominal scenario doses for the expected value optimization for random errors with fixed standard deviation for the number of fractions (a) *n* = 30 and (b) *n* = ∞; and (c) line doses for *n* in {1, 5, 30, ∞} taken along the line shown in Fig. 1.

Nominal scenario total and beam doses for the CVaR optimization for random errors with uncertain standard deviation. The other robust methods resulted in similar dose distributions (root mean square differences from CVaR below 0.02).

Nominal scenario total and beam doses for the CVaR optimization for random errors with uncertain standard deviation. The other robust methods resulted in similar dose distributions (root mean square differences from CVaR below 0.02).

DVH families for the robust methods for random errors with uncertain standard deviation over 100 realizations of random standard deviations and random errors in 30 fractions. The optimizations were performed with *n* = ∞. The dashed lines correspond to the nominal scenario DVHs.

DVH families for the robust methods for random errors with uncertain standard deviation over 100 realizations of random standard deviations and random errors in 30 fractions. The optimizations were performed with *n* = ∞. The dashed lines correspond to the nominal scenario DVHs.

DPHs for the robust methods for random errors with uncertain standard deviation based on 1000 simulations of random standard deviations and random errors in 30 fractions. The optimizations were performed with *n* = ∞.

DPHs for the robust methods for random errors with uncertain standard deviation based on 1000 simulations of random standard deviations and random errors in 30 fractions. The optimizations were performed with *n* = ∞.

Nominal scenario line doses taken along the line shown in Fig. 1 for the robust methods for random errors with uncertain standard deviation for the number of fractions *n* in {1, 5, 30, ∞}.

Nominal scenario line doses taken along the line shown in Fig. 1 for the robust methods for random errors with uncertain standard deviation for the number of fractions *n* in {1, 5, 30, ∞}.

(a) Total nominal scenario dose for the CVaR optimization for systematic errors and random errors with fixed standard deviation. The number of fractions *n* = ∞. The other robust methods resulted in similar heterogeneous dose distributions, which is reflected in inset (b), the line doses taken along the line shown in Fig. 1.

(a) Total nominal scenario dose for the CVaR optimization for systematic errors and random errors with fixed standard deviation. The number of fractions *n* = ∞. The other robust methods resulted in similar heterogeneous dose distributions, which is reflected in inset (b), the line doses taken along the line shown in Fig. 1.

Total nominal scenario doses for the robust methods for systematic errors and random errors with uncertain standard deviation. The number of fractions *n* = ∞.

Total nominal scenario doses for the robust methods for systematic errors and random errors with uncertain standard deviation. The number of fractions *n* = ∞.

Nominal scenario beam doses when random errors with uncertain standard deviation are handled by the robust methods. The number of fractions *n* = ∞.

Nominal scenario beam doses when random errors with uncertain standard deviation are handled by the robust methods. The number of fractions *n* = ∞.

DVH families for the robust methods for systematic errors and random errors with uncertain standard deviation over 100 realizations of systematic errors, random standard deviations, and random errors in 30 fractions. The optimizations were performed with *n* = ∞. The dashed lines correspond to the nominal scenario DVHs.

DVH families for the robust methods for systematic errors and random errors with uncertain standard deviation over 100 realizations of systematic errors, random standard deviations, and random errors in 30 fractions. The optimizations were performed with *n* = ∞. The dashed lines correspond to the nominal scenario DVHs.

(a) Worst (lowest target , highest OAR ), mean, and best (highest target , lowest OAR ) case trade-off curves of the methods accounting for systematic errors and random errors with uncertain standard deviation and *n* = ∞, for the target importance weight in [100, 1000]; (b) DPHs of the methods accounting for systematic errors and random errors with uncertain standard deviation and *n* = ∞; and (c) DPHs of the methods accounting for systematic errors only. The three figures are resulting from the robust methods evaluated over 1000 simulated treatments of systematic errors, random standard deviations, and random errors in 30 fractions.

(a) Worst (lowest target , highest OAR ), mean, and best (highest target , lowest OAR ) case trade-off curves of the methods accounting for systematic errors and random errors with uncertain standard deviation and *n* = ∞, for the target importance weight in [100, 1000]; (b) DPHs of the methods accounting for systematic errors and random errors with uncertain standard deviation and *n* = ∞; and (c) DPHs of the methods accounting for systematic errors only. The three figures are resulting from the robust methods evaluated over 1000 simulated treatments of systematic errors, random standard deviations, and random errors in 30 fractions.

Nominal scenario line doses taken along the line shown in Fig. 1 for the robust methods for systematic errors and random errors with uncertain standard deviation. The number of fractions *n* = ∞.

Nominal scenario line doses taken along the line shown in Fig. 1 for the robust methods for systematic errors and random errors with uncertain standard deviation. The number of fractions *n* = ∞.

Doses and DVH families for the conventional method with a 5 mm margin. (a) Total dose; (b) and (c) beam doses. (d) DVH family over the 89 systematic error scenarios; (e) over 100 realizations of random standard deviations and random errors in 30 fractions; and (f) over 100 realizations of systematic error as well as random standard deviations and random errors in 30 fractions. The dashed lines correspond to the nominal scenario DVHs.

Doses and DVH families for the conventional method with a 5 mm margin. (a) Total dose; (b) and (c) beam doses. (d) DVH family over the 89 systematic error scenarios; (e) over 100 realizations of random standard deviations and random errors in 30 fractions; and (f) over 100 realizations of systematic error as well as random standard deviations and random errors in 30 fractions. The dashed lines correspond to the nominal scenario DVHs.

## Tables

Notation for the different types of uncertainties.

Notation for the different types of uncertainties.

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