## Highlights

- •We determined how motion management strategies influence treatment margins.
- •A simple recipe was found to determine the margin when intrafraction motion is constrained.
- •For a continuous drift, the margin was well approximated by assuming the target is at ¾ of the drift.
- •Our analysis can be used to study the benefit of different motion management strategies.

## Abstract

### Background and purpose

### Materials and methods

*c*σ, with σ the standard deviation of the distribution and

*c*a free parameter. Using Monte Carlo simulations, we determined how motion management changed the required margin. The analysis was performed for different number of treatment fractions and different standard deviations of the underlying random and systematic errors.

### Results

*c*σ, the relative change in the margin was equal to 0.3

*c*. This result held for both models, was independent of σ or the number of fractions and naturally generalizes to the situation with a residual (systematic) error.

### Conclusion

## Keywords

## 1. Introduction

Intven MPW, de Mol van Otterloo SR, Mook S, Doornaert PAH, de Groot-van Breugel EN, Sikkes GG, et al. Online adaptive MR-guided radiotherapy for rectal cancer; feasibility of the workflow on a 1.5T MR-linac: clinical implementation and initial experience. Radiother Oncol 2021;154:172–8. https://doi.org/10.1016/j.radonc.2020.09.024.

- Bertholet J.
- Knopf A.
- Eiben B.
- McClelland J.
- Grimwood A.
- Harris E.
- et al.

*Phys Med Biol.*2019; 64https://doi.org/10.1088/1361-6560/ab2ba8

## 2. Materials and methods

### 2.1 Model description

*σp*= 3.2 mm. We assumed systematic errors to be absent and random intra-fraction errors to be Gaussian distributed with width

*σ*. We assumed an online intervention at a threshold

*c*to change the error distribution to a truncated Gaussian, with width

*σ*and truncation at ±

*cσ.*(See supplement A). We considered a margin adequate when the minimal target dose is 95 % of the prescribed dose in 90 % of the population. Table C1 in the supplementary material provides an overview of the notation used.

#### 2.1.1 Model 1

*σ*and truncation

*c*. This can be interpreted as a random residual displacement after image guidance or plan adaptation, with no additional motion after the irradiation starts. The truncation can be interpreted as a threshold used as a check, just prior to irradiation, what residual displacement would still be acceptable. Such an approach is common practice in many image guidance workflows, with or without online adaptation.

#### 2.1.2 Model 2

*σ*and truncation

*c*. We assumed the full dose to be delivered at a constant rate during this movement and ignored interplay effects. This model is applicable for intra-fraction motion during irradiation. While the precise results will depend on the trajectory (we assumed a straight line) and speed (we assumed to be constant), these can be expected to be higher order corrections. The interpretation of the truncation in this case is a threshold used in intra-fraction motion monitoring.

#### 2.1.3 Variation 1

#### 2.1.4 Variation 2

### 2.2 Analytic approximations

*N*, the error

*σ*effectively results in a ‘random’ component σ

*N*and a ‘systematic’ component Σ

*N*[

*σ*was replaced with the standard deviation of the truncated distribution:

*σC*(see supplement A). Equivalent to equation (1) this resulted in

*σN,c*. The margin

*mc*was given by

*c*was determined by equation (B.7) from supplement B, independent of

*N*. α

*c*= 2.5 when no truncation was present and changed to 2.4, 1.8 and 1.0 for

*c*equals 3, 2 and 1, respectively. The random factor β followed from equation (B.6). β = 1.64 for a prescription of 95 % and β = 0.84 for a prescription of 80 %.

*σN*and Σ

*N*are correlated via equations ((1), (2)), the margin is determined by those patients with the smaller blur, causing equation (3) to give an under-estimation of the margin for this problem. The Gaussian blur is also incorrect in the presence of truncation since the Gaussian convolution of a truncated Gaussian is not anymore Gaussian, but more peaked. This resulted in a different functional form, causing equation (3) to give a slight over-estimation of the margin for this problem.

### 2.3 Monte Carlo simulations

### 2.4 Experiments

- 1.Using the Monte Carlo code, we determined the margin required for a range of parameter values. σ = {1, 2.5, 5, 7.5} mm;
*N*= {1, 2, …, 20};*c*= {1, 2, 3, inf}. We also determined the accuracy of the analytic approximation from equation (3). For model 2 we determined parameter ε by minimizing the relative distance between equation (3) and the Monte Carlo results for σ = {1, 2.5, 5} mm,*N =*{1,3,5,20} and no truncation (*c =*inf). - 2.We determined a margin recipe for the relative change of the margin as a function of the truncation
*c*, for the different values of σ and*N*. We considered a linear fit*mc*/*m*= ω*c*and determined ω for the fit of σ = {1, 2.5, 5} mm,*N =*{1,3,5,20}*.*

*m0*required for

*c =*0 was 2.5Σ. The fit used was the equivalent of the linear fit without systematic errors:

## 3. Results

### 3.1 Determination of required margins

*c*from < 5.3 mm without truncation to < 0.4 mm with

*c*= 1 for the parameters studied (Fig. 1; Tables 1 (accuracy of approximation) and

*C*2 (full numerical results)). This effect was considerably less pronounced for model 2, where the results where < 2.5 mm and < 0.6 mm respectively (Fig. 2; Tables 1 and C3).

*N,*and the truncation parameter

*c.*Results are shown for model 1 (discrete displacement) and model 2 (continuous movement).

MC result (mm) | analytic model (mm) | absolute difference (mm) | relative difference | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

model 1 | σ (mm) | N = 1 | N = 3 | N = 5 | N = 20 | N = 1 | N = 3 | N = 5 | N = 20 | N = 1 | N = 3 | N = 5 | N = 20 | N = 1 | N = 3 | N = 5 | N = 20 |

c = 1 | 1 | 1.0 | 0.7 | 0.6 | 0.4 | 1.0 | 0.6 | 0.5 | 0.3 | 0.0 | 0.1 | 0.1 | 0.1 | 0.02 | 0.15 | 0.18 | 0.24 |

2.5 | 2.4 | 1.7 | 1.5 | 0.9 | 2.4 | 1.6 | 1.3 | 0.8 | 0.0 | 0.1 | 0.1 | 0.1 | 0.02 | 0.07 | 0.10 | 0.11 | |

5 | 4.8 | 3.7 | 3.2 | 2.4 | 4.8 | 3.5 | 3.0 | 2.1 | 0.0 | 0.1 | 0.2 | 0.3 | 0.00 | 0.04 | 0.06 | 0.12 | |

7.5 | 7.2 | 5.8 | 5.3 | 4.2 | 7.2 | 5.7 | 5.1 | 3.8 | 0.0 | 0.0 | 0.3 | 0.4 | 0.00 | 0.01 | 0.05 | 0.10 | |

c = 2 | 1 | 1.9 | 1.2 | 1.0 | 0.7 | 1.9 | 1.2 | 1.0 | 0.6 | 0.0 | 0.0 | 0.0 | 0.1 | −0.01 | 0.03 | 0.01 | 0.13 |

2.5 | 4.7 | 3.4 | 3.0 | 2.1 | 4.6 | 3.3 | 2.8 | 1.9 | 0.0 | 0.1 | 0.2 | 0.2 | 0.01 | 0.04 | 0.06 | 0.10 | |

5 | 9.2 | 7.9 | 7.1 | 5.7 | 9.2 | 7.5 | 6.7 | 5.0 | 0.0 | 0.4 | 0.4 | 0.7 | 0.00 | 0.05 | 0.06 | 0.12 | |

7.5 | 13.8 | 12.7 | 11.8 | 9.8 | 13.8 | 12.2 | 11.0 | 8.6 | 0.0 | 0.5 | 0.7 | 1.2 | 0.00 | 0.04 | 0.06 | 0.12 | |

c = 3 | 1 | 2.5 | 1.6 | 1.4 | 0.8 | 2.4 | 1.6 | 1.3 | 0.8 | 0.0 | 0.1 | 0.1 | 0.1 | 0.00 | 0.04 | 0.06 | 0.08 |

2.5 | 5.9 | 5.0 | 4.4 | 3.0 | 6.0 | 4.4 | 3.8 | 2.6 | −0.1 | 0.6 | 0.6 | 0.4 | −0.01 | 0.12 | 0.14 | 0.14 | |

5 | 12.0 | 11.6 | 11.0 | 8.5 | 12.0 | 10.0 | 8.9 | 6.7 | 0.0 | 1.7 | 2.1 | 1.7 | 0.00 | 0.14 | 0.19 | 0.20 | |

7.5 | 18.0 | 18.3 | 17.8 | 14.7 | 18.0 | 16.1 | 14.6 | 11.4 | 0.0 | 2.2 | 3.3 | 3.3 | 0.00 | 0.12 | 0.18 | 0.22 | |

no truncation | 1 | 2.6 | 1.8 | 1.5 | 0.9 | 2.5 | 1.6 | 1.3 | 0.8 | 0.0 | 0.1 | 0.1 | 0.1 | 0.01 | 0.08 | 0.09 | 0.09 |

2.5 | 6.3 | 5.5 | 5.0 | 3.5 | 6.3 | 4.6 | 4.0 | 2.8 | 0.0 | 0.9 | 1.0 | 0.7 | −0.01 | 0.16 | 0.20 | 0.21 | |

5 | 12.6 | 12.9 | 12.7 | 9.8 | 12.5 | 10.5 | 9.4 | 7.1 | 0.1 | 2.4 | 3.3 | 2.7 | 0.01 | 0.18 | 0.26 | 0.27 | |

7.5 | 18.7 | 20.3 | 20.3 | 17.3 | 18.8 | 16.9 | 15.4 | 12.1 | −0.1 | 3.3 | 4.9 | 5.3 | −0.01 | 0.16 | 0.24 | 0.30 | |

model 2 | |||||||||||||||||

c = 1 | 1 | 0.6 | 0.4 | 0.3 | 0.2 | 0.7 | 0.4 | 0.3 | 0.2 | −0.1 | 0.0 | 0.0 | 0.0 | −0.21 | −0.08 | −0.04 | 0.03 |

2.5 | 1.4 | 0.9 | 0.8 | 0.5 | 1.8 | 1.1 | 0.9 | 0.6 | −0.4 | −0.2 | −0.2 | −0.1 | −0.28 | −0.27 | −0.22 | −0.13 | |

5 | 3.0 | 2.0 | 1.8 | 1.2 | 3.6 | 2.5 | 2.1 | 1.4 | −0.6 | −0.5 | −0.4 | −0.3 | −0.21 | −0.25 | −0.21 | −0.23 | |

7.5 | 4.8 | 3.5 | 3.0 | 2.1 | 5.4 | 4.0 | 3.5 | 2.5 | −0.6 | −0.6 | −0.5 | −0.4 | −0.12 | −0.16 | −0.18 | −0.19 | |

c = 2 | 1 | 1.0 | 0.7 | 0.6 | 0.4 | 1.4 | 0.9 | 0.7 | 0.4 | −0.4 | −0.2 | −0.1 | 0.0 | −0.39 | −0.27 | −0.22 | −0.12 |

2.5 | 2.8 | 1.9 | 1.6 | 1.0 | 3.5 | 2.3 | 2.0 | 1.3 | −0.6 | −0.5 | −0.4 | −0.3 | −0.22 | −0.26 | −0.23 | −0.29 | |

5 | 6.5 | 4.7 | 4.2 | 2.9 | 6.9 | 5.3 | 4.6 | 3.3 | −0.5 | −0.6 | −0.5 | −0.4 | −0.07 | −0.12 | −0.11 | −0.14 | |

7.5 | 10.5 | 8.4 | 7.4 | 5.6 | 10.4 | 8.6 | 7.7 | 5.8 | 0.1 | −0.2 | −0.3 | −0.3 | 0.01 | −0.03 | −0.03 | −0.05 | |

c = 3 | 1 | 1.4 | 0.8 | 0.7 | 0.4 | 1.8 | 1.1 | 0.9 | 0.5 | −0.4 | −0.3 | −0.2 | −0.1 | −0.30 | −0.35 | −0.30 | −0.19 |

2.5 | 3.8 | 2.7 | 2.2 | 1.5 | 4.5 | 3.1 | 2.6 | 1.7 | −0.7 | −0.4 | −0.4 | −0.3 | −0.17 | −0.14 | −0.18 | −0.20 | |

5 | 8.8 | 7.6 | 6.4 | 4.3 | 9.0 | 7.1 | 6.2 | 4.5 | −0.2 | 0.5 | 0.2 | −0.2 | −0.02 | 0.07 | 0.02 | −0.06 | |

7.5 | 14.2 | 13.1 | 11.5 | 8.1 | 13.5 | 11.4 | 10.3 | 7.8 | 0.7 | 1.6 | 1.2 | 0.3 | 0.05 | 0.13 | 0.11 | 0.04 | |

no truncation | 1 | 1.4 | 0.9 | 0.8 | 0.5 | 1.9 | 1.2 | 1.0 | 0.6 | −0.4 | −0.3 | −0.2 | −0.1 | −0.30 | −0.31 | −0.26 | −0.18 |

2.5 | 4.1 | 3.0 | 2.6 | 1.6 | 4.7 | 3.3 | 2.8 | 1.8 | −0.6 | −0.3 | −0.2 | −0.2 | −0.15 | −0.09 | −0.09 | −0.16 | |

5 | 9.3 | 8.5 | 7.5 | 4.8 | 9.4 | 7.4 | 6.6 | 4.8 | −0.1 | 1.1 | 0.9 | 0.0 | −0.01 | 0.13 | 0.12 | 0.00 | |

7.5 | 15.0 | 14.5 | 13.3 | 9.0 | 14.1 | 12.1 | 10.8 | 8.3 | 1.0 | 2.4 | 2.5 | 0.6 | 0.06 | 0.17 | 0.19 | 0.07 |

*N*increases (Fig. D1).

### 3.2 Relative change of the margin in the presence of truncation

*mc*/

*m*was approximately linear in

*c*in the regime of interest, justifying the fit

*mc*/

*m*= ω

*c.*The fit for ω resulted for model 1 in an average value of 0.32 (range: 0.28–0.35; Fig. 3) and for model 2 in 0.31 (range 0.28–0.35; Fig. 4), which we approximated for simplicity with ω

*=*0.3 for both models.

*mc*/

*m*, evaluated over the range

*c*= {0,3}, was ≤ 0.12 and ≤ 0.15 for model 1 and model 2 respectively (Table 2). The upper quartile error for

*mc*was ≤ 0.10 mm and ≤ 0.04 mm for model 1 and model 2 respectively (Table C4).

*mc / m)*between the Monte Carlo calculation and linear fit

*mc / m*= 0.3

*c,*for different values of the standard deviation σ and number of fractions

*N*. The difference is calculated over the range of the truncation parameter

*c =*{0…3} and the reported values are the median (lower quartile; upper quartile).

Model 1 | σ (mm) | N = 1 | N = 3 | N = 5 | N = 20 |
---|---|---|---|---|---|

1 | 0.09 (0.07; 0.12) | 0.08 (0.06; 0.09) | 0.07 (0.04; 0.08) | 0.09 (0.06; 0.12) | |

2,5 | 0.09 (0.06; 0.12) | 0.01 (0.00; 0.02) | −0.00 (-0.01; 0.00) | −0.01 (-0.02; 0.00) | |

5 | 0.09 (0.06; 0.11) | −0.00 (-0.01; 0.01) | −0.04 (-0.04; −0.03) | −0.04 (-0.05; −0.03) | |

7,5 | 0.09 (0.06; 0.11) | 0.01 (-0.01; 0.02) | −0.02 (-0.03; −0.02) | −0.05 (-0.05; −0.04) | |

Model 2 | |||||

1 | 0.09 (0.07; 0.10) | 0.12 (0.07; 0.15) | 0.11 (0.07; 0.14) | 0.10 (0.06; 0.12) | |

2,5 | 0.05 (0.03; 0.07) | 0.01 (0.01; 0.02) | 0.00 (0.00; 0.01) | 0.01 (0.01; 0.02) | |

5 | 0.06 (0.01; 0.08) | −0.04 (-0.05; −0.03) | −0.05 (-0.06; −0.04) | −0.02 (-0.05; 0.00) | |

7,5 | 0.05 (0.01; 0.07) | −0.03 (-0.05; −0.01) | −0.04 (-0.06; −0.04) | −0.02 (-0.05; 0.01) |

*mc*/

*m*was approximately independent from σ and

*N (*for

*N*large enough), which is evident from the graphs. In the supplement (sections B.3.1 and B.3.2) we provide a heuristic argument for this.

*=*3 mm,. the linear fit (equation (4)) again resulted in ω

*=*0.3 (Fig. D3).

## 4. Discussion

*mc*/

*m = 0.3c*, independent of the standard deviation and number of fractions (for

*N*> 1). This relation held for both models of intra-fraction motion we studied. To appreciate this finding, consider a treatment given in 5 fractions, with intra-fraction motion characterized by σ = 5 mm. In this case, a margin of 12.7 mm is required. If motion management is introduced, such that all movement is limited to be < 10 mm (

*c*= 2), the new margin is easily determined to be 7.6 mm.

*mc*/

*m*showed discrepancies of up to 0.15 for σ = 1 mm, the margin

*m*in this case was only 2–3 mm and these errors were thus < 1 mm. Therefore, we believe our results ensure sufficient coverage (with the same caveats as the Van Herk recipe). The exception was for

*N = 1*and relatively large σ. In that case the error could be up to 2 mm for the parameters studied.

*N =*1. In this case all errors were effectively systematic and their contribution to the margin via equation (3) could be calculated exactly for all values of

*c*(using equations (A.4) and (B.7) from the supplementary material). For larger values of

*N,*the analytic approximation in the case of no truncation showed errors of < 5.3 mm for model 1 for the parameters studied. The lack of correspondence is known and substantial improvements are feasible [

*c*equals 3, 2 and 1 respectively for model 1. This improvement was not solely due to the fact that σ

*c <*σ, as could be seen from the values of σ

*c*provided in the figures. For example, the approximation for

*c =*2 with σ = 7.5 mm (σ

*c*= 5.9 mm) was considerably better compared to σ = 5 mm without truncation. The underlying reason for this was probably that the analytic approximation used the fully convolved dose distribution to reach a closed form result. With a finite number of fractions, the probability that this was a reasonable assumption increases when there are less outliers in the underlying distribution, i.e. in the presence of truncation. The analytic approximation when considering continuous motion (model 2) was considerably better compared to model 1 (see Fig. 1, Fig. 2). The likely reason here was that the motion provides an additional blur and therefore improved the approximation of the fully convolved dose distribution.

*N*also included a systematic component Σ

*N*and the dependency of

*mc*/

*m*was independent of

*N*, a systematic component in the drift does not change the simple linear dependency of the relative margin on

*c*.

*mc*/

*m = 0.3c*to be sufficient. Based on the results, treatment margins can be determined when motion management strategies are applied. Moreover, our analysis can be used to study the potential benefit of different motion management strategies for different treatment sites. This allows to discuss and determine the most appropriate strategy for margin reduction.

## Declaration of Competing Interest

## Appendix A. Supplementary data

- Supplementary Data 1

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