Coping with interfraction time trends in tumor setup.

PURPOSE: Interfraction tumor setup variations in radiotherapy are often reduced with image guidance procedures. Clinical target volume (CTV)-planning target volume (PTV) margins are then used to deal with residual errors. We have investigated characterization of setup errors in patient populations with explicit modelling of occurring interfraction time trends.
METHODS: The core of a "trendline characterization" of observed setup errors in a population is a distribution of trendlines, each obtained by fitting a straight line through a patient's daily setup errors. Random errors are defined as daily deviations from the trendline. Monte Carlo simulations were performed to predict the impact of offline setup correction protocols on residual setup errors in patient populations with time trends. A novel CTV-PTV margin recipe was derived that assumes that systematic underdosing of tumor edges in multiple consecutive fractions, as caused by trend motion, should preferentially be avoided. Similar to the well-known approach by van Herk et al. for conventional error characterization (no explicit modelling of trends), only a predefined percentage of patients (generally 10%) was allowed to have nonrandom (systematic + trend) setup errors outside the margin. Additionally, a method was proposed to avoid erroneous results in Monte Carlo simulations with setup errors, related to decoupling of error sources in characterizations. The investigations were based on a database of daily measured setup errors in 835 prostate cancer patients that were treated with 39 fractions, and on Monte Carlo-generated patient populations with time trends.
RESULTS: With conventional characterization of setup errors in patient populations with time trends, predicted standard deviations of residual systematic errors ( Σ res ) after application of an offline correction protocol could be underestimated by more than 50%, potentially resulting in application of too small margins. With the new trendline characterization this was avoided. With the novel CTV-PTV margin recipe with an allowed 10% of patients having nonrandom errors outside the margin, the observed percentage was 10.0% ± 0.2%. When using conventional characterization of errors and the van Herk margin recipe, on average 58.0% ± 24.3% of patients had errors outside the margin, while 10% was prescribed. For populations with no time trends, the novel recipe simplifies to the generally applied M = 2.5 Σ + 0.7 σ formula proposed by van Herk et al.
CONCLUSIONS: In populations with time trends in setup errors, the use of trendline characterizations in Monte Carlo simulations for establishment of residual errors after a setup correction protocol can avoid application of erroneous margins. The novel margin recipe can be used to accurately control the percentage of patients with nonrandom errors outside the margin. In case of daily image guidance of patients with multiple targets with differential motion, the recipe can be used to establish margins for the targets that are not the primary target for the image guidance (e.g., nodal regions). Probabilistic planning might be improved by using trendline characterization for modelling of setup errors. Population analyses of interfraction setup errors need to take into account potential time trends.

Introduction

In fractionated radiotherapy, tumor setup errors at the linac are often mitigated with image‐guided corrections.1, 2, 3 For planning, a Clinical target volume (CTV)–planning target volume (PTV) margin4, 5 is used to cope with residual errors. Both for estimating the expected impact of a setup correction protocol on treatment accuracy and for establishment or validation of margin recipes, Monte Carlo (MC) simulations may be performed using a characterization of the setup errors in the patient population.

It is generally assumed that setup errors occurring in a fractionated treatment of a patient can be described with normal distributions and that they can be characterized by the mean (systematic) error during the treatment and day‐to‐day variations around the mean (random errors). Observed systematic and random errors are used to derive population parameters that characterize the error distributions along the principal directions, that is, anterior–posterior, superior–inferior, left–right. In this paper, this much applied characterization of setup errors in a patient population is designated as the “conventional characterization”6 (see Appendix A for equations).

Daily tumor setup relative to the treatment unit isocenter may gradually change during a fractionated treatment, resulting in an interfraction time trend in setup.7, 8, 9, 10, 11, 12, 13, 14 Existence of interfraction time trends has been reported by several groups for different cancer sites. El Gayed et al.7 reported trend motion of 4–11 mm for 2 rectal and 3 prostate patients out of 10 patients per cancer site. Hanley et al.8 found statistically significant trends in 10 out of the 50 prostate patients with a range of 2–7 mm. Stroom et al.9 compared 15 prostate patients treated in prone position with 15 treated in supine position and found time trends in rectum diameter and prostate translations. van der Heide et al.10 investigated prostate treatment with fiducials for 453 patients receiving a 35 fraction treatment. They found total motion of 3.1 mm in AP and 1.7 mm in SI direction. Namysl‐Kaletka et al.11 analyzed 57 patients with gastric cancer treated with 25 or 28 fractions. They reported 1 and 1.6 mm total trend motion in LR and SI direction, respectively. Gangsaas et al.12 showed caudal trend motion of up to 11 mm (average 3.2 mm) for 30 patients with laryngeal cancer. Penninkhof et al.13 showed more than 5 mm total tumor bed trend motion in 20% of breast cancer patients treated with a simultaneously integrated boost technique.

Such time trends are not explicitly considered in the conventional characterization. Rather, they are implicitly treated as part of the random error. However, a time trend motion is clearly deterministic, with a gradual, cumulative shift of the tumor in the 3D dose distribution during the fractionated treatment. This deterministic motion may have an impact on the performance of offline setup correction protocols, with corrections based on setup measurements in the first fractions. Not explicitly accounting for time trends in the CTV‐PTV margin may result in underdosage of tumor edges in substantial numbers of consecutive fractions. For example, for a patient with no systematic setup error, a time trend in LR direction can result in a systematic underdose in the left tumor edge in each of the first 50% of fractions, and an underdose in the right tumor edge in all subsequent fractions. Existing rather simple TCP models suggest that the order of fractions with underdose would not be important. However, to the best of our knowledge there is no evidence that systematically underdosing the same part of the tumor in many fractions at a row, and compensating it with adequate dose delivery in the other fractions, would be equivalent to a random ordering of fractions with underdose and adequate dose. There are many examples in the radiotherapy literature showing that time patterns in dose delivery can indeed matter.

In this paper, we investigated the explicit modelling of interfraction time trends in tumor setup errors, using so‐called trendline characterizations of setup errors observed in patient populations. Trendline characterizations were compared to conventional characterizations regarding accuracy of Monte Carlo (MC) predicted residual setup errors in case a no‐action‐level (NAL) protocol2 or an extended NAL (eNAL) protocol3 was used for setup corrections. We developed a novel CTV‐PTV margin recipe that assumes that deterministic underdosing of tumor edges due to time trends should preferentially be avoided. The approach was highly similar to van Herk’s derivation of his well‐known margin recipe,4 aiming at equality of the recipes in the limit of no time trends in the population.

The investigations included synthetic patient populations with time trends and a database with daily setup errors measured in a large population of prostate cancer patients that experienced time trends (“Erasmus database”).

Materials and methods

Trendline characterization of setup errors to model interfraction time trends in setup errors

In contrast to a conventional characterization of setup errors in a population (Appendix A), a trendline characterization explicitly models occurring time trends.3 To this purpose, for each patient p the setup errors in the fractionated treatment along each of the principal axes are characterized with a linear trendline, fitted through the daily measured setup errors (see Fig. 1), and defined by the slope a (mm/fraction), and middle position m (mm). The latter is the setup error half‐way the fractioned treatment according to the fitted trendline, that is, the mean of the trendline tumor positions in fraction 1 and the last fraction F. It can be easily proven that this m equals the mean setup error in the fractionated treatment as used in the conventional characterization. In the remainder of this paper, setup errors according to the trendline are designated “trendline errors.” Daily deviations from the trendline are now defined as the random errors (see Fig. 1). This leads to the following parameters defining a trendline characterization, with M and also used in a conventional characterization: with the mean setup error of patient in the fractionated treatment

Figure 1

Setup errors for an example patient with a time trend. The straight dashed line is the fitted trendline. Each fraction the total setup error (blue dot) is the sum of the error according to the dashed trendline (denoted as “trendline error,” see black arrow as example) and the random error defined as the daily deviation from the setup according to the trendline (see red arrow). MD is the patient’s maximum trendline setup error, used for calculation of clinical target volume (CTV)–planning target volume (PTV) margin for nonrandom errors (Section 2.F). m is the trendline error in the middle of the fractionated treatment which equals the systematic tumor setup. a is the trendline slope, F is the total number of fractions. [Color figure can be viewed at http://wileyonlinelibrary.com]

the overall mean setup error in the population

the standard deviation describing the distribution of systematic (i.e., mean) setup errors in the population

with the trendline slope calculated for patient , and the number of patients in the population

the population mean of the trendline slopes:

the standard deviation describing the interpatient variation in the trendline slopes:

with the standard deviation of random errors relative to the trendline observed for patient

the population standard deviation describing random errors relative to the trendlines (see also Fig. 1):

the standard deviation describing the variation of in the population15:

The Erasmus database with prostate setup errors

Daily setup errors of 835 prostate cancer patients treated between November 2007 and May 2017 at the Erasmus MC Cancer Institute with 39 fractions of 2 Gy were used to build the database. Setup deviations in these patients were measured with kV/MV crossfire imaging of four implanted gold markers.10, 16, 17, 18 Setup errors were quantitated as marker center‐of‐mass displacements along the three principal directions, realizing that the mechanism for motion could in some cases also be rotations or deformations. The database was filled with setup errors that would have occurred in case no correction protocol would have been applied, that is, applied (a priori) setup corrections prior to imaging were subtracted from measured setup errors.

Synthetic patient populations with setup errors

Synthetic populations were created by choosing concrete values for the parameters in a trendline characterization and then using a MC approach to randomly create 39 fraction treatments for 10 000 patients (see Section 2.D for details). Parameters used for generation of synthetic setup errors along the three principal axes were: , , , , , and .

Simulations of the NAL and eNAL protocol (Sections 2.E and 3.B) were performed per principal direction, in line with the clinical application of these protocols.

Two types of synthetic populations were created for investigations on the CTV‐PTV margin: isotropic (using the same distributions of trendline parameters for all three directions) and anisotropic (with different, randomly chosen distributions of trendline parameters for the three directions). By including all possible combinations of preselected parameters (above), a total number of 144 isotropic populations was generated. For generating the anisotropic populations, the same parameters were used as for isotropic populations but randomly chosen for each of principal directions (without replacements) resulting also in 144 anisotropic populations.

Monte Carlo (MC) generation of setup errors in a patient population

As described above, setup errors in a population are generally described by a conventional characterization. As discussed in this paper, alternatively, a trendline characterization can be used if time trends are (potentially) present. However, in both cases, when using the characterization parameters in a MC experiment for generating a population of, for example, 10 000 new patients, the characterization parameters for these 10 000 patients will not be equal to the original parameters. The problem can be illustrated for a conventional characterization using a simple example: the random setup errors for a particular patient in the 39 fractions treatment are randomly drawn from a gaussian distribution , with SD randomly drawn from the distribution . Due to the finite number of fractions (39), the mean of the drawn ‘random’ errors for the patient will in general not be equal to zero, that is, effectively the drawn errors are not completely random as they have a systematic component. Basically, this is caused by de‐coupling of error sources in the characterization. Something very similar occurs with a trendline characterization: due to the finite number (F = 39) of drawn random errors, both the mean setup error of the patient and the slope of the trendline will in general be different from the drawn m and a, respectively. To avoid errors in the MC simulations, corrections described in Appendix B were always performed for drawn random errors.

NAL and eNAL offline protocols for correction of interfraction setup errors

The NAL2 and eNAL3 protocols are briefly summarized in Appendix C. In this study, we investigated for conventional and trendline characterizations the accuracy of MC simulated predictions of residual systematic setup errors for the NAL and eNAL protocols in patient populations with time trends. For both characterizations, , the standard deviation describing the population residual systematic errors after NAL or eNAL was established.

A novel CTV‐PTV margin recipe to account for time trends in setup errors

For the derivation of the margin recipe , based on the conventional characterization of setup errors, the margins for systematic and random setup errors ( and , respectively) were independently established. Similarly, in the proposed margin recipe for a population with time trends, the margin contribution related to the trendlines, defined by a slope and a middle position (the latter equaling the patient’s systematic setup error see Section 2.A), and the contribution from the random errors around the trendlines are treated separately (see Fig. 1). Actually, in the van Herk recipe is replaced by [see Eq. (5)], while a new term, , is derived for coping with the trendline errors to replace the term. Equivalent to the work by van Herk et al.,4 this new term is derived by requiring that for 90% of patients, the full CTV is within the PTV for 100% of the (nonrandom) trendline errors.

The margin component related to trendline errors, , is a 3D vector. In order to calculate its components, , for each principal direction, i, a procedure similar to that used by van Herk et al.4 for deriving is used, assuming for each direction a spherical 3D situation. In other words, for establishment of it is assumed that the distributions of trendline errors for all three directions, k, are the same as for direction i. This procedure would be performed for each axis i based on the population parameters: M, , and [see Eqs. (1), (2), …, for definitions].

randomly select for a large number of patients, p, the trendlines, that is, select and for the three principal directions, k, from the gaussian distributions and ;

determine for each patient, p, the maximum setup deviations following from the trendlines for the three principal directions as (see Fig. 1, note: the distributions are folded‐Gaussians);

determine for each patient the length of the vector defined by the : ;

establish as the 90th percentile value of the distribution of ‐values.

After calculation of for the three principal directions, i, margins in any direction are derived from the three‐dimensional ellipsoid defined by the .

Validation of the proposed margin recipe

The recipe for calculation of the margin component for coping with trendline errors as described in the previous section was validated for all 288 synthetic populations. For each of the populations we assessed for which percentage of the 10 000 patients all trendline errors were within the calculated margin. According to the design requirements (previous section) this should be 90%, so only 10% of patients can have one or more trendline errors outside the calculated margin. Mathematically, a trendline error with components t, of a patient p, is within the calculated margin if

For comparison, for all synthetic populations (most of them with time trends, see Section 2.A) we also established the percentage of patients with all trendline errors within the margin as calculated with the van Herk recipe. For a population with time trends, the true random errors are quantified by [Eq. (5)], yielding a margin for random errors, (Section 2.E). However, in the van Herk approach, trendline errors are treated as random errors, resulting in , with defined in Eq. (A1). As , the prescribed margin for random errors in the van Herk approach is slightly larger than actually required for the true random errors. For this reason, for the van Herk approach, we established for each of the synthetic populations the percentage of patients with all trendline errors inside an ellipsoid defined by the margin components .

An analytical expression for the novel CTV‐PTV margin

Section 2.F describes a numerical procedure for deriving the CTV‐PTV margin, given the trendline characterization of the setup errors in the patient population. For populations with M = 0 and M0, that is, assuming that on average the patients’ systematic setup errors and trendline slopes are zero, we also derived an analytical expression for the margin. To this purpose, margins calculated with the method presented in Section 2.E were fit to Eq. (7). The least square method as implemented in the SciPy package was used to establish the values for the fitting parameters , , and .with , the standard deviation describing the distribution of trend motions during one‐half of the fractions.

Origin of time trend errors

We propose a statistical method to determine whether observed trends in a population are caused by limited numbers of fractions or and by other (e.g., physiological) causes. First, for a large number of patients (106) setup errors are randomly generated for fractionated treatments, using M, , , and SD (i.e., ignoring trendline parameters). For all simulated patients, trendlines are then fitted. Next, the distribution of trendline slopes obtained from the simulations is compared to the distribution of slopes derived from the original (i.e., clinical) data using the Kolmogorov–Smirnov two‐sample test.

Results

Characterization of setup errors in the Erasmus database

Figure 2 shows for the three principal directions the distributions of total trend motion in the fractionated treatments. Absolute total trend motion for 10% of patients was larger than 2.6, 5.2, and 5.3 mm for left–right, superior–inferior, and anterior–posterior directions, respectively. Table 1 shows both the conventional characterization and the trendline characterization for the setup errors in the Erasmus database. Results of the test proposed in Section 2.I showed that observed trends in the Erasmus database are indeed larger than expected from the finite number of fractions (P < 0.001).

Figure 2

Distributions of total trend motion, defined as the trendline error of a patient in the last fraction minus the trendline error in the first fraction, in the Erasmus database along the principal directions. [Color figure can be viewed at http://wileyonlinelibrary.com]

Table 1

Conventional and trendline characterizations of uncorrected setup errors in the Erasmus database. See Eqs. (1)–(6) and (A1)–(A2) for definition of the parameters. M and are part of both characterizations. Trendline slope parameters, M, are given in mm/fraction. All other values are given in mm.

	Conventional characterization
			Trendline characterization
	σ	SD_SD	M	Σ	M_a	Σ_a	σ^′	SDSD^′
Left–right	1.93	0.71	−0.32	2.50	0.002	0.046	1.86	0.69
Superior–inferior	2.64	0.68	−0.97	3.37	−0.042	0.075	2.45	0.60
Anterior–posterior	2.77	0.80	−0.54	3.47	−0.019	0.083	2.59	0.72

Monte Carlo simulations of residual setup errors for NAL and eNAL

For patient populations with time trends we investigated the impact of using a conventional characterization of setup errors for establishment of the distribution of residual systematic errors, , instead of the more precise trendline characterization. Simulations were performed both for synthetic populations and for the measured errors in the Erasmus database. Always, the NAL/eNAL protocols were also simulated by directly using the fraction setup errors, that is, not using any characterization as an intermediate step. The latter simulations reflect the ground truth regarding the reduction of systematic setup errors with NAL/eNAL. For synthetic populations, a schematic overview of the investigations is provided in Fig. 3.

Figure 3

Schematic overview of the investigations on Monte Carlo simulated residual setup errors for the no‐action‐level (NAL) and extended NAL off‐line correction protocols for synthetic patient populations. * parameter values are different.

NAL for synthetic populations

For all synthetic populations, the investigations demonstrated that the simulations based on trendline characterization did indeed accurately predict , that is, the values were close to the ground truth values. In contrast, the use of conventional characterization did often result in significant deviations in estimated . For all populations with , the simulation based on the conventional characterization overestimated the reductions in systematic setup errors with the NAL protocol. For N = 3, that is, imaging in the first three fractions, results for 12 out of 144 synthetic populations are summarized in Fig. 4. For these populations, the mean difference between the simulated based on the trendline characterization and the ground truth value was . For simulated based on conventional characterization of trendline errors the difference was .

Figure 4

Simulated residual systematic setup errors for the no‐action‐level protocol for 12 synthetic populations. In all simulations, imaging in only the first three fractions was assumed. For all populations: M = 0 mm,  mm and . are standard deviations describing distributions of trendline slopes. are standard deviations describing distributions of random errors around trendlines. Direct simulation: no intermediate characterization used (ground truth), Conventional Monte Carlo (MC)/Trendline MC: MC simulation based on a conventional/trendline characterization of the population setup errors. [Color figure can be viewed at http://wileyonlinelibrary.com]

eNAL for synthetic populations

Similar to the simulations for the NAL protocol, estimated with the use of trendline characterization agreed very well with the ground truth (mean difference for N = 3: ). Different from NAL, for eNAL simulations done with conventional characterization underestimated the positive impact of the protocol with a mean difference in of . Figure 5 shows results for the 12 out of 144 synthetic populations. As explained in the M&M section, eNAL was developed to reduce residual errors in populations with time trends better than NAL. This is indeed observed when comparing the curves for Direct simulation/Trendline MC in Fig. 5. with those in Fig. 4.

Figure 5

Simulated residual systematic setup errors for the extended no‐action‐level protocol for 12 synthetic populations. In all simulations, imaging in the first three fractions was followed by image acquisition in the first fraction of each following week. For all populations: M = 0 mm,  mm and . are standard deviations describing distributions of trendline slopes. are standard deviations describing distributions of random errors around trendlines. Direct simulation: no intermediate characterization used (ground truth), Conventional Monte Carlo (MC)/Trendline MC: MC simulation based on a conventional/trendline characterization of the population setup errors. [Color figure can be viewed at http://wileyonlinelibrary.com]

NAL and eNAL for the Erasmus database

Also for the Erasmus database, the NAL simulations based on the trendline characterization clearly agreed best with the ground truth (Table 2). However, the agreement was less good than observed for the synthetic populations (above). The eNAL simulations both for conventional and trendline characterization agreed well with the ground truth.

Table 2

Residual systematic setup errors,, for the no‐action‐level (NAL) and extended NAL protocols applied to the Erasmus database. All values are given in mm. Direct simulation: no intermediate characterization used (ground truth), Conventional Monte Carlo (MC)/Trendline MC: MC simulation based on a conventional/trendline characterization of the population setup errors.

	NAL			eNAL
	Direct simulation	Trendline MC	Conventional MC	Direct simulation	Trendline MC	Conventional MC
Left–right	1.4	1.2	1.1	0.8	1.1	1.0
Superior–inferior	2.0	1.7	1.3	0.8	1.1	1.0
Anterior–posterior	2.1	1.9	1.4	0.8	1.1	1.0

Required margins for trendline errors

Using the Python code provided in Data S1, margins were calculated for a wide range of parameter values. The results are provided in Data S2 as look‐up tables.

Results of fitting Eq. (7) to data with M0 and M0 are presented in Table 3. Differences between calculation with the Python code and Eq. (7) are negligible. In case of no time trends (), Eq. (7) reduces to the van Herk formula for systematic setup errors with equal ‐values (compare columns 2 and 6 in Table 3).

Table 3

, and : fit parameters for Eq. (7) as a function of the required percentage of patients (percentile) inside . : for 441 combinations of and mean differences with ranges in , calculated with exact simulation with Python code and with Eq. (7). For comparison, the last column contains α‐values given by van Herk et al.4

Percentile	α	δ	γ	ΔMargin (mm)	α by van Herk et al.
80	2.15	−0.94	0.86	2.65 × 10−5; [−0.005, 0.004]	2.16
85	2.31	−1.08	0.67	4.33 × 10−5; [−0.006, 0.005]	2.31
90	2.50	−1.27	0.52	8.19 × 10−5; [−0.008, 0.007]	2.50
95	2.79	−1.54	0.38	1.11 × 10−4; [−0.015, 0.012]	2.79
99	3.37	−2.09	0.26	1.75 × 10−4; [−0.016, 0.019]	3.36

Validation of the proposed margin recipe for trendline errors

The simulations described in Section 2.G demonstrated that in the 288 synthetic populations (Section 2.C), of patients had one or more (nonrandom) trendline errors outside the margin calculated with the proposed novel recipe (y‐axis Fig. 6), which is very close to the required 10.0%. In most populations, the van Herk margin for nonrandom errors (, see Section 2.G) was clearly too small, with 58%±24% of patients with one or more trendline errors outside (x‐axis Fig. 6). The largest percentages with the van Herk approach were found for the largest population and values. The markers inside the depicted circle in Fig. 6 belong to the simulated isotropic synthetic populations that did not have time trends. They show that our simulations for the van Herk protocol (x‐axis) are indeed correct, that is, they resulted in an expected 10% of errors outside the margin. Moreover, they demonstrate that for populations without time trends, the proposed novel recipe agrees with van Herk’s recipe (compare x‐axis with y‐axis).

Figure 6

For the 288 synthetic patient populations and the Erasmus database, percentages of patients with trendline error(s) outside the three‐dimensional clinical target volume (CTV)–planning target volume (PTV) margin for nonrandom errors. X‐ axis: percentages for margins according to van Herk et al., Y‐axis: percentages according to the proposed novel recipe. [Color figure can be viewed at http://wileyonlinelibrary.com]

Table 4 shows calculated margins for the Erasmus database and percentages of patients with trendline error(s) outside. For the novel margin recipe this was 9.6%, while the van Herk recipe resulted in 23.7%. Margins calculated with the van Herk recipe were up to 2.2 mm too small to guarantee that not more than 10% of patients would have trendline error(s) outside.

Table 4

Calculated margins and percentages of patients with one or more trendline errors outside, in case no setup protocols applied. To calculate total margins, is added to the margin for nonrandom errors, see Section 2.G.

	Margin — van Herk recipe nonrandom/total	Margin — proposed recipe nonrandom/total
Left–right	6.3/7.6 mm	7.5/8.8 mm
Superior–inferior	8.6/10.3 mm	10.8/12.5 mm
Anterior–posterior	8.8/10.6 mm	11.0/12.8 mm
% patients outside 3D margin	23.7%	9.6%

Discussion

For the much applied NAL protocol for setup corrections it was demonstrated that MC simulations based on a conventional characterization overestimated the reduction in systematic setup errors. This is attributed to the fact that in a conventional characterization there is no explicit modelling of the time trends. These results demonstrate that the use of a conventional characterization for simulation of setup protocols in a patient population with trends may be potentially dangerous as it may point at required margins that are smaller than needed. As expected in populations with time trends, the eNAL protocol could better reduce setup errors than NAL. However, also for eNAL, predicted residual errors were inaccurate when simulations were based on conventional characterization.

Based on the proposed trendline characterization of setup errors, a novel CTV‐PTV recipe for nonrandom errors was developed, ensuring that deterministic underdosing of tumor edges in multiple consecutive fractions as a result of trend motion was avoided. Different from the approach proposed by van Herk et al.,4 the nonrandom errors were not only characterized by mean setup errors in fractionated treatments but also by the slopes of fitted trendlines. Similar to the van Herk approach, the margin for nonrandom errors was defined by prescribing that only 10% of patients could have a non‐random error outside the calculated margin. In the absence of time trends in the population ( and ) the two margins are equal. The proposed recipe describes a numerical procedure for obtaining the margins. We derived an analytical margin formula [Eq. (7)] in case of zero mean population slopes and zero mean translational setup errors. We provided a Python code (Data S1) as well as look‐up tables (Data S2) for if this is not the case.

For the proposed margin recipe we have adopted the generally applied approach of separating the total margin in two components, one for non‐random errors (described by trendlines) and the other for random errors, that is, where is used to cope with the blurring of planned dose distributions due to random setup errors. In this paper we have used the well‐known expression, .4 On the other hand, we are aware of other recipes for calculating , which may be more appropriate, e.g. for lung tumors or SBRT19, 20, 21, 22. If considered appropriate, the applied can be substituted by any other recipe. It will not impact the proposed mechanism for calculation of .

To the best of our knowledge there is no radiobiological or clinical literature that confirms or denies a need for avoiding systematic underdosing of tumor edges in multiple consecutive fractions that can result from trend motion. Therefore, there is possibly no need, or not always a need, to (fully) apply the enlarged margins following from the proposed CTV‐PTV margin recipe. On the other hand, as the recipe uses distributions of combined errors, MD (Section 2.F), resulting from systematic errors (m) and trends () (so no separate distributions of errors resulting from and from , i.e., no addition of margins), the margin increase compared to the van Herk approach may in practice be limited (see, e.g., Table 4; 7.6 mm compared to 8.8 mm). Nevertheless, as margin increases can result in increased OAR doses, there can be arguments to choose for the original van Herk margin recipe. Alternatively, margins could be calculated with the novel recipe while allowing some underdose in specific areas of the extended PTV that would otherwise result in unacceptable enhancement of OAR doses. In any way, systematic radiobiological studies are warranted to guide future clinical decisions regarding the margin in case of time trends. As described elsewhere in this section, also if enlarged margins would not be needed, there are still other arguments to apply trendline characterizations instead of the conventional fractionation in case time trends occur in the patient population.

Both in conventional and in trendline characterizations of setup errors there is a decoupling of random and nonrandom errors. This decoupling may result in erroneous conclusions from MC simulations. We have proposed correction schemes to avoid these issues (Section 2.D).

As explained in Section 2.D, even in populations that would not have systematic setup errors or time trends in case of an infinite number of treatment fractions, such errors and time trends are to be expected if the same patients would be treated with a (more realistic) finite number of fractions. A test has been proposed to find out whether observed time trends in a population are related to the finite number of fractions or whether there are other (physiologic) causes for observed trends (Section 2.I).

For the investigated synthetic patient populations, MC simulations using trendline characterization resulted in highly accurate residual setup errors after NAL corrections. For N = 3 the difference in with the ground truth was . For the Erasmus database, ground truth values were up to 0.3 mm larger than those obtained with MC simulations based on trendline characterization, depending on direction (see Table 2). Part of the explanation for enhanced ground truth residual errors may be found in Fig. 7, showing values calculated with Eq. (A1) for patients’ standard deviations in setup errors calculated over three fractions (). It shows for the superior–inferior and anterior–posterior directions enhanced standard deviations at the start of treatment. In the ground truth simulations for NAL this is expected to result in increased as the mean setup errors determined in the first three fractions, used for set‐up corrections in the following fractions, may deviate more from the true mean set‐up errors due to the larger random errors at the start of treatment. To investigate this further, we also performed direct simulations with set‐up errors that were inverted in time, that is, , , etc. For the inverted fraction order, the achieved indeed reduced: from 1.4 to 1.2 mm, from 2.0 to 1.6 mm, and from 2.1 to 1.8 mm for left–right, superior–inferior, and anterior–posterior directions, respectively. The new do better agree with the MC predictions for trendline characterization, that is, 1.2, 1.7, and 1.9 mm, respectively (see Table 2). The observed enhanced variation in setup errors at the start of treatment could be related to patients being more nervous at the start, rather than at later moments in the fractionated treatment.

Figure 7

Population of setup errors for patients in the Erasmus database. For each patient, the standard deviation for each fraction, f, was established from the measured setup errors in the fractions f‐2, f‐1, and f. For each fraction, f, these patient‐specific standard deviations, SD(f), were then combined into a [see Eq. (A1)] as presented along the y‐axis. [Color figure can be viewed at http://wileyonlinelibrary.com]

Currently, many patients are treated with daily online setup corrections.23, 24 Obviously, this approach can reduce the occurrence of time trends. However, in case of differential motion between various targets this may not be the case. For example, time trends have been observed for the lumpectomy cavity in breast cancer patients12 for primary larynx tumors13 and for lung.14 However, for these patient groups, the surrounding nodal targets do not move with the primary target. Therefore, correction of time trends in these targets based on daily setup corrections can induce effective time trends in the setup of the surrounding nodes. The proposed margin recipe can then be used for margin definition for the nodes. In that case, setup errors of nodes have to be described relative to the tumor center of mass. Such a procedure has to be further evaluated and verified prior to clinical implementation.

Probabilistic/robust planning does not need margins. However, it is based on distributions of geometric uncertainties in patient populations.25, 26, 27 To the best of our knowledge, explicit inclusion of time trends in probabilistic planning has not yet been investigated. It would add an extra complexity to the plan generation. It would also need rethinking the probability requirements for CTV coverage in case the probability of deterministic underdosing parts of the CTV in multiple consecutive fractions needs to minimized. On the other hand, we hypothesize that using trendline characterizations of setup errors in patient populations that have time trends may improve the robustness of the generated plans. Further research is needed to investigate the full consequences of explicit inclusion of time trends in probabilistic planning.

Conclusions

The conventional characterization of tumor setup errors in patient populations, using only distributions of random and systematic errors, has limitations for populations with interfraction time trends. We have investigated the use of the trendline characterization that explicitly models these time trends. Trendline characterization resulted in more accurate simulated residual setup errors after the application of offline setup correction protocols, avoiding the application of erroneous margins. The proposed novel CTV‐PTV margin recipe for nonrandom errors can be used to avoid or reduce underdosing of tumor edges in multiple consecutive fractions caused by trend motions. In the limit of the absence of time trends in a population, the margin recipe reduces to the well‐known van Herk recipe for systematic errors. In case of image guided therapy for patients with multiple targets with differential motion for daily correction of trend motion of one of the targets, the proposed recipe may be used to calculate margins for the other targets. In probabilistic planning, use of trendline characterization may result in more robust treatment plans. Population analyses of interfraction setup errors need to include the potential occurrence of time trends.

Conflict of interest

Erasmus MC Cancer Institute has research collaborations with Elekta AB (Stockholm, Sweden) and Accuracy Inc (Sunnyvale, USA). This work was not part of these collaborations.

Data S1. Python script for margin calculation.

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Data S2. Tables containing margin sizes for different parameters.

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