Sensitivity of dose‐volume indices to computation settings in high‐dose‐rate prostate brachytherapy treatment plan evaluation

Abstract Purpose To investigate the variation in computed dose‐volume (DV) indices for high‐dose‐rate (HDR) prostate brachytherapy that can result from typical differences in computation settings in treatment planning systems (TPSs). Methods Five factors were taken into account: number of dose‐calculation points, radioactive source description, interpolation between delineated contours, intersections between delineated organ contours, and organ shape at the top and bottom contour using either full or partial slice thickness. Using in‐house developed software, the DV indices of the treatment plans of 26 patients were calculated with different settings, and compared to a baseline setting that closely followed the default settings of the TPS used in our medical center. Studied organs were prostate and seminal vesicles, denoted as targets, and bladder, rectum, and urethra, denoted as organs at risk (OARs), which were delineated on MRI scans with a 3.3 mm slice thickness. Results When sampling a fixed number of points in each organ, in order to achieve a width of the 95% confidence interval over all patients of the DV indices of 1% or less, only 32,000 points had to be sampled per target, but 256,000 points had to be sampled per OAR. For the remaining factors, DV indices changed up to 0.4% for rectum, 1.3% for urethra, and 2.6% for prostate. DV indices of the bladder changed especially if the high‐dose‐region was (partly) located at the most caudal contour, up to 8.5%, and DV indices of the vesicles changed especially if there were few delineated contours, up to 9.8%, both due to the use of full slice thickness for the top and bottom contour. Conclusions The values of DV indices used in prostate HDR brachytherapy treatment planning are influenced by the computation settings in a TPS, especially at the most caudal part of the bladder, as well as in the seminal vesicles.


| INTRODUCTION
High-dose-rate (HDR) brachytherapy is widely applied in the treatment of prostate cancer. 1 An important tool in the evaluation and comparison of HDR prostate brachytherapy treatment plans are dose-volume (DV) indices that describe the volumes of different organs receiving a certain dose. DV indices are an essential part of recent recommendations and guidelines for HDR prostate brachytherapy 1,2 as well as prospective treatment planning, 3 making an understanding of the accuracy of such indices essential. Potential uncertainties in these indices should be taken into account when taking final decisions.
Many clinical and physical factors leading to uncertainties in the DV indices in brachytherapy in general have already been investigated. 4 These include changes in geometry between treatment planning and delivery, the source strength calibration, and inter-and intra-observer variability in image delineations of targets, organs at risk (OARs), and catheters. However, in addition to these uncertainties, settings in the algorithm for the computation of DV indices can also lead to differences.
A fundamental setting is the number of dose-calculation points.
The computation of DV indices is usually performed by calculating the dose in a number of points in the region of interest (ROI, either a whole organ or part of it) and assuming that these points are representative for the entire volume. These dose-calculation points can for instance be placed in a regular grid spanning the ROI. However, the use of random sampling was argued to be superior for calculating DV indices. 5 The number of points and their placement 6 influences the values of the DV indices.
Another setting that can be varied is the source description, consisting of the dosimetric and geometrical data on the radioactive source. The dose in a point is typically calculated following the TG-43 model, to which the source description is an input. 7 The source description is based on previously done Monte Carlo simulations and measurements. It determines the dose in each dose-calculation point and therefore the DV indices. The data in the source description is not exact, and improvements in simulations and measurements over time lead to updates in the source description such as mHDR-v2, 8 mHDR-v2r, 9 and mHDR-v2c, 10 revealing an inherent uncertainty in the source description.
A different type of setting is found in the representation of organs. The usual input of two-dimensional contours does not uniquely define three-dimensional organs. Therefore, the represented shape of the organ between two contours depends on the interpolation algorithm used. In the first reported DV histograms (DVHs) calculation method, 11 the contours drawn on the two-dimensional slices of the medical images were considered to fill the volume spanned by the scan, i.e., the delineated contour on the 2D slice was used for the entire slice thickness. Smoother organ surfaces can be obtained by using continuous interpolation. 12 Apart from the interpolation between contours, another setting in the organ representation is that the intersection between two organs can be considered to be part of both organs or of only one of them. Furthermore, the organ shape beyond the top and bottom contour can be defined by partial or full slice thickness. An example of a setting with continuous interpolation in combination with top and bottom contour cut-off is shown in Fig. 1, together with an illustration of the setting for the intersection between two organs.
In general, variations in the settings for the computation of DV indices lead to different values and hence potentially to different decisions regarding treatment plans, i.e., to direct clinical impact. Moreover, a comparison of plans between different TPSs is difficult if the DV computation settings are different. The differences in DV indices between phantom-based values and TPSs, 13,14 as well as between different commercial TPSs 15 have been investigated before. However, the effects of different settings in the computation of DV indices on the DV-index values of actual clinical treatment plans were not analyzed in such a manner. The aim of this study is to investigate the variation in computed DV indices that can result from typical differences in settings for the case of HDR prostate brachytherapy.

| METHODS
There are two types of DV indices: volume and dose indices. Volume indices, i.e., the sub volume of an organ that receives at least (or at most) a specific dose, are useful for describing the volume of the tumor that receives a sufficiently high dose. Dose indices, i.e., the lowest dose to the most irradiated sub volume of a certain size of an organ, are useful for describing the amount of radiation delivered to OARs, as well as to targets. In this article, we use the following notation: V a x% : the volume of organ a that receives at least x% of the planning-aim dose. D a xcm3 : the lowest dose to the most irradiated x cm 3 of organ a.

2.B | Clinical software
The TPS in which the clinically accepted plans for the patient group were created was Oncentra Brachy (version 4.3 or 4.5, Elekta AB., Stockholm, Sweden). In the TPS, DV indices were computed and evaluated in the "Brachy Planning" module.

2.C | DV computation algorithm
Software for computing the DV indices of each patient used in this article was in-house developed and validated with Oncentra Brachy.
Validation was performed by calculating the dose in a fixed set of points in both systems. For one patient case, 5000 dose calculation points were equally distributed over the 5 ROIs (Table 1). Excluding points for which the distance to the active part of the source was less than 0.5 mm, the difference was below 0.08% of the prescribed dose.
The input of our in-house developed software was the following information: • Treatment date of the patient for determining the source strength.
• Delineated contours for the ROIs.
• Catheters information, including coordinates of the implanted catheters and source dwell positions and dwell times.
• Source information, including TG-43 data describing the source.
In the following sections we describe components of the software that play a key role in computing DV indices.

2.C.1 | Dose-calculation points
The placement of dose-calculation points was done by uniform random sampling inside an ROI. To this end, for each ROI, a bounding box was created which completely enclosed the ROI. Next, points were sampled in this box uniformly randomly and only points which were within the ROI were accepted, i.e., rejection sampling was used. Sampling was continued until the desired number of points inside the ROI (allowing points to be on the surface of the ROI) was reached.
In the TPS used in our medical center, random sampling is performed with a fixed seed for the random number generator, essentially making the algorithm deterministic. Moreover, a fixed number of sample points per ROI is used. This approach introduces a dependency of the precision of a DV index on the volume it pertains to, both for volume indices and for dose indices, in the following way.
For volume indices, when a total of n points is sampled in an ROI, of which a fraction p consists of the volume corresponding to the DV index, then the number of points inside the volume of the DV index follows a binomial distribution. The probability that k of the n sampling points will be inside the volume of the DV index is equal to P X ¼ k ð Þ¼ n k p k 1 À p ð Þ nÀk with an average of μ ¼ np and a variance of The closer p is to 0.5, the larger the variance. Hence, the variance of the DV indices is based on how close the volume of the DV index is to 50% of the organ volume. The volume of the urethra can either be excluded from or included in the volume of the organ it intersects, namely the prostate.
T A B L E 1 DV indices and clinical criteria used for treatment planning. All patients involved in this study were treated at our medical center based on these criteria. Volume criteria V are relative to the total organ volume, dose criteria D are relative to the planning-aim dose.

Targets OARs
Prostate Seminal vesicles Bladder Rectum Urethra For dose indices, when sampling a number of points per ROI, the variance depends on the number of sample points that are in the volume of the DV index. This means that the variance of these DV indices is based on the number of sample points per cm 3 .
We considered the impact of the number of sample points used for the dose calculation on the precision of the DV indices. Following the approach of the TPS used in our medical center, a fixed number of sample points per ROI was used. However, in order to eliminate the dependency of the precision of dose indices on the relevant volumes, we additionally considered using a fixed number of sample points per cm 3 for dose indices.

2.C.2 | Radioactive source description
Dose calculation was based on the update of the AAPM Task Group No.43 dose formalism. 16 The radial dose function and anisotropy function were based on previously done Monte Carlo simulations of the 192-Iridium source. In clinical treatment planning, the mHDR-v2 source description 8 was used. Because of the small design change made by the manufacturer after this first study, resulting in a small change to the source used in clinical practice, new dosimetric data has been provided, 9 resulting in the mHDR-v2r source description. Both studies were then taken into account in the publishing of a consensus file, the mHDR-v2c source description. 10 We considered the impact of using each of these three different source description files.

2.C.3 | Contour interpolation
A straightforward way of defining a three-dimensional volume from two-dimensional contours made on individual slices is to assume that each contour fills the volume in the z-direction spanned by the slice (i.e., slice thickness). This approach assumes that MRI (or computed tomography, CT) scan slices represent usually an average over the slice thickness. To obtain smoother organ surfaces, an interpolation algorithm can be used. In order to study the influence of interpolation, we applied shape-based interpolation using a chamfer distance, 12 which is the interpolation method implemented in our clinically used TPS (Fig. 1).
The algorithm used for interpolation between contours of an ROI used a volume grid. 12 For the interpolation between two contours at height z 1 and z 2 , a two-dimensional grid was placed on each of the contours. For each point in a slice, the smallest Euclidean distance to the contour in that slice was calculated, where the distance is positive if the point is inside the contour and negative otherwise. Next, linear interpolation was performed between each pair of corresponding grid points on the two contours to obtain the value of that grid point at height z = (z 1 + z 2 )/2. Finally, we used the marching squares algorithm 17 to obtain the contour at height z.
For all patients involved in this study, the grid spacing in the clinically used TPS was set to "auto spacing", giving a spacing of 0.82 mm. The same spacing was used in our software. The interpolation algorithm was used for all pairs of consecutive contours on the MRI slices. This way, an interpolated contour was added half-way between each pair of delineated contours. After this, each contour was assumed to fill half the volume that the slice spanned.

2.C.4 | Including or excluding contour intersection
The intersection between two contours can be assumed to be

2.E | Analysis
When computing DV indices, we studied the influence of five factors.
1. The number of dose-calculation points per ROI used in random sampling.
3. Whether interpolation was used between pairs of consecutive delineated contours.

4.
Whether the urethra was considered to be part of the intersecting organs.

Full or partial slice thickness inclusion at the top and bottom
contour.
For a given number of dose-calculation points, we defined a baseline setting that closely followed the default settings in the clinical TPS. Specifically, in the baseline setting, the mHDR-v2 dosimetric data was used, the urethra was considered part of the prostate and bladder, contour interpolation was used, and partial slice thickness was used at the top and bottom contour.
Because the number of dose-calculation points was not a categorical variable, we first studied this factor separately, using the baseline settings for the other factors. By considering the number of points that were actually located inside an organ, the result was independent of the bounding box that was used for sampling points in that organ.
For a given number of dose-calculation points, the DV indices were computed 100 times using a pseudo-random number generator with different random seeds (the Mersenne Twister 19937 19 ). The variance was used to calculate the width of a 95% confidence interval (CI). Since the sampled points follow a binomial distribution which rapidly converges to a normal distribution for many dose-calculation points, a normal distribution was assumed. The result was averaged over all patients. By using a large number of dose-calculation points, the true influence of the remaining factors could be studied. We fixed the number of dose-calculation points to 256,000 per target, and the number of dose-calculation points in OARs to 2,560 dose-calculation points per cm 3 . First, we studied only the impact of changing the dosimetric data from mHDR-v2 to either mHDR-v2r or mHDR-v2c.
Then, we studied the impact of the remaining three factors, which resulted in a total of eight possible settings. All results were compared to the baseline setting. For the most influential factors, the influence compared to the baseline setting was tested using a paired statistical test, selected based on the data. The significance threshold was set at 0.01. Normality of the data of two variables was tested using a Q-Q plot; symmetry of the data was tested using a boxplot of the difference between two variables.

| RESULTS
The width of the 95% CI of each DV index as a function of the number of dose-calculation points when considering the baseline setting is shown in Fig. 3. When the number of dose-calculation points is fixed per target, the ordering of the confidence interval of volume indices of targets from large to small is V prostate 150% , V prostate 200% , V vesicles 80% , V prostate 100% , i.e., ordered on how close on average the volume of the DV index is to 50% of the organ volume. When the number of dose-calculation points is fixed per cm 3 , the total number per OAR depends on the OAR volume. The average delineated volume of each of the ROIs in the baseline setting is shown in Table 2.
The differences over all patients for the setting of the dosimetric data of the source with respect to the baseline setting for DV indices of prostate, seminal vesicles, and OARS, are very small. The maximum difference was observed for the seminal vesicles with 0.94% (Supplementary material, Fig. S1).
The differences in the other settings are shown in Fig. 4.
Because the urethra was delineated as the urinary catheter, there could be overlap between the ROI delineated as the urethra, and the bladder. The most sensitive DV indices were found to be the V vesicles 80% , and the D bladder  For the V prostate 150% and the V prostate 200% , the relative volume receiving over 150% or 200% of the planning-aim dose is used. However, the absolute volume receiving over 150% or 200% of the planning-aim dose was in practice independent of whether the urethra was included in the prostate. Since the DV index was calculated with respect to the total volume, the larger this total volume, the larger the difference in DV index between including and excluding the urethra. Since the volume receiving over 150% of the planning-aim dose includes the volume receiving over 200% of the planning-aim dose, the V prostate 150% was more sensitive to urethra exclusion than the , there was a statistically significant median decrease using full slice thickness (−2.6%) compared to partial slice thickness (P < 0.001).

| DISCUSSION
In this study, the influence of computation settings on the resulting DV indices of clinically optimized HDR prostate brachytherapy plans was investigated. These settings were related to number of dose-calculation points, dosimetric data (source models), and organ representation, and can differ between TPSs 2,15 . Differences in DV indices of up to 9.8% were observed.

4.A | Dose-calculation points
The study showed that a large number of dose-calculation points is required for the DV indices of the OARs to be accurate (i.e., have little uncertainty). When sampling a fixed number of points in an organ, in order to achieve a width of the 95% CI of 1% or less, only 32,000 points have to be sampled per target, but 256,000 points  have to be sampled per OAR. This is due to the use of dose indices of an absolute volume which is small compared to the total volume of the OAR, such as the D bladder 1cm3 . The large number of dose-calculation points that is required for the DV indices of OARs to reach high accuracy is in accordance with previous studies, 6 where it was recommended not to use the D min and the D max to describe dose distributions because of their large sensitivity to the number of dose-calculation points.
More dose-calculation points result in a more accurate result, but also a slower calculation. The uncertainty of grid sampling versus random sampling has been studied before and is in general even higher. 5

4.B | Radioactive source description
The maximum difference resulting from different dosimetric data of the source 8-10 was observed for the seminal vesicles with 0.94%, making this uncertainty in the range of the uncertainty of the dosecalculation points. A newer version of the dosimetric data can be assumed to be better, but the influence of this setting is negligible.

4.C | Organ intersections
The setting in our study that influenced the DV indices of the prostate the most was whether or not to include the urethra in the pros- This setting should be carefully considered by a medical center, before designing treatment plans based on a certain clinical protocol.
Excluding the urethra from the prostate not only influences the DV index values, it also affects the clinical dose aims used during treatment planning that apply to the urethra. If the urethra is included in the prostate, then the requirement V prostate 100% >95% comprises the dose to the urethra as well. However, if the urethra is excluded from the prostate, then the only aim would be D urethra 0:1cm3 <110% of the planningaim dose. Due to uncertainties, it could then be prudent to define a lower limit on the dose, e.g., on D urethra 0:1cm3 . For the seminal vesicles, the sensitivity of the DV indices to these settings could be explained by the small target volume, in combination with the large surface of the top and bottom contour. This is also because usually only the base of the vesicles is delineated. For the bladder, the most irradiated 1 cm 3 and 2 cm 3 were often exactly at the bottom contour. The importance of the 3D reconstruction algorithm at the outer slices has been noted before in a phantom study. 15 Still, settings differ between different TPSs. Oncentra Brachy uses interpolation in combination with partial slice thickness. Vitesse

4.E | Clinical impact
A limitation of this study is that it is a single-center study. The described treatment plans were optimized in our clinical TPS (i.e., Oncentra Brachy) with its default settings and could give different results if the treatment plans had been optimized in other TPSs.
However, our finding that computation settings can influence the DV index values is general. The clinical relevance of this uncertainty in the DV index values depends on the total planned dose. If patients receive brachytherapy next to external beam radiotherapy, the deviations are a smaller part of the total dose than if brachytherapy is given as monotherapy. The study is limited to prostate brachytherapy, but similar results may occur even for different treatment sites. This study has been performed retrospectively. However, computation settings may influence the optimization process inherent in treatment planning, be it manual or automated. 21,22 It would be interesting to also consider the magnitude of this influence on the optimization process and the outcome thereof. Moreover, treatment plan optimization could possibly be adapted to account for this influence by applying robust optimization to these uncertainties. Treatment plan optimization whereby the influence of different computation settings is accounted for, is suggested as future work.

| CONCLUSION S
The values of DV indices used in prostate HDR brachytherapy treatment planning are influenced by the computation settings in a TPS, especially at the most caudal part of the bladder, as well as in the seminal vesicles, potentially to an extent that it could influence decisions on final treatment plan construction.

ACKNOWLEDG MENTS
This work is part of the research program IPPSI-TA with project number 628.006.003, which is financed by the Netherlands Organi-  confidence interval associated with the uncertainty related to the sampling of dose-calculation points. Each boxplot shows the distribution 10 over all patients (median at 50%, box from 25% to 75%, whiskers at 0% and 100%).