Robust and optimal dose distribution for brain metastases with robotic radiosurgery system: recipe for an inflection point

Purpose. This study aims to establish a robust dose prescription methodology in stereotactic radiosurgery (SRS) and stereotactic radiotherapy (SRT) for brain metastases, considering geometrical uncertainty and minimising dose exposure to the surrounding normal brain tissue. Methods and Materials. Treatment plans employing 40%–90% isodose lines (IDL) at 10% IDL intervals were created for variously sized brain metastases. The plans were constructed to deliver 21 Gy in SRS. Robustness of each plan was analysed using parameters such as the near minimum dose to the tumour, the near maximum dose to the normal brain, and the volume of normal brain irradiated above 14 Gy. Results. Plans prescribed at 60% IDL demonstrated the least variation in the near minimum dose to the tumour and the near maximum dose to the normal brain under conditions of minimal geometrical uncertainty relative to tumour radius. When the IDL-percentage prescription was below 60%, geometrical uncertainties led to increases in these doses. Conversely, they decreased with IDL-percentage prescriptions above 60%. The volume of normal brain irradiated above 14 Gy was lowest at 60% IDL, regardless of geometrical uncertainty. Conclusions. To enhance robustness against geometrical uncertainty and to better spare healthy brain tissue, a 60% IDL prescription is recommended in SRS and SRT for brain metastases using a robotic radiosurgery system.


Introduction
Stereotactic radiosurgery (SRS) is recommended for patients lacking systemic therapy options or those with 1-4 metastatic brain tumours, excluding cases of small-cell lung cancer (Vogelbaum et al 2022).This approach achieves local control exceeding 90%, with infrequent recurrence.However, radiation-induced brain necrosis is observed in 5%-10% of patients surviving beyond five years post-treatment (Higuchi et al 2018).The probability of radiation brain necrosis is associated with the volume of normal brain irradiated above a specific dose, and the volume of normal brain receiving over 12 or 14Gy in SRS is a substantial risk indicator (Korytko et al 2006, Minniti et al 2011, Inoue et al 2013, 2014, Keller et al 2017).To mitigate this risk, lowering the percentage isodose lines (%IDL) to the prescribed level has proven effective (Xuyao et al 2020).Decreasing the IDL percentage steepens the dose gradient, thereby lessening the volume of normal brain irradiated above a specific dose (Dai et al 2019).According to ICRU Report 91, stereotactic radiotherapy (SRT) employs steep dose gradients, high doses, and few fractions, and the impact of setup errors is more significant than that of conventional radiotherapy (Wilke et al 2019).Nevertheless, no previous study has clarified the impact of geometrical uncertainties on IDLpercentage prescriptions in SRS and SRT.Therefore, this study aims to define a robust and optimal prescription method in SRS and SRT for brain metastases concerning geometrical uncertainty, while reducing the dose to the surrounding normal brain.This paper is structured as follows: (i) section 2.4 discusses geometrical uncertainties in clinical practice utilising a robotic radiosurgery system.
(ii) The effects of uncertainty on various IDL percentage prescriptions are elaborated in section 3.
(iii) section 4.1 outlines the determination of a robust and optimal IDL percentage prescription method for metastatic brain tumours in SRS and SRT.The necessity for margins based on tumour size in SRS and SRT is proposed in section 4.1 and appendix B.

CyberKnife
The CyberKnife (CK) M6 System (Accuray Inc., Madison, WI, USA) was employed in this study.The CK refers to a system for stereotactic radiosurgery and stereotactic radiotherapy (Adler et al 1998).It comprises a linear accelerator mounted on a six-axis robotic manipulator, a six-axis robotic couch, and an image guidance system.The system continuously acquires live x-ray images during treatment, adjusting the beam's position and direction to compensate for any misalignment between target and patient coordinates.Three collimation systems are available for treatment: 12 fixed circular collimators (fixed), the variable circular aperture (Iris), or the InCise TM multileaf collimators (MLC).

Target and OAR creation
The impact of geometric uncertainty on the dose distribution varies with the dose gradient, as the dose gradient is a function of position.In SRS for metastatic brain tumours, the gradient's steepness is inversely proportional to tumour size (Reynolds et al 2020).
Accordingly, this study modelled four spherical singlebrain metastatic tumour diameters (1, 2, 3, and 4 cm) of varying sizes within a simulated patient's CT scan image, designating them as the clinical target volume (CTV) (figure 1).Previous studies have indicated that assigning a CTV-to-planning target volume (PTV) margin in SRS for brain metastatic tumours increases adverse events without significantly affecting local control or overall survival (Nataf et al 2008).Hence, a CTV-to-PTV margin was not applied in this study.
For intracranial lesions, an explicit organ at risk (OAR) was not defined, and the surrounding normal brain tissue was considered the implicit OAR (Wilke et al 2019).To evaluate the maximum dose to the normal brain immediately adjacent to the tumour, a 1 mm thick volume of interest (VOI) was created around the CTV, serving as the OAR.Additionally, to assess the normal brain V14-the volume of normal brain receiving at least 14 Gy-a normal brain model was created for each tumour, subtracting the tumour from the whole brain volume.

Planning procedure
The CK treatment planning system, Precision version 3.3.1.3,was utilised for treatment planning in this study.For the CTV diameters, different collimators were employed: a fixed collimator for the 1 cm CTV, the Iris collimator for 2 and 3 cm CTVs, and an MLC for the 4 cm CTV.Based on the homogeneity of brain tissue, the Ray-tracing algorithm (Wilcox and Daskalov 2008) was applied for the fixed/Iris collimators, and the finite size pencil beam (FSPB) algorithm (Jelén et al 2005) was used for the MLC, including during optimisation.All plans were generated using the FULL for Treatment Path Set and the SKULL for Synchrony Method.Plans with 40%-90% IDL prescriptions were formulated for each CTV at 10% intervals.A total of six plans, each with different IDL percentages, were created for the four tumour radii.The inner-shell planning method (Li et al 2015) was employed, where the IDL percentage was adjusted by altering the dose gradient using four shells created around the CTV.For the 1 cm CTV, the inner-shell dimension was reduced by 2.5 mm.The minimum dose receiving 98% of the CTV (CTV D98) was established to be at least 21 Gy (Matsunaga et al 2018, Moraes et al 2019).Although fractionated irradiation is commonly adopted for CTV diameters of 3 cm or larger, this study applied single fractionation across all tumour diameters to enable comparative analysis.

Robustness analysis and Plan evaluation
The robustness analysis in this study focused geometrical uncertainties arising from patient-derived intrafractional motion and machine-derived mechanical accuracy, assuming these factors to be independent.Intrafractional motion was estimated using skull position logs from the CK system, gathered during treatments (Okamoto et al 2016) of 144 patients who underwent SRS and SRT at our institution from 2020 to 2022.These patients were immobilised using a type-S thermoplastic mask(CiVCO Radiotherapy, Orange City, IA, USA).Mechanical accuracy of the CK was evaluated based on irradiated positions obtained from monthly end-to-end (E2E) tests (Pantelis et al 2018) conducted at our institution between 2016 and 2022, with results calculated for each collimation system.Table 1 in the paper details the geometrical uncertainties considered in this study, while table 2 presents elements of the robustness analysis, following the method described by Yock et al (2019).
Dose distributions for the uncertainty scenarios were calculated using the static dose cloud approximation.These uncertainties were represented by a three-dimensional probability distribution, simulating 1000 scenarios for each plan.Intrafractional motion effects were approximated in dose calculations by shifting each voxel relative to the nominal dose distribution (Unkelbach et al 2018), which is defined as the static dose plan.The dose-grid resolution of the nominal dose was set to 0.5 mm.The resampled dose distribution was equivalent to the per-beam error-calculated dose distribution in terms of the dose for each voxel and dose-volume histogram (DVH) parameter (Sharma et al 2012).SRS with the CK system automatically detects and corrects for patient motion during treatment, with the number of image-guided corrections per treatment set at 40.This figure is typical for intracranial treatment at our institution and was selected to ensure convergence of the error function.The robustness analysis in this study primarily focused on dose changes at the tumour-normal brain boundary, specifically analysing the minimum dose for the CTV and the maximum dose for the OAR.In this study, the mean and maximum doses for the CTV were excluded from the analysis due to substantial variations between plans with different IDL percentages.
Similarly, the mean and minimum doses for the OAR were not included.Following ICRU Report 83 (Hodapp 2012), CTV D98(the near minimum dose) was utilised to quantify the minimum dose received by CTV, and OAR D2(the near maximum dose) was used to quantify the maximum dose received by OAR.
Additionally, the analysis encompassed normal brain volumes irradiated above 14 Gy, a recognised risk indicator for radiation-induced brain necrosis.As noted, normal brain volumes receiving over 12 or 14 Gy are indicative of this risk.Xuyao et al (2020) reported identical changes in V12 and V14 with dose gradients.Therefore, in this study, only the higher dose of V14 was evaluated as an indicator of radiationinduced brain necrosis.

Results
A total of 24 plans with varying IDL percentages (40%-90%, at 10% intervals) for the four tumour radii were generated to evaluate robustness against geometrical uncertainty.

Robustness analysis of CTV D98
The plans were designed to ensure that D98 was at least 21 Gy, with actual values ranging from 21.00 to 21.18 Gy for all tumour sizes and IDL percentages.
Figure 2 illustrates the differences between the average CTV D98 from the robustness analysis and the CTV D98 of the nominal dose plan, the latter referring to the static dose plan.For tumour diameters of 4, 3, and 2 cm, the 60% IDL demonstrated the smallest deviation in CTV D98.A reduction in IDL percentage from 60% to 40% resulted in a positive expansion in CTV D98 change.Conversely, increasing the IDL percentage from 60% to 90% led to a negative expansion in CTV D98 change.For a tumour diameter of 1 cm, CTV D98 decreased across all IDL percentages, with the minimal change observed at 40% IDL.Additionally, an increase in IDL percentage was associated with a decrease in dose relative to the nominal dose plan.

Robustness analysis of OAR(adjacent normal brain) D2
Figure 3 illustrates the differences between the average OAR D2 from the robustness analysis and the OAR D2 of the nominal dose plan.For tumour diameters of 4, 3, and 2 cm, the smallest change in OAR D2 was observed at 60% IDL.Decreasing the IDL percentage from 60% to 40% resulted in a positive expansion in OAR D2 change.In contrast, increasing the IDL percentage from 60% to 90% led to a negative expansion in OAR D2 change.For a tumour diameter of 1 cm, the change in OAR D2 was minimal at 40% IDL, with the dose decreasing relative to the nominal dose plan as the IDL percentage increased.

Robustness analysis of Normal brain V14
Figure 4 presents the results of the robustness analysis for normal brain V14.V14 was lowest at 60% IDL for all tumour sizes.The V14 Gy decreased rapidly from 90% IDL to 60% IDL and increased more gradually from 60% IDL to 40% IDL.This pattern was consistent between the nominal dose plans and the resampled dose plans.

Uncertainty versus IDL percentage prescription
The observation that both CTV D98 and OAR D2 increased at 40% and 50% IDL suggests a more pronounced effect on these parameters when the VOIs shift towards the dose-increasing side.This implies that the dose gradient from the prescription IDL was steeper in the direction of increasing dose compared to the decreasing dose direction, as depicted in figure 5(a).The minimal change in both CTV D98 and OAR D2 at 60% IDL suggests that the increasing and decreasing dose gradients cancel each other out.Essentially, at the 60% IDL prescription, the dose gradients in both increasing and decreasing directions are balanced (figure 5(b)), indicating a gradient shift from increasing to decreasing at the prescription IDL.This characteristic is indicative of an inflection point.The decrease in both CTV D98 and OAR D2 at 70%, 80%, and 90% IDL indicates a more substantial effect on these parameters when the VOIs shift towards the dose-decreasing side.The observation that the dose gradient from the prescription IDL is steeper in the dose-decreasing direction compared to the dose-increasing direction is illustrated in figure 5(c).Interestingly, while the applied uncertainty constitutes a random error, both the CTV D98 and the OAR D2 display systematic errors.These systematic errors manifest as a positive bias at 40% IDL, intensifying into a negative bias at 90% IDL.However, at 60% IDL, the systematic error is negligibly small.Statistically, this phenomenon is identified as systematic error stemming from random bias (Berendsen 2011).
Practically, if the curvature of a function f (x) is nonzero in a region subject to random errors, f will exhibit systematic errors.A zero curvature implies that the second derivative of the function is zero.A Gaussian distribution is characterised by an inflection point at 60% of its maximum value.The inflection point of a Gaussian distribution is defined by its width parameter σ.Detailed discussions on this are provided in appendix A. The 60% IDL prescription can thus be conceptualised as a prescription to the inflection point of a Gaussian distribution, where the tumour's centre corresponds to the centre of the Gaussian distribution and its width parameter equates to the tumour radius.This approach is henceforth referred to as the 'inflection point prescription'.Van Herk et al (2000) noted that a Gaussian distribution represents an idealised dose distribution and suggested that SRT likely offers the closest approximation to this ideal, a hypothesis supported by this study.Further backing this concept is the observation by Wang et al (2021) that CK transverse dose profiles for small fields are well approximated by a Gaussian distribution.
The inflection point prescription is expressed in coordinates with the centre of the tumour as the origin as follows: where x is a scalar variable, r tmr is the position vector whose endpoint is the tumour edge, and D Pre is the prescribed dose at the tumour edge.| | r tmr denotes the tumour radius.The second term on the right-hand side of equation (1) corresponds to the right-hand side of equation (A5) (appendix A).The 2D and 3D distributions were obtained using the bivariate and trivariate Gaussian distributions shown in appendix A, respectively.
The inflection point prescription facilitates the consideration of uncertainties represented by a Gaussian distribution.According to Van Herk et al (2000), the impact of Gaussian distribution-induced uncertainties on dose distribution is expressed through convolution.The convolution of Gaussian distributions produces another Gaussian distribution, where the width parameter is derived from the sum of squares of the individual distributions' width parameters.Consequently, the influence of uncertainty is minimal when the tumour diameter is large compared to the uncertainty, but it becomes significant when the tumour diameter is small relative to the uncertainty.
Our findings indicate that the inflection point prescription had a minor impact on uncertainty for tumour diameters of 4, 3, and 2 cm.However, for a 1 cm tumour diameter, the uncertainty led to changes in CTV D98 by approximately -1.8% and in OAR D2 by -0.5%.Further simulations were conducted to assess the effect of the ratio between the width parameter of the inflection point prescription and the magnitude of the uncertainty on dose changes, with results detailed in appendix B. These simulations revealed that the impact of uncertainty is negligible when it is less than 1/20 of the tumour radius.Therefore, for identical uncertainties, the larger the tumour, the smaller the margin required when applying a CTV-to-PTV margin.This underscores the necessity for margins based on tumour size in SRT.However, it is important to note that the aforementioned principles do not apply to uncertainties characterised by distributions other than Gaussian (Unkelbach et al 2018).
After examining the mathematical aspects and uncertainties dose distribution shape, the focus shifts to clinical implications.Clinically, optimizing by increasing the minimum dose to the tumour and decreasing the maximum dose to the normal brain is preferred.However, this study reveals a conflict between these two objectives.With a 40% IDL prescription, CTV D98 coverage increases, with the exception of 1 cm tumours, where a concomitant increase in adjacent OAR D2 is also observed.In contrast, both parameters decrease with a 70% IDL prescription.This might be a clinically favoured approach, particularly when adjacent OARs include critical organs like the brainstem or optic nerve.However, it should be noted that an IDL percentage exceeding 70% may reduce the maximum dose in adjacent OARs but could potentially increase the volume of normal brain V14.

Dose gradient of inflection point prescription versus normal brain V14
The lowest values of normal brain V14 were observed at the 60% IDL for all tumour sizes.The steeper the dose gradient from 21 to 14 Gy, the lower the normal brain V14.The inflection point in a Gaussian distribution marks the location of maximum change in the dose distribution.For more details, see appendix C. The inflection point prescription creates the steepest dose gradient at the tumour-normal brain boundary, thereby resulting in a lower normal brain V14 compared to other IDL-percentage prescriptions.This attribute makes the inflection point prescription particularly advantageous in reducing the irradiation of normal brain tissue, aligning with the overarching goal of sparing the implicit OAR in SRS and SRT.

Conclusion
The administration of a 60% IDL prescription consistently yielded the lowest volume of normal brain V14, irrespective of geometrical uncertainties.This prescription demonstrated stability in the CTV D98 and OAR D2, particularly when geometrical uncertainties were minor in comparison to the tumour radius.The dose distribution associated with the 60% IDL prescription exhibited a close resemblance to a Gaussian distribution, aligning with the inflection point of the Gaussian curve.This alignment minimises susceptibility to systematic errors arising from random bias.Incorporating this inflection point prescription into SRS and SRT for brain metastases, particularly when utilising a robotic radiosurgery system, significantly enhances precision in the face of uncertainties.This approach also reduces the risk of radiationinduced complications in the brain.The study highlights that the impact of uncertainties on the minimum tumour dose is contingent upon the tumour radius under the inflection point prescription.Pronounced alterations were observed in cases where the tumour radius was relatively small compared to the uncertainty.Conversely, negligible changes were noted in scenarios involving larger tumours.The inflection point prescription emerges as a pivotal standard for establishing margin guidelines in stereotactic radiotherapy.
Therefore, the value of this function at σ is approximately 60% of the maximum value.
For the bivariate Gaussian distribution, the PDF is described by: In the case of a Gaussian distribution, the inflection points are determined as follows: Upon further simplification: Normalisation constants are disregarded, as they have no impact on the resulting value.Consequently, this function attains its maximum value, denoted as h, at zero the centre of the function.

Figure 1 .
Figure 1.Simulated CT images of an actual patient feature four tumours with differing diameters.All tumours share the same centre position and were constructed using the 3D brush tool in MIM Maestro (MIM Software Inc., Cleveland, OH, USA).(a) In the sagittal view, tumours are represented by lines of varying colours: red for 4 cm, yellow for 3 cm, cyan for 2 cm, and magenta for 1 cm.(b) The transverse view maintains colour coding analogous to those in the sagittal view.

Figure 2 .
Figure 2. Variations in CTV D98 are displayed for each tumour diameter at different %IDL prescriptions.The figures result from a robustness analysis conducted on the nominal dose plan.These plots show the mean values for 1000 random samples.The error bar represents three standard deviations of the mean.(a) 4 cm diameter CTV.(b) 3 cm diameter CTV.(c) 2 cm diameter CTV.(d) 1 cm diameter CTV.A red dashed line signifies no change in dose relative to the nominal plan.Abbreviations: %IDL, % isodose line; CTV, clinical target volume.

Figure 3 .
Figure 3. Fluctuations in OAR D2 are shown for each tumour diameter at various %IDL prescriptions.The robustness analysis used the nominal dose plan as its basis.These plots show the mean values for 1000 random samples.The error bar represents three standard deviations of the mean.(a) 4 cm diameter CTV.(b) 3 cm diameter CTV.(c) 2 cm diameter CTV.(d) 1 cm diameter CTV.A red dashed line marks no alteration in dose compared to the nominal plan.Abbreviations: %IDL, % isodose line; CTV, clinical target volume; OAR, organs at risk.

Figure 4 .
Figure 4. Variations in normal brain V14 are analysed across different tumour diameters and %IDL prescriptions.A blue dashed line represents V14 in the nominal dose plan, whereas an orange line denotes V14 as determined through robustness analysis.These plots show the mean values for 1000 random samples.The error bar represents three standard deviations of the mean.(a) 4 cm diameter CTV.(b) 3 cm diameter CTV.(c) 2 cm diameter CTV.(d) 1 cm diameter CTV.Abbreviations: %IDL, % isodose line.

Figure 5 .
Figure 5.The schematic provides insights into the distribution of different %IDL doses.An arbitrary position is denoted by the horizontal axis X.A solid black line maps the dose distribution, and a solid blue line delineates the dose gradient at that point.Dashed lines serve to demarcate both the prescription points and the boundary between the OAR and CTV.(a) 40% IDL: The dose gradient is steep in the direction of dose increase.(b) 60% IDL: Dose gradients are equal in both dose-increasing and dose-decreasing directions.(c) 80% IDL: The dose gradient is steep in the direction of dose decrease.Abbreviations: %IDL, % isodose line; CTV, clinical target volume; OAR, organs at risk.

(
x, y are the random variables, X, Y are the mean values and centres of the distributions, and σ x , σ y are the standard deviations, serving as the width parameters for each distribution.The correlation between x and y is assumed to be zero.Analogous to the univariate Gaussian distribution, the inflection points occur where the second derivative equates to zero:

Table 1 .
Overview of uncertainties in stereotactic radiosurgery using the CyberKnife system.Intrafractional motion and accuracy are treated as independent variables.Systematic errors are not included in the analysis but are noted for reference.Abbreviations: MLC, multileaf collimators; LR, left-right; SI, superior-inferior; AP, anterior-posterior.

Table 2 .
Elements of uncertainty scenarios and their dosimetric effects.
This table details the various elements contributing to uncertainty in dosimetric outcomes.Abbreviations: MLC, multileaf collimators; LR, left-right; SI, superior-inferior; AP, anterior-posterior; CTV, clinical target volume; OAR, organ at risk.