Generation of prescriptions robust against geometric uncertainties in dose painting by numbers.

Abstract Background. In the context of dose painting by numbers delivered with intensity-modulated radiotherapy, the robustness of dose distributions against geometric uncertainties can be ensured by robust optimization. As robust optimization is seldom available in treatment planning systems (TPS), we propose an alternative method that reaches the same goal by modifying the heterogeneous dose prescription (based on 18FDG-PET) and guarantees coverage in spite of systematic and random errors with known standard deviations Σ and σ, respectively. Material and methods. The objective was that 95% of all voxels in the GTVPET received at least 95% of the prescribed dose despite geometric errors. The prescription was modified by a geometric dilation of αΣ for systematic errors and a deconvolution by a Gaussian function of width σ for random errors. For a 90% confidence interval, α = 2.5. Planning was performed on a TomoTherapy system, such that 95% of the voxels received at least 95% of the modified prescription and less than 5% of the voxels received more than 105% of the modified prescription. The applicability of the method was illustrated for two head-and-neck tumors. Results. Systematic and random displacements larger than αΣ and σ degraded coverage. Down to 62.8% of the points received at least 95% of prescribed dose for the largest considered displacements (5 mm systematic translation and 3 mm standard deviation for random errors). When systematic and random displacements were smaller than αΣ and σ, no degradation of target coverage was observed. Conclusions. The method led to treatment plans with target coverage robust against geometric uncertainties without the need to incorporate these in the optimizer of the TPS. The methodology was illustrated for head-and-neck cancer but can be potentially extended to all treatment sites.

In the last decade, voxel by voxel heterogeneous dose prescription, also known as dose painting by numbers (dpBn) [1], has gained interest in the radiotherapy community [1][2][3][4][5][6]. A heterogeneous dose prescription within the tumor volume in accordance with a given tumor phenotype, instead of uniform dose, is expected to improve the treatment outcome in terms of local control and/or side effects [1]. However, dpBn suffers from limitations that fall into three categories: 1) biological relevance and soundness of the heterogeneous prescription, based for instance on functional imaging like positron emission tomography (pEt); 2) ability of the treatment delivery method to match the prescription with an actual dose distribution; 3) geometric uncertainties in the broad sense (patient setup, tumor motion, etc).
At the moment, the first point has not been completely addressed and important questions are still raised in the literature [1][2][3][4][5]. As in Witte et al. [6], we assume that a prescription may be derived from pEt images and needs to be delivered with minimal, controlled distortion.
Regarding the second issue, several groups have already established the technical feasibility of dpBn prescriptions for advanced treatments modalities (vMAt, tomotherapy) using structure-based treatment planning systems [7,8]. In such systems, it is possible to closely approximate dpBn using a sufficient number of sub-contours within the volume of interest.
However, the literature is sparse regarding the inclusion of patient-related geometric errors. Usual margin recipes are valid for one or several uniform dose prescriptions [9], but not for dpBn.
In this study, we assume a non-uniform dose escalation in the tumor volume (the gtv). the objective is to find a methodology such that planned dose distributions are robust against geometric errors. In Witte et al. [6], geometric uncertainties are directly taken into account during the dose optimization process. this approach is termed 'robust optimization' and has been implemented by several groups for uniform prescriptions [10][11][12][13]. However, such robust optimization techniques are rarely available in commercial tpSs.
Illustrated with two head-and-neck cases, this study proposes a methodology that provides robust dpBn treatment plans, without specific optimization algorithms in the tpS. Our methodology has therefore the potential to popularize robust dpBn. It accounts for geometric uncertainties by appropriately modifying the heterogeneous prescription. Although the chosen treatment modality is tomotherapy, the methodology also applies to any other treatment modality relying on photon beams, provided that geometric errors do not exceed certain limits that are discussed further below and also that the treatment modality can accurately deliver heterogeneous prescriptions like those encountered in dpBn.

Material and methods
Consider a volume candidate to dpBn (the gtv) with a prescription D p (r) defined in all points r of the Ct geometry. the proposed workflow to deal with geometric uncertainties is illustrated in figure 1 of Supplementary Appendix 1 'Mathematical formalism of the robust prescription' (available online at http:// informahealthcare.com/doi/abs/10.3109/0284186X. figure 1. QvHs for patient 1 for shifted and blurred planned doses D Planning with respect to the original prescription Dp within gtv pEt . the robust prescriptions D RP were computed with (aΣ, s) mentioned in top of each figure. the legend of each figure details the systematic displacements ("dis") that were simulated (no displacements, left, right, anterior, posterior, superior, inferior). Q-planning refers to QvH for D Planning with respect to D RP within ptv pEt . 2014.930171). the method consists in calculating a modified prescription D Rp (r), where RP stands for robust prescription, such that the planned dose distribution D Planning (r) is consistent with the original prescription D p (r) and robust against geometric errors, with regards to target coverage.
Quantification of treatment planning quality for non-uniform dose prescriptions for the volume candidate to dpBn, treatmentplanning quality is quantified using the decumulative quality-volume histogram (QvH) introduced by vanderstraeten et al. [2], where the quality is defined in every voxel as the ratio of the planned dose to the prescribed dose. for the tomotherapy system, deveau et al. [7] showed that the following quality criteria should be achievable using appropriate planning parameters: 1) more than 95% of the points receive at least 95% of the local prescription [(V Q  0.95 ) planning  95%]; 2) less than 5% of the points receive more than 105% of the local prescription [(V Q  1.05 ) planning  5%]; and 3) the mean quality value (Q mean ) planning is close to 1 within 1%. the subscript 'planning' indicates that the metric quantifies the tpS ability to reproduce the prescription.

Robust prescription
Like in Witte et al. [6], a plan is considered to be robust when the planned dose is superior or equal to the prescribed dose in all points of the gtv within a certain tolerance (tumor coverage objective), in spite of geometric errors. Considering our planning objective [(V Q  0.95 ) planning  95%], this amounts to requiring that (V Q  0.95 ) (aΣ,s)  95%, i.e. robustness against systematic shifts of aΣ and random shifts with standard deviation s (see [9] and Supplementary Appendix 1 available online at http:// informahealthcare.com/doi/abs/10.3109/0284186X. 2014.930171). As ensuring robustness of tumor coverage necessarily entails excess dose, the criterion (V Q  1.05 ) (aΣ,s)  5% no longer applies.
the methodology proposed in this study relies on the hypothesis of local shift invariance of the dose distributions, which is generally accepted for external photon therapy [9,14]. the presented methodology aims at satisfying ( Robustness against systematic errors within a 90% confidence interval is ensured by dilating the original prescription D p (r) by an ellipsoid with vector radius a∑ (a  2.5). A ptv pEt is subsequently defined as the gtv pEt plus a margin of a∑. for the random errors, the dilated prescription is deconvolved with a gaussian function (zero mean and standard deviation s), which corresponds to the assumed distribution of random errors. An exact solution to the deconvolution problem exists only if the prescription is the result of a convolution with a similar gaussian function having a width greater or equal to s. due to various physical limitations, pEt images may fulfill this requirement. In particular, their rather poor resolution is often approximately characterized with a gaussian point-spread function.
dilation is carried out prior to deconvolution. the order of these two operations is important to properly counteract the effect of random errors on the planned dose distributions (blurring does not have the same effect depending on whether the dose distributions reproduce original or dilated prescriptions). the effects of the aforementioned operations on the prescription are illustrated on a clinical case in Supplementary Appendix 2 (available online at http:// informahealthcare.com/doi/abs/10.3109/0284186X. 2014.930171) 'Illustrations of original, dilated and robust prescriptions on Ct geometries' (available online at http://informahealthcare.com/doi/abs/ 10.3109/0284186X).

Preparation and execution of treatment planning tests
the proposed robust method was tested on two consecutive patients treated for locally advanced head-and-neck squamous cell carcinoma using a simultaneous integrated boost protocol (SIB). the prophylactic and the therapeutic ptvs were created from the Ctv delineated on the planning Ct according to published guidelines [15][16][17] and expanded by a 4 mm margin using a generic margin recipe [9]. for prophylactic and therapeutic ptvs, uniform doses of 56 and 70 gy were prescribed, respectively. the volume subject to dpBn (the gtv pEt ) was automatically segmented on fdg-pEt images with the gradient-based method previously implemented and validated by our group [18]. the subscript pEt indicates that the external contour of the gtv pEt stems from the fdg-pEt image. the fdg-pEt images were acquired with a philips gemini pEt-Ct camera; the point-spread function was approximately gaussian, with a fWHM of 7.5 mm, corresponding to a standard deviation of 3.2 mm. this gives the theoretical assurance that pEt images from this camera may be deconvolved with a gaussian function having a standard deviation up to 3.2 mm. Accurate dilation was guaranteed by resampling the pEt images to 1 mm 3 voxels with trilinear interpolation. dilation itself used a nearest neighbor interpolator, leading to an accuracy of 0.5 mm. the validity of the present method does not depend on any particular segmentation method, provided the image may be deconvolved.
Consistently with vanderstraeten et al. [2] and Witte et al. [6], the fdg uptake was converted linearly to a non-uniform dose-escalation from 70 (minimal uptake) to 86 gy (maximal uptake). treatment planning was performed on a tomotherapy tpS with a gpU architecture, whose accuracy is comparable to the extensively validated CpU algorithm [19][20][21]. Unless otherwise mentioned, the tpS parameterization of deveau et al. [7] was followed (slice width of 1 cm; modulation factor of 3.0; and pitch of 0.43). the dose calculation grid was set in "fine" mode (about 2  2 2 mm 3 resolution).
Except for the boost volume (ptv pEt ), the clinical constraints were D 95  95% for target volumes, D mean inferior to 30 and 26 gy for ipsilateral and contralateral parotids, respectively, and D 2 inferior to 35 gy for the spinal cord extended with a 5 mm margin.
the ptv pEt was divided into seven equally spaced sub-contours to approximate the dpBn prescription. for each volume enclosed in the i th subcontour (i going from 1 to 7), the minimum dose ] . Another constraint was that D mean between contours i and i  1 equaled the average of the minimum and maximum constraints. for the last contours, the maximum dose constraint equaled (D RP ) max .

Evaluation of the robustness of the non-uniform dose distributions to geometric displacements
table I reports four couples of (aΣ, s) values used to compute the robust prescriptions D RP . for all cases, Σ x  Σ y  Σ z  Σ and s x  s y  s z  s. the robustness of the dose distributions was evaluated by: 1) shifting the dose distributions by a displacement T in six directions (left, right, anterior, posterior, superior, and inferior); and 2) blurring the dose distributions with a gaussian distribution with standard deviation s d . Subscript d was used to distinguish s d from s used to compute the robust prescription. Symbols (T, s d ) denoted the values of the displacements. Random ones were limited to s d lower or equal to 3 mm, thus lower than the resolution of the pEt camera (3.2 mm standard deviation). two sets of robustness tests were considered.
first, three scenarios evaluated tumor coverage for each patient. the associated values of (T, s d ) are detailed in table I. the three scenarios can be met in, respectively: 1) head-and-neck treatments using the best imaging tools available [22]; 2) thoracic treatments with baseline shift correction [23]; and 3) the idealized case of random errors only. to enhance readability of the results, only the scenarios involving displacements (T, s d ) larger than or equal to (aΣ, s) were considered.
In the second set of tests, the robustness was more finely tested for D Rp prepared with (aΣ, s)  (2.5 mm, 1 mm). translations of 2.5, 3.12, 3.75, 4.37 and 5 mm were applied (which corresponds to values of Σ of 1, 1.25, 1.5, 1.75 and 2 mm, respectively). Random errors s d were kept at 1 mm. Ideally, the plan should be robust to displacements of maximum (2.5 mm, 1 mm).

Results
for the first set of tests, constraints on organs at risk were easily met for each pair (aΣ, s) (see table II). figure 1 shows examples of QvHs for patient 1 for shifted and blurred D Planning with respect to D P within gtv pEt , for the scenarios detailed in table I. We focus on a possible crossing of the curves corresponding to each scenario and the one named Q planning at V Q  0.95, which is the threshold the robust prescription method aims at. When translations T exceed aΣ, there is always at least one translation direction for which (V Q  0.95 ) (T, s d ) is lower than the value of (V Q  0.95 ) planning that was obtained to quantify treatment planning quality and listed in table II. However, (V Q  0.95 ) (T, s d ) is superior to the value of (V Q  0.95 ) planning that was obtained to quantify treatment planning quality when translations T are smaller than aΣ.
the only case where random errors do impact robustness of treatment planning by more than 1% [(V Q  0.95 ) (T, s d ) ] is for a random error with 3 mm of standard deviation and no systematic error. In figure  1a and b, target coverage is preserved when s  3 mm, whereas the QvH is shifted towards lower values when s  0 mm. Moreover, there is no significant excess dose to the gtv pEt observed if no systematic translation occurs (V Q  1.05 ) (T, s d ) [ 5% in gtv pEt for (T, a d )  (0 mm, 3 mm)].
As the margin was used for systematic errors, excess dose was inherently planned to the gtv pEt as soon as aΣ departed from zero. for (aΣ, s) of (5 mm, 3 mm), (V Q  1.05 ) (T, s d ) reached a maximum of 53%.
these observations are confirmed quantitatively for both patients in table III. table Iv reports the results of the second set of tests, where (aΣ, s) is fixed to (2.5 mm,1 mm). for systematic displacements larger than twice aΣ, target coverage is clearly degraded, down to 94.5% and 92.2% for patients 1 and 2, respectively. However, target coverage is preserved for translations up to 3.75 and 3.12 mm for patients 1 and 2, respectively.

Discussion
Overall, the results shown in table II confirmed the ability of tomotherapy to closely approximate voxel by voxel prescriptions, following similar guidelines as those proposed by deveau et al. [7]. despite our efforts during planning, we could not perfectly satisfy (V Q  0.95 ) planning  95% and (V Q  1.05 ) planning  5% for (aΣ, s) of (5 mm, 3 mm) in patient 2. the likely cause is deconvolution, which sharpens the transition between dose levels and thus complicates the task of Robust prescriptions D RP were computed for the listed couples of values (aS, s). for all cases, S x  S y  S z  S. dose to the parotids and to the pRv of the spinal cord are also reported. * pitch was reduced to 0.287 and modulation factor increased to 3.5. deconvolution might still lead to dose distributions