

ORIGINAL ARTICLE 

Ahead of print publication 


An uncertaintyincorporated method for fast beam angle selection in intensitymodulated proton therapy
Natarajan Ramar^{1}, Samir Ranjan Meher^{2}
^{1} Philips Health Systems, Philips India Limited, Bengaluru, Karnataka; Department of Physics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, Tamil Nadu, India ^{2} Department of Physics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, Tamil Nadu, India
Date of Submission  31Mar2021 
Date of Decision  21Jul2021 
Date of Acceptance  12Aug2021 
Date of Web Publication  14Jan2022 
Correspondence Address: Samir Ranjan Meher, Department of Physics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, Tamil Nadu India
Source of Support: None, Conflict of Interest: None DOI: 10.4103/jcrt.jcrt_530_21
Aim: We propose a novel metric called ψ – score to rank the Intensity Modulated Proton Therapy (IMPT) beams in the order of their optimality and robustness. The beams ranked based on this metric were accordingly chosen for IMPT optimization. The objective of this work is to study the effectiveness of the proposed method in various clinical cases. Methods and Materials: We have used Pinnacle TPS (Philips Medical System V 16.2) for performing the optimization. To validate our approach, we have applied it in four clinical cases: Lung, Pancreas, Prostate+Node and Prostate. Basically, for all clinical cases, four set of plans were created using Multi field optimization (MFO) and Robust Optimization (RO) with same clinical objectives, namely (1) Conventional angle plan without Robust Optimization (CA Plan), (2) Suitable angle Plan without Robust Optimization (SA Plan), (3) Conventional angle plan with Robust Optimization (CARO Plan), (4) Suitable angle Plan with Robust Optimization (SARO Plan). Initial plan was generated with 20 equiangular beams starting from the gantry angle of 0°. In the corresponding SA Plan and SARO Plan, the beam angles were obtained using the guidance provided by ψ – score. Results: All CA plans were compared against the SA plans in terms of Dose distribution, Dose volume histogram (DVH) and percentage of dose difference. The results obtained from the clinical cases indicate that the plan quality is considerably improved without significantly compromising the robustness when the beam angles are optimized using the proposed method. It takes approximately 10–15 min to find the suitable beam angles without Robust Optimization (RO), while it takes approximately 2030 min to find the suitable beam angles with RO. However, the inclusion of RO in BAO did not result in a change in the final beam angles for anatomies other than lung. Conclusion: The results obtained in different anatomic sites demonstrate the usefulness of our approach in improving the plan quality by determining optimal beam angles in IMPT.
Keywords: Beam angle optimization, intensity modulated proton therapy, intensity modulated radiation therapy, objective function, robust optimization
> IntroductioN   
Intensitymodulated proton therapy (IMPT) is gradually becoming the treatment modality of choice for various difficult clinical situations and pediatric cases, wherein intensitymodulated radiation therapy (IMRT) is found to be lacking the ability to produce required level of intensity modulation. This advantage of IMPT stems from the fact that the intensity in IMPT can be modulated in both directions (Lateral as well as depthwise). An issue in employing proton beams in a clinical setup is its susceptibility for various uncertainties.^{[1],[2],[3]} Methods have been developed to incorporate the plan quality robustness constraints directly in the IMPT optimization process, which is known as Robust Optimization (RO).^{[4],[5],[6]} Typical treatment planning process of RO involves the following steps: (1) a number of proton beams and their angles are manually selected by the planner, (2) initial multifield optimization (MFO) is performed to obtain the optimal dose distribution, (3) the planner inputs the range and setup errors, (4) RO is performed to obtain the dose distribution accounting for range and setup errors, and (5) the planner optionally performs robustness analysis to confirm if the plan is robust against the range and setup errors.
An important concern in this workflow is that it is impossible for the planner to know which combination of beams is optimal in terms of both the plan quality and robustness. The beam angles and number of beams that result in a better plan quality need not necessarily be robust in terms of delivery uncertainties and vice versa. There are many methods that provide optimal solution to the beam angle optimization (BAO) problem for IMRT.^{[7],[8],[9],[10],[11],[12],[13]} Many studies have highlighted the importance of the selection of suitable beam angles in IMPT,^{[14],[15],[16]} and a few methods have been proposed to optimize beam angles in IMPT.^{[17],[18],[19]} On the other hand, there has been growing interest in finding the solution in a shorter time.^{[11],[12],[13],[20]} Moreover, some studies have emphasized on the need to optimize the beam angles in IMPT considering both the aspects, that is, plan quality and robustness.^{[17]}
In this work, we have adapted a fast BAO method that was originally proposed for IMRT in our previous work^{[21]} for the use in IMPT considering plan quality and robustness. Since the incorporation of robustness constraints in the beam angle selection problem significantly increases the overall time taken to optimize the beam angles, it is important to assess the figure of merit of the same. Moreover, there is no work that has been reported on the impact of the incorporation of robustness constraints into the beam angle selection algorithm on the final beam angle configuration. The purpose of this work is twofold: first, to demonstrate the feasibility of arriving at an optimal beam angle configuration incorporating uncertainty constraints using the proposed algorithm for different anatomies: lung, pancreas, prostate + node, and prostate; second, to examine the impact of the incorporation of the robustness constraints on the final beam angle configurations.
> methods and materials   
Recently, a beam angle selection algorithm for staticbeam IMRT was proposed, which uses a novel objective functionbased scoring method called ψ– score to determine the suitable beam angles^{[21],[22]} as described in Equation 1.
The targettocritical organ objective function ratio (hereafter called as ψ – score for the sake of simplicity) for a beam indexed as i is given as follows:
Here, OFV stands for objective function value and OAR stands for organ at risk.
The terms [μiμ] and [ΦΦi] indicate the increase in the OFV of target volume and reduction in the OFV of OARs, respectively, when i beam is removed from the optimized IMRT plan. The reason for adding the term “1+” in the denominator is to avoid becoming infinite when [ΦΦi] tends to zero.
Here,
In Equations (1) and (2a2d), μ is the OFV considering only the target volume, μi is the modified OFV considering only target volume when the dose contribution of the i^{th} beam is removed from the optimized IMRT plan, Φ is the OFV considering only the OARs, and Φi is the modified OFV considering only the OARs when the dose contribution of the i^{th} beam is removed from the optimized IMRT plan.
Furthermore, D_{p }is the prescribed dose to target or OARs, D_{o }is the obtained dose to target or OARs, and D^{i}_{o }is the modified dose to target or OARs when the dose contribution of the i^{th} beam is removed from the optimized IMRT plan. The W parameters are weights, i.e., W_{target} is an importance factor (i.e., weight) for the target volume, W_{OAR1},…,W_{OARx} are the respective importance factors (i.e., weights) for the OARs designated OAR 1,…,OAR x (where the limiting case of x=1, i.e., a single OAR, is contemplated). Equations (1) and (2a2d) further employ the following additional indices and count values: index k denotes a voxel in target volume; count n denotes a total number of voxels in target volume; index m denotes a voxel in a given OAR (i.e., in a given critical organ); count x denotes the number of OARs (i.e., number of critical organs) considered in the optimization; and counts n1,n2,…nx denotes the number of voxels in, OAR1, OAR2,…,OARx, respectively.
It is to be noted that the equations can also be stated in terms of dosevolumebased objective function instead of dosebased objective function. The optimization algorithm used in Pinnacle TPS uses a dosevolume based objective function.
In Equation 1, the calculation happens per beam denoted by i. Essentially, Equation 1 is based on the assertion that if a Beam's dose contribution from an optimal placement (gantry angle) is removed from an optimized plan, it will result in a larger increase in OFV of target volume than the increase in OFV corresponding to OAR. Conversely, if a Beam's dose contribution from a nonoptimal or suboptimal placement is removed from the optimized IMRT plan, it will result in a lesser increase in the OFV of target volume than the increase in OFV corresponding to OAR. The ratio between increase in the OFV of target volume and reduction in the OFV of OAR constitute the proposed ψ – score, which will be commensurate with the optimality of beam in terms of producing the desired dose.^{[21],[22]} Fundamentally, ψ – score calculates the intrinsic freedom available for depositing the required dose to tumor in a given angle. Hence, a low ψ – score indicates lesser freedom for depositing the dose to tumor and vice versa. We exploit this property of ψ – score to differentiate an optimal beam from a suboptimal one.
In the present work, we have repurposed Equation 1 for IMPT context based on the assumption that the abovementioned assertions to qualify an optimal beam for IMRT will also be applicable for IMPT beams.
[Figure 1] illustrates the proposed beam angle selection algorithm. We have created Pinnacle scripts to automate below mentioned workflow, which has significantly reduced the manual effort and time taken.
We have used Pinnacle TPS (Philips Medical System V 16.2) for performing IMPT optimization and RO. To validate our approach, we have applied it in four clinical cases: lung, pancreas, prostate with lymph nodes, and prostate. Basically, for all cases, four set of plans were created with same clinical objectives, namely, (1) conventional angle plan without RO (CA plan), (2) suitable angle plan without RO (SA plan), (3) Conventional angle plan with RO (CARO Plan). and (4) Suitable angle plan with RO (SARO Plan). In the CA plans (with and without RO), the number of beams and their angles were decided based on the existing class solutions for each anatomy.^{[16],[17],[18],[19],[23],[24]} In the corresponding SA plans (with and without RO), the beam angles were obtained using the guidance provided by the algorithm explained in our previous paper.^{[21]} We have used twenty equiangular beams starting from the gantry angle of 0^{o} as input to the BAO algorithm. For SA plans, we have performed MFO for candidate beams to calculate ψ – score, whereas for SARO plans, MFO followed by RO was performed for all candidate beams to calculate ψ– score. In order to maintain an effective comparison, the number of beams was kept the same in CA, CARO, SA, and SARO plans. [Table 1] provides the setup error and range errors used in different anatomies to perform RO. Subsequently, we have studied the impact of the incorporation of RO on the final beam angles resulting from BAO algorithm and also the total time taken to get optimal beam angles. In all plans, a computed tomography slice thickness of 3 mm and a dose grid resolution of 0.3 cm was used. The tumor volumes and OARs were segmented by qualified radiation oncologists. We used IBA spot scanning Machine to generate all IMPT plans. The following hardware configuration was used in the study: X6–2 Professional (Solaris V.11) with two Intel Xeon CPU E52699 v4 @ 2.20 GHz, RAM of 384 GB.
Dosimetric results were quantitively evaluated by calculating Homogeneity index (HI) and target coverage for all plans.^{[24]}
HI = D5/D95, where D5 is dose to 5% of volume of target and D95% is dose to 95% of volume of target, and Target coverage (D95%) = Dose to 95% volume of target.
Beam angles used in CA, SA, CARO, and SARO plans for the clinical cases were listed in [Table 2]. The beam angles were obtained from the ψ – score plots in [Figure 2]. The percentage reduction in dose for OARs and target coverage are shown in [Table 3] and [Table 4], respectively.  Table 2: The beam angles used in this study plans for the clinical cases
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 Figure 2: Shows the selected beam angles used in suitable angle and suitable angleRO Plan. The arrows indicate the selected suitable beam angles from the respective ψ score plots
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In the Lung case, CA plan includes one posterior and one right posterior oblique (180° and 234°). In case of SA plan, angles obtained from the algorithm include one right posterior oblique and one right anterior oblique (234°_{,} 306°). In SA RO plan, angles obtained from the algorithm include one right posterior oblique and one right anterior oblique (216°_{,} 288°). However, the beam angles obtained from the algorithm for SA and SARO plans remain the same in pancreas and prostate cases.
Dose distribution and dose volume histogram (DVH) comparison of all four clinical cases are shown in [Figure 3] and [Figure 4], respectively. In [Figure 3], left side indicates the dose distribution between CA plan and SA plan, on the right side indicate the dose distribution between CARO plan and SARO Plan. In [Figure 4], Left side indicates the DVH comparison between CA plan and SA plan, on the right side indicate the comparison of CARO and SARO plan.  Figure 3: Axial dose distribution of conventional angle, suitable angle, conventional angleRO and suitable angleRO plans for lung, pancreas, prostate + nodes and prostate cases. Left: Conventional angle plan versus suitable angle plan and right: Conventional angleRO Plan versus suitable angleRO plan. Dotted lines indicate the gantry angles
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 Figure 4: Dose volume histogram comparison of conventional angle (dotted line) and Suitable angle (solid line) plans for lung, pancreas, prostate + nodes and prostate cases. Left: Conventional angle plan versus suitable angle plan and right: Conventional angleRO plan versus suitable angleRO plan
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Lung case
In both SA and SARO plan, dose distribution is demonstrated better compared to CA and CARO plan. We observed that the spinal cord and esophagus sparing were better with SA plans with equal target coverage. There were less dose difference in bronchial tree sparing and slight increase in rest Rt lung dose.
Pancreas case
In both SA and SARO plan dose distribution are demonstrated better compared to CA and CARO plan. We observed that the liver, spleen, and rest left kidney sparing were better with SA plans with equal target coverage. There was less dose difference in rest bowel sparing.
Prostate + node and prostate case
In both SA and SARO plan, dose distribution is demonstrated better compared to CA and CARO plan. We observed that the rectum, left femur, and right femur sparing were better with SA plans with equal target coverage. There was less dose difference in bladder sparing.
[Figure 5] shows the comparison of target dose homogeneity of robustly optimized plans (CARO and SARO) under setup and range uncertainties for all clinical cases. [Figure 6] shows the comparison of target dose coverage of robustly optimized plans (CARO and SARO) under setup and range uncertainties for all clinical cases. It is evident from [Figure 5] and [Figure 6] that the fluctuations in HI and target dose coverage (95%) are similar between CARO and SARO plans. There was a considerable difference in lung case.  Figure 5: The comparison of target dose homogeneity of robustly optimized plans (CARO and SARO) under setup and range uncertainties for all clinical cases
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 Figure 6: The comparison of target dose coverage (95%) of robustly optimized plans (CARO and SARO) under setup and range uncertainties for all clinical cases
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> Discussion   
In this work, we have used a novel method for selecting optimal beam angles in IMPT.^{[21]} The results indicate that the plan quality for the plans employing the optimal beam angle configuration using the proposed algorithm is considerably improved as compared to the plans employing the conventional beam angles. The reduction in OAR mean dose is about 10%–40% and 15%–60% in SA plan and SARO plan, respectively, depending on the treatment site while obtaining equal target coverage as compared to their CA counterparts.
On the computational front, it took around 10–15 min to find the suitable beam angles in SA plans using the proposed BAO algorithm, which includes the time taken for initial optimization with all candidate beams and the time taken for selecting angles from the ψ – score. When RO was included in the optimization (that is, in SARO plans), it took around 20–30 min to find the suitable beam angles. In general, the final beam angles obtained using the BAO algorithm in SA plans were different from the conventional beam angles (anatomyspecific class solution) used in CA plans, and especially for lung case, this difference was significant. However, we did not observe any difference in the final beam angles between SA and SARO plans for prostate (with and without nodes) and pancreas cases. There was a considerable difference in the final angles for SA and SARO plan for lung. This result indicates that the inclusion of RO may not impact the optimality of the beam angles significantly for anatomies other than lung. Considering the additional time taken to determine the optimal angles when RO is included in the process, it is suggested to include RO in the beam angle selection process judiciously in situations, wherein the uncertaintyinduced errors would be significant.
One way to reduce the time taken for finding the optimal angles is to restrict the feasible angle range based on the tumorOAR geometry. This will reduce the number of candidate beams used in the initial optimization and thereby reduce the overall time taken in the angle selection process. For instance, in an offcentrically located tumor such as in lung cases, we can avoid sampling the candidate beams in the farther side. An additional advantage of this approach would be a possible reduction in the interplay effect,^{[25]} which might improve the efficiency of the proposed method in finding optimal angles.
The impact of the number of iterations on the final beam angles obtained from BAO was studied by varying the total number of iterations from 10 to 50 with an interval of 10 in the lung case. SA plan could not be distinguished from the other angles using the resulting ψcurves when the number of iterations was 10, 20, and 30 iterations. When the optimization was performed with 40 iterations, the resulting ψcurve became effective enough to distinguish SA from the rest of the angles.
Although the proposed algorithm has been tested in coplanar geometry, in theory, it is applicable in noncoplanar geometry as well. However, the inclusion of noncoplanar geometry will significantly increase the number of candidate beams and in turn increase the interplay effect.^{[25]} Since BAO is a nonconvex optimization problem, it is highly difficult to arrive at a global optimum solution in a clinically relevant timeframe. The algorithm presented in this work is only aimed at providing an intuitive first guess to the clinical users in a short time so that the amount of manual effort and time involved in finetuning the plan quality can be potentially reduced.
Although significantly lesser number of beams are employed in IMPT as compared to IMRT, there is still a need to use the optimal number of beams in IMPT. The emphasis for IMPT for using optimal number of beams comes from the delivery standpoint rather than the plan quality standpoint. There are methods available to find optimal number of beams in IMRT.^{[26]} We can use similar such methods to find out optimal number of beams in IMPT as well. Alternatively, we can use ψ – score plot itself to determine the optimal number of beams using the shape of the ψ– score plot. Moreover, there are studies that report a reduction of number of beams when the beam angles are optimized.^{[27]} It would be interesting to study if the same is applicable for IMPT. We will be addressing these research questions in our future communications.
Unlike IMRT, IMPT plans can employ far fewer number of beams to produce acceptable plan quality. Typically, an IMPT plan employs two to three beams. There are studies that demonstrate no significant differences in CTV coverage and plan robustness when the beam number is increased.^{[28],[29]} Moreover, after the advent of uncertaintyincorporated IMPT optimization techniques, the need to add more IMPT beams merely for the sake of improving plan robustness has been alleviated in many clinical situations. For instance, more beams were traditionally used to mitigate the end range uncertainty in conventional IMPT. However, in the uncertaintyincorporated IMPT, range error can be directly included during optimization itself that eliminates the need for additional beams.
We have used a novel metric called ψ– score to rank the IMPT beams in the order of their optimality and robustness. The results obtained in different anatomic sites demonstrate the usefulness of our approach in improving the plan quality by determining optimal beam angles in IMPT. The study shows that there is considerable increase in the time taken to obtain the final beam angles if RO is incorporated in the workflow. However, the inclusion of RO in BAO did not result in a change in the final beam angles for anatomies other than lung. Hence, it is recommended to weigh the benefit of including RO in the BAO workflow.
Acknowledgment
I would like to thank Dr. R. Vaitheeswaran, Philips Health System, for his valuable guidance/Discussion.
Financial support and sponsorship
Nil.
Conflicts of interest
There are no conflicts of interest.
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[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6]
[Table 1], [Table 2], [Table 3], [Table 4]
