Beam flatness modulation for a flattening filter free photon beam utilizing a novel direct leaf trajectory optimization model

Abstract Flattening filter free (FFF) linear accelerators produce a fluence distribution that is forward peaked. Various dosimetric benefits, such as increased dose rate, reduced leakage and out of field dose has led to the growth of FFF technology in the clinic. The literature has suggested the idea of vendors offering dedicated FFF units where the flattening filter (FF) is removed completely and manipulating the beam to deliver conventional flat radiotherapy treatments. This work aims to develop an effective way to deliver modulated flat beam treatments, rather than utilizing a physical FF. This novel optimization model is an extension of the direct leaf trajectory optimization (DLTO) previously developed for volumetric modulated radiation therapy (VMAT) and is capable of accounting for all machine and multileaf collimator (MLC) dynamic delivery constraints, using a combination of linear constraints and a convex objective function. Furthermore, the tongue and groove (T&G) effect was also incorporated directly into our model without introducing nonlinearity to the constraints, nor nonconvexity to the objective function. The overall beam flatness, machine deliverability, and treatment time efficiency were assessed. Regular square fields, including field sizes of 10 × 10 cm2 to 40 × 40 cm2 were analyzed, as well as three clinical fields, and three arbitrary contours with "concave" features. Quantitative flatness was measured for all modulated FFF fields, and the results were comparable or better than their open FF counterparts, with the majority having a quantitative flatness of less than 3.0%. The modulated FFF beams, due to the included efficiency constraint, were able to achieve acceptable delivery time compared to their open FF counterpart. The results indicated that the dose uniformity and flatness for the modulated FFF beams optimized with the DLTO model can successfully match the uniformity and flatness of their conventional FF counterparts, and may even provide further benefit by taking advantage of the unique FFF beam characteristics.


| INTRODUCTION
The bremsstrahlung distribution of megavoltage x rays produced by the target in a medical linear accelerator (Linac) is a strongly forward peaked intensity distribution. The variation in both energy and intensity across the beamline is compensated for by introducing a flattening filter (FF) in the beamline. The physical filter is developed to produce a nominally flat beam at a designated depth below the patient surface to accommodate the uniform dose requirement of conventional treatments. However, the FF design itself has some major drawbacks for dosimetry. Studies have shown that the FF contributes to the majority of treatment head scatter, which results in an increase in patient skin dose. 1,2 It is also energy-dependent and machine-type-dependent, which is not ideal for machine design, dosimetry, and treatment planning modeling. 1,2 More importantly, introducing the FF into the beam path significantly decreases the original dose rate due to the attenuating effect, and subsequently increases the beam delivery time. In the 1990s, several groups studied flattening filter free (FFF) high-energy photon beams. The main focus for using FFF beams at that time was on boosting the dose rate for radiosurgery since dose uniformity is not a concern for radiosurgery treatment. 3 In addition to the dose rate increase, it was concluded that FFF beams offer some dosimetric advantages compared to FF beams, such as lower out-of-field dose, lower head scatter magnitude, and less spectrum variation for various field sizes. [4][5][6][7][8][9][10] Those features can help simplify the beam model and ultimately improve dose calculation accuracy. 11,12 In recent years, the focus of FFF studies and application has shifted to intensity modulated radiotherapy (IMRT) treatment techniques, as intensity modulation (IM) techniques inherently do not require a flat beam profile for beam modulation, and the dosimetric advantages of the FFF beam can be utilized in IMRT planning. 11,13 All contemporary linear accelerators now offer the options for beam delivery using both FF and FFF modes. This allows the user to choose either mode to tailor to various treatment techniques. For example, users can choose FF beams to deliver 2D/3D plans for conventional treatment and choose FFF beams for arc-therapy and/or IMRT delivery. However, the preservation of both modes not only complicates the gantry head design, it also increases the workload for machine maintenance as well as for initial commissioning and routine quality assurance (QA). Hypothetically, if all of the current treatment delivery techniques could be facilitated using FFF beams, it would be ideal to completely remove the FF from the gantry head, which would simplify both the machine design and the physics/engineering QA. Nevertheless, the realization is that the removal of the FF from the machine would also restrict the capability to deliver conventional treatments that require flat beam delivery, for example, 2D/3D plan beams. We are, therefore, motivated to study an alternative method of producing a flat photon beam without using a FF.
To that end, a seemingly straightforward idea would be to use IMRT methods to generate flat dose maps (perpendicular to the beam axis). To the best of our knowledge, no such attempts have been published. However, our previous study provides some insight as to whether or not current IMRT delivery techniques (e.g., stepand-shoot, sliding window) are suitable to achieve such a goal. 14 Although the study suggests that reasonably flat dose maps can be achieved in most cases, some practical issues still need to be resolved prior to the full implementation of a FFF-only machine.
From one perspective, step-and-shoot delivery can generate the best beam flatness with loose segmentation number restriction. However, its delivery efficiency can be as much as five times worse than the conventional beam for the same field size. From another perspective, if the segment number was restricted extensively for the optimization in order to obtain better delivery efficiency, the flatness generated by the modulated FFF beams would become unacceptable for some cases, especially for those with large field sizes. Compared to step-and-shoot delivery, the sliding window technique appears to be a more promising delivery method for keeping both flatness and efficiency to a clinically acceptable level, according to our study. 14  Section II.F. Finally, the optimization weighting factors, beam delivery and evaluation methods for the flat beam production are described in Section II.G.

2.A | FFF beam and MLC characteristics
The FFF beam model of a Versa HD (Elekta Inc. Stockholm Sweden) machine was used for flat beam profile generation. Final dose maps of the modulated FFF beam were compared to its counterpart generated by open FF beams. The energy of the tested beam was 6 MV with a nominal dose rate 600 MU/min for the FF beam and 1400 MU/min for the FFF beam. The unit features the "Agility" head, which has a 160 leaf MLC (80 leaf pairs), and a 0.5 cm leaf-width projection at isocenter, with a maximum leaf speed of 6.0 cm/s and interdigitating capability. The MLC requires a 6 mm leaf gap for all leaf pairs during dynamic delivery. The leaf extension limit from a movable carriage is 15 cm into the opposing plane. Each bank of the movable carriages cannot pass the center line of the beam. The leaf extension limit for each individual bank is 20 cm, which is also the maximum distance between any two leaves from the same leaf bank. (1998). [16][17][18] However, this approach can potentially degrade the original optimized dose distribution generated by the FMO, due to the conversion process, since the optimized fluence map can be compromised by the restricted leaf movement. Such a drawback can severely alter the flatness of the beam for our application. The degradation effect has been thoroughly investigated in our previous study. The desire to mitigate the degradation of the optimal fluence due to the two-step process led us to pursue a direct optimization approach.

2.B | Direct leaf trajectory model
The DLTO model was initially introduced by Papp and Unkelbach (2014) for VMAT optimization. 19 Rather than optimizing the fluence as the traditional FMO two-step approach does, DLTO is able to In the above optimization model, variable d i has been successfully transferred to variable t nj , the effective beam on time. t nj can then be determined by the relationship of the arrival/departure times between the leading leaf and the trailing leaf of the same pair, as indicated in Eq. (4).
In Eq. (4), r in nj and r out nj represent the arrival and departure times of the leading leaf at the boundary between bixel j-1 and j for leaf pair n, respectively. Similarly, l in nj and l out nj represent the arrival and departure times of the trailing leaf at the boundary between bixel j-1 and j for leaf pair n, respectively. The arrival and departure times are depicted in Fig. 1, for one leaf pair trajectory, traversing bixel j.
To ensure that the breakpoints in the piecewise linear leaf trajectories are properly ordered and the trailing leaf is always behind the leading one, Eq. (5) and Eq. (6) were also added into the constraints.
These MLC timing constraints enforce monotonic leaf motion in the DLTO model and allow us to directly optimize leaf trajectories, along with finding the optimal dose distribution. This model is a convex optimization problem, as the objective function is convex and the constraints are linear. Convex optimization problems can be solved efficiently, in general, and global optimal solutions are guaranteed.
In principal, DLTO can be applied to the proposed problem to acquire the modulation to achieve a flat beam using the FFF beam.

2.C | Dynamic MLC delivery constraints
The DLTO model can be extended to take into account the additional dynamic delivery machine constraints while keeping the linearity of the constraints intact. The idea can be described as follows. As variable j represents the distance and location indicator for each leaf in the trajectory, we can intuitively utilize j to introduce distancebased constraints for the MLC. The added constraints include minimum leaf gap, maximum leaf travel distance into the opposing plane, maximum leaf travel of adjacent leaves in the same bank, and equal beam off times for all leaf pairs. Each of these is described separately in this section.
To incorporate the minimum leaf gap of opposing leaves during delivery, denoted as Lgap, the general model of Eq. (6) can be rewritten as Eq. (7) to directly enforce the leaf gap requirement. Such a constraint imposes the minimum gap throughout the entire trajectory, and also keeps the trailing leaf behind the leading one. The range of subscript, j, in Eq. (7) should be confined by leave's maximum travel distance across the centerline for both banks. Thus, both the starting bixel j; j start and ending bixel j, j end are restricted to be less than the maximum leaf travel distance, as indicated in Eq. (8).
Left Bank : j start ≤ Travel Constr:and j end ≤ Travel Constr: Right Bank : j start ≥ À ð ÞTravel Constr:and j end ≥ À ð ÞTravel Constr: Equation (9), implements the leaf carriage constraint. The constraints will be enforced only if a potential violation is found due to field shape and size. In that manner, we can customized the size of the constraints for each individual model to acquire better computation efficiency. This is achieved by preprocessing and evaluating the difference in starting and ending locations for each leaf trajectory for a given field shape. A potential violation arises if the difference in starting positons, difference in ending positons, or difference between the start and end positons is greater than the maximum allowed distance. If a potential violation is found, then constraints will be added to the model for each potential. The functions of the introduced restrictions are similar to those of Eq. (8), as it limits the range of the bixel indicator j, for a given leaf bank.
If j Sn À j Sm j j >Carr:Constr: or j Sn À j Em j j >Carr:Constr: or j En À j Em j j Then l out nj ≤ l out in which λ 1 and λ 2 are weighting factors for dose uniformity and delivery efficiency, respectively.

2.E | Tongue and groove effect
To To minimize the T&G effects, total time associated with T&G effect from both banks was added into our objective function for minimization. The objective function will then be rewritten in the where λ 3 is the weighing factor tongue and groove effect term.

2.F | Trajectory map conversion for beam delivery
Prior to the delivery of the modulated FFF beams, the trajectory maps had to be converted from "times" to "positions".  The measurements for profiles were performed using the IC Profiler (SunNuclear Inc. Melbourne, FL). The plane for dose comparison was defined at 10 cm depth with a 100 cm SAD setup throughout the study.
The first portion of the assessment focuses on the capabilities of flat beam generation for the fields with a variety of geometries using our optimization model. 2D dose discrepancy between modulated FFF beams and their FF beam counterparts for square fields of various sizes were analyzed using gamma analysis. Clinical fields including whole brain, asymmetrical spine, mantle and three other artificial fields with concave contours were also assessed to test the robustness of the model. All field opening contours are shown in Fig. 4.
Tongue and groove effect correction was examined for some special cases. Comparisons between the optimization with and without the tongue and groove effect are listed in the results section to show the efficacy of the proposed method. Delivery time is also listed for all fields used. Computation time was compared for all fields used.
The impact of the optimization parameters are evaluated below in the results section.
F I G . 4. Contours utilized throughout the study, including three arbitrary contours with concave "features", (a-c), as well as three clinical contours: (d) mantle field, (e) whole brain field and (f) asymmetric spine field.  Table 2. The definition of flatness is in accordance with IEC60976. 25 The relative central axis profiles are shown in Figs. 5, 6, and 7. Figure 5 displays the profile comparisons for both the crossline and inline for square fields of 10 × 10 cm 2 , 20 × 20 cm 2 , 30 × 30 cm 2 , and 40 × 40 cm 2 . Figures. 6 and 7 show the profile comparisons for the whole brain contour and an arbitrary concave "hourglass" contour shape, respectively.

3.B | Tongue and groove effect correction
The T&G effect control scheme described in Section II.E was employed for the fields with severe T&G effect after the initial modulation, for example, the mantle field. Figure 8 illustrates the crossline central axis profile difference of a mantle field with and without T&G effect correction. The flatness of the T&G effect-corrected mantle field is also recorded in Table 2.

3.D | Computation assessment
Computation times for modulation generation were compared for all cases presented in the study, and are displayed in Table 4. The table includes overall calculation times and specific interior point optimization times. 26 For all fields tested, except for the 40 × 40 cm 2 square field, the computation times are less than 30 s. The 40 × 40 cm 2 square field takes 80 s for optimization. The mantle field computation with the T&G correction requires a bit more computation time than the same field without T&G correction.
The correlations between the weighting factors of Eq. 14 and delivery performance are displayed in Fig. 9 and Fig. 10. Figure 8 T A B L E 1 Gamma comparison passing rates using 3%/3 mm criteria for modulated FFF beams and open FF reference beams for various field sizes.

Field
Gamma passing rate(%) 3%/3 mm  depicts the trend lines of the gamma passing rate and the relative "dip dose" difference (indicated in Fig. 8), along with the λ 1 / λ 3 ratio variation for the mantle field. Figure 10 illustrates the trend lines of the Gamma passing rate and delivery times along with the λ 1 / λ 2 ratio variation for the mantle field.

| DISCUSSION
All modulated flat beams exhibited comparable flatness to FF open beams, as indicated in Table 1, Table 2  tested. This suggests that our modulation method not only performs well for fields with regular field shapes, but also properly functions for fields with irregular field shapes. The 1D dose profile comparison results (quantitative flatness summarization in Table 2) present a similar indication. It is also worth pointing out that the profile discrepancies between the modulated FFF beams and their corresponding FF beams is clinically acceptable, while still being qualitatively apparent on measurements, especially for fields with irregular shapes (e.g., Arb SW3 field).  Table 2 implies a comparable result. Since the T&G effect control term was added into the objective function (the original objective function only has the flatness control term), the weighting factors for the flatness control term and the T&G effect control term need to be properly weighted to achieve an optimal trade off.
According to Fig. 9, the optimal clinical value of the λ 1 over λ 3 ratio should be chosen to be in the range of 0.1 to 1 in order to maintain flatness while minimizing the dose profile "dips" attributed to the T&G effect. Our final result for the mantle field uses a value of 0.1 as the ratio of λ 1 over λ 3 , optimized with the T&G effect control term in the model. An interesting observation is that the T&G effect is only noticeable in the irregular concave-shaped field contour (e.g., Mantle field). For the majority of the regular fields, the T&G effect is negligible, even when the beams are optimized with the absence of the T&G effect control term in the objective function.
As discussed in the introduction, beam flatness is not the only criteria that can be used to evaluate our method. Delivery efficiency is another key factor to assess the feasibility and practicality for the beam delivery. According to lated and open beam profiles, as was discussed previously. Figure 10 suggests that the weighting of the efficiency control term in the objective function can be very low when compared to the flatness control term, as the variation of average delivery time across all weighting schemes is very small, at just 1.7 s. Just the inclusion of the delivery efficiency control term in the objective function provides the benefit of simpler modulation and shorter delivery time. We chose 1000 as the λ 1 to λ 2 ratio for all of the cases that were tested.
Regarding the computation efficiency of the optimization algorithm (as shown in F I G . 9. Specifically for the mantle field, an evaluation of the tradeoff between Gamma passing rate and tongue and groove effect, due to the optimization weighting factors λ1 and λ3, the flatness control term and the tongue and groove control term, respectively.
F I G . 1 0 . An evaluation of the tradeoff between the Gamma passing rate and the delivery time, averaged across all fields analyzed, due to the optimization weighting factors λ 1 and λ 2 , the flatness control term and the delivery time control term, respectively.
POTTER ET AL.
| 151 direct SW based IMRT and VMAT optimization is possible using our model structure.

| CONCLUSION
We have developed a SW modulation model to generate conventional flat beams while operating in FFF mode to achieve a machine design that permits the complete removal of the flattening filter.