Determining efficient helical IMRT modulation factor from the MLC leaf‐open time distribution on precision treatment planning system

Abstract Purpose Since the modulation factor (MF) impacts both plan quality and delivery efficiency in tomotherapy Intensity Modulated Radiation Therapy (IMRT) treatment planning, the purpose of this study was to demonstrate a technique in determining an efficient MF from the Multileaf Collimator (MLC) leaf‐open time (LOT) distribution of a tomotherapy treatment delivery plan. Methods Eight clinical plans of varying complexity were optimized with the highest allowed MF on the Accuracy Precision treatment planning system. Using a central limit theorem argument a range of reduced MFs were then determined from the first two moments of the LOT distribution. A step down approach was used to calculate the reduced‐MF plans and plan comparison tools available on the Precision treatment planning system were used to evaluate dose differences with the reference plan. Results A reduced‐MF plan that balanced delivery time and dosimetric quality was found from the set of five MFs determined from the LOT distribution of the reference plan. The reduced‐MF plan showed good agreement with the reference plan (target and critical organ dose‐volume region of interest dose differences were within 1% and 2% of prescription dose, respectively). Discussion Plan evaluation and acceptance criteria can vary depending on individual clinical expectations and dosimetric quality trade‐offs. With the scheme presented in this paper a planner should be able to efficiently generate a high‐quality plan with efficient delivery time without knowing a good MF beforehand. Conclusion A methodology for deriving a reduced MF from the LOT distribution of a high MF treatment plan using the central limit theorem has been presented. A scheme for finding a reduced MF from a set of MFs that results in a plan balanced in both dosimetric quality and treatment delivery efficiency has also been presented.


| INTRODUCTION
Helical tomotherapy (HT) delivers radiation therapy through synchronization of the binary Multileaf Collimator (MLC) leaf-pair openings, gantry rotation period, and couch longitudinal speed. Highly conformal dose distributions can be achieved through intensity modulation of the HT radiation field. The HT radiation field is divided into 51 projections per gantry rotation (7.06°of gantry rotation per projection).
Each projection is further divided into 64 beamlets representing each of the 64 MLC binary leafs (i.e., the leaf being either open or closed).
The leaf-open time (LOT) of each beamlet that intersects a target volume determines the instantaneous radiation dose delivered from it through the projection arc or fraction thereof. Intensity-modulated radiation therapy (IMRT) is achieved by varying the LOT of each beamlet with inverse-planning optimization of the treatment plan.
Inverse-planning optimization for Tomotherapy requires the planner to choose a modulation factor (MF) that is defined as where LOT max is the maximum LOT and LOT mean is the average of all beamlet LOTs. 1 The MF is a parameter that influences the complexity to derive an initial MF (2.1 and 1.8, respectively) and upper limit MF (2.6 and 2.2, respectively) specific to the treatment site. 14 The MF has a direct impact on treatment delivery time. Because the linear accelerator dose rate, couch speed, and gantry period are constant during helical treatment delivery, the total time for "beamon" delivery is a product of number of gantry rotations and gantry period, Total delivery time ¼ active gantry rotations Â gantry period: (2) The number of gantry rotations is determined by the pitch and the length of cranial-caudal treatment volume plus jaw width. The gantry period is equal to 51 × LOT max , unless LOT max is <235 ms, in which case the gantry period minimum has been reached at 11.8 s.
Therefore, for gantry periods above 11.8 s, Total delivery time ¼ 51 Â LOT max Â active gantry rotations: A high MF value can allow the optimizer to generate beamlets with long LOTs that have minimal impact on the dose distribution. 15 It has been suggested for complex plans that planners start with a high MF to achieve a good conformal plan and then reduce the MF until the dosimetric qualities of the plan degrade to clinically unacceptable values. 9,16 This paper presents a heuristic approach for determining a MF from the first two moments of the LOT distribution of a plan optimized with the highest allowed MF. A technique is then used to determine a set of MFs for subsequent "reduced-MF" plan calculations to find a balance between dosimetric quality and treatment delivery time.  Figure 1 shows the Precision TPS "Dx Vx

| MATERIALS AND METHODS
Value" table that allows a user to observe specific dose-volume values of a plan and comparative differences with a reference plan.

2.A | MF determination from LOT distribution
In the planning system the LOT histogram is described by three parameters; its mean, mode, and standard deviation. In our approximation we have set the mean of our un-normalized Gaussian equal to the mean of the LOT distribution and have taken the standard deviation of LOT distribution as its standard deviation.
Using this approximation to the LOT distribution, as one is justified to do by the central limit theorem, one can easily calculate a cutoff LOT max value that eliminates the upper tail of LOTs using the following equation, where LOT Mean,final is the mean of the LOT distribution after final dose calculation, and LOT Std,final is the standard deviation after final dose calculation, and z critical is the critical z-value corresponding to percentage of upper LOTs that would be filtered from a true Gaussian distribution ( Table 2). This LOT max value was then used in Eq. (1) to determine the MF for subsequent "reduced-MF plan" calculations. However, to maintain LOT Max cutoff value, and therefore the intended reduced delivery time, the mean of the optimization LOT distribution was used to calculate MF, that is,   Table 4 lists the reduced MF as a function of the z critical score using Eq. (5) for each reference plan and associated LOT standard deviation along with the estimated time of delivery using Eq. (3); the entry in bold is the lowest MF that maintained ROI dose-difference tolerances with the reference plan. Also listed in Table 4 are the fraction of LOT events in the LOT max bin. These values are slightly higher than those listed in Table 2, especially for more complex plans, but do seem to indicate that the central limit theorem is valid in describing the LOT distribution.       sparing (b,d).

3.B | SIB plans
what is remarkable is that lowest LOT-generated MF with dosevolume ROIs within dose-difference tolerances is at a z critical = 1.28, resulting in the dominant LOT max bin in the LOT distribution. Figure 7 shows the DVH comparison of the MF(z critical = 1.28) plan with the reference plan calculated at MF = 5.0. Very good agreement is seen with the PTVs and critical structures shown. The largest difference is 2 Gy seen with the spinal cord V30 but this was not controlled by an optimization dose-volume critical structure constraint, only the maximum dose was. Figure 8 shows the reference and reduced-MF plan LOTs for each of the HN plans. For case 5 the reduced MF is 2.31. For case 6.a, however, a reduced-MF plan that had ROI dose differences within tolerances could not be achieved below a z critical score of 2.33. This plan was particularly complex because of proximity of PTV to the left orbit as shown in Fig. 9 and the desire to limit the dose as much as possible to the right eye. The resulting reduced MF for case 6.a was 3.58. Plan 6.b was the same as plan 6.a with number of optimization iterations taken to 2000. The LOT distribution for reference plan 6.b shown in Fig. 8(e) clearly shows how spread out the LOT distribution becomes compared to the 500 iteration plan [ Fig. 8(c)]. While there was marginal improvement with reference plan 6.b compared to 6.a (e.g., PTV69.96 D 2% was 1 Gy cooler), the resulting reduced MF value was 4.13 and the difference in delivery time between the two reduced-MF plans was approximately 100 s.

| DISCUSSION AN D CONCLUSION
Using the central limit theorem, we have derived a methodology that allows a planner to arrive at a reduced MF from the LOT distribution of a high MF treatment plan. A scheme for finding a reduced MF from a set of MFs that results in a plan balanced in both dosimetric quality and treatment delivery efficiency has also been presented. Negative LOTs are forced to "zero value" during optimization which can effectively cutoff the lower part of the LOT distribution [e.g., the GYN reference plan 3 LOT distribution shown in Fig. 6(a)] thereby skewing the LOT distribution of the reference plan from a normal distribution. The upper tail of the LOT distribution is mostly unaffected unless the plan requires an MF that lies above the currently possible maximum MF that can be set in the planning system, as is evident in the buildup of LOT max in the more complex SIB plans presented [e.g., 1.2% for the LOT max bin in GYN reference plan 4 shown in Fig. 6(c)]. Aggressive OAR sparing and beamlet blocking can skew LOT distributions also, as evident in the prostate plans where aggressive sparing of the rectum produced bimodal LOT distributions [Figs. 5(c) and 5(g)]. However, even in these cases our method generating MFs from the LOT distribution shows utility even though the distribution looks far from a standard Gaussian distribution.
The fraction of LOT events in the maximum bin ranged between 5% and 15% with an average value of 8.9% for the final efficient MFs determined in this study. This would suggest an expected range for the fraction of LOT events in the maximum bin for plan evaluation of treatment delivery efficiency. Further studies with a more comprehensive dataset would be needed to validate this approach.
The LOT distribution expands with number of optimization iterations as seen in Figs. 3 and 10; this has an impact on the MF derived from the LOT distribution. One reason to limit each step in the progressive MF reduction scheme to 50 iterations was to minimize the impact of an ever-widening LOT distribution when trying to improve differences with adjustments to ROI weighting and penalty values. It was found from experience that 50 iterations per step MF reduction was sufficient in allowing the LOT distribution to adjust.
The other reason was to present an efficient scheme for finding good MF that a planner could utilize. We did not evaluate different MF reduction schemes since the focus of this study was on presenting a proof of concept that a good MF can be determined from the LOT distribution based on a general argument involving the central limit theorem. Further investigation of different MF reduction schema might produce interesting insights and more efficient planning schemes.

CONFLI CTS OF INTEREST
The authors declare no conflicts of interest.