Fundamental properties of the delivery of volumetric modulated arc ...

Fundamental properties of the delivery of volumetric modulated arc therapy (VMAT) to static patient anatomy D. Rangaraj, S. Oddiraju, B. Sun, L. Santanam, D. Yang, S. Goddu, and L. Papiez Citation: Medical Physics 37, 4056 (2010); doi: 10.1118/1.3453575 View online: http://dx.doi.org/10.1118/1.3453575 View Table of Contents: http://scitation.aip.org/content/aapm/journal/medphys/37/8?ver=pdfcov Published by the American Association of Physicists in Medicine Articles you may be interested in Feasibility of a unified approach to intensity-modulated radiation therapy and volume-modulated arc therapy optimization and delivery Med. Phys. 42, 726 (2015); 10.1118/1.4905373 Characterization of a novel 2D array dosimeter for patient-specific quality assurance with volumetric arc therapy Med. Phys. 40, 071731 (2013); 10.1118/1.4812415 External beam pulsed low dose radiotherapy using volumetric modulated arc therapy: Planning and delivery Med. Phys. 40, 011704 (2013); 10.1118/1.4769119 Evaluation of volumetric modulated arc therapy for cranial radiosurgery using multiple noncoplanar arcs Med. Phys. 38, 5863 (2011); 10.1118/1.3641874 Treatment planning for volumetric modulated arc therapy Med. Phys. 36, 5128 (2009); 10.1118/1.3240488

Fundamental properties of the delivery of volumetric modulated arc therapy „VMAT… to static patient anatomy D. Rangaraja兲 Washington University School of Medicine, St. Louis, Missouri 63110

S. Oddiraju Washington University School of Medicine, St. Louis, Missouri 63110 and University of Missouri, Columbia, Missouri 65201

B. Sun, L. Santanam, D. Yang, and S. Goddu Washington University School of Medicine, St. Louis, Missouri 63110

L. Papiezb兲 University of Texas Southwestern Medical Center, Texas 75390

共Received 21 January 2010; revised 12 April 2010; accepted for publication 20 May 2010; published 13 July 2010兲 Purpose: The primary goal of this article is to formulate volumetric modulated arc therapy 共VMAT兲 delivery problem and study interdependence between several parameters 共beam dose rate, gantry angular speed, and MLC leaf speed兲 in the delivery of VMAT treatment plan. The secondary aim is to provide delivery solution and prove optimality 共minimal beam on time兲 of the solution. An additional goal of this study is to investigate alternative delivery approaches to VMAT 共like constant beam dose rate and constant gantry angular speed delivery兲. Method: The problem of the VMAT delivery is formulated as a control problem with machine constraints. The relationships between parameters of arc therapy delivery are derived under the constraint of treatment plan invariance and limitations on delivery parameters. The nonuniqueness of arc therapy delivery solutions is revealed from these relations. The most efficient delivery of arc therapy is then formulated as optimal control problem and solved by geometrical methods. A computer program is developed to find numerical solutions for deliveries of specific VMAT plan. Results: Explicit examples of VMAT plan deliveries are computed and illustrated with graphical representations of the variability of delivery parameters. Comparison of delivery parameters with that of Varian’s delivery are shown and discussed. Alternative delivery strategies such as constant gantry angular speed delivery and constant beam dose rate delivery are formulated and solutions are provided. The treatment times for all the delivery solutions are provided. Conclusion: The investigations derive and prove time optimal VMAT deliveries. The relationships between delivery parameters are determined. The optimal alternative delivery strategies are discussed. © 2010 American Association of Physicists in Medicine. 关DOI: 10.1118/1.3453575兴 Key words: VMAT, IMRT, RapidArc, static target, DMLC delivery, arc therapy, optimization I. INTRODUCTION Intensity modulated radiation therapy 共IMRT兲 using rotational methods has gained great popularity and interest in recent times.1–4 The idea of rotational therapy was initially explored by Brahme et al.5 as a proof of concept. Later, more detailed analytical investigations at the level of treatment planning were performed by Cormack and co-worker,6–8 Tulovsky et al.,9 Papiez and Ringor,10 and Ringor and Papiez.11 The feasibility of linear accelerator based intensity modulated arc therapy for clinical use was studied by Yu et al.12 in 1995. In this work, they proposed clinical use of arc therapy with field shape modulated per gantry angle on linear accelerator to produce conformal dose distribution. This method did not gain much interest until the single arc version of the same, called volume modulated arc therapy, was proposed by Otto15 because of its simplicity and delivery efficiency compared to the method proposed by Yu et al. Along with these developments, commercial availability of the volumetric 4056

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modulated arc therapy 共VMAT兲 version of rotational delivery such as Varian’s RapidArc and Elekta’s VMAT products developed considerable momentum in clinical implementation of this technique by early adopters. Even with all these developments, VMAT is still in its infancy. The complete understanding of VMAT delivery, VMAT specific optimization and treatment planning approaches, has not been thoroughly studied. It is clear that more work is needed to understand the VMAT treatment and its potential for delivering modulated dose distribution as stated by Bortfeld13,14 and Webb.16,17 In this work, we clarify main aspects of VMAT delivery using commercial available linear accelerator technology. The questions we are addressing in this work are the following: 共a兲共b兲

What are the relationships between the VMAT delivery parameters? How to choose the optimal VMAT delivery solution

0094-2405/2010/37„8…/4056/12/$30.00

© 2010 Am. Assoc. Phys. Med.

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共c兲

Rangaraj et al.: The delivery of VMAT modulated arc therapy

among several solutions available to deliver the treatment plan? How to choose alternate solutions to deliver the treatment plan and what are their characteristics?

Additionally, we provide proof for optimality of our solution and analytical relationship with pictorial representation of the delivery parameters. Also, we qualitatively compare our delivery solutions to a commercial delivery solution 共RapidArc兲. We avoided quantitative conclusions because of the unavailability of complete details of the commercial solution 共RapidArc兲 in terms of constraints, algorithm, and minor practical issues. In this work, we do not aim to resolve issues involved with the understanding of VMAT planning and optimization. In other words, we do not address the questions such as 共a兲 how to improve dose distribution of VMAT plans, 共b兲 how to design optimization techniques that do not compromise plan quality for delivery efficiency, 共c兲 are VMAT plans universally better than conventional IMRT, etc. Rather, our starting point is that the VMAT treatment plan is generated. Our interest is to clarify and quantify all possible realizations of a given VMAT plan when parameters of delivery vary so that the treatment plan is delivered as computed by the treatment planning system. The goal of this paper is to derive relations between parameters of delivery 共gantry speed dg / dt, dose rate dm / dt, and MLC leaf velocities dz / dt兲 that deliver the intended treatment plan without any compromise and without violating delivery parameters’ constraints. The secondary goal is to derive the most time efficient delivery under typical hardware constraints explicitly formulated in our investigations that minimize the beam on time 共fastest delivery possible兲 given the treatment plan is not violated. Additionally, we have explored alternative delivery approaches to the VMAT plan treatments such as constant gantry angular speed 共CGAS兲 and constant beam dose rate 共CBDR兲 deliveries. We have utilized five Varian RapidArc plans generated for clinical use for our investigation. There is no scientific reason for choosing treatment plans from the Varian Eclipse treatment planning system and we could have very well used plans from any other planning system. Our aim is to reveal the properties of the VMAT delivery and not make direct comparison against any commercial vendor’s delivery. We have qualitatively compared our results to Varian delivery parameters.

II. MATERIALS AND METHODS In VMAT delivery, radiation is delivered to the patient by rotating the gantry of a linac around the patient few times during which the radiation beam is continuously ON. During this process, several delivery parameters could be varied such as 共1兲 the dose rate or fluence rate, 共2兲 the MLC aperture shape, and 共3兲 the gantry rotation speed. The allowable limits of these variable parameters become the delivery constraints for the delivery. The MLC aperture shape is determined during treatment planning process as a function of Medical Physics, Vol. 37, No. 8, August 2010

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discrete gantry angle position. Also, the dose or monitor units 共MU兲 to be delivered for each MLC aperture are determined during this stage and this define a treatment plan which becomes an invariant for the delivery. Below we describe in detail the delivery invariant, delivery parameters, and constraints, and derive a relationship between delivery parameters to be maintained to realize the delivery invariant 共the treatment plan兲. II.A. VMAT treatment plan: Delivery invariant

In our study, we use five clinical treatment plans generated using Varian Eclipse RapidArc treatment planning system version 8.5 for prostate carcinoma. The starting point for this delivery problem is the treatment plan. These treatment plans were prepared to be delivered on Varian linac. The DICOM RT plans are exported and the MLC aperture shape and the gantry position as a discrete function of monitor units are extracted. This defines a treatment plan which should be treated as an invariant of the treatment delivery, i.e., it cannot be changed during delivery without violating the dosimetric integrity of the plan. For RapidArc plans, the number of MLC apertures is 177, which are spaced every 2° for one full arc. II.B. Notation introduction and mathematical formulation of VMAT treatment plan

Before getting into the details of the delivery parameters and their relationships, we define the treatment plan using mathematical notations. The VMAT treatment plan contains uniquely shaped MLC apertures A, defined for each gantry angle g 关denoted by A共g兲兴 and uniquely determined monitor unit 共dose output兲 function M describing the number of monitor units associated with each MLC aperture indexed appropriately by gantry angle g 关denoted by M共g兲兴. It is easy to notice that a given sequence 共A共g兲 , M共g兲兲 results in a uniquely defined dose distribution in the patient. Thus, the requirements to deliver a given treatment plan results in precise delivery of the sequence of apertures A共g兲 and monitor unit function M共g兲. Note that notation A共g兲 for aperture at gantry angle g is symbolic. For MLC, the aperture is defined by opening of leaves of MLC assembly. It consists of positioning of left and right leaf pairs that define the aperture A共g兲 at gantry angle g. Thus, if the number of leaf pairs involved is K, then A共g兲 is equivalent to a sequence of 2K numbers that indicate left zL,k共g兲 and right zR,k共g兲 leaf positions of pairs k = 1 , 2 , . . . , K of MLC leaves at gantry angle g. Since the delivery controlling parameter in MLC is leaf velocity, it would be rather useful to define MLC apertures as a function of MLC leaf velocities rather than MLC leaf positions. We notice that the treatment plan can be defined through derivatives dA共g兲 / dg of apertures A共g兲 rather than apertures A共g兲 themselves. First, if we know dA共g兲 / dg, denoted by V共g兲, we can determine A共g兲 by integrating over all gantry angles, provided initial condition for A共g兲 is known 共at g = 0, for example兲. Moreover, we note that similarly as A共g兲, the quantity of V共g兲 = dA共g兲 / dg is equivalent to a sequence of 2K numbers that indicate left dzL,k共g兲 / dg and right

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dzR,k共g兲 / dg leaf velocities 共relative to the gantry angle parameter change兲 of pairs k = 1 , 2 , . . . , K of MLC leaves at g that are directly controllable variables in DMLC delivery. Summing up, we can state that VMAT arc treatment plan is uniquely defined by pair 共A共g兲 , M共g兲兲 or, equivalently, by pair 共V共g兲 , M共g兲兲.

II.C. VMAT delivery parameters

It is well known that delivery parameters for VMAT therapy are gantry angular speed 共dg / dt兲, MLC leaf velocities 共dz / dt兲, and beam dose rate 共dm / dt兲. All these parameters are allowed to vary with time while satisfying the constraints imposed by hardware limitations 共see Sec. II D兲. These are the typical delivery parameters for Varian linac and thus only these are considered. Other constraints such as gantry acceleration and collimator rotation speed etc. are not considered, as these are not the commonly stated constraints for RapidArc treatment plans used in this work. The problem of VMAT delivery is to relate these parameters to each other at each instant during treatment so that treatment plan 共V共g兲 , M共g兲兲 is delivered exactly as planned. The derivation of the relationship between the parameters as well as insight into their geometrical interpretation and understanding of their properties are presented in the following sections.

II.D. Delivery parameter constraints

To better formulate the problem, the hardware constraints present in the existing photon therapy delivery systems for VMAT are to be considered. Typical constraints for clinical linear accelerators equipped with MLC systems include 共a兲共b兲

共c兲

Limitation ␻max for the maximum speed of gantry rotation. Limitation rmax for the maximum value of beam dose rate. rmax is the maximum value for quantity 关dm / dt兴, where m共t兲 denotes cumulative number of monitor units delivered for time t and 关dm / dt兴 is derivative of m共t兲 which interpret the number of monitor units delivered per unit time t. In other words the limitation of beam dose rate means that 关dm / dt兴 belongs to interval 关0 , rmax兴. Limitation vmax for the maximum physical speed of leaves denoted ␯physical = 关dz / dt兴, where z denotes position of leaf measured relative to the isocenter along the direction of leaf motion. Thus, vmax is the maximum value for quantity 关dz / dt兴, where z共t兲 denotes position of leaf at t. In other words the limitation for leaf velocity means that 关dz / dt兴 belongs to interval 关 −vmax , vmax兴.

The delivery constraints such as maximum gantry speed, maximum MLC leaf velocity and maximum dose rate were set in the Eclipse treatment planning system and the same were used for our algorithm. Medical Physics, Vol. 37, No. 8, August 2010

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II.E. Relation between delivery parameters for a given arc treatment plan

To deliver the VMAT treatment plan, the gantry should be rotated around the patient, for example, from g = 0 to g = 360, while preserving MLC apertures as functions of gantry angle g and preserving beam weights 共monitor units兲 associated with each aperture. Moreover, delivery parameters of VMAT treatments should not violate machine constraints discussed in Sec. II D 关items 共a兲–共c兲兴. 共i兲

The condition to achieve beam weight M共g兲 at g during delivery of arc treatment plan can be expressed in terms of gantry speed dg / dt and beam dose rate dm / dt at g as M共g兲 =

dm dt dm 共g兲 = 共g兲 · 共g兲. dg dt dg

共1兲

Thus, setting beam dose rate dm / dt at g as dg dm 共g兲 = M共g兲 · 共g兲 dt dt

共2兲

assures that for gantry rotating with angular speed dg / dt at g, the number of monitor units associated with aperture A共g兲 is M共g兲 as required by the VMAT treatment plan. 共ii兲 The condition for preserving MLC aperture A共g兲关or, equivalently, preserving the speed of change V共g兲 of beam aperture at g兴 can be expressed in terms of parameter dg / dt and leaf velocity vL,k = dzL,k共g兲 / dg and vR,k共g兲 = dzF,k共g兲 / dg for left and right leaves of leaf pair k as follows: vL,k共g兲 =

␯phys共g兲 dzL,k dt dzL,k 共g兲 = 共g兲 · 共g兲 = L,k dg dt dg 关dg/dt兴

共3兲

␯phys共g兲 dzR,k dzR,k dt 共g兲 = 共g兲 · 共g兲 = R,k , dg dt dg 关dg/dt兴

共3⬘兲

and vR,k共g兲 =

phys phys 共g兲共共dzR,k / dt兲共g兲 = vR,k 共g兲兲 dewhere 共dzL,k / dt兲共g兲 = vL,k notes the physical speed 共cm/s兲 of left or right leaf of pair k. Thus, setting the left and right leaf physical speed of each pair k at g to value

phys 共g兲 = vL,k共g兲 · vL,k

dg 共g兲 dt

共4兲

dg 共g兲 dt

共4⬘兲

and phys 共g兲 = vR,k共g兲 · vR,k

assures that for gantry rotating with angular speed dg / dt at g, the speed of change V共g兲 of beam aperture is defined by vL,k共g兲 and vR,k共g兲 as required by the arc treatment plan.

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Plane v = vmax

(x, y=ax,v=b2x)

vmax Plane x = 0

P*

(x, y=ax,v=b1x)

v = b1 x Plane y = 0

P*

rmax y = dm/dt

ωmax Plane y = rmax

x = dg/dt

y=ax Plane x =

ωmax Plane v = - vmax

FIG. 1. Geometric representation of the relation among delivery parameters of a VMAT plan for a given fixed g. Note that the solid line represents set of delivery parameters for leaf of pair number 1 and that the dashed line represents set of delivery parameters for leaf of pair number 2. Projections of both lines on plane 共x , y兲 is the same line and projections of both lines 共solid and dashed兲 on 共x , v兲 plane are different lines.

II.F. Geometric representation of relationship between delivery parameters

We like to note that in Eqs. 共2兲 and 共4兲, beam dose rate and leaf velocities at g appear as linear functions of independent variable dg / dt at g. Denoting three parameters of delivery 共dg / dt , dm / dt , dz / dt兲 at g as variables x, y, and v respectively, we can rewrite Eqs. 共2兲 and 共4兲 as y = M共g兲 · x

共5兲

and v = v共g兲 · x.

共6兲

For any fixed gantry angle g the relations among arc delivery parameters realizing treatment plan 关A共g兲, M共g兲 or V共g兲, M共g兲兴 are given by a straight line 共x , y = M共g兲 · x , v = v共g兲 · x兲 in three dimensional 共3D兲 space, where coefficients M共g兲 and v共g兲共for fixed g兲 are constants describing the direction of this half-line in space. In Fig. 1, x, y, and v axes represent gantry speed dg / dt, beam dose rate dm / dt, and one leaf velocity 共say left leaf of pair 1, i.e., z = xL,1兲, respectively. Thick solid half-line represents points in 3D that realize treatment plan 共A共g兲 , M共g兲兲关or, equivalently, treatment plan 共V共g兲 , M共g兲兲兴. This line is given as 共x , y = M共g兲 · x , v = v共g兲 · x兲 and it projects on 共x , y兲 plane along line y = a · x and it projects on plane 共x , z兲 along line v = b · x 共for concise notation, we denote constants M共g兲 and V共g兲 given by treatment plan 共V共g兲 , M共g兲兲 by a and b, respectively兲. Points of thick solid half-line inside the box B = 兵共x , y , z兲 ; 0 ⬍ x ⬍ ␻max0 ⬍ y ⬍ rmax , −共vmax ⬍ z ⬍ vmax兲其 define parameters of delivery that can be applied with hardware constraints imposed by delivery system, while points of thick solid half-line outside of box B denote parameters of delivery that are unachievable for linear accelerators satisfying constraints discussed at points a–c above. Figure 1 shows Medical Physics, Vol. 37, No. 8, August 2010

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clearly that large number of deliveries is feasible for a given treatment plan 共A共g兲 , M共g兲兲 or, equivalently, treatment plan 共V共g兲 , M共g兲兲. The figure also shows that most efficient delivery 共one that takes least amount of time to complete arc therapy for given plan兲 is defined by parameters of point Pⴱ defined by intersection of half-line 共x , y = M共g兲 · x , v = v共g兲 · x兲 with the boundary of box B. The point of intersection of half-line with box boundary has clearly the largest values of all parameters involved in delivery. In particular, the largest value of coordinate x 共angular speed of gantry兲. Thus, if for each subsequent gantry angle g, we chose parameters of delivery belonging to points of intersection of half-line with box boundary we will keep the angular speed of gantry angle at largest possible value at each g consistent with delivering treatment plan 共A共g兲 , M共g兲兲 or, equivalently, treatment plan 共V共g兲M共g兲兲. This will result in fastest completion of the arc therapy treatment plan. Figure 1 is not complete as it does not take all the leaves into consideration 共please note, however, that to illustrate the dependence of the set of parameters for second pair of leaves, the dashed halfline is displayed兲. Appendix A provides details of the geometric interpretation of parameter selection for multiple leaves. The above heuristic justification of the optimality condition for arc delivery is supplemented with the rigorous proof in Appendix B. II.G. Delivery under special constraints

The understanding of relations between delivery parameters facilitates designing arc deliveries that can satisfy additional constraints and restrictions of the radiation delivery systems that may be imposed by hardware constraints or simply implemented as beneficial for patient for specific clinical circumstances. In the following, we consider the deliveries that require constant gantry speed and also deliveries that require constant beam dose rate. 共a兲

CGAS delivery: The easiest understanding of how we can design the fastest delivery for VMAT with constant gantry speed technique we need to look again at Fig. 1. First, we note that the line defining admissible delivery parameters 共for given gantry position g兲 begins at the center of coordinates and successively moves away from plane x = 0, reaching, at some instant, the plane x = ␻max. This reaching of plane x = ␻max may take place inside box B or outside of it. If this happens inside box B, the gantry speed 共for given g兲 is allowed to rotate with the maximum speed ␻max. If this happens outside box, B the gantry speed is not allowed to rotate with the maximum speed ␻max. We note that if plane x = ␻max is allowed to shift toward origin 共this is equivalent to decreasing values of ␻max兲 it will finally reach position close enough to origin 共say when x = ␻c that the line defining admissible delivery parameters will cross the plane x = ␻c earlier than it can cross planes y = rmax or v = −vmax or v = vmax兲. We may notice that when value of parameter ␻c is decreased appropriately, it will reach value that will be equal to the gantry speed ␻c that is the largest constant gantry speed 共valid for all

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共b兲


gantry positions g兲 sustainable over the whole duration of delivery 关with beam dose rate given at each g as M共g兲ⴱ ␻c and speeds of L , i and R , i leaves given at each g as vL,i共g兲ⴱ ␻c and vR,i共g兲 ⴱ␻c, respectively 共for i = 1 , 2 , . . . , K兲兴. We can call delivery based on constant gantry speed ␻c as defined above the optimal 共minimizing time of delivery兲 constant gantry speed 共CGAS兲 delivery for arc therapy. CBDR delivery: To design the fastest delivery for VMAT with constant beam dose rate technique we turn again to Fig. 1. The line defining admissible delivery parameters 共for the given gantry position g兲 begins at the center of coordinates and successively moves away from plane y = 0, reaching at some instant the plane y = rmax. This reaching of plane y = rmax may take place inside box B or outside of it. If this happens inside box B, then beam dose rate 共for the given g兲 is allowed to be run with the maximum intensity rmax. If this happens outside of box B, then beam dose rate 共for given g兲 is not allowed to achieve its maximum admissible intensity rmax. We note that if plane y = rmax is allowed to shift toward origin 共this is equivalent to decreasing values of rmax兲, it will finally reach position close enough to origin 共say when y = rc that the line defining admissible delivery parameters will cross the plane y = rc for any g earlier than it can cross planes x = ␻max or v = −vmax or v = vmax兲. We may noticed that when the value of parameter rc is decreased appropriately, it will reach value that will be equal to the beam dose rate rc that is the largest constant beam dose rate sustainable over the whole duration of delivery 关with gantry speed given at each g as x = y / rc and speeds of L , i and F , i leaves given at each g as v = vL,i共g兲共y / rc兲 and v = vF,i共g兲 ⫻共y / rc兲, respectively 共for i = 1 , 2 , . . . , K兲兴. We can call delivery based on constant beam dose rate rc described above as the optimal 共minimizing time of delivery兲 CBDR delivery for arc therapy.

Please note that the word optimal in the previous sentence means that minimization is done under additional constraints 共constant gantry speed and constant beam dose rate兲 imposed over previously discusses optimal delivery that admitted these parameters to be variable rather than constant. Satisfying these additional constraints composes the space of admissible for search solutions less numerous, making new optimal solutions suboptimal in the complete set of solutions unconstrained by additional limitations of constancy for gantry speed, constant beam dose rate. II.H. RapidArc delivery analysis using dynalogs

The delivery parameters of the RapidArc plans such as gantry speed profile and dose rate profile cannot be determined directly from the Eclipse planning system. These parameters are computed at the Varian linac console. In this work, we used the dynalog files to determine the gantry speed profile and dose rate profile for the Varian RapidArc delivery. The MLC dynalog file is created by the MLC conMedical Physics, Vol. 37, No. 8, August 2010

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troller and a gantry dynalog file is created by the linac console. MLC dynalog files contain information regarding gantry position, MLC position, and fractional dose delivered every 50 ms of the delivery. Gantry dynalogs contain the gantry position as a function of MUs intended to deliver 共segmented treatment table in Varian’s system兲 and also the actual gantry position as a function of MUs delivered during treatment. More details and accuracy of such log files can be found in the literature and in Varian manual.18 From this information, we could compute the gantry speed profile, dose rate profile, and MLC leaf velocity as a function of time and gantry angular speed. The treatment delivery time can also be determined from this information. We used dynalog files and time displayed at the linac console to determine the delivery time for Varian delivery.

III. RESULTS III.A. Optimal VMAT delivery „variable dose rate and variable gantry speed…

Deliveries for optimized VMAT inverse planned treatments for five prostate patients 共derived from Varian RapidArc planning system 8.5兲 have been designed following the method described in this paper. Aperture shapes A共g兲 indexed by gantry angles and the function M共g兲 expressing monitor units to be delivered as a function of gantry angle have been determined by treatment plans. These records constitute the invariant for designing the arc delivery procedure as described in Sec. II. The system constraints for delivery parameters have been set for these following cases: 共i兲 Maximum gantry speed of 5.5 deg/s; 共ii兲 Maximum leaf velocity of 2.25 cm/s; and 共iii兲 Maximum dose rate of 600 MU/min. The constraints we used are from the Varian treatment planning system in which RapidArc treatment plans used in this work were generated. While we verified the accuracy of these values from RapidArc manual, their technical support, and in the treatment planning system settings we are not aware of any other additional constraint imposed on the planning or on the delivery. Figure 2 shows the gantry angular velocity profile comparing the Varian delivery and optimal delivery derived from our algorithm as a function of gantry angle 关Fig. 2共a兲兴 and as a function of time 关Fig. 2共b兲兴. Our delivery is optimal 共see Appendix B for proof兲 given the constraints we considered and the comparison against Varian delivery is for qualitative analysis only. For the optimal delivery, we see that there is gantry velocity modulation during one segment of the delivery process and that most of the time the gantry angular speed is allowed to keep its maximum value and practically no modulation of this parameter is needed. Figure 2 shows explicitly that maximum gantry velocity of 5.5 deg/s is not violated during the course of the delivery. Figure 2共b兲 indicates times of termination of optimal delivery 共66.15 s兲 and Varian RapidArc delivery 共78.0 s兲. The delivery time for Varian delivery is derived from dynalog files and verified

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FIG. 2. Angular gantry velocity profile comparing the Varian RapidArc delivery and the Optimal VMAT delivery for plan 1 共with 467 MU兲. 共a兲 represents dependence of gantry angular speed versus gantry angle while 共b兲 represent dependence of gantry angular speed vs time. The solid line shows the gantry speed profile for the Optimal VMAT delivery during the course of the treatment. The dotted line represents the gantry speed profile for the Varian RapidArc delivery determined using dynalogs. Note the faster termination of the Optimal VMAT delivery relative to RapidArc on 共b兲. The black dot shows the termination of the delivery and we can see Optimal VMAT finishes at 66.15 s, while Varian delivery is 78 s. Note all the constraints on the Varian delivery is unknown and so the graph presented is for qualitative comparison only. Note maximum gantry speed constraint of 5.5 deg/s is not violated in both deliveries. Medical Physics, Vol. 37, No. 8, August 2010

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with the time on the linac console after delivery is completed. Since we do not have information about other constraints that Varian delivery might have we cannot determine Varian delivery is suboptimal. But from our results it can be seen that Varian delivery might have additional constraint on planning or delivery. These details have been discussed in Sec. IV. Figure 3 shows the corresponding dose rate profile comparison between the Varian delivery 共derived from dynalogs兲 and optimal delivery as a function of gantry angle 关Fig. 3共a兲兴 and as a function of time 关Fig. 3共b兲兴 for the same treatment, demonstrating that dose rate maximum of 600 MU/min is reached over some portion of the treatment while being below this value at all other instances during treatment. Figure 4 shows the maximum value of the fastest leaf velocity among all leaves of MLC used for optimal delivery. We see again that maximum leaf velocity is reached over some portion of the treatment delivery, while at other times it is being kept below the maximum velocity constraint of 2.25 cm/s. Figure 5 shows the beam dose rate profile for optimal CGAS delivery 共dg / dt = ␻c = 4.29 deg/ s in this case兲 for the same plan as a function of time. For comparison, we also show in Fig. 5 the beam dose rate profile for optimal VMAT delivery when all delivery parameters are allowed to vary. From Fig. 2, we can see that for optimal VMAT delivery the smallest gantry speed for the entire delivery is 4.29 deg/s and that turns out to be the value to be used for optimal CGAS delivery. Figure 6 shows the gantry angular speed profile for optimal CBDR therapy 共dm / dt = rc = 192 MU/ min in this case兲 for prostate plan as a function of time. For comparison, we also show in Fig. 6 the gantry angular speed profile for optimal VMAT when all delivery parameters are allowed to vary. From Fig. 3, we can see that, for optimal VMAT delivery the lowest dose rate for the entire delivery is 192 MU/ min and that turns out to be the value to be used for optimal CBDR delivery. Table I shows the comparison of delivery time for all arc treatment cases investigated. Table II shows arc treatment times for five VMAT plans delivered with different techniques, including optimal VMAT delivery, Varian RapidArc delivery, and CGAS and CBDR techniques. As expected, we see consistently among all deliveries that the optimal VMAT provides the minimal treatment time delivery. Optimal CGAS delivery techniques are generally faster than optimal CBDR delivery techniques and slower than Varian RapidArc delivery. Still it is worth pointing out that the CGAS delivery is faster for plan 2 than Varian RapidArc. IV. DISCUSSION Investigating the features of VMAT delivery we have found solutions that are essential for clinical aspects of the treatment. The formulation of the problem of VMAT delivery enables us to enlighten several aspects of this type of delivery. First, it shows that there exists infinite number of arc delivery solutions that realize the same VMAT treatment plan. This allows us to search for deliveries that are characMedical Physics, Vol. 37, No. 8, August 2010

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terized by specific features, beneficial for clinical objectives in the family of all possible deliveries. In particular, we may visibly determine the delivery that is most efficient 共minimizes time of irradiation兲 but we can also clearly design deliveries that have other advantages for any given clinical case. One aspect of our solution may be of special importance—the rigorous determination of the most efficient VMAT delivery. The optimal delivery solutions provided shows shorter treatment time relative to RapidArc delivery provided by Varian. The explanation of this is not apparent. First our proof of the optimality is based on precise assumptions about the constraints for delivery imposed by machine hardware. We assume that complete set of these constraints are maximum gantry speed, maximum dose rate. and maximum leaf velocity. These seem to be all constraints that linear accelerators impose on arc therapy delivery and these constraints are explicitly admitted by Varian. Nevertheless, we cannot be entirely sure that the set of constraints that we impose is a truly complete set of limitations and that machine, mechanical or control characteristics do not impose additional constraints on the delivery. If these additional constraints exist, then comparing our and Varian deliveries is not legitimate as they refer to different sets of circumstances. In these circumstances, our algorithm would require further adjustments to be directly comparable to Varian RapidArc. However, this is not the intent of this work. The solutions of CGAS and CBDR delivery might be of interest for certain scenarios. If the machine is mechanically not robust in terms of control mechanics and cannot vary the gantry speed 共because of the design or age of the machine兲, then CGAS would be a good option. Similarly, if the linear accelerator is not able to modulate the dose rate effectively 共because of the design or electron gun technology兲, then CBDR would be a viable option to deliver such plan. The compromise with both these techniques compared to VMAT plan is the delivery time. But given that constraint of constant gantry speed or constant dose rate the solutions provided are optimal. Other constraints such as collimator rotation speed and gantry acceleration constraints were not considered in this work. Elekta linear accelerators have ability to perform collimator rotation during delivery and this would be an interesting parameter to consider. Gantry acceleration constraint has not been explicitly admitted by the commercial manufactures but this would be an important constraint for delivery of VMAT plans to moving targets. Also, even for static target a maximum and minimum acceleration constraint would provide smoothness of the delivery which would be desirable from mechanical stability point of view. Both of these problems could be formulated and studied but was left out to keep this manuscript clear from understanding the relationship between main delivery parameters. In this work we have investigated delivery for prostate VMAT treatments and thus, a question of how these observations would be extended to other treatment sites or other treatment techniques would be worthwhile to comment. First, we may expect that plans requiring more modulations

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FIG. 3. Shows beam dose rate variation profile comparing the Varian RapidArc delivery and the Optimal VMAT delivery of Plan 1 共with 467 MU as shown兲. 共a兲 represents dependence of beam dose rate vs gantry angle, while 共b兲 represents dependence of beam dose rate vs time. The solid line shows the beam dose rate profile during the course of the Optimal VMAT treatment. The dotted line shows the dose rate profile for the Varian RapidArc delivery determined from dynalogs. Note the faster termination of the Optimal VMAT delivery relative to RapidArc on 共b兲. Note all the constraints on the Varian delivery are unknown so it cannot be concluded that optimal delivery is faster than Varian delivery. Note maximum dose rate constraint of 600 MU/min is not violated in both deliveries. Medical Physics, Vol. 37, No. 8, August 2010

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FIG. 4. Graph represents maximum leaf velocity among all leaves of MLC assembly for the Optimal VMAT delivery of Plan 1. Note that most of the time the fastest leaf of MLC can travel with maximum admissible speed of 2.25 cm/s but it is not violated during the entire delivered.

of parameters during delivery 共more variable changes of apertures and beam intensity with angle兲 should benefit from treatment delivery time aspects. On the other hand, in case of hypofractionated treatments, the gain in efficiency of optimal treatment will likely be insignificant due to natural slowing

FIG. 5. The graph shows the dose rate profile 共shorter curve兲 for optimal constant 共given in this case by value 4.29 deg/s兲 gantry angular speed 共CGAS兲 delivery and dose rate profile for the Optimal VMAT delivery 共longer curve兲 as function of time for plan 1. We note that the largest dose rate for the Optimal VMAT is kept at maximal admissible value over interval from 10 to 20 s forcing other parameters of delivery 共including gantry speed兲 to decrease below the maximal admissible value in this time interval. In contrast, during CGAS delivery the largest dose rate is consistently kept below the maximal admissible value, allowing gantry speed to be kept constant at all time. Medical Physics, Vol. 37, No. 8, August 2010

FIG. 6. The graph shows the gantry angular speed profile 共lower and longer curve兲 for optimal constant 共given in this case by value 192 MU/min兲 beam dose rate 共CBDR兲 delivery and the gantry angular speed profile for the Optimal VMAT delivery 共upper and shorter curve兲 as function of time for plan 1.

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TABLE I. Delivery times comparison between the Varian RapidArc deliveries vs optimal VMAT deliveries. The Varian RapidArc delivery time was determined from the treatment performed on the machine looking at the console and also verifying that again the treatment time determined from dynalogs.

Prostate plan No. 1 2 3 4 5

MU

Treatment time 共s兲

Varian treatment time 共s兲

Percent time gain 共%兲

467 292 594 637 719

66.15 64.96 66.96 68.57 74.98

78 76.8 79.2 80.4 85.8

17.9 18.2 19.3 17.24 14.42

down of gantry motion and leaf motions necessitated by the need to deliver large number of monitor units per unit gantry angle.

V. CONCLUSION The VMAT delivery problem has been formulated and the relationship among the delivery parameters are determined and studied. The solution for optimal VMAT delivery is provided and the optimality is proved. Our optimal VMAT delivery solution is characterized by shorter treatment time when machine constraints include the limitations on maximal gantry rotation speed, maximal admissible beam dose rate, and maximal admissible leaf speeds. We have compared the delivery parameters and the delivery time with Varian RapidArc delivery. The delivery parameters were indirectly determined from the delivery log files. Our approach clarifies the mutual dependence of delivery parameters in arc therapy which allows designing arc delivery methods for given plan that customizes delivery strategies. The need for special delivery strategies may be imposed by delivery systems or demanded by clinical circumstance. We investigate two special cases of delivery, one based on

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constant gantry angular speed and the other based on constant beam dose rate. Solutions for optimal CBDR and CGAS deliveries are provided. ACKNOWLEDGMENTS The authors would like to thank Dr. James Monroe for his help in generating the RapidArc plans and providing the information. Also they would like to thank him for providing the dynalog files of the delivery and fruitful discussions. The authors would like to thank the referees for their comments and critique which has substantially improved the clarity of the manuscript. Dr. Rangaraj and this work were partially supported by Varian Medical Systems. APPENDIX A: DESCRIPTION OF ALGORITHM AND NUMERICAL SOLUTION First, we notice that Fig. 1 does not provide complete representation of conditions and relations that underlie the algorithm for optimal delivery of VMAT therapy. The incompleteness of Fig. 1 representation comes from the absence of relationships for leaves other than leaf 共L , k = 1兲. The straightforward generalization from the case represented by Fig. 1 to situation when all leaves are portrayed would require writing not three linear equations x , y = a · x , z = b · x in three dimensional space but 2K + 2 linear equations x , y = a · x , v1 = b1 · x , v2 = b2 · x , . . . , v2K = b2K · x in 2K + 2 dimensional space where coefficients 共b1 , b2 , . . . , bK兲 are simply velocities vL,k共g兲 and vR,k共g兲 of leaf motions at g for all leaves of MLC assembly for arc treatment plan under consideration. The graphical representation of such a situation is not possible. We can imagine, however, that resulting from these equations straight line in 2K + 2 dimensional space will be stretching from the origin of Cartesian coordinate system to the boundary of the 2K + 2 dimensional box B共2K+2兲 defined by 2共2K + 2兲 planes perpendicular to all axes 共two perpendicular planes per each axis兲 of which first two are perpendicular to the x axis, one defined by x = 0 and the other

TABLE II. Treatment delivery times for five VMAT plans with the Optimal VMAT algorithm, the Varian RapidArc delivery, the optimal constant beam dose rate delivery and the optimal constant gantry angular speed delivery. For the optimal constant gantry speed delivery and for the optimal constant dose rate delivery the maximum constant value of the gantry angular speed and the maximum constant value of the beam dose rate have been determined and used over the whole time of delivery. Therefore treatment times are optimal for each defined sets of constraints of the delivery scenario investigated. Delivery technique/treatment time 共s兲

Plan No.

MU

Optimal delivery

1 2 3 4 5

467 292 594 637 719

66.15 64.96 66.96 68.57 74.98

a

Varian RapidArc delivery

Constant gantry speed delivery

Constant dose rate delivery

78 76.8 79.2 80.4 85.8

83.22 (4.29 deg/s)a 66.11 (5.43 deg/s) 81.14 (4.43 deg/s) 88.59 (4.03 deg/s) 99.17 (3.60 deg/s)

145.89 (192 MU/min)b 131.12 (179 MU/min) 108.24 (314 MU/min) 110.85 (344 MU/min) 104.35 (413 MU/min)

Maximum gantry speed that can be kept constant. Maximum dose rate that can be kept constant.

b

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defined by x = ␻max. In a similar way, the third and fourth planes of box B共2K+2兲 are perpendicular to the y axis, defined by y = 0 and the fourth defined by y = rmax. All other planes of box B共2K+2兲 are grouped in pairs of planes perpendicular to axes v1 , . . ., and v2K, appropriately, with one of each pair being set at −vmax and the other at +vmax along each subsequent z axis. The line in 2K + 2 space will be crossing the boundary of the box B共2K+2兲 in point Pⴱ共2k+2兲 similarly as it does in the 3 dimensional case illustrated in Fig. 1. The algorithmic method of finding point Pⴱ共2k+2兲 is the same as finding point Pⴱ in three-dimensional space, i.e., we find first points common between line 共x , y = a · x , v1 = b1 · x , z2 = b2 · x , v2K = b2K · x兲 and boundary of box B共2K+2兲 in 2K + 2 space. This is point Pⴱ共2k+2兲. The program searching for this solution is straightforward to write and solutions are computed very fast. The other solution for finding coordinates of point Pⴱ共2k+2兲 that is less convenient computationally but is viable for visualization in three dimensional space. In this approach we have to represent formulas for all leaf speeds as mapping from x to only one axis, z axis. However, instead, of one line v = bⴱx on plane 共x , v兲, as in case for one leaf speed representation, in this case 2K lines v1 = b1 · x , . . . , v2K = b2K · x will have to be used. Each line would be graphed on plane 共x , v兲 as successfully representing equations v1 = b1 · x , . . . , v2K = b2K · x, where i = 1 , 2 , . . . , K. This representation will result in 2K lines 共x , y = a · x , v = bL,i · x兲共x , y = a · x , v = bR,i · x兲, where i = 1 , 2 , . . . , K in three dimensional space 共x , y , v兲. Each line from the set of 2K lines 共x , y = a · x , v = bL,i · x兲 , . . . , 共x , y = a · x , v = bR,i · x兲 will cross the boundary of three dimensional box B at point Pⴱi, i = 1 , . . . , 2K 共notice that these lines can move up or down depending on the sign of bi兲. In the set of 2K points Pⴱi, i = 1 , 2 , . . . , 2K, we look for one that has the lowest value of the coordinate x. Let us denote this coordinate as xⴱmin. We notice that a plane defined by x = xⴱmin will determine all parameters of optimal delivery of arc therapy. To this end, we find coordinates on axes y and v where plane x = xⴱmin crosses all lines y = a · x and v = bL,i, v = bi,L, v = bR,i · x, where i = 1 , 2 , . . . , K. Explicitly, these coordinates define gantry speed as xⴱmin, beam dose rate as y = M共g兲 · xⴱminn and speed of leading and following leaves from pairs i = 1 , 2 , . . . , K as v = vL,i共g兲 · xⴱmin and v = vR,i共g兲 · xⴱmin, appropriately. These coordinates give largest possible values for all treatment parameters 共x , y , v1 , . . . , v2K兲 at g that are admissible for arc plan 共V共g兲 , M共g兲兲. In particular, so found delivery will maximize the speed of gantry rotation at g 共as xⴱmin兲 without violating arc plan 共V共g兲 , M共g兲兲 and integrity of constraints 共a兲–共c兲 for all leaves. APPENDIX B: PROOF OF OPTIMALITY For convenience, we take the inverse 共reciprocal兲 of the gantry speed dg / dt as control parameter. In other words, function dt / dg, dependent on g, is considered as variable 共control兲 of our optimization problem. We denote dt / dg as a共g兲. Medical Physics, Vol. 37, No. 8, August 2010

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Assuming a共g兲 = 共dt / dg兲共g兲 is the inverse of angular velocity at g, we can interpret a共g兲dg as time interval dt that passes when gantry rotating with speed 1 / a共g兲 moves from g to g + dg. This allows us to write 兰2兿 0 a共g兲dg = ta and interpret ta as time required for a full gantry rotation from g = 0 to g = 2␲. We took a = 共a1 , a2 , . . . , an兲 as a control vector and define i=n ai = ta, where 共⌬g兲 = 2␲ / n. We search for aⴱ 共⌬g兲兺i=1 ⴱ = 共a1 , a2ⴱ , . . . , anⴱ兲 such that mina ta = taⴱ The constraints for a共g兲 = al = a共l · 2␲ / n兲 , l = 1 , 2 , . . . , n are a共g兲 ⱖ amin, where amin = 1 / ␻max with ␻max being the maximal 共mechanical兲 admissible speed of the gantry. This means that each component a1 , a2 , . . . , an of a vector a is larger than amin. a共g兲 ⱖ M共g兲 / rmax, where M共g兲 is given from plan and rmax is the maximal admissible beam dose rate. This means that each subsequent component a1 , a2 , . . . , an of a vector a is larger than

共i兲

共ii兲

M

冉冊冉冊 2␲ n

rmax

M 2·

,

2␲ n

rmax

冉冊

M n· , ... ,

2␲ n

rmax

.

共iii兲 a共g兲 ⱖ vL,k共g兲 / vmax and a共g兲 ⱖ vR,k共g兲 / vmax, where vL,k共g兲 and vR,k are given from the plan and vmax is the maximal 共mechanical兲 admissible speed of the leaves of MLC. This means that each subsequent component ai of a vector a is larger than any vk,L

冉冊冉冊 2␲ n

vmax

vk,L 1 ·

,

2␲ n

vmax

, ... ,

and any vk,F

冉冊冉冊 2␲ n

Vmax

vk,F 2 ·

,

2␲ n

vmax

冉冊

vk,L n ·

vmax

冉冊

vk,F n ·

, ... ,

2␲ n

2␲ n

vmax

for k = 1 , 2 , . . . , K. The minimization of ta requires that as small numbers ai n in sum 兺i=1 ai are substituted as possible without violating constraints 共i兲–共iii兲. As constraints 共i兲–共iii兲 are independent of each g 共i.e., at each l · 2␲ / n, l = 1 , 2 , . . . , n兲, the minimal value at each g for 共a兲 can be found as the largest of

冢

冉冊冉冊冉冊冣

M l· amin,

2␲ n

rmax

vk,L l ·

,

2␲ n

vmax

vk,F l ·

,

2␲ n

vmax

,

where k = 1 , 2 , . . . , K. This choice completes optimization of the problem of minimization of time of delivery for VMAT. a兲

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