INSTITUTE OF PHYSICS PUBLISHING
`
`Phys. Med. Biol. 48 (2003) 1075–1089
`
`PHYSICS IN MEDICINE AND BIOLOGY
`
`PII: S0031-9155(03)57398-5
`
`Inverse planning for intensity-modulated arc therapy
`using direct aperture optimization
`
`M A Earl, D M Shepard, S Naqvi, X A Li and C X Yu
`
`Department of Radiation Oncology, University of Maryland School of Medicine, Baltimore,
`MD 21201, USA
`
`E-mail: mearl001@umaryland.edu
`
`Received 17 December 2002
`Published 1 April 2003
`Online at stacks.iop.org/PMB/48/1075
`
`Abstract
`Intensity-modulated arc therapy (IMAT) is a radiation therapy delivery
`technique that combines gantry rotation with dynamic multi-leaf collimation
`(MLC). With IMAT, the benefits of rotational IMRT can be realized using
`a conventional linear accelerator and a conventional MLC. Thus far, the
`advantages of IMAT have gone largely unrealized due to the lack of robust
`automated planning tools capable of producing efficient IMAT treatment plans.
`This work describes an inverse treatment planning algorithm, called ‘direct
`aperture optimization’ (DAO) that can be used to generate inverse treatment
`plans for IMAT. In contrast to traditional inverse planning techniques where the
`relative weights of a series of pencil beams are optimized, DAO optimizes the
`leaf positions and weights of the apertures in the plan. This technique allows
`any delivery constraints to be enforced during the optimization, eliminating the
`need for a leaf-sequencing step. It is this feature that enables DAO to easily
`create inverse plans for IMAT. To illustrate the feasibility of DAO applied to
`IMAT, several cases are presented, including a cylindrical phantom, a head and
`neck patient and a prostate patient.
`
`1. Introduction
`
`A fundamental goal of radiotherapy is to deliver a conformal dose of radiation to the tumour
`while simultaneously minimizing the dose to healthy tissue. Intensity-modulated radiation
`therapy (IMRT) is an improvement over conventional techniques in that it has the ability to
`better sculpt the dose distribution around critical structures. Consequently, the sparing of
`critical structures and normal tissue is improved, allowing physicians to escalate the dose
`to the tumour. IMRT delivery techniques can be divided into two classes: fixed gantry and
`rotational. Fixed-gantry IMRT is achieved by delivering overlapping fields from fixed beam
`directions. The field shapes either remain constant during the delivery of radiation or can
`
`0031-9155/03/081075+15$30.00 © 2003 IOP Publishing Ltd Printed in the UK
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`dynamically change during radiation delivery. Rotational IMRT is achieved by dynamically
`moving the leaves of the MLC and the gantry during radiation delivery.
`Intensity-modulated arc therapy (IMAT) was first proposed by Yu (1995) as an alternative
`rotational IMRT delivery technique to tomotherapy (Mackie et al 1993, 1995, Carol et al
`1997). With tomotherapy, a binary MLC is used to modulate the intensity of radiation as the
`source rotates about the patient. IMAT is performed with a conventional linear accelerator,
`where the leaves of the MLC change shape as the gantry rotates during delivery. The treatment
`is composed of ‘arcs’, each with a start and stop angle. The MLC field shape changes
`continuously during gantry rotation from the beginning to the end of each arc. To achieve
`a modulated intensity distribution from a given delivery angle, multiple overlapping arcs are
`employed. A key advantage of IMAT is that the delivery is achieved using a conventional
`linear accelerator and a conventional MLC. Therefore, IMAT treatments can be delivered
`using existing equipment in most radiation oncology departments. It should be noted that to
`deliver IMAT plans, the linear accelerator must be equipped with the capability for dynamic
`delivery.
`Traditionally, IMRT treatment plans are generated with a two-step process (Bortfeld et al
`1994, Chui et al 1994, Galvin et al 1993, Webb 1994, Xia and Verhey 1998). First, a set of
`pencil beam intensities is optimized for each specified beam direction. The resulting optimized
`intensity maps are then sent to a leaf-sequencer which determines the set of MLC shapes to
`be used for delivery. It is in this step where any delivery constraints of the treatment unit
`are enforced. The output of the two-step procedure is a set of MLC aperture shapes and
`their corresponding weights, or monitor units. For fixed-field IMRT, no further sequencing is
`needed. However, for IMAT additional sequencing is required to produce deliverable plans.
`To make the dose calculation feasible in IMAT, the arcs are approximated by a series
`of equispaced static beams (Li et al 2001). Applying the aforementioned two-step approach
`to IMAT requires a more complex leaf-sequencing algorithm that links the aperture shapes
`between consecutive angles. This significantly complicates the leaf-sequencing step and makes
`production of IMAT treatment plans using this conventional two-step approach extremely
`cumbersome. Generally, eight to ten arcs are required to achieve three intensity levels
`(Yu 1995).
`Multiple investigators have successfully implemented IMAT using a forward planning
`approach (Yu et al 2002, Wong et al 2002, Ramsey et al 2001). Recently, Yu et al (2002)
`reported on a clinical implementation of IMAT whereby field shapes were generated based
`upon beam’s eye view of the tumour and critical structures. The arc weights were then
`manually changed to achieve the desired dose distribution. Ma et al (2001) proposed a more
`sophisticated method whereby the weights of the arcs in prostate cancer treatments were
`optimized using a gradient search method. Cotrutz et al (2000) introduced a similar technique
`where several arcs were defined, each with the same blocking of the central portion of the field,
`but with varying overall aperture size. The weights of the arcs were then optimized to achieve
`the desired dose distribution. Although some inverse planning was employed in the latter two
`approaches for beam weight optimization, the field shapes were still defined manually. To our
`knowledge, there is no fully automated inverse planning system reported for IMAT.
`In this work, we propose a method that simultaneously optimizes both the field shapes
`and the arc weights. We are able to generate inversely planned IMAT treatment plans using a
`technique that we call ‘direct aperture optimization’ (DAO). In a previous work (Shepard et al
`2002), we discussed the application of the DAO technique to fixed-field step-and-shoot IMRT.
`The key feature of DAO is that the MLC leaf positions and beam weights are the parameters
`that are optimized, as opposed to pencil beam weights. Using this technique, all of the delivery
`constraints are enforced during the optimization, thereby eliminating the need for a separate
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`Input:
`(1)
`(2)
`(3)
`
`Number and range of arcs
`Prescription
`Optimization parameters
`
`Calculate pencil beams
`
`Start with BEV of target for
`each beam direction
`
`Select change
`
`Does change
`satisfy delivery
`constraints?
`
`Yes
`
`Calculate dose resulting
`from change and the
`corresponding objective
`function
`
`Reject change
`
`No
`
`No
`
`Reached finishing
`criteria?
`
`Yes
`
`Export dose and plan
`for evaluation and
`delivery
`
`Does objective
`function improve?
`
`Yes
`
`Keep change
`
`No
`
`Generate random number
`x between 0 and 1
`
`No
`
`Is x less than
`probability dictated
`by simulated
`annealing?
`
`Yes
`
`Figure 1. Flow chart of optimization procedure.
`
`leaf-sequencing step. The elimination of the leaf-sequencing step significantly simplifies the
`generation of inverse-planned IMAT treatment plans.
`To illustrate the concept, we present the results of an inverse-planned IMAT case for a
`pseudo-prostate target on a cylindrical phantom, a head and neck case and a prostate case.
`In addition, we present results of the delivery of an inverse-planned IMAT plan to a c-shape
`target in a cylindrical phantom.
`
`2. Materials and methods
`
`2.1. Overall scheme
`
`To generate a DAO inverse plan for an IMAT treatment, the user specifies: (1) the number and
`angular range of the arcs, (2) the prescription parameters and (3) the optimization parameters.
`The software will then: (1) calculate the dose for each beam for the patient-specific CT data,
`(2) perform the optimization and (3) output the results for analysis/delivery. Each step is
`described in this section. An overview of the entire procedure is shown in figure 1.
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`2.2. Dose calculation
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`M A Earl et al
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`◦
`
`. For the cases presented
`The dose is calculated for a set of equispaced beams spanning 360
`◦
`in this work, 36 beams with an equal spacing of 10
`were used. This spacing convention is
`the compromise used in reference (Yu et al 2002) for the clinical implementation of IMAT.
`Each of these beams is divided into 1 cm × 0.5 cm pencil beams. A separate dose calculation
`is performed for each pencil beam, and the 3D dose matrix for each pencil is stored on disk.
`For example, if we are calculating an 8 × 8 field, we would have 8 × 8 × 2 × 36 = 4608
`separate 3D dose matrices. Since the dose calculation is somewhat time consuming, we store
`the dose matrices on disk so that the dose calculation only needs to be performed once. Every
`time an additional optimization is performed, the pencil beam dose distributions are available
`immediately.
`The dose is calculated using a home-grown algorithm in which photon interactions are
`randomly sampled, and energy kernels are randomly propagated.
`In essence, the method
`superimposes energy deposition kernels using the Monte Carlo method. We will give a brief
`description of the technique here, but a more detailed account of the technique will be described
`in a later publication.
`Histories are tracked on a photon-by-photon basis. A history starts by randomly sampling
`the starting position of the photon spatially within the field. The energy of the photon is then
`randomly sampled from the energy spectrum of the beam. An interaction point is randomly
`selected based upon radiological path length, and a monoenergetic dose spread array (DSA)
`corresponding to the energy sampled is propagated from that point along a randomly selected
`direction. The DSAs calculated by Mackie et al (1988) are used. Hundreds of thousands of
`histories are run for each pencil beam. Instead of propagating secondary electrons as in full-
`blown Monte Carlo, the technique propagates the dose kernel. The use of random sampling
`allows easier computer parallelization and explicit sampling of photon energy obviates the
`need for any depth hardening or off-axis corrections. In addition, no TERMA calculation
`is required. Because generation of IMAT treatment plans requires the calculation of a large
`number of pencil beam dose distributions, the calculation time for each pencil beam is a
`significant consideration. Using this technique enables us to generate the required dose
`matrices for IMAT in a reasonable time without a significant sacrifice in dose calculation
`accuracy.
`Because the dose due to each pencil beam has limited spatial extent, many of the voxels
`receive little or no dose. To reduce the amount of hard disk space required to store each beam,
`we keep the dose information only for voxels that receive more than some fraction of the
`maximum dose. Typically, we set this fraction to 0.25%. This compression technique reduces
`the amount of space required to store the pencil beams by approximately a factor of 500. After
`each pencil beam calculation, the data are stored on disk to be used in the optimization phase
`of the procedure. The pencil beam calculation is performed only once. Depending on the
`voxel size, field size and patient size, the time required to compute all of the pencil beams for
`IMAT cases ranges from 3 to 12 h on 12 750 MHz Pentium III CPUs. The disk space required
`to store the beams ranges from 250 to 500 MB.
`It should be noted that the proposed inverse planning algorithm is independent of the
`pencil beam dose calculation engine. A more accurate dose engine, such as Monte Carlo,
`may be used with the cost of longer computing time. For example, we have examined the
`EGS4/DOSXYZ system (Nelson et al 1985, Ma et al 1998) and have found that the computing
`time is increased by a factor of 20 to 30 as compared with our dose engine. An alternative
`Monte Carlo code is the EGS4/MCDOSE system (Ma et al 2002) which has a significant
`computing time reduction compared to DOSXYZ.
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`2.3. Optimization
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`In the prescription, the user specifies the angular range of the arcs to be included in the
`optimization. The only constraint on the start and stop angles of the arc paths is the beam angles
`for which dose was calculated. To produce intensity modulation, two or more overlapping
`arcs for each arc path are needed.
`The optimization is performed using simulated annealing (Kirkpatrick et al 1993). The
`starting MLC shape conforms to the beam’s eye view of the tumour. The optimization is an
`iterative procedure containing many thousands of iterations. At each iteration, a parameter
`is randomly selected. Applying DAO to IMAT, the parameters are the leaf positions and arc
`weights, or beam weights if the gantry speed is allowed to vary. The parameter is then changed
`an amount sampled from a Gaussian distribution, whose width decreases with iteration number
`according to the schedule
`σ = 1 + (A − 1)
`
`(1)
`
`step
`0
`
`1
`(nsucc + 1)1/T
`step
`where A is the step size at the start of optimization, T
`is the parameter that dictates the
`0
`rate at which σ decreases (or cooling rate) and nsucc is the number of successes, which will be
`described later in the text.
`Intuitively, the final field shapes should be significantly different from the initial BEV of
`the tumour. By making large steps initially, the field will deviate from the BEV more quickly
`and, therefore, become closer to the desired solution. At the end of the optimization, smaller
`steps are required to fine-tune the aperture shapes. The magnitude of the initial step size A
`varies according to the starting size of the field. For starting field sizes around 20 × 20 cm2,
`we typically set A to 5 cm. For starting fields around 8 × 8 cm2, A is typically set to 3 cm.
`These are just sample values: other values could be chosen. Since A is simply the starting
`width of the Gaussian distribution, step sizes smaller and larger than A are possible. It should
`be noted that since pencil beams are used, only discrete changes in the MLC leaf positions are
`allowed. The smallest change in leaf position is equal to the resolution of the pencil beams.
`For instance, in the case where 1 cm × 0.5 cm pencil beams were calculated (1 cm corresponds
`to the width of our MLC leaves), steps of 0.5 cm, 1 cm, 1.5 cm, etc. are allowed.
`Since beam weights are continuous variables, it is not necessary to force the change to
`be discrete as we do with changes in leaf position. Instead, the size of the change is sampled
`directly from a Gaussian distribution. At the start of the optimization, the weights of all beams
`are set to 1.0. The starting width of the Gaussian distribution for the weight change is typically
`set 0.20, although other values can be chosen. As with the leaf position variables, the width
`of the Gaussian decreases with increasing iteration in the optimization.
`After each change in leaf position, the new aperture shape is checked to determine if it
`satisfies the delivery constraints. The delivery constraints are divided into two categories:
`MLC constraints and IMAT constraints. The MLC-based constraints are the same as those
`encountered in static beam delivery: they are imposed by the physical limitations of the MLC
`leaves in forming an aperture. IMAT-based constraints are dynamic: they are imposed by the
`limitations of the speed of leaf motion and the speed of gantry rotation. MLCs manufactured
`by different vendors have different constraints. For instance, with Elekta’s MLC, both the
`opposed leaves and the opposed-adjacent leaves cannot come within 0.8 cm of one another.
`As a result, interdigitation of the opposing leaves is not allowed.
`For IMAT, leaves between consecutive angles within an arc are constrained to move a
`distance of
`
`d = vl θ
`
`ω
`
`(2)
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`where vl is the leaf travel, ω is the gantry angular rotation speed and θ is the angular
`◦
`separation between consecutive angles, typically 10
`. In some cases, the gantry speed can
`be slowed and the dose rate decreased to accommodate large leaf steps between consecutive
`angles. Since radiation is delivered continuously between consecutive angles and the leaf
`positions are interpolated between consecutive angles, the calculated and delivered dose
`distributions can be somewhat different. In general, we found that a maximum leaf step of 3 cm
`was sufficient to provide a continuous delivery with no plan degradation.
`The Elekta machines used in our department require a constant dose rate and a constant
`gantry rotation speed in the delivery of IMAT plans. This constraint is imposed in the
`optimization by requiring that all angles composing an arc have the same relative weight.
`However, the full potential of IMAT can be better realized if either the dose rate or the gantry
`rotation speed can be varied during delivery. To illustrate the effect of this constraint, we will
`present two plans for the pseudo-prostate case for each of the two scenarios: equal weight
`constraint and full IMAT potential.
`The selected change is checked against the constraints outlined above and discarded if any
`constraint is violated. If the change satisfies the delivery constraints, the new dose distribution
`is calculated. Because we are employing pre-computed pencil beams stored on disk, the
`calculation of the new dose is a simple addition or subtraction of the affected pencil beams.
`This technique leads to a very fast iteration time: approximately 1000 iterations per minute
`on a 750 MHz Pentium III. The new dose is then used to calculate the objective function.
`Using the newly calculated dose reflecting the change in field shape or weight, the new
`objective function is calculated. The objective function is a single number which represents
`the deviation from the goals of the treatment plan. More details of the objective function will
`be given in the following section. If the new objective function value is smaller (less deviation
`from the goals of the treatment plan) than the previous one, the change is accepted and a new
`change is chosen. However, if the objective function is larger (more deviation from the goals
`of the treatment plan), the change will be accepted with a certain probability given by the
`relation
`
`P = B
`
`1
`(nsucc + 1)1/T
`prob
`is the cooling rate and nsucc is the number of successes.
`where B is the initial probability, T
`0
`A success is defined as either a change that results in a lower objective function value or a
`change that satisfies the probability constraint.
`
`prob
`0
`
`(3)
`
`2.4. Objective function
`
`The objective functions used in this work were based on a simple least-square function. Two
`types of least-square objective functions were used, a ‘range-based’ and a DVH-based.
`(cid:1)
`(cid:2)
`(cid:1)
`(cid:2)
`For the ‘range-based’ objective function, a set of dose ‘ranges’ is defined for each
`(cid:1)
`(cid:2)
`high
`d low
`structure i. A ‘range’ r(i) has a low dose
`, a high dose
`, a prescription dose
`d
`r (i)
`r (i)
`d p
`and a weight wr (i). Voxels belonging to the ith structure, whose dose d falls in range
`r (i)
`high
`r(i) between dlow
`r (i) , contribute
`r (i) and d
`(cid:1)
`(cid:2)2
`d − d p
`wr (i)
`j
`to the objective function. The total objective function is given by
`nstruct(cid:3)
`i(cid:3)
`i(cid:3)
`(cid:1)
`(cid:1)
`(cid:2) × (cid:1)
`nvoxel
`dj − dlow
`− dj
`dj − d p
`
`1
`nvoxel
`i
`
`ranges
`
`n
`
`r (i)=1
`
`i=1
`
`wr (i)
`
`H
`
`j=1
`
`(cid:2) × H
`
`r (i)
`
`high
`r (i)
`
`d
`
`(cid:2)2
`
`r (i)
`
`(4)
`
`(5)
`
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`(6)
`
`1
`
`2
`
`12
`
`
`
`
`H (x) =
`
`where H (x) is the Heaviside step function given by
`0
`: x < 0
`: x = 0
`: x > 0.
`1
`For instance for the target volume with a prescription dose of 1.00 (100%), one might
`= 0.0, d
`= 0.95, d p
`= 1.00 and w1 = 20.0; (2) d low
`=
`high
`specify two dose ranges: (1) d low
`= 1.00 and w2 = 20.0. Range 1 prevents underdosage of the target
`= 5.00, d p
`1
`1
`high
`1.10, d
`2
`2
`high
`while range 2 prevents overdosage. The d
`value of 5.00 in range 2 simply represents some
`2
`dose very much higher than we would deliver, essentially stating that any voxels with a dose
`above 1.10 will be penalized with the parameters specified in range 2.
`For a critical structure such as a spinal cord that has already received significant radiation,
`= 0.0, d
`= 0.20, d p
`= 0.0 and w1 = 1.0;
`high
`we might specify two dose ranges: (1) d low1
`= 0.0 and w1 = 1000.0. Range 1 provides a relatively
`= 0.20, d
`= 5.00, d p
`1
`1
`high
`(2) d low
`1
`1
`1
`small penalty for portions of the cord which receive a dose that is less than 20% of the PTV
`prescription dose. Range 2 provides a very large penalty for any portion of the cord greater
`than 20% of the PTV prescription dose. This type of range preferentially makes voxels inside
`the cord less than 20% of the PTV prescription dose.
`The DVH-based objective function has been outlined in other works (Bortfeld et al 1997),
`so only a brief description will be given here. The DVH constraints take the form ‘no more
`than x% of the volume is to exceed y% of the dose.’ For each DVH constraint, only a subset
`of the voxels is penalized within the given structure. The subset includes those voxels whose
`∗
`∗
`dose falls between Dmax and D
`is the current dose to at least
`, where Dmax is the y% and D
`x% of the volume. The form of the penalty to these voxels is the same as that in equation (4).
`
`1
`
`3. Results and discussion
`
`3.1. Pseudo-prostate
`
`For this case, a PTV and critical structure simulating a prostate case were drawn in a cylindrical
`phantom of 20 cm diameter. We calculated 36 equispaced 8 × 8 beams each composed of 128
`10 mm × 5 mm pencil beams. The phantom was divided into 2 mm × 2 mm × 5 mm voxels.
`The dose calculation time for one beam direction was about 70 min on one 750 MHz Pentium
`III CPU. Using 12 CPUs, the calculation for all 36 beams was about 3.5 h.
`In total, six arcs were employed: three arc paths, each with two overlapping arcs. The
`to −40
`, and −110
`to −170
`◦
`◦
`◦
`◦
`◦
`◦
`arc paths were 170
`to 110
`, 40
`(we use IEC convention in
`◦
`−1 and a maximum leaf-travel speed ofs
`our clinic). A minimum gantry-rotation speed of 10
`
`−1 were applied to the optimization. Figure 2 shows two axial slices of the final plan.
`3 cm s
`The cumulative dose volume histogram (DVH) for the PTV and critical structure is shown in
`figure 3.
`Figure 4 shows a sample sequence of aperture shapes for an arc of this case. The shapes
`◦
`shown represent the apertures belonging to angles separated by 10
`. The leaf travel direction
`is horizontal and each leaf step is 0.5 cm. Examining the figure reveals that the maximum leaf
`travel between consecutive angles is no more than 3 cm.
`
`3.2. Head and neck
`Like the pseudo-prostate case, we calculated 36 equispaced 8 × 8 beams each composed of
`128 10 mm × 5 mm pencil beams. The phantom was divided into 3 mm × 3 mm × 4 mm
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`Figure 2. Sample axial slices with isodose lines for the pseudo-prostate case.
`
`Figure 3. Dose volume histogram for the pseudo-prostate case.
`
`voxels. The total dose calculation time for all 36 beams was about 3 h on 12 750 MHz Pentium
`III CPUs. Three arc paths were employed, each with two overlapping arcs. The angular range
`to −50
`and −70
`to −170
`◦
`◦
`◦
`◦
`◦
`◦
`of the paths was 170
`to 70
`, 50
`. The minimum gantry-rotation
`◦
`
`−1 and a maximum leaf-travel speed of 3 cm ss −1 was applied.
`speed was 10
`
`There were several clinical goals in this case. First, for the patient to retain the ability to
`salivate after completion of the treatment, the dose to the parotid glands was to be kept below
`30% of the PTV prescription of 70 Gy. Since this was recurrent disease and the patient had
`already received a significant dose to the spinal cord in previous treatments, the cord dose
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`Figure 4. Sample sequence of aperture shapes for an arc of the pseudo-prostate case. Steps are
`denoted by the vertical lines and are separated by 0.5 cm. The maximum leaf travel between
`consecutive angles is always 3 cm or less.
`
`was required to be below 20% of the PTV prescription dose. Finally, no more than 30% of
`the mandible was allowed to receive 50% of the PTV prescription dose. Since one of the
`features of IMAT is that the higher dose regions in normal tissue are minimized at the expense
`of creating larger low dose regions, this case was well suited for IMAT.
`An axial and sagittal slices are shown in figure 5. The 95, 50, 30 and 20% isodose lines
`are shown. Also shown are the PTV, the parotid glands, the mandible and the spinal cord. The
`sharp dose fall-off from the 95% isodose line to the 50% isodose line is evident, and the dose
`to the parotid glands is below the specification. A cumulative dose volume historgram for this
`case is shown in figure 6.
`
`3.3. Prostate
`For the prostate case, the CT data were divided into voxels of 4 mm × 4 mm × 3 mm size.
`Two bilateral arcs were applied, each with three overlapping arcs. The angular range of the
`and −30
`to −150
`◦
`◦
`◦
`◦
`paths were 150
`to 30
`. As with the previous two cases, the leaf travel
`−1 and the minimum gantry-rotation speed was set to 10
`◦
`−1.s
`speed was 3 cm s
`
`Figure 7 shows an axial slice of the final dose distribution. As with any prostate IMRT
`case, a specific clinical goal of this case was to keep the dose to the rectum to an acceptable
`level while maintaining uniform dose to the PTV. Figure 8 shows the cumulative DVH for
`the PTV and rectum, illustrating the level to which these clinical goals were achieved. The
`‘hot-spot’ in the rectum was around 98% of the PTV dose prescription and 50% of the entire
`volume received no more than 20% of the prescription dose. For the PTV, 98% of the volume
`received at least 95% of the prescription dose.
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`Figure 5. Sample axial and sagittal slices with isodose lines for the head and neck case.
`
`Figure 6. Dose volume histogram for the head and neck case.
`
`3.4. Variable gantry rotation or dose rate
`
`For all of the cases presented,we restricted the gantry rotation speed to be constant,or similarly,
`for the dose rate to be constant. This is reflected in the fact that during the optimization, the
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`Figure 7. Sample axial slice with isodose lines for the prostate case.
`
`Figure 8. Dose volume histogram for the prostate case.
`
`relative weights of segments within a single arc are forced to be constant. However, changing
`either the gantry speed or dose rate during gantry rotation will help realize the true potential
`for IMAT. To study this, we relaxed the constant weight constraint within a given arc for the
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`Figure 9. DVH comparison of weight constraint versus no weight constraint for the pseudo-prostate
`case. An improvement is seen for sparing of the critical structure.
`
`pseudo-prostate case. For the ‘full IMAT potential case’, the weights of the beams belonging
`an arc are independent and are allowed to be different from one another. This essentially
`simulates the effect of allowing the gantry speed to vary or the dose rate to change during
`delivery.
`Figure 9 shows the DVH comparison for the pseudo-prostate case. A noticeable
`improvement in the critical structure DVH is evident, while the PTV gains only slightly.
`The relaxed constraint on the weights of the fields within an arc can be compensated in the
`‘equal-weight’ scenario by adding additional arcs. To illustrate this, for the pseudo-prostate
`case we add one additional arc to each arc path. The comparison between the ‘varying-weight’
`and the ‘additional-arc’ cases is shown in figure 10. It is apparent that some of what was gained
`by relaxing the weight constraint can be compensated in the ‘equal-weight’ scenario by adding
`arcs. The cost of this is the additional delivery time required to deliver the additional arcs.
`
`3.5. IMAT delivery: c-shape target
`
`To demonstrate the efficiency and accuracy of delivery, we present an IMAT plan to which
`we applied the DAO algorithm described above. Figure 11 shows the geometry and arc
`configuration for the case. A c-shape target was placed in the centre of a cylindrical volume.
`Four arc-paths were employed to achieve the desired dose distribution. The two arc-paths on
`the left side of the volume were each composed of two overlapping arcs. These two arc paths
`had angular ranges of −30
`to −80
`and −100
`to −150
`◦
`◦
`◦
`◦
`. The arc-paths on the right-hand
`side of the volume were simple arcs with no overlapping segments. The angular ranges of
`◦
`◦
`◦
`◦
`these arcs were 150
`to 100
`and 80
`to 30
`.
`The optimized plan was output to an Elekta SL20 linear accelerator and delivered to
`a cylindrical phantom. The dose distribution of an axial slice was measured with Kodak
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`Figure 10. DVH comparison of no-weight constraint versus additional arc and no weight constraint
`for the pseudo-prostate case. Some of the loss that was observed by imposing the equal-weight
`constraint was recovered by adding additional arcs.
`
`Figure 11. Configuration of the arc delivery to the cylindrical phantom. The target has a ‘c’ shape
`and surrounds the critical structure.
`
`EDR2 film. The plan was delivered in approximately 12 min. Figure 12 shows a comparison
`between the calculated and measured doses. For both cases, the 90% and 50% isodose lines
`are plotted. For the calculated distribution, the effect of approximating the dose calculation
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`Figure 12. Calculated dose distribution and measured dose distribution for c-shape target in
`cylindrical phantom. The 90% and 50% isodose lines are shown as white lines.
`
`with equispaced beams is evident. The film measurement shows a smoother dose distribution
`than the calculated distribution due to the discrete beam approximation in the calculated
`distribution. Despite this, the 90% and 50% isodose lines are in good agreement. The
`◦
`agreement between the two distributions is an indication that the 3 cm separation between 10
`consecutive angles provides an adequate approximation to a continuous delivery.
`
`4. Conclusion
`
`The full potential of IMAT has gone largely unrealized due to the lack of inverse planning
`techniques which can elegantly produce such plans. Current commercially available IMRT
`systems generally require a separate leaf-sequencing step. Since IMAT imposes constraints
`in addition to the physical limitations of the MLC, leaf sequencing becomes extremely
`complicated. To bypass this problem, we have applied the DAO technique to generate inverse
`plans for IMAT. Since DAO incorporates all delivery constraints into the optimization, the
`leaf-sequencing step is eliminated, thereby significantly simplifying the optimization process.
`We have demonstrated the feasibility of applying DAO to IMAT by presenting several
`plans. All plans show that IMAT is able to achieve conformal dose distributions to the tumour
`with only six to ten arcs. The small number of arcs enables a short delivery time of 10 to
`20 min. We have also shown that we can deliver an IMAT plan on a conventional linear
`accelerator which matches the calculated dose distribution.
`
`Acknowledgment
`
`This research was supported by National Institute of Health Grant no. R29CA66075.
`
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