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`Non-coplanar beam direction optimization for intensity-modulated radiotherapy
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`2003 Phys. Med. Biol. 48 2999
`
`(http://iopscience.iop.org/0031-9155/48/18/304)
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`INSTITUTE OF PHYSICS PUBLISHING
`
`Phys. Med. Biol. 48 (2003) 2999–3019
`
`PHYSICS IN MEDICINE AND BIOLOGY
`
`PII: S0031-9155(03)59031-5
`
`Non-coplanar beam direction optimization for
`intensity-modulated radiotherapy
`
`G Meedt, M Alber and F N¨usslin
`
`Department of Medical Physics, Universit¨atsklinikum T¨ubingen, Hoppe-Seyler Strasse 3,
`72076 T¨ubingen, Germany
`
`E-mail: msalber@med.uni-tuebingen.de
`
`Received 3 February 2003, in final form 16 June 2003
`Published 3 September 2003
`Online at stacks.iop.org/PMB/48/2999
`
`Abstract
`An algorithm for the optimization of the direction of intensity-modulated
`beams is presented. Although the global optimum dose distribution cannot be
`predicted, usually a large number of equivalent beam configurations exists. This
`degeneracy facilitates beam direction optimization (BDO) through a number of
`possible approximations and because the target set of good beam configurations
`is very large. Usually, the target volume is accessible through a finite number
`of paths of little resistance, which are defined by the properties of the objective
`function and the global optimum dose distribution. Since these paths can be
`occupied by a finite number of beams, it is reasonable to assume that a minimum
`number of beams for a configuration that is degenerate to the global optimum
`exists. Efficiency of the BDO will be characterized by detecting this degeneracy
`threshold. Beam configurations are altered by adding and deleting beams.
`A fast exhaustive (up to 3500 non-coplanar orientations) search finds beam
`directions that improve a configuration. Redundant beams of a configuration
`can be identified by a fast criterion based on second-order derivative information
`of the objective function. This offers a fast means of iteratively substituting
`redundant beams from a configuration. Inferior stationary states can be evaded
`by adding more beams than the desired number to the current configuration,
`followed by the subsequent cancellation of superfluous beams. The significance
`of BDO is examined in a coplanar and a non-coplanar test case. The existence
`of a threshold number for the minimum configuration and its dependence on the
`complexity of the problem are shown. BDO outperforms manual configurations
`and equispaced coplanar beam arrangements in both example cases.
`
`1. Introduction
`
`there has been continuous development of inverse planning
`During the last few years,
`algorithms for intensity-modulated radiotherapy (IMRT). These tools are designed to
`
`0031-9155/03/182999+21$30.00 © 2003 IOP Publishing Ltd Printed in the UK
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`G Meedt et al
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`determine fluence profiles for a given configuration of beams such that the goal of shaping
`high dose regions to the target volume (TV) is achieved while organs at risk (OARs) are
`avoided. Usually, the physics of dose deposition does not allow us to realize the ideal
`dose distribution. Hence, optimization methods have been devised which utilize an objective
`function (F) that translates the clinical trade-off strategies with respect to the TV and the OARs
`into mathematical terms. The optimum fluence profile (cid:2) is obtained as the minimizer of
`F by
` (cid:2) = arg min
`(1)
`where the dose distribution (D( ) = T ) is a function of the fluence profiles and the
`energy deposition operator T. The linear operator T describes the energy absorption per mass
`ηi , i = 1, . . . , N, such that =(cid:1)
`and fluence unit in the patient volume. The parameter space of the fluence optimization can
`be reduced to (φi )i=1,...,N ∈ RN
`0 if there exists a decomposition of into a fluence base
`=1 φi ηi. The φi are the weights of the fluence elements ηi.
`In many cases, a large number of fluence weight profiles (φ)i=1,...,N can be found which are
`i=1,...,N with respect to the objective function F (Alber et al
`equivalent to the optimum (φ)(cid:2)
`2002).
`In the following, this is referred to as degeneracy of the fluence optimization.
`Because of this degeneracy, the question arises if any reasonable arrangement of beams,
`e.g. a sufficiently large number of evenly spaced coplanar beam directions, will always be
`degenerate to the global optimum (Bortfeld and Schlegel 1993, Stein et al 1997). From a
`different point of view, the high degeneracy could be exploited to find a smaller number of
`well-chosen (non-coplanar) beams to reach similar (or superior) dose distributions.
`This can be analysed only by including the degrees of freedom of gantry (γ ) and table
`(ϑ ) angles in the optimization. While a large number of fluence optimization engines have
`been proposed, the combination of fluence optimization with beam direction optimization has
`rarely been addressed. Two main reasons can be stated. Firstly, determining an optimum
`configuration of fluence-optimized beams chosen from the set of all possible directions results
`in a very large parameter space (γ , ϑ, u, v) → . Every direction defined by the solid angle
` = (γ , ϑ ) corresponds to a unique set of fluence elements whose superposition spans the
`fluence profile with the field coordinates (u, v). The optimum fluence weight profile of each
`beam is strongly dependent on the other beams of the given configuration. Changes within a
`beam configuration invalidate previously optimized fluence profiles.
`Secondly, an objective function depending on discrete gantry and table angles assumes
`multiple local minima and thus poses a challenge to algorithms. Often, stochastic methods
`such as simulated annealing are utilized which are suitable for escaping local minima
`(Stein et al 1997) or exhaustive search methods are used (S¨oderstr¨om and Brahme 1993, 1995,
`Rowbottom et al 1999, Pugachev et al 2001).
`Automatic optimization of beam directions may resort to brute force sampling of the search
`space. Depending on the coarseness of discretization in terms of beam angles (particularly in
`the case of non-coplanar configurations), this results in a large number of beam configurations.
`In order to evaluate each configuration, an optimization of fluence profiles has to be performed,
`leading to intractably large computational tasks. Decoupling the optimization of beam angles
`from fluence profiles was suggested (Pugachev et al 2000). Fluence profiles are then calculated
`independently up front, e.g. by filtered backprojection methods, and used as a substitute
`measure of potential of a beam orientation. Any similar approach necessarily neglects the
`interaction between fluence elements of different beam directions in the optimization process.
`A human planner, who is bound to optimize beam configurations by experience or
`intuition, is faced with the difficulty of anticipating the interdependence of optimum fluence
`profiles from different directions when setting the beams. Geometrical aspects of the volumes
`
`N i
`
`(F (D( )))
`
`
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`Non-coplanar beam direction optimization for intensity-modulated radiotherapy
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`of interest may suggest exclusion or a preference for certain orientations. The intuitive potential
`of beam directions has been attempted to capture by detecting ‘paths of least resistance’
`(Gokhale and Hussein 1994) or windows of opportunity, defined as the minimum overlap
`between the projections of TV and OARs (Das et al 1997) within the patient volume. These
`concepts try to express the ‘target coverage capacity’ of beam directions by substitute measures,
`where the objective function should be the sole means of dose evaluation.
`In the following development, the intuition of ‘path of least resistance’ is formalized with
`respect to the objective function.
`It is crucial to include the individual dose constraints to
`OARs as well as the prescription to the TVs in the process of selecting beam directions: often
`the tolerance dose of some OAR has to be exploited in order to achieve the target dose while
`it is not predictable which OAR and to what extent. Furthermore, it is not known up front
`which fluence-modulated beam can spare an OAR or how its acceptable tolerance dose is
`apportioned between beams. Hence, in order to extend the concept of ‘path of least resistance’
`by the element of dose prescription, it is necessary to describe the geometry of the objective
`function of a given dose distribution rather than the patient geometry or the geometry of the
`dose distribution alone.
`To achieve this, an objective density f (D((cid:3)x)) corresponding to each voxel (cid:3)x of the volume
`(cid:2)
`of interest (Alber and N¨usslin 1999, Alber et al 2002) is defined such that
`f (D((cid:3)x)) dx3.
`F (D( )) =
`
`(2)
`
`V
`
`(cid:2)
`
`∂F (D( + wη))
`∂w
`
`(3)
`
`Note, that in OARs, f is a monotonically increasing function, while on TVs, f is a cup-
`convex function that assumes a minimum at the prescribed dose. The path of least resistance
`can thus be identified by small values of f (D((cid:3)x)) along the track of a beam. Let η(γ , ϑ )
`be a fluence element of some shape (e.g. the fluence distribution of a circular aperture) and
`T η be its corresponding dose distribution and w its weight. For a given dose distribution D,
`considering the derivative of the objective function with respect to the weight w,
`∂f (D((cid:3)x))
`=
`T η((cid:3)x) dx3.
`∂D((cid:3)x)
`In this picture, the resistance of the path taken by η is the integral of the effect density
`weighted with the dose distribution of the probing fluence element T η. Note that this resistance
`depends on the dose distribution D and will naturally vary with varying D. Also, note that the
`probing fluence element need not contribute to D.1
`A fluence optimization algorithm will face conflicts if certain OAR constraints or the
`maximum dose constraints in the TV prevent the delivery of the prescription dose to the TV.
`A path of high resistance marks conflict zones of already exhausted tolerance dose of some
`OAR. Both OARs in the direct track of a beam as well as OARs adjacent to the TV can cause
`conflicts due to the finite penumbra. The smaller the conflict zones become, the more paths of
`little resistance open up. This means that in an iteratively evolving beam configuration, initial
`paths of little resistance may shift, grow or disappear.
`If beams impinging from other directions find a path of little resistance to resolve an
`existing conflict, the target becomes more accessible. In the picture of paths of little resistance,
`a certain minimum number of beams is necessary to reduce the conflict zones of a dose
`distribution and open up maximum access, but since the number of paths of little resistance
`is finite, the objective function will saturate and configurations become degenerate if all paths
`are occupied by beams. It follows that any optimum dose distribution employs the maximum
`
`1 This integral is the analogue to the backprojection in image reconstruction.
`
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`G Meedt et al
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`Figure 1. Search space S of the non-coplanar beam search. Every dot on the unit sphere denotes
`a candidate beam direction.
`
`number of paths of little resistance available to resolve the conflict posed by the interplay of
`patient geometry and the physics of dose deposition. While some conflicts can be avoided by
`changing beam directions or allowing for a higher number of beams to spread the dose among
`the OARs, other conflicts may not be resolvable. These unavoidable conflicts will drive the
`fluence optimization and determine the eventual selection of beam directions.
`
`2. Methods
`
`The selection of a suitable and sufficient set of directions for intensity-modulated beams is
`not strictly an optimization problem. The requirement for a ‘good’ configuration of beam
`directions only arises because the number of beams is finite, so that the task becomes finding
`a set of beam directions that produce a dose distribution which cannot be improved with
`reasonable effort.
`As the dose optimization problem for IMRT is very highly degenerate, it can be required
`that a beam configuration be found whose optimum dose distribution is equivalent to the global
`optimum with respect to the objective function. The search space for this task is enormous:
`a discretization of the search space as in figure 1 allows approximately 5 × 1015 different
`configurations of five beam directions. Clearly, this means also that there is a large number of
`almost equivalent configurations of a given number of beams, and the greater the number of
`beams allowed, the easier it will be to find a sufficient configuration.
`The concept presented here comprises three elements: construction of a configuration of
`beam directions, evaluation of this configuration and verification that it is not a local optimum
`(see figure 2). The initial beam configuration 1 = { 1, . . . , n} is achieved by building
`up n beams sequentially, n being the desired number of beams. This first stage starts with
`one beam direction and iteratively adds a single new beam direction to the configuration. At
`each iteration, the phase space is searched for a beam that finds the path of least resistance
`according to equation (3). The search space for this task is still large but substantially smaller
`than finding all n orientations out of all possible directions simultaneously. Once a new beam
`direction has been found and integrated into 1 the fluence profiles of all beams are optimized.
`
`Page 5 of 22
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`Non-coplanar beam direction optimization for intensity-modulated radiotherapy
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`Figure 2. Beam direction optimization flowchart.
`
`This is performed by the in-house planning system Hyperion (Alber et al 2000). Hyperion
`maximizes the equivalent uniform dose (EUD) to the TV while strictly obeying EUD
`constraints applied to the OARs and maximum dose constraints for the TVs. The dose
`computation method for the fluence profile optimizations as well as the beam direction searches
`is a pencil beam algorithm with density-dependent depth scaling and realistic penumbra
`modelling. Based on the result of the fluence profile optimization, the new beam search can
`be executed again.
`During the process of building up the initial configuration, a beam selected at the beginning
`can become ill-adapted or redundant. Although a beam is rarely completely redundant, its
`replacement can improve the configuration. A selection criterion for dispensable beams is
`described in section 2.2. The deletion of a beam from a configuration is followed by a re-
`optimization of the fluence profiles and a search for a new supplementary beam, again followed
`by an optimization of fluence profiles. Beams are replaced until no further improvement is
`achieved or beams to be taken away had been found previously by the beam search algorithm
`(configuration 2)2. This terminates the second stage of the search.
`This state can be understood as convergence of the BDO process. In order to exclude
`a stationary solution of low quality being found, a set of more than one beam can be added
`simultaneously, followed by the cancellation of redundant beams, resulting in configuration
` 3 (stage three).
`If the configuration 3 equals 2, this configuration is assumed to be
`degenerate to the optimum solution. This configuration corresponds to a stationary state for
`which additional beams merely increase the degree of degeneracy of the solution space. In
`the case 2 (cid:4)= 3 beam replacement is performed again on 3 ( 3 → 2).
`The rationale for the simultaneous addition of multiple beams (for example, one could run
`the beam search with different choices for the first beam and then choose 3 = 1
`∪ 2
`∪···) is
`2
`2
`that both the search for a new beam and the search for a redundant beam may deliver less than
`optimum results. The principal reason is that both search criteria operate on the basis of a dose
`distribution which may differ significantly from the global optimum. Also, to accelerate the
`search based on both criteria, approximations are made. However, the use of approximations
`seems justifiable because even rigorous brute force searches may be misleading. The key
`hypothesis of the presented algorithm is that in the light of degeneracy, there is neither a single
`→
`→ 2
`
`2 It is also possible that the search converges to a limit cycle which passes through configurations 1
`··· → 1
`1
`1, although this has not been observed yet.
`
`1
`
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`‘best’ beam to be added or deleted, nor a ‘best’ configuration that needs to be found. This
`opens up some scope for error in the various stages of the search, as long as the result is
`verified, which is attempted in the third stage of the search.
`
`2.1. Add supplementary beam
`The search space S of possible beam directions is defined by the discretization of either
`gantry angle alone or the discretization of solid angle (table, gantry) in the non-coplanar case.
`Directions are defined by evenly spaced points on a unit sphere (figure 1). Approximately
`3500 non-coplanar beam orientations are checked each time during the beam search process.
`Table–gantry collisions are excluded.
`Based on a given beam configuration of n beams from angles { 1, . . . , n} with fluence
`profiles = { 1, . . . , n}, a beam direction (cid:2) best capable of improving the set has to be
`found. Imaginable as a brute force method is checking any possible direction by adding
`it to the current configuration and performing a complete fluence profile optimization of all
`beams:
`
` (cid:2) = arg min
`min
` ∈S
`{ i}i=1,...,n∩ ( )
`The orientation which leads to the smallest objective value can be identified as the optimum
`new direction with respect to the current configuration of beams.
`This approach would respect the interdependence between optimum fluence profiles and
`beam directions, yet checking all directions within the search space takes an enormous amount
`of time (proportional to the fineness of the beam angle discretization). In order to accelerate
`the exhaustive search of equation (4), the requirement of a full fluence optimization for all
`possible (n + 1) beam configurations has to be replaced with a substitute. Here, the intuitive
`concept of the ‘path of least resistance’ can be brought into use. Usually, an optimum dose
`distribution of n intensity-modulated beams can be expected to exploit the paths of little
`resistance accessible to it, and exhaust the tolerance of the irradiated OARs where necessary.
`Any significant improvement on this dose distribution can only be achieved if a beam
`direction can be found which avoids these areas in the patient where the normal tissue tolerance
`is not already depleted and which impinges on a relatively cold spot in the target volume.
`Usually, only a part of the cross section of a beam will yield a positive contribution. Therefore,
`we propose to represent a beam by a fluence element which is large enough to cover the cold
`spot, but small enough to find the paths of little resistance between the conflict volumes.
`Turning back to equation (3), such a direction can be found if η(γ , ϑ, (cid:3)z) is a probing fluence
`element (e.g. a small circular field of some centimetres in diameter or a small rectangular field
`if a coarse MLC shall be approximated) and T η its corresponding dose distribution, which
`aims from angle (γ , ϑ ) at the centre of the cold spot (cid:3)z.
`The respective cold spot can be detected by identifying the maximum integral of
`equation (2) with regard to the volume of a sphere with the dimension of the probing beamlet
`for each centre (cid:3)z within the TV. Hence, a substitute to equation (4) can be found as
` (cid:2) = arg min
`[min
`F (D( (cid:2)) + wT η( ))].
` ∈S
`w(cid:1)0
`Note that the optimization of the weight of the supplementary beamlet is performed relative
`to the fixed optimum dose distribution of configuration 1. This ‘reduced’ optimization can be
`performed within 0.2 to 0.5 s for each direction including computation of T η and approximates
`the ‘brute force’ evaluation (equation (4)) of a huge search space within reasonable times.
`Alternatively, neglecting higher orders of F, the potential of a new beam direction can be
`
`[
`
`F (D( , ( )))].
`
`(4)
`
`(5)
`
`Page 7 of 22
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`Non-coplanar beam direction optimization for intensity-modulated radiotherapy
`
`3005
`
`Figure 3. Results of the beam search of a sample head and neck case after optimizing fluence
`◦
`gantry angle): solid line—brute force,
`profiles of three coplanar equispaced beams (0, 120, 240
`dotted line—optimization of the probing beam within given dose distribution and dashed line—
`‘probing derivative’.
`
`calculated by the first derivative of the objective function with respect to the weight of the
`probing beam.
`
`(cid:5)
`
`(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)
`
`.
`
`(6)
`
`(cid:3)
`
`∂F (D(cid:2), wT η( ))
`∂w
`
` (cid:2) = arg min
` ∈S
`w=0
`Note that this derivative will be negative for the most promising beam. OARs within
`the beamline contribute positive terms to the integral equation (3) while TVs add negative
`contributions. Positive or zero ‘probing derivatives’ thus display poor potential for new beam
`directions.
`In contrast, negative derivatives indicate a higher potential of candidate beam
`directions. Hence the minimum ‘probing derivative’ for all possible directions within the
`search space characterizes the optimum new beam direction. In figure 3, the findings for the
`three methods (equations (4), (5) and (6)) concerning an example head and neck case after
`fluence optimization of three equispaced coplanar beams are compared. Note the analogous
`behaviour of all curves. The minima found by the brute force method can be well detected by
`the optimization of the weight of the probing beam relative to the dose distribution of the three
`fluence-optimized beams. Testing beam directions by calculating the ‘probing derivative’
`reproduces the results of the brute force method closely.
`
`2.2. Delete worst beam
`
`The iterative growth of the beam configuration can reduce the relative importance of directions
`selected earlier. The role of these beams (i.e. filling a ‘path of little resistance’) may have
`been taken by other beams, or the high degeneracy may make them redundant so that their
`elimination can be compensated for by the remaining configuration.
`In the following, we
`propose a selection criterion for the least important beam which is based on a measure of
`degeneracy of the objective function. The brute force method of quantifying the relative
`importance of a beam in the sense that the elimination of the least important beam causes the
`smallest deterioration of the objective function is to run a fluence profile optimization for all
`(n − 1) beam subconfigurations of the current n-beam configuration:
` = { 1, . . . , n}
` red = arg min
`[ min
`F (D( ))].
` i∈
` ( \ i )
`The beam ( red) whose cancellation leads to the smallest change of the objective function
`value represents the beam which is best compensated for by the remaining beams. For an
`
`(7)
`
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`n-beam plan this method results in n fluence optimizations at this stage of the BDO, which
`can be quite time consuming.
`The degeneracy of the solution space can be related to the curvature of the objective
`function around the minimum. It has been shown that the objective function is flat in the vast
`majority of directions, indicating that a perturbation of the fluence profile in these directions
`will not affect the objective function value (Alber et al 2002). The set of all directions in which
`the objective function is flat shall be called ‘zero-curvature space’, ZCS ⊂ RN . If a beam
`lies entirely in ZCS, its elimination will not greatly affect the quality of the dose distribution
`because it can be compensated for by changing the fluence profiles of the other beams. The
`criterion described below quantifies the component of a fluence profile of a single beam which
`does not lie in the ZCS, i.e. the magnitude of the component that cannot be compensated by
`other beams.
`Let Hij be the curvature tensor (the Hessian matrix) of F at the optimum,
`Hij = ∂ 2F
` = (φ1, . . . , φN ).
`(8)
`The eigenvectors of Hij with eigenvalue 0 span the ZCS. Let k = (0, 0, . . . ,0,
`φm+1, . . . , φm+nk , 0, . . . , 0) be the fluence weight profile of beam k, starting from index
`m + 1 and having nk fluence element weights. Let Vk + Zk = k be a decomposition of k
`such that Zk ∈ ZCS, Vk ∈ RN \ ZCS. Zk is a linear combination of eigenvectors of Hij
`k H = H Zk = 0.
`(cid:7)
`k H k =(cid:6)
`with 0 eigenvalue, i.e. ZT
`We define as quantity of interest the curvature-weighted norm of k:
`H (Vk + Zk)
`k + ZT
`V T
` T
`= V T
`k H Vk
`
`k
`
`(9)
`
`∂φi ∂φj
`
`but
`
`(10)
`
`lhs = ∂ 2F (D( + k))
` ∈ R.
`∂ 2
`This number becomes large if the fluence profile k has large components in the directions
`of the eigenvectors of large eigenvalues, so that a variation (e.g. a rescaling) of k causes a
`large change in the objective function. Hence, we define the substitute beam deletion criterion
`as
`
`.
`
`(11)
`
` red = arg min
`∂ 2F (D( + k))
` k∈
`∂ 2
`In figure 4, this criterion is compared to the brute force criterion of equation (7) for an
`equispaced coplanar 15-beam configuration of the first example case below. The rather good
`agreement is a consequence of the high level of degeneracy of this configuration: the removal
`of any single beam will not require large compensations on the side of the other beams, so
`that the curvature tensor around the initial minimum differs only slightly from the tensor at
`the minimum of the subconfigurations.
`For smaller configurations with less capacity for compensating beam removals, the
`criterion is less accurate. For the second example case below, an intermediate non-coplanar
`three beam configuration (in stage 2) was supplemented with a beam randomly chosen from
`the unit sphere. Then, the beam deletion criteria were applied. There was agreement in 72/100
`cases. While the curvature criterion removed the latest beam in 79/100 cases, the brute force
`method did so in only 67/100 cases. This tendency could be inherent in the curvature criterion
`which favours beams with a smaller overall weight. However, since the deletion criterion is
`never applied to very small configurations 2 in stage 2, it can be assumed to be reasonably
`accurate.
`
`(cid:8)
`
`(cid:9)
`
`Page 9 of 22
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`
`Non-coplanar beam direction optimization for intensity-modulated radiotherapy
`
`3007
`
`Figure 4. Comparison of the brute force beam deletion criterion equation (7), based on re-
`optimization of all 14 beam subconfigurations of an equispaced coplanar 15-beam configuration,
`of the example below, and the second-order criterion equation (11). Both sets of figures of merit
`are scaled to the relative maximum extent.
`
`Figure 5. Coplanar test case.
`
`3. Results
`
`3.1. Coplanar beam direction optimization
`
`When considering an elongated symmetry of the patient or the TV geometry, coplanar
`configurations can be expected to be sufficient to reach optimum dose distributions. The
`coplanar test case is a carcinoma of the larynx with suspected microscopic disease in the
`lymph nodes of the cervix and upper mediastinum. The TV overlaps with the left and right
`lungs while being in close vicinity to the spinal cord. It extends into the head where it lies
`adjacent to both parotid glands (figure 5).
`A six- and seven-beam BDO plan is compared to a manually optimized coplanar nine-field
`configuration and an equispaced coplanar 15-field configuration, which is considered a good
`standard setting. Figure 6 shows normalized plots of the objective functions during the search
`for new beams. 360 coplanar beam orientations are examined during each beam search of the
`
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`Figure 6. Coplanar beam search: first (solid) and second (dotted)—left; third (solid) and fourth
`(dotted)—right.
`
`Table 1. Setting of constraints: prescriptions and results. The spinal cord is considered as a
`serially organized organ with an EUD constraint set to 32 Gy. The right lung is treated as a parallel
`organ. 25% of the organ volume may receive more than 20 Gy.
`
`Dose
`constraint
`
`Spinal cord
`serial EUD (Gy)
`32.0
`
`Left parotid
`parallel (%) 15 Gy
`15
`
`Right lung
`parallel (%) 20 Gy
`25.0
`
`Prescription
`TV: EUD (Gy)
`70.0
`
`BDO 6
`BDO 7
`15 equispaced
`Manual 9
`
`31.8
`31.9
`31.9
`31.9
`
`3.1
`3.2
`0.2
`0.8
`
`21.0
`19.5
`15.7
`15.4
`
`70.0
`70.3
`70.3
`70.2
`
`BDO. The direction leading to the smallest value of the objective function is identified as the
`optimum new direction.
`The plots are taken during the first stage of the BDO process when the initial beam
`configuration is built up sequentially. The first beam search is based on no previous dose
`◦
`) are discernible.
`distribution. Two apparent minima of the objective function (at 340 and 180
`The second beam search takes into account the dose distribution achieved after fluence
`◦
`). It leads to a manifold of local minima
`optimization of the first beam (gantry set at 178
`which are no longer predictable when considering the first beam search alone. Regarding the
`graphs of the objective function for the third and the fourth beam searches a similar situation
`becomes apparent. Beam directions regarded as ‘good’ orientations during the beam search
`after selection of two beams can lose any potential in the search for the fourth beam.
`Furthermore, beam directions previously perceived as ‘poor’ can gain potential in later
`stages. The interdependence of fluence profiles and beam directions prevents any prediction
`of the eventual usefulness of beams when building up the initial configuration.
`All different beam configurations could achieve the clinical dose prescription in the target
`volume (table 1). The constrained optimization demands that all constraints be obeyed. The
`objective function is only effective if the dose in an OAR reaches the applied constraint. In
`this respect all plans are equivalent with respect to the employed constraints. Yet, more beams
`will stress OARs less since the dose is spread between the beams. The redistribution of dose
`to larger volumes at intermediate or low dose level is only penalized if active dose–volume
`constraints are present. For maximum dose constraints, the objective function is blind to these
`effects so that differences in the dose–volume histograms (DVHs) are a mere coincidence.
`The DVHs (figure 7) of the six-beam BDO plan and the nine-beam manual plan display
`the equality of both plans for the target volume. The seven-beam BDO plan could reach a
`
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`

`
`Non-coplanar beam direction optimization for intensity-modulated radiotherapy
`
`3009
`
`Figure 7. Dose–volume histograms of the coplanar BDO plans, the manually optimized plan and
`the evenly distributed 15-beam plan.
`
`Table 2. Manual choice of gantry orientations versus BDO. Diagonally opposed beam directions
`are given in parentheses.
`
`9 man
`BDO 6
`BDO 7
`BDO 8
`15 equi.
`
`0
`(178)
`359
`(178)
`0
`
`32
`
`(211)
`(216)
`
`63
`(240)
`
`75
`74
`79
`74
`72
`
`135
`
`160
`
`109
`
`89
`
`137
`
`(264)
`
`(288)
`
`(312)
`
`(340)
`
`200
`205
`201
`
`(24)
`
`225
`226
`227
`226
`(48)
`
`263
`
`264
`
`285
`
`328
`323
`
`285
`288
`
`321
`(144)
`
`188
`192
`
`higher dose conformity to the target volume. The equispaced plan showed inferior results
`to all other plans in terms of target coverage. This could be due to the fact that only five
`beams of the 15-beam plan are in the union set of beam orientations picked by BDO plans, see
`table 2. An even greater number of equispaced beam directions would be necessary to cover
`all available paths of least resistance.
`Some of the beam directions found by the BDO are similar to the directions of the
`manually optimized configuration (table 2), where diagonally opposed beams as well as beam
`◦
`are counted as equivalent. However, the number and
`angles with a difference of less than 5
`direction of similar directions are dependent on the number of beams one is willing to accept.
`
`Page 12 of 22
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`

`
`3010
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`G Meedt et al
`
`Figure 8. The non-coplanar test case.
`
`This suggests that a certain set of candidate beam directions exists which represent all paths
`of little resistance. The BDO picks beam directions from this set which are most suitable
`subject to the desired number of beams. While a six-beam BDO plan results in four beam
`orientations (five if counting diagonally opposed beams) that are analogous to the manually
`optimized plan, the seven-beam plan has five similar beam directions. Optimizing eight beam
`directions reproduces four (six, respectively) orientations of the manually chosen plan. The
`◦
`gantry angle which appeared good by intuition was not selected by the BDO algorithm for
`160
`any plan. The six-beam BDO required a total of 11 direction searches while th