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INSTITUTE OF PHYSICS PUBLISHING
PHYSICS IN MEDICINE AND BIOLOGY
Phys. Med. Biol. 46 (2001) 2637–2663 PII: S0031-9155(01)24104-9
Comparative behaviour of the Dynamically Penalized
Likelihood algorithm in inverse radiation therapy
planning
Jorge Llacer1, Timothy D Solberg2and Claus Promberger3
1EC Engineering Consultants, LLC, 130 Forest Hill Drive, Los Gatos, CA 95032, USA
2Department of Radiation Oncology, University of California, Los Angeles, CA 90095, USA
3BrainLAB AG, Ammerthalstrasse 8, 85551 Heimstetten, Germany
E-mail: jllacer@home.com, Solberg@radonc.ucla.edu and promberg@brainlab.com
Received 19 April 2001, in final form 25 June 2001
Published 20 September 2001
Online at stacks.iop.org/PMB/46/2637
Abstract
This paper presents a description of tests carried out to compare the behaviour
of five algorithms in inverse radiation therapy planning: (1) The Dynamically
Penalized Likelihood (DPL), an algorithm based on statistical estimation
theory;(2)anacceleratedversionofthesamealgorithm;(3)anewfastadaptive
simulated annealing (ASA) algorithm; (4) a conjugate gradient method; and
(5) a Newton gradient method. A three-dimensional mathematical phantom
and two clinical cases have been studied in detail. The phantom consisted
of a U-shaped tumour with a partially enclosed ‘spinal cord’. The clinical
examples were a cavernous sinus meningioma and a prostate case.
algorithms have been tested in carefully selected and controlled conditions
so as to ensure fairness in the assessment of results. It has been found that
all five methods can yield relatively similar optimizations, except when a very
demanding optimization is carried out. For the easier cases, the differences
are principally in robustness, ease of use and optimization speed.
more demanding case, there are significant differences in the resulting dose
distributions. The accelerated DPL emerges as possibly the algorithm of
choiceforclinicalpractice. Anappendixdescribesthedifferencesinbehaviour
between the new ASA method and the one based on a patent by the Nomos
Corporation.
The
In the
1. Introduction
In November 1997, the first author of this paper published a description of the Dynamically
Penalized Likelihood (DPL) method of inverse therapy planning (Llacer 1997). It responded
0031-9155/01/102637+27$30.00© 2001 IOP Publishing LtdPrinted in the UK2637
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2638 J Llacer et al
to the need, already pointed out by Powlis et al (1989) and by Bortfeld et al (1990),
to treat voxels corresponding to organs at risk (OAR) in a manner different from those
of the planning target volume (PTV) by specifying for the former a desired maximum
dose, for example, but otherwise allowing those voxels complete freedom to receive
any dose lower than that maximum. Since that time, the algorithms of Bortfeld et al
(1990, 1997) and Spirou and Chui (1998) have become perhaps the two most recognized
analytic (non-stochastic) inversion methods being used in treatment planning.
its first publication, the DPL has evolved considerably and we feel that it is now
ready to be compared to those two algorithms and to a suitable simulated annealing
method.
It is recognized that it is practically impossible to set up the conjugate gradient (CG)
method of Spirou and Chui and the Newton gradient (NG) method of Bortfeld et al in the
same manner as in the Memorial Sloan-Kettering Cancer Center and in DKFZ-Heidelberg,
respectively, even with the extensive help that has been received from these authors. For that
reason, this work has been specifically directed to test the actual mechanisms for solving the
inverseproblemin a set of conditionsthat is as close as possible to the conditionsunderwhich
the DPL inversion engine is operating successfully in the BrainLAB’s Intensity Modulated
Radiation Therapy/Surgery (IMRT/IMRS) software package (BrainSCAN 2001). These
conditions can be summarized as follows:
Since
1. The algorithm has to be fast enough so that optimizations using dose matrices including
complete scattering effects can be carried out.
through a Federal Drug Administration (USA) approved algorithm will not produce
any significant differences between what was expected and the actual outcome of an
optimization.
2. TheoncologisthastobeabletospecifythedesiredOARdosevolumehistograms(DVHs)
bydefininganumberofpointsinthecorrespondingDVHgraphs. OARvoxelsmayreceive
lower doses than the oncologist’s specifications.
3. The specification of DVHs for the PTV has been considered an over-constraint on the
problem. Indeed, a specified set of DVH curves for the OARs and for the PTV is
likely to be, to a smaller or larger extent, contradictory. Instead, the algorithm has to
provide the most ‘compact’ PTV DVH that is compatible with the requirements placed
on the OARs. That includes, of course, the smallest possible under-dosing of the PTV
voxels.
4. The algorithm should be able to incorporate a filtering method, or smoothing constraint,
inside the optimization loop, so that the resulting beam fluences are the best possible for
a certain degree of smoothness in the beam profiles to be delivered.
In this manner, a verification step
ThesimulatedannealingmethodthatappearstobeinusebytheNomosCorp.,asdescribed
in their patent (Nomos Corp. 2000), is basically out of contention for the purposes of this
comparison. The use of non-analytical DVHs as target functions makes that process much
slower than the new adaptive simulated annealing (ASA) algorithm to be described below, it
cannotincorporatea filter inside the optimizationloop and offers no apparentadvantagesover
the ASA. The appendix will describe our implementation of the Nomos algorithm (NM) and
the differences in behaviour in relation to the ASA.
This paper will initially describe the five algorithms used in the detailed comparison,
their specific implementation, the phantom and medical cases that have been studied, give
results in terms of quantitative and qualitative characteristics of the optimizations, discuss
some particularities of the different algorithms and draw some conclusions.
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Comparative behaviour of the DPL algorithm 2639
2. Algorithms
2.1. DPL and accelerated DPL algorithms
The DPL is a variant of the maximum likelihood estimator (MLE) method of statistical
parameter estimation. The basic foundation for the use of the MLE in treatment planning
optimizationwas describedinLlacer(1997)andthecurrentformofthe DPLhasbeengivenin
Llacer (2000). The relationship between the MLE-DPL method and a minimum least squares
solution will be shown here and the final iterative formula, with the addition of filtering inside
the optimization loop, will be given.
It will be useful to start with a description of the target functionfor the MLE, first derived
for PET image reconstruction by Shepp and Vardi (1982), adapted to our problem. The joint
probability of obtaining a vector of doses d in the voxels of a PTV, when a vector of beamlet
fluences a deposits energy in those voxels is given by
P(d|a) =
?
i∈D
exp(−hi)(hi)di
di!
(1)
where hi =?
desired in a specific voxel of the PTV and Fijare the dose matrix elements, dose delivered by
beamlet j to voxel i per unit beam fluence.
Equation (1) holds strictly for the numbers of photon interactions that deposit energy in a
set of voxels, which follow Poisson statistics. Note that hiis the mean of each corresponding
Poisson distribution and the values of d are limited to non-negative integers in a strict
interpretation of Poisson statistics. An extension to dose values that are not integers is
straightforward (Vardi and Lee 1993).
When the number of photons is very large, as is the case in radiation therapy, the Poisson
distribution is almost identical to a Gaussian distribution of variance equal to its mean, except
at the region near di= 0, as there are no negative values of diin the Poisson case and there
will be a negative region in the Gaussian distribution, even if small.
It is instructive, then, to write a target formula equivalent to equation (1) for a Gaussian
distribution
jFijaj is the mean dose received by voxel i, ajis the fluence of beamlet j,
iis theindexforeachvoxelinthePTV,D is theregionthatincludesallPTVvoxels,dithedose
PG(d|a) =
?
jFijajis the mean dose received by voxel i and also the variance of
each Gaussian in the product.
The MLE does not attempt to maximize equation (1) as a function of the parameters ai,
but its logarithm
i∈D
?
1
√2πhi
exp
?
−(hi− di)2
2hi
??
(2)
where, as above, hi=?
L(d|a) =
?
i∈D
[−hi+ dilog(hi) − log(di!)].
(3)
If we then look at the corresponding equation for the Gaussian case we have
LG(d|a) =
?
i∈D
?
−1
2log(2πhi) −(hi− di)2
2hi
?
.
(4)
Because of the slowly varying log terms, a maximization of equation (4) is closely equivalent
to a minimization of its quadratic terms. It follows that a maximization of the log likelihood
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2640 J Llacer et al
functionof equation (3) is approximatelya minimizationof the quadraticdifferencesbetween
thedesireddosesin eachPTVvoxelandtheactualdosethat will bedelivered,weightedbythe
inverseof the actual dose received. This means that the MLE method tries to satisfy better the
desired doses delivered to voxels that receive lower dose than those that receive higher dose.
It does that optimization process without attempting to find negative values of the beamlet
weights, which have no meaning in Poisson statistics or in the physical problem that we are
trying to solve. The iterative formula for the DPL algorithm, derived from the maximization
of equation (3), including filtering terms is given by
a(k+1)
j
= a(k)
j
1
qj
?
i∈D
Fij
di
h(k)
i
+
?
−si
i∈S
?
h(k)
i
?
>0
βiFij
si
h(k)
i
− α
a(k)
j
− λη
?
η∈Nj
a(k)
η
n
(5)
where a(k)
voxels, siis the maximum dose desired in a specific voxel of an OAR, α is the filter parameter
that controls the degree of smoothness, η ∈ Nj is the neighbourhood of pencil beam j to
be considered for filtering, ληare the weight parameters for the neighbouring beam fluences,
qj=?
solutions that yield beamlet weights that are substantially different from their neighbours.
The exponent n in the outer brackets corresponds to an acceleration parameter that must
be equal to 1.0 for the DPL algorithm (DPL1) as derived from the MLE target function by the
expectation–maximization algorithm (Shepp and Vardi 1982). The accelerated form of the
DPL,whichwillbelabelledDPL2,allowsanexponentn > 1.0whenderivedbythesuccessive
approximation method (Hildebrandt 1974). The convergence to a single broad maximum is
assured for the DPL1 (Shepp and Vardi 1982), while convergence of the DPL2 has not been
proven theoretically and, in practice, depends on the magnitude of the acceleration exponent
for a particular type of problem.
Excludingthefilteringterm,thecorrectionbetweeniterationsconsistsoftwosummations,
the first over the voxels in the PTVs (i ∈ D) and the second over the voxels in the OARs
(i ∈ S), but only for those voxels that receive a dose hiat iteration (k) that is larger than
the desired dose si. The desired dose in the PTV is given by di. One recognizes the form
of equation (5) as belonging to an MLE iterative step, but with the number of terms in the
second summation changing dynamically as required to satisfy a desired dose distribution in
the OARs. The iterative function of the algorithm is, in effect, an adaptive function.
It has been demonstrated in a paper by Llacer et al (1989) that an MLE solution is
equivalent to a series of iterations in which the results of an iteration formally fulfil the role
of the best prior information available before the start of the next iteration. If we take the
results at iteration (k) and use equation (5) to calculate the optimization at iteration (k + 1)
and then permanently fix the number of terms in the second summation of equation (1), the
calculation of (k + 2), (k + 3), etc would proceed as a totally normal MLE estimation which
leads to a stable single maximum (Shepp and Vardi 1982). Instead, we do not fix the number
of terms in the second summation and take the results of iteration (k + 1) as prior information
to calculate the results of iteration (k + 2), etc with a variable number of terms. At each one
of the iterations the algorithm is provided with a starting set of beam values that is the best
knowledge up to that point about the solution and it could continue with the same number of
j
is the fluence of beamlet j at iteration (k), S is the region that includes all OAR
i∈DFij+?
i∈S
(hi−si)>0Fijis a normalizationfactor and βiare weights that determinethe
relative importance of the conditions set for voxel i of the OAR. The filtering terms penalize
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Comparative behaviour of the DPL algorithm2641
terms in the second summation to a stable maximum. A sequence of those iterations has to
lead to a stable single maximum and it does.
2.2. Quadratic cost function algorithms
The CG, NG and ASA algorithms use the following quadratic cost function:
B(a) =
Nt
?
i
(hi− di)2+ βi
Noar
?
(hi−si)>0
i∈S
(hi− si)2+ α
Nbeam
?
j
a(k)
j
− λη
?
η∈Nj
a(k)
η
2
.
(6)
The notation is identical to that of equations (1) and (5).
FortheCGalgorithm,theconjugategradientfunctionsofNumericalRecipesinCprovide
the framework for the iterative procedure. The gradient vector at the beginningof an iteration
is calculated from equation (6) and a ‘line minimization’ procedure is carried out to find the
pointxminalongthatdirectionthatleadstothelowestcostfunction. Followingtheprescription
of Spirou and Chui (1998), if any of the vector components (beamlet weights) are negative
at the point of lowest cost, the magnitude of xminis decreased until all the components are
positive or zero. If a component was zero in the previous iteration and has become negative
in the current one, it is simply returned to zero and the computationcontinues with that value.
Refer to Spirou and Chui (1998) and to Numerical Recipes in C (1988) for details of the
conjugate gradient calculation.
For line minimization, Spirou and Chui (1998)have indicated that an exact value for xmin
can be obtained for a quadratic cost function without having to use the ‘linmin’ routine from
Numerical Recipes that is more general and can be very slow. The exact solution for xmin
has been used for the work reported here, with considerable computation time saving over
‘linmin’.
Following Bortfeld et al (1990), the iterative formula derived from equation (6) for the
NG method is
γ
Nqjj
a(k+1)
j
= a(k)
j
−
?
i∈D
Fij
?
h(k)
i
− di
?
+
?
−si
i∈S
?
h(k)
i
?
>0
βiFij
?
h(k)
i
− si
?
− α[?T(?a)]j
(7)
where N is the number of therapy beams (ports) used in the planning, γ is a relaxation
parameter, qjj =?
been possible to leave the inverse of the Hessian out of the normalization terms qjj. A more
complete formulation renders the iterative function much more complex, with necessarily
slower calculation of the iterative process.
In the ASA method, the cost function of equation (6) and its first partial derivatives are
used to calculate the acceptability of a test change in a randomly selected beam fluence.
The use of partial derivatives of the analytic cost function equation (6) greatly speeds up the
solution when compared to a cost function that uses the non-analytic desired DVH curves
directly. When testing whetherto accept or reject a small changein a randomlyselected beam
weight, the magnitude and sign of that change is multiplied by the values of the first partial
derivatives as a first-order approximationto the change in cost function. If that change in cost
functionis acceptable, the small changein beam weight is accepted. The methodologyfor the
i∈D(Fij)2+?
i∈S
(hi−si)>0βi(Fij)2and ?T? is the Hessian of the filtering
matrix obtained from the last summation of the cost function. For moderate filtering, it has
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2642J Llacer et al
use of simulated annealing in inverse therapy planning is well known. Webb (1995)has given
a most complete description of the method, including discussions on a series of issues that
affect its performance. The ASA method is a direct application of that methodology to the
cost function of equation (6), including the adaptive nature of the OAR terms. Extensive
tests have been made with the adjustable algorithm parameters in order to be able to report
the apparently best results obtainable in the optimization cases studied. The values of those
parameters will be given below. In describing the results of the ASA method, one ‘iteration’
will mean one pass through all the randomly arranged dose matrix columns. The random
arrangement is different for each iteration. The optimization results depend, of course, on the
initial seed for the randomnumber generator,but the effect is sufficiently small that any of the
results obtained with different seeds could be reported here without affecting the conclusions.
3. Implementation
3.1. Specification of maximum desired OAR doses
One of the conditions that has been required of the tested algorithms is that an oncologist
should be able to specify the maximum desired OAR DVHs by defining a number of points
in the corresponding DVH graphs. It is then necessary to define a relationship between those
DVH points and the maximum desired OAR doses siin equations (5), (6) and (7). When
there are voxels in a particular OAR that receive excessive dose, and there are too many of
them, Bortfeld et al (1997) choose to apply a penalty to those voxels that receive the smallest
excess dose. Spirou and Chui (1998) apply the penalty to those voxels that, when sorted in
ascending order of dose received, exceed the maximum allowed volume. The procedure used
for the work reported in this paper is more closely related to the latter than to the former. It is
based on the observation that the OAR voxels that receive highest dose at some point in the
iterative process are likely to be the voxels that will also receive the highest dose after the next
iteration. The procedure can be described as follows: before an iteration, a ranking of the
doses received by each voxel in an OAR is done in terms of the dose that they receive at the
end of the previous iteration. The fraction of voxels that are desired to have doses between
the maximum allowable dose and the next point in the DVH to the left of that maximum is
selected from the ranked list starting from the maximum. The desired doses sithat will be
used for the next iteration for those selected voxels will then be the doses that they receivedin
the ranking, scaled to fit between the maximum desired dose and the dose for the first point to
its left. The same procedureis used for the voxels that have to fit between the first DVH point
to the left of the maximum and the second point, selecting the set of highest ranked voxels
still unassigned, and so forth. The procedure is repeated after each iteration of the algorithm,
as there is always some rearranging of the voxel order between iterations. The programs are
set to allow up to 4 points to describe a desired DVH in a graph.
3.2. Optimization procedure
The procedureto carry out an optimizationhas been made identical in the DPL1, DPL2, ASA
and CG methods.
1. One iteration of the MLE is carried out for the only purpose of bringing the beamlet
fluences, initially set exactly to 1.0, to the proper level for the dose required in the PTV.
2. The average of the beam fluences obtained in step 1 is used as a starting point to optimize
the PTV only. In this step the OARs are totally disregarded and no filtering is done. A
fixed number of iterations is adequate for that purpose.
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Comparative behaviour of the DPL algorithm 2643
3. The beam fluences from the PTV-only optimization are used to start the full PTV and
OAR optimization. The results of the PTV-only optimization are needed as a starting
point for the assignment of the desired OAR doses siin equations (5), (6) and (7). These
results containthe OAR dose values that, when rankedandscaled, will be used in the first
iteration of the full optimization, as described in section 3.1.
For the NG method, as suggested by Bortfeld in private communication, the PTV-only
optimization is started from beamlet fluences equal to 0.0 and a relaxation factor γ = 1.0.
At the end of the second iteration, the relaxation value is scaled by the ratio of the desired
dose in the PTV to the average dose at the end of that second iteration. The scaled value
of γ is maintained throughout the optimization. Step 3, above, is carried out as in the other
methods.
As indicated above, the ASA method has a number of internal parameters that need to be
experimented with until the apparently best results are obtained. For the problems studied in
this paper, the following values have been chosen:
1. Initial grain fraction, the initial relative size of change in a beamlet weight that will be
tested for acceptance = 0.05.
2. Smallest grain fraction = 0.0005 at iteration 600. Grain fraction decreases linearly
between the first iteration and the 600th, remaining constant after that. None of the
solutions presented here have reached the 600th iteration.
3. Grain0 is the highest beamlet weight after the first MLE iteration of the PTV (step 1,
above) multiplied by the grain fraction.
4. kT0, is the temperature that would cause a positive change in the cost function to be
accepted with a probability of 0.02 when the highest beamlet weight changes by Grain0.
5. kT is temperature of the simulated annealing process, given by kT0/log(1 + iteration
number), starting at iteration 1.
The random number generator used in the ASA optimizations is Subroutine ran2, from
Numerical Recipes in C (1988).
In all the methods tested, the weights βiin equations (5), (6) and (7) have been varied
between0.1and10.0forvoxelscorrespondingtodifferentOARs. Inorderto limitthenumber
of figures and results in this paper to a reasonable value, only the results with βi= 1.0 will
be given in detail. The effect of changing those parameters in the different algorithms will be
discussed in the conclusions.
The acceleration exponent n of the DPL2 algorithm has been set equal to 2.0 for the
PTV-only optimizations and 1.7 for the full optimizations, except when some values of βi
have been above approximately 5.0, in which case n has been reduced to between 1.25 and
1.4, as required for stability.
3.3. Stopping rule
Thebestwaytostoptheiterationsconsistentlyforthefivemethodstestedinvolvedmonitoring
the normalized absolute value of the derivative of the average dose in each OAR. When all
the normalized derivatives, averaged over 10 iterations, are below a certain convergence
threshold, the iterative process stops. Values of the threshold as small as 0.0005 and 0.0002
have been used for the work reported here, with significant improvements in the underdosing
tail of the PTV in the latter iterations. Larger values may be sufficiently good for clinical
use.
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3.4. The fluence map complexity (FMC)
The user-supplied filter parameter α controls the strength of the smoothing or filtering
operation. Because of the different nature of the algorithms tested, the same value of the
parameter does not correspond to a similar effect in the optimization by different methods.
A measure of smoothness in the beam fluence profiles has been devised that has been found
useful in setting the values of α in such a way that the results of different algorithms can be
compared. This measure, called the fluence map complexity (FMC), can perhaps be an initial
step in determining the desirability of a therapy plan for delivery by multi-leaf collimators.
The FMC responds to both the differences between adjacent beam weights and the existence
of some excessively large beam weights in the peripheryof the field in an otherwise relatively
uniform beam map.
Based ontheformofthefilteringterms inequations(5)and(6),an FMC has beendefined
by
FMC =
1
?
j
aj
?
?
?
?
??
j
aj− λk
?
k∈Nj
ak
2
.
Thesummationunderthesquarerootcontainsthesametermsas thefilterinthetargetfunction
of the DPL. Each term goes to zero if the fluence ajis equal to its two lateral neighbours,
which have λk= 0.5 assigned to them. For a beam in the periphery, λkof its only lateral
neighbour is set to 1.0. Because of the quadratic form, a single very high fluence carries a lot
of weight in the summation. The square root and the normalization by the sum of all fluences
are intended to lead to numbers that are reasonable for comparisons.
3.5. Coding concerns
The algorithms have been implemented in such a manner that the high-speed processing
characteristics for data in the cache of the Intel Pentium III architecture could be utilized.
The advantage of that coding is felt principally in the DPL and NG algorithms that have two
matrix–vector multiplications per iteration, in the calculation of the hivalues of equations
(5) and (7). These operations, projections of the current beam weights onto the PTV and
OAR volumes, can be up to five times faster when the outer loop of the two-loop code is
done over the matrix columns. In that way, all the arithmetic operations that can be carried
out with one column of the dose matrix F while it is in the cache memory are completed
before the processing unit calls the next column from RAM. The speed improvement is most
importantin large problems,because matrix–vectormultiplicationsbecomedominantoverall
othercalculations. The optimizationtimes reportedbelow correspondto the inversionprocess
carried out by a single Pentium III Xeon 1GHz processor with a 256 Kbyte cache, exclusive
of dose matrix calculation and disk I/O. The complete dose matrices are in RAM during
inversion. Note that the backprojectionoperation of the errors in equations (5) and (7), which
is a multiplication by the transpose of the matrix F, is naturally done in the favourable order
for the loops.
Although the dose matrices have been calculated in single precision floating point
arithmetic, the algorithmic calculations have been carried out mostly in double precision
in order to avoid possible effects due to small differences between large numbers or ratios
betweennearlyidenticalnumbers. Theincreaseincomputationtimefordoubleprecisionwith
the Pentium III Xeon architecture is a few percent at worst for the problems solved.
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Comparative behaviour of the DPL algorithm 2645
Figure 1. Three-dimensional view of the phantom used for the first set of optimization tests. A
U-shaped tumour partially surrounds a cylindrical ‘spinal cord’ in a 20-cm diameter water-
equivalent phantom.
4. Cases studied
4.1. Mathematical phantom
Figure 1 shows a 9-plane U-shaped lesion with a ‘spinal cord’ OAR. The water-equivalent
phantom consists of a 20 cm diameter cylinder in the centre of 100 × 100 voxel planes,
each voxel being of dimensions 0.25 × 0.25 × 0.5 cm. The PTV and OAR regions are
approximately in the centre of the cylinder. Nine equally spaced beams or ports spanning 2π
were used for the optimization,each with pencil beams of 0.5× 0.5 cm nominalcross-section
at the entrance plane of the cylindrical water phantom. The dose delivered per unit fluence of
each beamlet to the phantom voxels was calculated using a public domain program from the
UniversityofWisconsinthatincludesall scatteringterms. Thecalculateddoseswerearranged
into a dose matrix containingas many columns as beamlets and as many rows as voxels are in
the PTV and OAR. The number of pencil beams involved in the optimization was 1002 and
the numbers of voxels were 4266 in the PTV and 1008 in the OAR. Figure 2 shows the DVHs
resulting from the PTV-only optimization and the desired OAR DVH.
4.2. Patient cases
ThecurrentversionofthedevelopmentalsoftwarefortheBrainSCAN(2001)softwarepackage
hasbeenusedtogeneratedosematricesforpatientcases intheformatneededforthe inversion
procedures. After an optimization, the resulting beam fluences have been placed back into
BrainSCAN to verify, examine and document the results. The DVHs and dose distributions
shownhereincludetheverificationstepinBrainSCAN,butnoleafsequencingortransmission
effects have been included. Two cases will be presentedhere, both fromUCLA, one is a brain
tumour and the other is a prostate case.
4.2.1. Cavernous sinus meningioma.
meningioma adjacent to the brain stem. The defined OARs were the brain stem, the left optic
Figure 3 shows the central plane of a cavernous sinus
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2646 J Llacer et al
Figure 2. DVHs resulting from the PTV-only optimization of the water phantom of figure 1. Four
points defining the desired OAR for the full optimization are also shown.
Figure 3. Plane containing the isocentre (cross) of a cavernous sinus meningioma case showing
two of the specified OARs (left optic nerve and brain stem) and the PTV.
nerve and the optic chiasm, the latter not visible in the plane of the figure. It is not possible
to define a single entry angle that can treat the full tumour without impacting an OAR. The
case was initially studied as a 14-beam conformal therapy case. An IMRS case with seven
non-coplanar beams has been devised for the tests. Table 1 gives the gantry and table angles
correspondingto the seven beams.
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Comparative behaviour of the DPL algorithm 2647
Table 1. Table and gantry angles for meningioma case.
Beam number Table angle (deg.) Gantry angle (deg.)
1
2
3
4
5
6
7
90
86
59
0
0
0
284
96
50
88
271
68
128
282
Figure 4. DVHs resulting from the PTV-only optimization of the cavernous sinus meningioma
case (thick lines) and points used to define the desired DVHs for the OARs (thin lines).
ThenumberofbeamletsselectedbytheBrainSCANsoftwarewas1105,ofapproximately
0.2×0.3cmcross-sectioneach. Thenumbersofvoxelswere1737inthePTV and5383inthe
OARs. After initial tests with the DPL software, a set of desired OAR DVHs was prescribed
as leading to a PTV DVH that would have less than 1% of the PTV voxels receiving less
than 90% dose. The DVHs resulting from a PTV-only optimization are shown in figure 4.
The points used to define the desired OAR DVHs are also shown in that figure joined by thin
broken lines.
4.2.2. Prostate.
which the prostate and the complete seminal vesicles are to be treated. The issues of whether
the location of the vesicles can be known with sufficient precision at treatment time and/or
the medical desirability for such a plan does not enter into consideration here. Figures 5(a)
and (b) show two different planes of the problem, one including the prostate and the other the
middle section of the vesicles. The femoral heads, the posterior wall of the bladder and the
anterior wall of the rectum have been designated as OARs. Eleven coplanar beams have been
specifiedwith a single isocentre. Table 2 shows the gantryangles forthe beams, with the table
angle remaining at 0 degrees.
This case has been prepared as an example of a difficult treatment plan in
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2648 J Llacer et al
Figure 5. Two planes of the prostate case showing the outlines of the four OARs (two femur heads
and bladder and rectum walls) and the PTV (a) at the level of the prostate and (b) at the level of
the seminal vesicles.
Table 2. Gantry angles for prostate case, table angle = 0.0.
Beam numberGantry angle (deg.)
1
2
3
4
5
6
7
8
9
10
11
210
230
260
280
310
0
50
80
100
130
150
The number of beams is rather high, but four of the beams have been added to a more
conventional plan for the main purpose of assisting in the irradiation of the vesicles, laterally
and partly through the femoral heads (beams 3, 4, 8 and 9). The addition of these four
Page 13
Comparative behaviour of the DPL algorithm 2649
Figure 6. DVHs resulting from the PTV-only optimization of the prostate case (thick lines) and
points used to define the desired DVHs for the OARS in the complete optimization (thin lines).
beams has brought out some interesting differences among the algorithms, as discussed in
section 5.3. The numberof beamlets is 4674,of approximatecross-section of 2 × 3 mm. The
numbers of voxels were 1260 in the PTV and 7950 in the OARs. Figure 6 shows the results
of the PTV-only optimization, along with the desired OAR DVHs shown by squares joined by
broken lines. The desired DVHs for the two femoral heads are identical and so are those for
the bladder and rectum walls. The principal aim was to reduce the doses in the bladder and
rectum walls significantly without underdosing the PTV by more than 1 or 2% of its volume
receiving less than 90% dose. No PTV volume was to receive less than 85% dose. In initial
tests, the demands for low dose at the high end of OAR DVHs of figure 6 were less strict than
shown in the figure. After successively becoming stricter with these demands, the desired
OAR DVHs shown in the figure were arrived at as being conditions in which the different
algorithms started to show differences in the optimization dose distributions.
5. Results
Of the many data resulting from the optimizations, the following set will be presented:
(a) PTV and OAR DVHs.
(b) Appearance of optimizations, looking for edges, hot spots, etc including some isodose
lines.
(c) Some beam profiles describing specific effects and the Fluence Map Complexity (FMC)
values.
(d) Number of iterations and optimization time. The times measured correspond to the
PTV-only optimization plus the full PTV and OAR optimizations.
When optimization results are very similar to each other, the dose distribution for only one of
the results will be shown.
5.1. Results with the mathematical phantom
In order to observe the behaviour of the algorithms, it appears useful to report first the results
without filtering, followed by filtered results.
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2650 J Llacer et al
Figure 7. Dose distributions resulting from the unfiltered optimizations of the mathematical
phantom, central plane. (a) DPL1 method, (b) ASA method. Results from the DPL2, CG and
NG methods are very similar to those of the DPL1. Phantom outlines are shown in broken lines,
isodose lines for 35, 55, 75 and 95% are shown in solid lines.
5.1.1. Unfiltered results.
0.0005, except for the DPL2, where a threshold of 0.00075 was sufficient to obtain the same
results as in the DPL1, and for the ASA algorithm, in which the threshold was decreased
to 0.0001 in an attempt to obtain better results. The DVHs from the DPL1 and DPL2 are
indistinguishablefrom each other and there are no substantial differences among the different
algorithms, except for the ASA results. DVHs show relatively small differences from those
of the filtered results, below. In all appearance, more iterations were needed for a better
optimization with the ASA, but both the PTV and OAR DVHs were oscillating at the point
where the algorithm stopped. Internal parameters of the ASA method were changed within
a reasonable range without being able to improve the results. They were finally left with the
values used in a large number of tests that have often been more successful. It appears to be
in the nature of the stochastic process that sometimes it will work better than others.
The dose distributions in the phantom are very similar in all the algorithms, except
the ASA. For that reason, only the distributions for the DPL1 and ASA will be shown here.
Figure7(a)showsthecentralplaneofthephantomwiththeoutlinesofthePTVandOARshown
inbrokenlines, andthe35,55,75and95%isodoselinesresultingfromtheDPL1optimization
in solid lines. Figure 7(b) shows the corresponding results for the ASA optimization. The
latter are not particularly good, with more distortion in the dose distributions than in the other
methods.
The beam profiles for port 0 (beamlets entering the phantom from below in figure 7)
are shown in figure 8. The FMC is indicated to the right of each image. The grey scale is
normalized to the maximum of all the profiles shown. The complex appearance of the ASA
results, with higher FMC, is typical of ASA unfiltered results. The high FMC of the NG
results is not due to excessive complexity in the interior of the beam maps, but to a few high
beam weights in the periphery of some maps, not visible in the port shown.
Table 3 gives optimization information for each of the five results.
The reported results correspond to a convergence threshold of
Page 15
Comparative behaviour of the DPL algorithm 2651
Figure 8.
optimizations. The FMC for each optimization is also shown. See text for an interpretation
of the high FMC of the NG results.
Beamlet weight maps for the 0 degree angle resulting from different unfiltered
Table 3. Optimization results for the mathematical phantom, unfiltered.
Method PTV-only iterations Full optimization iterationsTotal inversion time (min)
DPL1
DPL2
ASA
CG
NG
50
25
50
25
50
177
118
81
182
146
1.11
0.72
0.96
1.58
1.02
The NG-inversion time can be decreased in this problem by increasing the relaxation
factor γ. One has to be careful, however,as the procedurecan diverge,particularlyat the later
stages of the optimization, if that parameter is made too large.
5.1.2. Filtered results.
understood. The value of the filter parameter α that gives acceptable results for clinical use
has been established for the DPL to be between0.03 and 0.05with its internal normalizations.
For the other algorithms a value of α has been selected to bringthe FMC of the corresponding
optimizations to approximately the same number as in the DPL, when it has been possible.
The DVHs and dose distributions for the two DPL algorithms are virtually identical and
only one set of data will be shown for them. The resulting DVHs are shown in figure 9. There
has been some loss in quality with respect to the unfiltered results, as expected from filtering
and the ASA results for the OAR are still noticeably worse than in the other algorithms.
The dose distribution in the central plane of the phantom is shown in figure 10 for the
ASAmethod,whichis nowverysimilartothedistributionobtainedfromtheotheralgorithms.
Solid isodose lines for 35, 55, 75 and 95% are shown. The outlines of the PTV and OAR
are shown in broken lines. The beam profiles for port 0 are shown in figure 11, with the grey
scale normalized to the maximum of all the profiles shown. The four non-stochastic beam
profiles are quite similar to each other and the ASA profiles are substantially smoother than
before filtering but still quite complex. The FMC for the NG method is substantially higher
than in the other methods. Again, this is not due to lack of filtering in the interior of the beam
maps, but rather to some very high beam weights in the periphery of some maps. An increase
In inverse problems, the need for regularization or filtering is well
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2652 J Llacer et al
Figure 9. DVHs resulting from filtered optimizations of the mathematical phantom.
Figure 10. Dose distribution resulting from the filtered optimization of the mathematical phantom
given in the central plane. Results for all the methods studies are very similar and only those for
the ASA are presented. Phantom outlines are shown in broken lines and, isodose lines for 35, 55,
75 and 95% are shown in solid lines.
in filtering strength inside the loop would result in instability. Table 4 gives details of the
optimizations.
5.2. Results with patient cases
The results with patient cases will be given for optimizations with filtering, following the
guidelines discussed at the beginning of section 5.1.2.
Page 17
Comparative behaviour of the DPL algorithm2653
Figure 11.
optimizations. The FMC for each optimization is also shown. See text for an interpretation
of the high FMC of the NG results.
Beamlet weight maps for the 0 degree angle resulting from different filtered
Table 4. Optimization results for the mathematical phantom, filtered.
Method PTV-only iterations Full optimization iterationsTotal inversion time (min)
DPL1
DPL2
ASA
CG
NG
50
25
50
25
50
151
92
77
144
78
1.03
0.58
0.95
1.26
0.65
5.2.1.
algorithmstested,exceptfortheASAthatappearstoneedmoreiterations. Thatalgorithmwas
stopped at a point where there were some oscillations in the DVHs without any improvement
in the results. The DVHs are shown in figure 12.
Thereare minorbut noticeabledifferencesin the dose distributions achievedby the tested
algorithms, particularly in some relatively hot-spot configurations, with the results of the NG
case appearing somewhat under-filtered. The results from the DPL1 and DPL2 algorithms
are virtually identical to each other and the results of all the algorithms are quite similar to
each other. For that reason, only the dose distribution for the DPL1 optimization is shown in
figure 13, corresponding to the central plane of figure 3. Figures 14(a) and (b) show some
relative hot spots created at a plane 33 mm superior to the central plane by the DPL and the
NG algorithms. Seven isodose levels have been shown from 40 to 100%. The reasons for the
existenceof the hotter spot in figure 14(b)are discussed in detail in section 5.3.
Figure 15 shows the fluence maps generated by the four different algorithms for Beam
No. 1 in table 1, directed towards the tumour from the superior part of the cranium. The
avoidance of the optic chiasm is evident in all cases. Table 5 shows the optimization data for
the five methods.
It should be noted that the CG and NG methods, although they share the same cost
function, reach the final results by quite different paths. The NG, in particular, emphasizes
convergence of those beam weights that contribute lower dose to the target and this method
has not done as well for this particular example as it did with the mathematical phantom. The
Cavernous sinus meningioma.
The resulting DVHs are very similar for all the
Page 18
2654 J Llacer et al
Figure 12. DVHs resulting from optimization of the cavernous sinus meningioma case.
Figure 13. Dose distribution resulting from optimization of the cavernous sinus meningioma case
by the DPL1 method shown in the same plane as in figure 3. Seven isodose levels are shown
in patches of different colours from 40 to 100% dose. Results for the other methods are not
significantly different from those of the DPL1.
ASA optimization was stopped manually because of a small oscillation in the results without
further apparent improvement in the solution. The existence of some relatively hot beamlets
in the periphery of the NG-beam maps has contributed to the relatively high FMC result in
table 5. One of those beamlets can be seen in figure 15.
5.2.2. Prostate.
identical results for all the algorithms. For the more demanding desired OAR DVHs shown
in figure 6 some significant differences were observed. The resulting DVHs are shown in
figure 16. The curves for the DPL1 and DPL2 algorithms are almost identical and only those
for the DPL1 have been shown. The ASA results are very close to the DPL ones. The
CG results are visibly worse than those for the DPL. The CG procedure was ‘stuck’ at the
optimization results shown, with more iterations not changing the value of the cost function
in the first six significant figures. The NG results suffer from some overdosingin the PTV, not
improvingwithmoreiterations. Figure17showsthedosedistributionintheplaneoffigure5(a)
LessstrictdemandsontheOARDVHsthanshowninfigure6yieldedalmost
Page 19
Comparative behaviour of the DPL algorithm 2655
Figure 14. Some hot spots shown at a plane 33 mm superior to that of figure 13. See text for
interpretation.
Table 5. Optimization data for the meningioma case.
Method PTV-only iterations Full optimization iterations Total inversion time (min) Fluence map complexity
DPL1
DPL2
ASA
CG
NG
100
50
100
50
30
267
152
224
170
367
2.36
1.12
3.56
2.08
3.26
0.0048
0.0050
0.0048
0.0049
0.0069
for the ASA results and figure 18 shows the distributions in the plane of figure 5(b) for the CG
method. Differences among all the algorithms at those planes are minimal.
The beam maps corresponding to Beam No. 9 in table 2, one of the four beams added to
facilitate the treatment of the vesicles, are shown in figure 19. Each of the differentsections is
normalized to its own maximum (darkest). Optimization data are given in table 6.
The CG method shows a substantially longer optimization time than the other methods.
Thereare two principalfactors that affect inversiontime in the CG method: (1) Beam weights
becoming negative at the end of a line minimization step. When that happens, the weights
are corrected as indicated in section 2.2 and the CG action has to be restarted. The corrected
weights cannot be used to continue an ongoing calculation of conjugate gradients, as that
leads to instability. Without the fast convergence of the CG, the algorithm becomes a simple
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2656 J Llacer et al
Figure 15. Beamlet fluence maps for a port directly superior to the tumour for the cavernous sinus
meningioma case Beam 1 in table 1. The sparing of the optic chiasm is clear in all cases. Note an
outlying hot beamlet in the case of the NG optimization.
Figure 16. DVHs resulting from optimizations of the prostate case.
gradientmethodwith lineminimization. (2)Performanceofthelineminimizationalgorithms.
The one used here is probably as fast as it can be. As discussed above, it gives an exact value
xminin one single pass. The PTV-only optimization of the prostate case does not result in
any beams becoming negative during most of the calculation, 40 iterations are sufficient for
good convergenceand they are finished in 1.0 min. The subsequent full optimization leads to
negative beams immediately after starting the 115 iterations, which then take 20.11 minutes
to complete. A careful analysis of the factors leading to negative beams in the CG method
Page 21
Comparative behaviour of the DPL algorithm 2657
Figure 17. Results of optimization for the prostate case at the same plane as in figure 5(a) for the
ASA algorithm. The results for the other methods are not significantly different. Seven isodose
levels are shown in patches of different colours, from 40 to 100% dose, with the same colour scale
as in figure 13.
Figure 18. Results similar to those of figure 17, for the plane shown in figure 5(b).
Table 6. Optimization data for the prostate case.
Method PTV-only iterationsFull optimization iterations Total inversion time (min) Fluence map complexity
DPL1
DPL2
ASA
CG
NG
80
40
80
40
50
122
86
106
115
329
6.646
4.3
13.7
21.11
16.61
0.0025
0.0028
0.0030
0.0029
0.0130
has been carried out. In the dynamic implementation used in the current tests and with full
scatteringtermsin the dosematrix, it is possibleto avoidnegativebeamsonlybynot requiring
too much from the optimization. For example, in the case of the mathematical phantom of
section 4.1,requestinga maximumOAR doseof 80% and setting a seconddesired DVH point
to 50% dose at 40% volume will lead to very fast convergenceby avoiding the appearance of
negativebeams during the solution. The FMC for the NG algorithmis, again, higher than that
of the other methods because of peripheral hot beamlets.
5.3. Hot beamlets
Beamweightswithhigherintensitythandesiredcanresultfromanoptimization(hotbeamlets)
and may pose a problem in the normal tissues near the entrance to the patient, or elsewhere.
They occur in two distinct situations.
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2658 J Llacer et al
Figure 19. Beamlet fluence maps for Beam 9 of Table 2, a lateral port added to facilitate the
irradiation of the seminal vesicles. DPL1and DPL2results were practically identical. Each map is
normalized to its maximum. The high fluences at the bottom of the maps correspond to the upper
section of the seminal vesicles.
1. At entry or exit edges of the PTV. Pencil beams that contribute to PTV dose with their
penumbra may end an optimization with high weights so that that penumbra brings up
the delivered dose to the PTV, and
2. When a beamlet or beamlets from a particular port have a direct view of a portion of the
PTV, unobstructedby OARs, while beamlets fromotherports have to pass throughOARs
to reach that same PTV portion.
In the first case, filtering as described above corrects the problem to a large extent, although
the NG algorithm could not be filtered sufficiently inside the loop to achieve more desirable
results. In the second case filtering only diminishes the high fluxes slightly because the main
likelihood or cost terms of the target function dominate the solution. It is with respect to the
secondcase that a significant differencebetween the DPL and the gradient-basedmethodshas
been observed. It will be shown that the DPL is a more effective parameter estimator than the
other methods but this effectiveness may have to be ‘degraded’for clinical use. This example
corresponds to the prostate case described above, but with desired OAR DVHs that are less
strict than those of figure 6.
Consider figures 20(a) and (b), in which the top plane of the seminal vesicles is shown
for the DPL and the NG results, respectively. This plane is 9 mm superior to the one shown
in figure 18. The intensities of the beamlets involved appear at the bottom of the maps in
figure 19, as these maps are inverted with respect to the patient’s anatomy. The four lateral
beams have practically taken over the irradiation of the top of the vesicles but this effect is
substantially more pronouncedin the DPL case than in the others. The dose near the entrance
of the lower left beams in the figures is above 95% for the DPL and just above 75% for the
NG method. The corresponding result for the ASA is just above 55% and barely above 75%
for the CG. One can then ask the question whether the DPL result, which would be medically
Page 23
Comparative behaviour of the DPL algorithm 2659
Figure 20. Optimization results at the top plane of the seminal vesicles for the DPL methods
showing a high entrance dose (above 95%) for the lower left beam and for the NG optimization
with the high point of the entrance dose just above 75%. See text for interpretation.
unacceptable,is anartifact ofthat algorithmorthegradientalgorithmswouldeventuallyreach
the same result if allowed the continue their optimizations.
In order to find an answer to that question, extended optimizations with the gradient
algorithms were made to a stopping point with a convergence parameter 20 times smaller
than in the original calculation. The NG algorithm had no difficulty with small gradients and
reached results similar to those of the DPL (approximately 100% near the entrance to the
patient). TheASA didnotgo veryfardueto fluctuationsinthe results ofsuccessiveiterations.
Decreasing the size of the test fluences allowed the algorithm to go part of the way. The
CG also went a long way towards the DPL results, but the very small gradients slowed down
further convergence.
What the above results indicate is that a solution with the ‘hot beamlets’ has a lower
quadraticcost than one without them and, mathematically,is a better solution to the minimum
least squares problem. Although the DPL does not minimize the quadratic cost exactly, as
discussed in section 2.1, it easily finds mathematically optimum beam weights under the
criterion that take a very long processing time for the quadratic cost optimizations to find.
Investigation of a more ‘homogeneous’ medical case has also been carried out. In that
case, the PTV consists of a single tumour of roughly spherical shape in the brain, as opposed
to the prostate and the vesicles that have very different and complex geometry. In that case,
the gradient methods are nearly as effective in bringingup the hot beamlets as the DPL. Also,
in the more demanding OAR conditions of figure 6, both the DPL and the NG algorithms
exhibit similar hot beamlets.
This mathematical characteristic of the DPL may be undesirable in some cases. In the
BrainSCAN (2001) software, a user can apply hard constraints to the maximum fluences
Page 24
2660 J Llacer et al
allowed in any beam by limiting the beamlet values to some factor above what they would
have in a conformal-therapy plan. This strategy solves the problem of hot beamlets with
necessarily some loss in the quality of the resulting DVHs.
6. Conclusions
For the purpose of optimization in the specific conditions expressed in items 1 through 4 of
the introduction, this study indicates the following.
(a) TheDPL1methodworksflawlesslywithoutanyadjustablealgorithm-specificparameters,
althoughtheremaybemoreofaneedtoconstrainthebeamweightstoavoid‘hotbeamlets’
than in other algorithms. Changing the OAR weight parameters βiin a range of values
from 0.1 to 10.0, the resulting OAR DVHs adhere progressively more to the desired
values, with the PTV suffering from higher under-dosing in the upper left-hand section
of the DVH as the OAR weights increase.
(b) TheDPL2methodworksalmostidenticallytotheDPL1methodinapproximately60%of
theoptimizationtime. Adjustmentoftheaccelerationexponentnneedstobeunderstoodin
more detail, particularlywhen some OARs are assigned high weights in the optimization.
As indicated above, when those values are set above approximately 5.0, the exponent n
has to be reduced for stability.
(c) The ASA leads to beam maps that appear more complex than those of the non-stochastic
algorithms and may, therefore, be less desirable for delivery by MLCs. It may lead to
non-uniform dose distributions if used unfiltered. Optimization times are rather long.
Although OAR weights below 1.0 lead to lower adherence to the desired DVHs than
shown in this paper, increasing those weights above 1.0 does not result in substantially
higher adherence. The reason for this behaviour is not well understood at this time.
(d) The CG method is slower than the other methods in large optimization cases because of
the need to operate as a simple-gradient method with line minimization. It also appears
that, in demandingcases, the solution may reach a point in which further iterations do not
lead to an improvementof the optimization. Changingthe OAR weights between 0.1 and
10 leads to the expected results, as in the DPL1 case.
(e) TheNG methodmaybecomecumbersomeif a robustfilter inside the optimizationloopis
desired. As indicated by Bortfeld (personal communication), that is not a problem in the
implementation of this algorithm in Heidelberg. They stratify the beam weights into five
levels after the optimization and use a median filter ‘a posteriori’, leading to satisfactory
results (Kessen et al 2000). The observed higher values of the FMC due to the existence
of hot beamlets in the peripheryof the beam maps can certainly be avoided in practice by
either of two methods: removalfrom the optimizationof peripheral beamlets that deposit
very little energy in the PTV voxels or use of a filter (like the median filter) after the
optimization that will remove those high values from the beam maps. The NG method
appears to lead to less than optimum solutions in a very demanding case. Optimization
times can be relatively slow, depending on the problem. Decreasing the OAR weights
below 1.0 leads to the expected results, but using values larger than 1.0 can lead to strong
instability. If βivalues larger than 1.0 are needed, the values of the relaxation parameter
γ in equation (7) have to be reduced from those obtained by the process described in
section 3.2.
ItshouldbepointedoutthattheDPL1algorithmistheonlyoneofthosetestedthathasassured
convergenceto a unique solution without having to set intermediate negative beam weights to
zeroduringtheiterativeprocedure. OnepossibledisadvantageoftheDPLmethodologyisthat,
Page 25
Comparative behaviour of the DPL algorithm 2661
perhaps in the future, cost functions other than quadratic may become important. Gradient
methodswill be ableto handlethemas longas the‘targetparameter’(doseat the presenttime)
is linear with respect to beam intensities. Biological parameters, being extremely non-linear,
will have to be handledby optimizationmethods suitable for non-linearparameterestimation.
It is not clear whether setting negative intermediate results to zero during the iterations
of the gradient methods is important or not, since they can yield excellent results, but the
statement can be made that, with approximately 5 to 10% of the beam weights becoming
negativein the CG and NG methodsin thelatter part ofthe iterativeprocess, the results cannot
be precisely minimum least squares solutions, as the theories for those methods require the
existence of negative beam weights.
Use of the DPL2 in a clinical setting will require more extensive testing and development
to avoid the possibility of instability in the solution if the acceleration parameter is set too
high for specific values of OAR weights. The stochastic ASA method appears to suffer when
compared to the deterministic methods in that there is no way of knowing what is happening
during an optimization, there are many internal parameters to adjust and one cannot predict
whattheresultofchangingthemwillbe. TheDPLmethodisprotectedbyUSPatent5,602,892.
A patent for the ASA method has been applied for.
Acknowledgments
The authors thank C S Chui and S V Spirou for their assistance in setting up their conjugate
gradient method in the environment set out by the first author of this paper and correcting
early mistakes and/or misconceptions. Also, thanks are due to T Bortfeld for his assistance
in setting up the Newton gradient method and bringing it up to his standards. The work of
J Llacer has been supported, in part, by a Small Business Innovation Research Grant of the
National Cancer Institute, No CA76808.
Appendix
It has been mentioned in the introduction that a simulated annealing algorithm based on the
patent by the Nomos Corporation is basically out of contention for the purposes of this paper.
In this appendixan implementationof that algorithm(NM) will be discussed briefly and some
comparative results will be shown.
The NM algorithm uses the infrastructure devised for the ASA algorithm, including
avoidance of over-constraining the PTV and the dynamic character of the cost function. The
cost function of the NM algorithm is based on the desired DVHs: for the PTV it is a straight
vertical line at 100% dose and for the OARs they are specified by the different user-supplied
points, joined by straight lines. The desired DVHs are sampled in a number of vertical points
and quadratic or ‘absolute value’ distance cost functions are generated. It has been found that
the DVHs have to be formed with a relatively large number of bins and the sampling has to
be frequent if best results are to be obtained. For the problems tested, DVHs with 800 bins
have been used, sampled at 400 points. The quadratic cost function between the desired dose
at each of the sampling points and the actual dose at the current iteration has been found to
yield somewhat better results than an ‘absolute value’ function and it has been used. The
NM formulationcannotincludefilteringinside the optimizationloopand, therefore,its results
have to be compared to unfiltered results with the other algorithms.
Figure 21 shows the dose distribution obtained by the NM algorithm for the same plane
ofthe mathematicalphantomshownin figure7(b),to whichit canbe compared. Optimization
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2662J Llacer et al
Figure 21. Dose distributions resulting from NM optimization, central plane, to be compared to
figure 7. Phantom outlines are shown in broken lines and isodose lines for 35, 55, 75 and 95% are
shown in solid lines.
Figure 22. Dose distributions for the same plane as figure 17, carried out by the unfiltered NM
method.
time was 11.38min, comparedto 0.96min forthe ASA. Figure 22shows the dose distribution
obtained by the NM algorithm for the same plane of figure 5(a), showing no substantial
differences with the corresponding results by other algorithms. The NM iterative procedure
was stopped at iteration 300 although it had not reached the specified convergencepoint after
182.5 min of calculation. This figure can be compared to 19.65 min for the unfiltered ASA at
the desired convergencepoint.
It must be mentioned that the computationtime of the NM algorithm can be substantially
loweredbydecreasingthenumberofbinsintheDVHsandsamplingtheDVHslessfrequently.
These changes result invariably in lower quality of the PTV DVHs, particularly in the
underdosing part of the curve.
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