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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 UK 2637

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2638J 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 algorithm2639

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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2640J 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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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 algorithm2643

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 algorithm2645

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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2646J 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 algorithm2647

Table 1. Table and gantry angles for meningioma case.

Beam numberTable 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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2648J 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 number Gantry 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

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Comparative behaviour of the DPL algorithm2649

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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2650J 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

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Comparative behaviour of the DPL algorithm2651

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.

MethodPTV-only iterationsFull optimization iterations Total 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