Shaping of arbitrary dose distributions by dynamic multileaf collimation

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Shaping of arbitrary dose distributions by dynamic multileaf collimation

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Phys. Med. Biol.,

1988, Vol. 33, N o 11, 1291-1300. Printed in the UK

Shaping of arbitrary dose distributions by dynamic multileaf collimation P Kallman, B Lind, A Eklof and A Brahme Department of Radiation Physics, The Karolinska Institute and University Box 60204, S-l04 01 Stockholm, Sweden

of Stockholm,

Received 9 February 1988, in final form 20 April 1988 Abstract. Traditionally, the shaping of non-uniform dose distributions has been performed by using wedgesor compensating filters. The advent of high resolution multileaf collimators may largely eliminate the need for material attenuators for modification of the beam. This is achieved by a new technique for the shaping of arbitrary dose distributions by dynamic motion of the collimator leaves.By employing narrow elementary slit beams that correspond to the smallest possible opening of the multileaf collimator, the optimal density of such slit beams, i.e. opening density, can be determined automatically using a newly developed inversion algorithm. The present method has two major advantages (1) internal structures ( 2 ) the inthe field can be created, controlled solely by steering the collimator leaves, openingdensitydeterminedbythealgorithmnever gives rise tounderdosage:this is important from a radiobiological point of view.

1. Introduction

The advent of high resolution multileaf collimators has opened the route to efficient and accurate electron and photon beam shaping of irregular, individual patient fields. Initially, multileaf collimators were introduced to allow moving beam conformation therapy. The shape of the beam is then controlled dynamically to conform with the projection of the target volume on the effective radiation source as it rotates around the patient (Takahashi 1965). Recently the development and optimal setting of high resolution multileaf collimators for the generation of arbitrary irregular stationary fields was describedinsomedetail(Erahme1988a). In asubsequentpaperthe application of multileaf collimators in generalised non-uniform moving beam radiation therapytechniques was described(Brahme1988b).Thepresentwork is primarily devoted to the production of arbitrary dose distributions in stationary photon beams of irregular shapesby dynamic multileaf collimation. However, the approachis equally applicable to electron, proton or neutron beams. Independent collimator jaw motions (Svensson et al 1977) can be used to produce dynamic wedge effects in uniform photon beams (Kijewsky et a1 1978). In a similar way, a multileaf collimator couldbe used in a dynamic modeto generatefairly complex dose distibutions. To achieve this a very powerful computational technique has been developed. One of the results is an iterative algorithm that determines the best way to move the individual collimator vanes in a multileaf collimator in order to generate a desired dose distibutionin the target volume. The mathematical technique has many similarities with the procedures developed previously to determine the optimal scanning pattern for an elementary photonor electron beam generating a required dose distibution in the patient (Lind and Brahme 1985, 1987). It has also been used to solve the centralinversionprobleminbrachy, and general moving beamradiation,therapy (Brahme 1988b, Brahme et a1 1988). 0031-9155/88/111291+ 10$02.50 (Q 1988 IOP Publishing Ltd

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The most elementary dose distribution that could be generated with a multileaf collimator is obtained when all leaves are closed, except for a single leaf pair which is infinitesimally opened. This collimator settingwill allow a very narrow line segment of photons (the length of which equals the leaf thickness) to leave the treatment head and produce the elementary collimated beam in the phantom. The optimal setting of the collimator is thus reduced to the problem of finding the opening density of such collimated beams which generates the desired beamprofile. This problem is analogous to those treatedin the references mentioned aboveand canbe solved using a constrained iterative inversion algorithm. This algorithm will be applied on a few simple geometric and more clinically oriented target volumes to illustrateits capability to form stationary beams of desired shapes and uniformity. 2. The elementary photon beam from a multileaf collimator Themost fundamentalor elementary photonbeamdosedistributionthatcan be produced with a multileaf collimator can be determined either by direct measurement or by calculation. Both these techniques have been employed here. The elementary 'slit' or multileaf collimated dose distribution has been calculated using the basic point spreadfunctionsformonoenergeticphotoninteractions(Ahnesjo er a1 1987). By integrating the point spread functions over the exponentially decreasing primary energy fluence from a narrow slit beam in water, the resultant dose distribution has been determined as shown in figure 1. The energy is 1.25 MeV (60Cosolid line) and 10 MeV photons, corresponding to about30 MV bremsstrahlung (broken curve). It is seen that the electron penumbra is larger in the latter case, whereas the background of scattered photons is larger at the lower energy.

1.0 '

I

l

I!

10-6

10

0

10

20

r (cm)

Figure 1. Elementary slit beams of a 1.5 m m x 12.5 mm cross section. The dose distributions pertain to a plane parallel to the direction of motion of the collimator leaves for 6oCo (full curve) and 10 MeV photons (broken curve).

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l

Figure 2. Isodose distribution for a 50 MV photon beam measured in water. Slit size 5 mm X 10 mm.

In figure 2 the longitudinal isodose diagram measured in a 5 mm x 10 mm wide 50 MV photon beam is shown. It is seenthat due to the long lateral range of the secondary electrons, the penumbra at this high photon energy is fairly large. At smaller beam sizes, the quasi-uniform central portion of the beam is reduced and so is the absorbed dose level (Dutreix et al 1965). At normal clinical field sizes the collimator scattered photons contribute negligibly to the patient dose (Nilsson andBrahme 1981) so for the present purposes they are assumed to be included as part of the incident photon spectrum. 3. The computational technique The desired dose distribution in a radiation beam is expressed here by D ( r ) ,where r is the vector, indicating a point in the patient. This dose distribution may be approximated by acontinuum of collimatedelementary slit beamseach giving thedose distribution H ( r ) , as discussed in the previous section. If the density of application of such slit beams is F ( r ) it should fulfil the following condition to be of clinical value in generating the desired dose distribution, D

I

D(*)= F(r')H(r-r')d3r'=F*H.

For simplicity it is assumed here thatH ( r ) is independent of the locationof the central axis of the slit beam, so equation (1) expresses a straight forward convolution operation (*). Because D and H are known functions, F is implicitly determined by this integral equation. If H is given by an infinitesimally thin slit beam the opening or irradiation

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density, F, will be essentially proportional to the desired incident energy fluence. In fact, equation (1) will generate exactly the desired energy fluence when H is expressed intermsoftheresultantdosedistributionforagivenincidentelementaryenergy fluence. Equation (3) below will therefore give the solution to the inverse problem of convolutiontypedoseplanningalgorithms defined by equation(1)(Ahnesjo et a1 1987, Mohan et a1 1986). An alternative approach to generating the dose distribution, D, is to scan an elementary electron pencil beam or photon bremsstrahlung beam, as discussed by Lind and Brahme (1985, 1987). This problem results in a relation which looks very much like equation (1) and the present problem could thus be solved by similar methods. Because H ( r ) is a fairly narrow dose distribution in the direction of motion of the leaves, D and F will be rather similar and a first approximation to F is obtained simply by setting

F. = D.

(2)

An improvement of the kth approximation of D is obtained by correcting Fk by the remaining difference between D and the kth approximation of D namely: Fk * H as given by Fk+,=C(Fk'Ca(D-F,*H)'}.

(3)

Thus, if Fk is too large equation (3) will reduce Fk+, and so on until equation (1) is fulfilled asaccuratelyaspossible(LindandBrahme 1987). The constants a and b can be chosen to speed up the convergence of the algorithm. To get fastest possible convergence, a should have a large value when H has a very narrow extension. The constant _b can be set to unity or smaller to maintain a high speed of convergence when F approaches the optimal distribution.In general Q and b could even be replaced by somemonotonicfunction G of theargumentinparentheses,toassure fastest convergence. The capital C is the constraint operator that sets the negative portions of its argument to zero. The merit of thepresentapproach is thatthesetting ofthecollimatorcanbe automatically calculated and adjusted to produce the desired isodose distribution in the target volume. This makes the present method much more powerful and accurate than the simple analytical technique developed previously (Brahme 1988a). However, the principal result from that study may still be applied to find the best orientation of the direction of motion of the leaves before employing the above algorithm. In the first approximation the direction of leaf motion should therefore be aligned with the direction of the smallest diameteror cross section of the target volume (Brahme 1988a). The algorithm will consider both the irregular shape of the target volume and the effect of photon and electron scatter near the radiation field edge. Furthermore, as the algorithm has the interesting property of never generating areas with underdosage (due to the positivity operator C in equation (3), Brahme er al 1988a), there is no risk of getting cold spots inside the target volume. This is an advantage from radiobiological point of view as anarrowoverdosage is generallyacceptable,whereasanequal underdosage may result in a local recurrence (Brahme 1984). To optimise the speed of calculation the width of each pixel has been chosen to equal the width of each collimator leaf (12 mm in the present study). This resolution is too crude along the directionof motion of the leaves so a non-uniform calculational grid is used throughout (figures 3-7). Generally, the resolution along the direction of leaf motions is 1.5 mm. This figure could be increased (or decreased) by a factor of two any desired number of times, thus decreasing (or increasing) the calculation time

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by a factor of about two each time. At 1.5 mm x 12 mm pixels and a 256 x 32 matrix thecalculationtimefor 30 iterations is about 10 minutes on a HP9000-350 SRX computer. 4. Results and discussion

To illustrate clearly the action of the algorithm it is first applied on the very simple uniform triangular target volume in figures 3 ( a ) and 4 ( a ) . The resultant collimator opening density with the 6oCophoton beam in figure 1 is shown in figure 3 ( b ) . The corresponding results with the 10 MeV beam (figure 4 ( b ) )are also shown. It can be seen, especially at the higher energy, that the irradiation density is fairly uniform in theinterior of the targetvolume,whereasneartheedgesa significantly increased irradiation density is required to get a uniform dosein the whole target volume (figures 3 ( c ) and 4 ( c ) ) . At the narrow corners of the triangle this effect is maximal due to the lack of scattered electrons and photons from the surroundings. The scattered dose outside the field is increased somewhat by the increased irradiation near thefield edges (figures 3( c) and 4 ( c ) )compared with a uniform irradiation of the entire target (figures

Figure 3. The desired dose distribution ( a ) is a uniform triangular field. The required opening density (h) is shown at a photon energy of 1.25 MeV. The resultant dose distribution in the target volume is presented ( d ) , is in (c). The under dosage with an ordinary uniform incident beam due to loss of scattered photons totally eliminated.

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Figure 4. The same case as in figure 3 using the 10 MeV elementary beam.

3 ( d ) and 4 ( d ) ) . However, this last technique will no longer generate a uniform dose in the target due to lack of equilibrium in the lateral scatter near the field edges. The present algorithm thus simultaneously corrects the incident beam for the scatter in the patient and determines the optimal setting of the leaf collimator for a given target volume. It is also clear that the correction in figure 3 mainly accounts for the effect of photon scatter whereas in figure 4 the scatter of high energy electrons dominates. In principle the desired opening density (e.g. F ( r ) in figure 3 ( b ) ) is a measure of the dose that should be delivered or rather the length of time the collimator should be opened locally (assuming constant accelerator dose rate). The irradiation could thus be accomplished by a moving elementary slit beam, the dwelling time of which is given by F. However, this may result in very long irradiation times because only a small area is irradiated at a time. A more efficient way to realise a desired irradiation density is therefore to divide F in a few large piece-wise monotonic sections. Such sections could be irradiated one at a time by opening the collimator over the peak value of eachsection and dynamically expanding the field size to low irradiation densityareas.Alternatively, the beamcouldstart with maximal field size foreach monotonic section and then narrowin and close the collimator on the peak dose value.

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If the incident irradiation density rate in a broad beam is non-uniform and equal to F , ( r ) (proportional to theenergy fluence rate 9 in the open beam) thelocal opening time of the collimator should be exactly equal to t ( r )= F ( r ) / F i ( r ) . In the general case this latter quantity thus determines the local opening time of the collimator. The speed of motion of the collimator leaves will therefore be proportional to the inverse value of the gradient (first derivative) of this function in the direction of motion of the leaves dr dt

-1

In figure 5 a more complex target volume including the axillary, subclavicular and supraclavicular lymph node areas for the treatmentof a mammary carcinoma is shown to illustrate the possibilities of the present technique. The regions with lymph vessels and localised nodes are given full dose. The desired dose distribution in the target volume is therefore as shown in figure 6(a).

Figure 5. Thetargetvolume(brokencurve)whenirradiatingtheregionallymphnodesformammary carcinoma. The full curve is the target volume used with multileaf collimation.

If the 6oCoslit beam from figure 1 is employed, the required opening density with the multileaf collimator, as determined with the algorithm, is given in figure 6(b). It is clear that the irradiation or opening densities have many characteristics similar to figure 3 ( b ) except that a higher opening density is required in the thin sections of the target volume. The resultant dose distribution in the patient is shown in figure 6(c) in close agreement with the desired dose distribution. The dose due to photon and electron scatter out of the field edges is again seen as a low skirt outside the target volume. A uniform incident beam will produce the dose distribution in figure 6(d), where the considerable dose falloff in the narrow sections of the target are clearly seen. In the final example a large tumour of the cervix with surrounding regional lymph nodes is the targetvolume with the desired dose distribution shown in figure 7 ( a ) (Brahme et a1 1988a). In this case, with a lower dose to the lymph nodes than to the solid central tumour mass the required opening density distribution is more complex and computer assistanceis really necessary (figure7 ( b)). The resultant dose distribution

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a1

. -. . a

Figure 6. For the desired uniform dose in figure 5 , ( a ) , both the traditional uniform beam ( d ) and the 6oCo photons ( c ) . Theresultantdosedistributionfrom presentirradiationtechniqueareappliedfor of the field, whereas the homogeneous irradiation ( d ) shows significant underdosage at the narrow parts present method ( c ) exhibits a perfectly homogeneous field.

(figure 7 ( c ) is again very close to the desired one. However, the dose distribution with an uncompensated uniform field (figure 7 ( d ) )happens in this case to be a fairly good compromise as a lower dose was desired in the peripheral parts of the target volume. Inconclusion,thedynamicmultileafcollimationtechnique is a very powerful method to generate arbitrary radiation beam dose profiles. It is also shown that the lack of electron or photon scatter equilibrium near beam edges can be compensated by anincreasedincidentfluence of primaryphotons.Thisimpliesthat most dose distributions that may be required in the target volume canbe producedby the dynamic multileaf collimation technique. Acknowledgments

The help of Pedro Andreo and Anders Ahnesjo in calculating elementary slit beam dose distributions basedon Monte Carlo generated point spread functions and financial support from Stockholm Cancer Society are gratefully acknowledged.

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Figure 7. Here the target volume is a cancer of the cervix with graded doses to the involved regional lymph nodes. The resultant dose distribution for a flat field is shown in ( d ) , and the result from dynamically moving the 6oCo slit beam according to the irradiation density in ( b ) is shown in ( c ) .

Resume Realisation de distributions de dose arbitraires par une collimation multilames dynamique. Traditionnellement, la realisation de distributions de dose non-uniformes rCsultait de I’emploi de filtres en coin ou de filtres compensateurs. La mise en oeuvre de collimateurs multilames a haute resolution peut, en grande partie, faire disparaitre le besoin d’attenuateurs pour modifier le faisceau. Cet objectif est atteint grlce B une nouvelle technique permettant de realiser des distributions de dose arbitraires par une commande dynamique des machoires du collimateur. En utilisant des faisceaux eltmentaires ttroits, en bande, correla ponderation optimale de chacun spondant B plus petite ouverture possible du collimateur multilames, des faisceaux elementaires peut ttre h a l u t e automatiquement B I’aide d’un algorithme d’inversion ricemment diveloppt. La mCthode presentee a deux avantages ( l ) , les structures internes du champ peuvent Otre crCees par la seule commande des lames du collimateur, ( 2 ) la pondtration des faisceux elementaires, determinCe par I’algorithme, ne conduit jamais a un sous-dosage, ce quiest important d’un point de vue radiobiologique.

P Kallman et a1 Zusammenfassung Formgebung willkiirlicher Dosisverteilungen durch dynamische Segmentkollimation. TraditionellwirddieFormgebungnicht-homogenerDosisverteilungen mit HilfevonKeil-oderKompensationsfiltern durchgefiihrt. Durch Verwendung von hochauflosenden Mehrsegmentkollimatoren kann die Notwendigkeit der Strahlmodifikation durch schwachende Materialien wegfallen. Dies wird erreicht zur FormgebungvonwillkiirlichenDosisverteilungen mit Hilfedynamischer durcheinneuesVerfahren Bewegung derKollimatorsegmente.DurchAnwendungvonengenSchlitzelementen,diederkleinst moglichen Offnung des Mehrsegmentkollimators entsprechen, kann die optimale Dichte solcher Schlitzstrahlen, d.h. die Offnungsdichte, automatisch bestimmt werden mit HilfeeinesneuentwickeltenInversionsalgorithmus. Die vorliegende Methode hat zwei Hauptvorteile, (1 j innere Strukturen im Feld konnen allein durch Steuerung der Kollimatorsegmente erzeugt werden, (2) die Offnungsdichte, die durch den Algorithmus bestimmt wird, fiihrt niemals zu einer Unterdosierung; dies ist vom strahlenbiologischen Standpunkt aus wichtig.

References Ahnesjo A, AndreoP and Brahme A1987 Calculation and application of point spread functions for treatment planning with high energy photon beams Acta Oncol. 26 49 Acta Radio[. Oncol. 23 379 Brahme A 1984 Dosimetry precision requirements in radiation therapy Brahme A 1988a Optimal usage of multileaf collimators in stationary beam radiation therapy Strahlentherapie 164 343 BrahmeA1988bOptimizationofstationaryandmovingbeamradiationtherapytechniques Radiother. Oncol. 12 129 Brahme A, Lind B and Kallman P 1988 Clinical possibilities with a new generation of radiation therapy equipment Proc. Int. Symp. Dosim. Radioth. IAEA-SM-298-67 (Vienna: IAEA) p 321 Dutreix J, Dutreix A and Tubiana M 1965 Electronic equilibrium and transition stages Phys. Med. Bid. 10 177 Kijewski P K, Chin L M and Bjarngard B E 1978 Wedge-shaped dose distributions by computer-controlled collimator motion Med. Phys. 5 426 Lind B and Brahme A 1985 Generation of desired dose distributions with scanned elementary beams by deconvolution method Proc. VII ICMP, Espoo, Finland p 953 Lind B and Brahme A 1987 Optimization of radiation therapy dose distributions using scanned electron and photon beams and multileaf collimators Proc. 9th Int. Conf: Comp. Rad. Ther. p 235 Mohan R, Chui C and Lidofsky L 1986 Differential pencil beam dose computation model for photons Med. Phys. 64 Nilsson B and Brahme A 1981 Contamination of high-energy photon beams by scattered photons Strahlentherapie 157 181 Svensson H,Johansson L, Larsson L G, Brahme A, Lindberg B and Reistad D 1977 A 22 MeV microtron for radiotherapy Acta Radiol. 16 145 Takahashi S 1965 Conformation radiotherapy, rotation techniques as applied to radiography and radiotherapy Acta Radiol. Suppl 242