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'A finite size pencil beam for IMRT dose optimization'—a simpler analytical function for the

finite size pencil beam kernel

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2006 Phys. Med. Biol. 51 L13

(http://iopscience.iop.org/0031-9155/51/6/L01)

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INSTITUTE OF PHYSICS PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY

Phys. Med. Biol. 51 (2006) L13–L15 doi:10.1088/0031-9155/51/6/L01

LETTER TO THE EDITOR

‘A finite size pencil beam for IMRT doseoptimization’—a simpler analytical function for thefinite size pencil beam kernel

Hui Lin, Yican Wu and Yixue Chen

Institute of Plasma Physics, Chinese Academy of Sciences, HeFei, Anhui, China

E-mail: linh@ipp.ac.cn

Received 16 August 2005Published 1 March 2006Online at stacks.iop.org/PMB/51/L13

AbstractA simple and finite-termed analytical function for the finite size pencil beamkernel was constructed. The dose cross-profile of a semi-infinite field withfield edge at x = 0 can be well fitted by the Boltzmann function. The pencilbeam cross-profile of width 2x0 can be obtained as the difference between twosemi-infinite fields shifted by 2x0. If the profile is centred about x = 0, it canderive from P(x+x0)−P(x−x0). The penumbra influence can be taken by thepenumbra tuning factor f. The parameters A1, A2, A3, A4, f can be obtainedby fitting depth–dose curves and cross-profiles for a set of square fields. Thetwo-dimensional dose distribution F(x, y, x0, y0, A1, A2, A3, A4, f1, f2) of apencil beam of width (2x0, 2y0) is defined by multiplication of two independentone-dimensional profiles.

Dose optimization for intensity modulated radiotherapy (IMRT) using small field elements(beamlets) requires the computation of a large number of very small fields of typically a fewmm to 1 cm in size. Even Monte Carlo methods are limited in their accuracy and speed forcalculation of beamlet dose distributions. Some dose calculation methods based on analyticalkernels such as fsPB (a finite size pencil beam) were specifically designed for the purposeof beamlet-based IMRT (Jelen et al 2005; hereafter ref. 1). The paper ‘A finite size pencilbeam for IMRT dose optimization’ by Jelen et al (2005) appears to be an interesting work. Inthe paper, a self-consistent analytical function of a finite size pencil beam kernel (fPBK) hasbeen constructed based on the idea that the dose distribution of a beamlet can be interpretedas the difference between two broad beams with an incremental change of, say, the position ofone leaf. However, the self-consistent cross-profile analytical function of fPBK is relativelycomplex to a certain extent for it is based on a relatively complex dose cross-profile analyticaland infinite-term subsection function of a semi-infinite field. For simplifying calculations,the authors have to truncate the sum after the second element in accumulated terms in

0031-9155/06/060013+03$30.00 © 2006 IOP Publishing Ltd Printed in the UK L13

L14 Letter to the Editor

-10 -5 0 5 100.0

0.2

0.4

0.6

0.8

1.0

----

rel.

dose

X (mm)

data from ref.1 the fitted curve

-10 -5 0 5 100.0

0.2

0.4

0.6

0.8

1.0

P(x+x0)from Ref.1 P(x+x0)generated by Equ. 1' P(x-x0)from Ref.1 P(x-x0)generated by Equ. 1' P(x+x0)-p(x-x0)generated by Equ. 2'

X (mm)

rel.

dose

(a)

(b)

Figure 1. (a) Cross-profile P(x) of a semi-infinite field with edge at x = 0. (b) Pencil beamcross-profile of width 2x0 as the difference between two semi-infinite fields shifted by 2x0.

equation (3). And some assumptions were made, such as weight factor ω was forced to1 to construct the primary penumbra function, the off-axis head scatter and phantom scatterdistributions function (equation (5)). In fact the dose cross-profile of a semi-infinite field withfield edge at x = 0 can be well fitted by the Boltzmann function, including x < 0 and x � 0(equation (1′)). The fitted R-value is better than 0.99 (figure 1(a)). In this letter, we wouldlike to construct a more simple and finite-termed analytical function to describe the finite sizepencil beam kernel based on the same idea as Jelen et al (2005).

Equation (1′) gives the Boltzmann function fitting well the dose cross-profile of a semi-infinite field with field edge at x = 0 (figure 1(a)):

P(x) = A1 − A2

1 + e(x−A3)/A4+ A2. (1′)

And the pencil beam cross-profile of width 2x0 can be obtained as the difference betweentwo semi-infinite fields shifted by 2x0 (figure 1(b)). If the profile is centred about x = 0, it can

Letter to the Editor L15

be derived from P(x + x0) − P(x − x0) as

P(x, x0, A1, A2, A3, A4) = A1 − A2

1 + exp([(x + x0) − A3]/A4)− A1 − A2

1 + exp([(x − x0) − A3]/A4).

(2′)

This equation can be easily solved and coded for finite terms.The penumbra influence can be taken by the penumbra tuning factor f as in

De Gersem et al (2001), i.e.,

P(x, x0, A1, A2, A3, A4, f ) = A1 − A2

1 + exp([f (x + x0) − A3]/A4)

− A1 − A2

1 + exp([f (x − x0) − A3]/A4). (3′)

The parameters A1, A2, A3, A4, f can be obtained by fitting depth–dose curves and cross-profiles for a set of square fields as in ref. 1.

As to two dimensions, the dose distribution F(x, y, x0, y0, A1, A2, A3, A4, f1, f2) ofa pencil beam of width (2x0, 2y0) is defined by multiplication of two independent one-dimensional profiles as De Gersem et al (2001) suggest, i.e.,

F(x, y, x0, y0, A1, A2, A3, A4, f1, f2) ={

A1x − A2x

1 + exp([f1(x + x0) − A3x]/A4x)

− A1x − A2x

1 + exp([f1(x − x0) − A3x]/A4x)

}·{

A1y − A2y

1 + exp([f2(y + y0) − A3y]/A4y)

− A1y − A2y

1 + exp([f2(y − y0) − A3y]/A4y)

}. (4′)

P (x, x0, A1, A2, A3, A4) and F(x, y, x0, y0, A1, A2, A3, A4, f1, f2) represent the dosedistributions of the one-dimensional and the two-dimensional pencil beams, respectively.They should satisfy the same normalization condition as equations (4) and (7) in ref. 1.

References

Jelen U, Sohn M and Alber M 2005 A finite size pencil beam for IMRT dose optimization Phys. Med. Biol. 50 1747–66De Gersem W, Claus F, DeWagter C, VanDuyse B and De Neve W 2001 Leaf position optimization for step-and-shoot

IMRT Int. J. Radiat. Oncol. Biol. Phys. 51 1371–88