Phys. Med. Biol., 1989, Vol. 34, No 3, 315-341. Printed in the UK

A two-dimensional pencil-beam algorithm for calculation of arc electron dose distributions

Kenneth R Hogstrom, Rajendra G Kurup,? Almon S Shiu and George Starkschall Department of Radiation Physics, The University of Texas M D Anderson Cancer Center, 1515 Holcombe Blvd, Houston, TX 77030, USA

Received 29 March 1988, in final form 17 August 1988

Abstract. A two-dimensional pencil-beam algorithm is presented for the calculation of arc electron dose distributions in any plane that is perpendicular to the axis of rotation. The dose distributions are calculated by modelling the arced beam as a single broad beam defined by the irradiated surface of the patient. The algorithm is two-dimensional in that the anatomical cross section of the patient and the skin collimators are assumed identical in parallel planes outside the plane of calculation. The broad beam is modelled as a collection of strip beams, each strip beam being characterised by its planar fluence, mean projected angular direction and a root-mean-square spread about the mean direction. Using these parameters, the dose distribution is calculated using pencil-beam theory. Examples of strip-beam parameters and resulting dose distributions for patient geometries are presen- ted. Features of the algorithm, which include (1) incorporation of pencil-beam theory for the calculation of dose in heterogeneous tissue, ( 2 ) run times of only about twice that of comparable-sized fixed electron fields and (3) the input requirement of only a single depth dose and four off-axis dose profiles of measured data, make the algorithm practical for clinical use.

1. Introduction

Algorithms calculating the dose distribution for arc electron beam radiotherapy typi- cally model the arced beam as a set of fixed beams at small angular spacings. Such algorithms, e.g. the algorithm by Leavitt et a1 (1985), use a set of measured dose profiles to calculate the fixed-beam dose distributions. In the present work a pencil-beam algorithm (Hogstrom 1987) for calculating the fixed-beam dose distributions was considered, as it represents the state of the art for electrons. This would have required additional algorithm development because most current pencil-beam algorithms, specifically that of Hogstrom et a1 (1984), model neither skin collimation nor the air-scattering in the large air gap between the final beam collimator and patient. Although both of these effects could be incorporated into the fixed-beam model, the arced-beam calculation would still require summing distributions at small angular spacings. Resulting calculation times would be approximately 20 times longer than a typical fixed-beam dose calculation, which, given present computing technology, is clinically undesirable.

At the onset of the present work, it was our objective to develop an algorithm that not only incorporated the accuracy of the pencil-beam models but also allowed

t Present Address: Loyola-Hines Department of Radiotherapy, Edward J Hines Jr, Veterans Administration Hospital, Hines, IL 60141, USA.

0031-9155/89/030315+27$0?.50 6 1989 IOP Publishing Ltd 315

316 K R Hogstrom et a1

computer run times comparable to those for a fixed electron field. We also desired the requirement of minimal input data to perform the calculation because of the difficulty in acquiring large data sets for arced electron beams. Consequently, a pencil-beam algorithm has been developed for arc electron therapy that treats the arced beam as a single broad beam defined by the irradiated patient surface within the skin collimation. In its implementation the two-dimensional (ZD) heterogeneity correction is employed, whereby the anatomy in parallel planes outside the plane of calculation is assumed to have the same cross section as that in the plane of calculation. As is subsequently shown, pencil beams can then be summed to form strip beams. Each strip beam can be described by an electron planar fluence and an angular distribution. The angular distribution is modelled as Gaussian, with the same mean projected angle and root-mean-square ( R M S ) spread about the mean projected angle as the actual distribution. This approximation is the same as that used in the moments method of Storchi and Huizenga (1985). The dose distribution for each strip beam is then calculated using the theory of Hogstrom et a1 (1981). The present paper derives the theory supporting such an algorithm and provides sufficient detail to allow one to implement the algorithm in a treatment-planning computer.

2. Theory

2.1. Description of model

In this section the equations required to calculate the dose distributions in a patient undergoing arc electron therapy are derived. The current method does not simulate the moving beam by summing multiple fixed-beam dose distributions; rather, it models the arced beam as a single broad beam defined by the skin surface inside the patient collimation. The broad beam can be constructed from many pencil beams, which are summed in the Y direction to form strip beams, as illustrated in figure 1. By knowing the dose distribution from each strip beam in the plane of calculation of the patient, the dose at any point can be determined by performing a sum of the dose contributions from each strip beam.

Because the objective of this derivation was to produce a clinically useful algorithm, pragmatic assumptions that would have little influence on the resulting dose calculation have been made when required. The geometry of the treatment machine is assumed to be conceptually the same as found in figure 1. The radiation source (scatter foil or scanning magnet) is assumed to be a virtual point source rotating about the isocentric axis of the machine gantry at a fixed distance, referred to as the source-axis distance (SAD). The primary collimator is assumed opened to a fixed setting, independent of the size and shape of the secondary collimator aperture, and such that the electron planar fluence passing through the secondary collimator is independent of the X coordinate and varies slowly in the Y dimension. It is assumed that the only material between the source and the secondary collimator is the ambient atmosphere, which is assumed homogeneous; the atmosphere between the secondary collimator and patient will be the same. These assumptions are significant to the propagation of the electron beam from the source to the patient.

The dose calculations are performed in planes perpendicular to the isocentric axis; the calculation may be made in multiple planes to form a three-dimensional dose distribution. When calculating the dose distribution in a particular plane, the patient anatomy and the skin collimation in parallel, offset planes are assumed identical; i.e.,

2~ pencil-beam algorithm for electron arc 317

Source

---"-

Primary c o l h a t o r (x-ray jaws1

Secondary collimator lcerrobend insert I

. / th strlp beam

.Pencil beam

Patient (skin) collimator

Isocentrlc axis

Figure 1. Schematic view of set-up for arc electron therapy modelled by this paper. Note the three levels of collimation; the secondary collimator can be irregular in shape. The irradiated area defined by patient collimation is modelled as a single broad beam comprised of a set of pencil beams, which are summed to form strip beams for the dose calculation.

the patient is assumed to be a right prism with a constant cross section defined by the plane of calculation. Thus, the algorithm can be referred to as having a 2~ heterogeneity correction.

Also, as discussed in detail by Huizenga and Storchi (1987), the electron beams are propagated through air using the Fermi-Eyges theory of multiple Coulomb scatter- ing. The distribution of the electron pencil beams within the patients is treated according to the theory of Hogstrom et a1 (1981). As will be shown in the derivation to follow, the 2~ assumption above reduces the equation to what is equivalent to a 'strip-beam' model (Werner et a1 1982).

The dose distribution at a point P in the patient is taken to be the sum of the electron and x-ray dose components:

D ( X , Y, Z ) = De(X, Y, 2) + D y ( X , Y, Z ) (1)

where Y is the coordinate of the plane of calculation and ( X , 2 ) are the coordinates of the point P in the plane. (Throughout this paper the convention of using upper-case X , Y and Z for the fixed, patient coordinate system is used.) In the present work, the plane of calculation is assumed to be perpendicular to the isocentre, so that the Y axis is parallel to the isocentric axis.

2.2. Electron dose distribution calculated from strip beams

The electron dose distribution in the plane Y is calculated as the sum of many strip beam dose distributions:

where D: is the electron dose distribution of the ith strip, N S is the number of strip beams and (xi, zi) are the coordinates of the point P in the reference frame of the ith strip beam. (Throughout this paper lower-case x, y and z are used for the moving-beam

318 K R Hogstrom et a1

coordinate systems; both the broad electron beams coming from the gantry and the strip beams on the skin surface are considered moving beams.) The origin of each strip beam is translated and its central axis (+z) rotated with respect to the patient coordinate system, requiring the moving coordinate system defined in figure 2 .

limit

I Plane Y

Figure 2. Schematic view of the plane of calculation. The strip beams, SE,, are defined by equally spaced angular segments along the patient contour. The model can accurately calculate with or without collimation at the end of the arc; in the latter case, strip beams are summed 15” beyond the arc limit.

A cross section of the plane of calculation and the ith strip beam is illustrated in figure 2 . That part of the patient’s surface being irradiated is divided into equal angular spacings about the isocentre. The limits of the strip beams are either between the skin collimator limits if skin collimation is present, or the angular extent of radiation at the uncollimated boundaries, as denoted by the divisions along the skin surface. The contour line segment of the ith interval translated to the upper and lower skin collimator limits in Y, referred to as ysi and Y I , respectively, forms the surface defining the ith strip beam (cf figure l ) .

Within a strip beam all distribution functions are assumed independent of x. The strip beam is characterised by a planar fluence Q; its angular distribution is assumed Gaussian because that functional dependence allows the incident angular distribution to be easily convolved with the Gaussian solution of the Fermi-Eyges pencil beam theory (Hogstrom et a1 1981, Werner et a1 1982). The Gaussian distribution is charac- terised by the mean direction, (Ox) i , and the R M S spread about the mean direction (a0,Ji . These three quantities are related to the zero, first and second moments of the actual electron fluence differential in angle (Huizenga and Storchi 1985). They are determined at the central position (by angle) of the strip beam and are assumed to be constant across the x dimension of the surface of the strip beam. The planar fluence is the number of electrons per unit area crossing a plane that (1) is parallel to isocentre, ( 2 ) is perpendicular to (ey)! and (3) contains the origin of the strip beam. The beam coordinate system is defined with its origin at the surface point midway in angle between the boundaries of that strip beam and oriented as follows: + z is in the direction of the mean direction of the ith strip beam, ; + y is in the same direction as the + Y patient axis; and +x is in the direction to form an orthogonal left-handed coordinate system.

2D pencil-beam algorithm for electron arc 319

The Gaussian approximation is reasonable so long as the incident angular distribu- tion is near Gaussian in shape and its width is small enough that the small-angle approximation is reasonable. The present algorithm was developed for the application of an electron beam rotated about a patient with a radius of curvature large compared with the width of the rotated beam. This is the case for treatment of chest wall in the United States, where the radius of curvature is typically 15 cm and the width of the secondary collimator is typically 5 cm at isocentre. From geometrical considerations alone, the angular distribution at a point for such a geometry is expected to be a square pulse with a width of *9.6" (i.e. a small angle). The multiple scattering in air makes the distribution more Gaussian in shape while increasing its overall width. The shape of this angular distribution will be discussed in greater detail in a later section.

The dose distribution of the ith strip beam can now be calculated using the theory of Hogstrom et a1 (1981). First, the electron planar fluence at point P can be calculated using Fermi-Eyges theory (Eyges 1948):

where Axi is the width of the ith pencil beam projected on the plane perpendicular to the mean direction, (OX);, of the strip beam. is the RMS value of the lateral spatial distribution projected onto the X Z plane for a pencil beam originating in the ith strip, S i , and being transported to depth 2 , . is the RMS value of the lateral spatial distribution projected onto the YZ plane for a pencil beam originating at S i and being transported to depth zi. The two RMS values are different since the u in the X Z plane includes the angular dependence due to multiple Coulomb scattering and that due to the geometrical contribution from arcing, whereas the u in the Y Z plane includes only the former. The nomenclature, definition and determination of all U

values can be found in appendix A. Note that the planar fluence of the ith pencil beam at the skin surface, Q,, is defined at the coordinates (R , , 8;, Y), where R, is the radial distance from isocentre, 8, is the polar angle measured from the Z axis and Y is the coordinate along the isocentric axis.

The integral over the x variable involves only the Gaussian term and is readily calculated:

where erf is the standard error function. The integral over the y variable is calculated by realising that @;(R, , B ; , Y) is a slowly varying function in Y over distances compar- able to ;;uy,l, so that

Next, the planar fluence is converted to dose to water at the point P. Dose is the

320 K R Hogstrom et a1

product of fluence and the mass stopping power. The factor F ( z i ) is defined as the product of the conversion factors from planar fluence to fluence and from fluence to dose. This factor F is assumed to depend on the mean beam energy at depth, which is directly related to the effective depth ( z , ~ ) along the central axis of the pencil beam at z i . Therefore, the electron dose contribution at point P from the ith strip beam becomes

D:(xi , K zi) = @ ; ( x i , Y, z t )F(zef l ( z i ) ) . ( 6 ) Substituting equation (3) for planar fluence, and using the results of equations (4) and ( 5 ) :

The following two sections will discuss the derivation of the final two terms.

2.3. Determination of conversion factor from planar juence to dose

The determination of the conversion factor from planar fluence to dose is one of two factors in the algorithm that allow the algorithm to be referenced to measured beam data. In arc electron therapy the secondary collimator aperture is normally long in the direction of the isocentric axis ( Y ) and narrow in the direction perpendicular to the beam's central axis within the plane of rotation ( x axis). Its x dimension is typically 4-5 cm when projected to isocentre; the reason for this range of values has been explained by Hogstrom and Leavitt (1987). Typically, 5 cm is defined as the reference width so that patient secondary collimator widths are 5 cm in the reference plane and vary outside that plane to account for change in patient radius of curvature. The algorithm is designed to determine F from a measured depth dose for the reference field size and for a standard geometry. A standard geometry is taken to be an 85 cm source-surface distance (SSD) for a 100 cm SAD, which corresponds to a radius of curvature of 15 cm, a value typically encountered during arc therapy.

Therefore consider the depth dose for a reference field size of width WXR by length WYR, defined at isocentre. The depth dose is measured at a reference source-surface distance, SSDR. The dose in water on central axis for the reference field is related to the planar fluence according to the earlier theory of Hogstrom et a1 (1981) for stationary rectangular electron beams by

Dref( 0, 0, z ) = 1 dy 1 dx Q. ("") ' +WYRZ/Z + W X R Z / 2

-WYRZ/Z - W X R Z / ? SSDR + z

where WXRZ = [(SSDR+Z)/SAD]WXR and WYRZ= [ (SSDR+Z) /SAD]WYR are the refer- ence field dimensions projected to depth z in the water phantom, and Q. is the central-axis value of planar fluence defined at the level of the secondary collimator. In principle, Q" should be a function of (x, y ) , but its dependence can be ignored in equation (8) because of the slow variation of Q" relative to the exponential term that is peaked about the central axis. :U,,, is the R M S value of the projected lateral spatial distribution of a pencil beam originating within the secondary collimator and being

20 pencil-beam algorithm for electron arc 32 1

transported to depth z in the water phantom. Note that the corresponding RMS value in the YZ plane is equal to that in the X2 plane because of symmetry of the multiple Coulomb small-angle scattering. SCD is the source to secondary collimator distance. Evaluating the integrals in equation (8) and solving for the planar fluence-to-dose conversion factor:

It should be noted that this factor is calculated only for the reference field size and standard geometry. It is subsequently used for other field sizes and geometries so that absolute dose is always referenced to the dose units of the reference field.

2.4. Determination of planarjuence of ith strip beam

To calculate the total planar fluence of the ith strip beam, the contribution from the electron beam for each position along the arc must be accumulated. This requires first calculating the planar fluence contribution from an arbitrary gantry angle to the central axis of the ith strip beam. For this purpose, the irregular shape of the secondary collimator has been simplified to one rectangular in shape, which is illustrated in figure 3. Note that the field length, WY (defined at isocentre), is unchanged, and that the field width, wx (also defined at isocentre), is taken to be the width of the irregular collimator projected to isocentre at the Y position of the plane of calculation, Y. The gantry position is specified by the azimuthal angle, e,, as illustrated in figure 4. The planar fluence contribution from the j th gantry position to the ith strip beam is again calculated using the Fermi-Eyges theory to transport the electron beam through air, giving

@u(xi,, Y, S S D ~ ) +WX.SSD, , /ZSAD +WY.SSD, , /ZSAD

dx‘ / dY’ - W Y , S S D , , / 2 S A D

r - 1 I

l l

I l

I I I I I I

l I I I + X

1 % I

*

Plane of calculation

I

L J

I I ,, I

Figure 3. Comparison of the irregular shape of the secondary collimator (full curve) projected to the SAD

with the rectangularly shaped approximation (broken curve) used by the calculation. Note that both have the same width ( W X ) at ihe plane of calculation ( Y = -4 ) .

322 K R Hogstrom et a1

where S S D ~ is the central-axis distance of the ith strip beam and xu is the off-axis distance of the ith strip beam relative to the j th gantry position. @;(,v’) is the planar fluence at the level of the secondary collimator at off-axis position y’ and for the j th gantry position. As Q0(y ’ ) is a slowly varying function of y‘ compared with the exponential term in y in equation ( lo) , it may be removed from the integral and set equal to the value at Y so that equation (10) evaluates to

@v(Xg, Y, SSDi,)

= @{( Y ) (-) SCD ’ SSDq

xL[erf(l 2 IWX(SSDii /SAD) + X q ) +er(

x l [e r f ( ’ 2 -jWY(SSDq/SAD) + Y ) + erf(

T W X ( S S D ~ / S A D ) - X ~

2, UX.1 v‘? ‘S: ux , t

Jz 2,ux,, v9 2, UX.1

~ W Y ( S S D ~ / S A D ) - Y ) ] . ( 1 1 )

?,ax,, is the RMS value of the projected lateral spatial distribution of a pencil beam originating within the secondary collimator and being transported through air to a plane containing the origin of the ith strip beam and perpendicular to the central axis of the j th gantry position.

The quantity Qv is the planar fluence (number of electrons/cm2) passing through a plane perpendicular to the central axis of the jth gantry position at the origin of the ith strip beam. What is desired is the planar fluence passing through a plane perpen- dicular to the central axis of the ith strip beam (figure 2). Therefore, Dij is multiplied by cos(( e x ) i - 0,) to approximate the appropriate planar fluence. Summing over gantry positions, the total planar fluence of the ith strip beam becomes

N

Q,( ei, R ~ , Y ) = C Qi j (xq , Y, S S D ~ ) - e,) j = I

where the continuum of arc positions has been approximated by N equally spaced gantry angles and ( is the mean angle of the ith strip beam in the plane of rotation. If the beam is assumed to have constant intensity during the angular rotation, then the total planar fluence at x = 0, y = Y in the rotating coordinate system of the beam is given by

Qo( Y ) = @{( Y ) N . (13 )

Combining equations ( 1 1-13),

x l [e r f ( 2 $ W X ( S S D ~ / S A D ) + xij ~ W X ( S S D ~ / S A D ) - xd

Jz ‘S: Ux,1 ) + erf ( v9 2, %,I ) l

2.5. Resulting strip-beam dose distribution

The resulting dose distribution for the ith strip beam can now be determined by

ZD pencil-beam algorithm for electron arc 323

substitution of equations (9) and (14) into equation (7), giving D ; ( x , , Y, zi) = W,

x @o( V / @ O

x l [ e r f ( )+e r ( )] , 2 v5 s:ux,t Jz t'ux,l A x i / 2 + x i A X , / 2 - xi

where the second term is the off-axis factor in y , the third term is the depth-dose dependence and the fourth and fifth terms are the x and y scatter dependence of the electron transport through the patient from the skin surface to the depth of calculation. The first term, the strip-beam weight, W,, is given by

x 1 [ erf( +WX(SSD,~/SAD) + x q ) +er(' ~ W X ( S S D ~ ~ / S A D ) - x q

2 Jz 2, ux.1 Jz 2, ux,t ) l where in the summation, the first term is the inverse-square dependence, the second term the factor converting one planar fluence to another and the third and fourth terms are the x and y scatter dependence of the electron transport through air from the collimator to the origin of the strip beam.

Equation (15) can now be summed in equation ( 2 ) to calculate the electron dose contribution at any point. Evaluation of equations (15) and (16) requires that the mean direction for the ith strip beam, (&),, and the RMS about the mean direction for the ith strip beam, [ ( u ; , ) , ] " ~ , be known. The latter term is used to determine ~,'u,,, (see appendix A.4).

2.6. Angular parameters of the ith strip beam

The angular parameters for the ith strip beam can be determined from the first and second moments of the angular distribution at the midpoint of the strip beam. The R M S angular spread is related to the first two moments by

(UZB,)i=((ex-e,)')i=(ef;),-(e.~)f. (17) The angular distribution of the electrons reaching the point of interest is a sum of the angular distribution from each of the broad beam contributions at the different gantry locations. Referring to the geometry in figure 4,

where P v ( x i j , y ) is the probability density of an electron from the fixed beam at the j th gantry position reaching the point of calculation defined by off-axis distance, x,,, and having angle y with respect to the central axis of the fixed beam. Since A 0 represents the angular width of the strip beam at a distance R, from isocentre,

324 K R Hogstrom et a1

Figure 4. Schematic view of geometry within plane of calculation for contribution of beam at jth arc position to ith strip beam. Electrons are at position x,, and are assumed to be angled at an angle y with respect to the central axis of the central axis of the beam at the jth gantry position.

R,AB cos( ei - e,) represents the width of the strip beam projected onto the perpendicular line from the point of calculation to the central axis of the fixed beam at angle 0,. In the limit of small A@, the product ofthis factor with PIJ(xij, y ) d y represents the number of electrons in the strip beam with a projected angle in the xz plane between y and y + dy, having arrived from the fixed beam at the jth gantry position. Substituting y = 0, - B,, equation (18) reduces to

The function PIJ(xiJ , y ) can be determined from the Fermi-Eyges theory of multiple Coulomb scattering. Our derivation of Pl,(xl , , y ) , not presented in this paper, gives the same result as equation ( 7 ) of Huizenga and Storchi (1987), which in our notation is

Z D pencil-beam algorithm for electron arc 325

where SSD, = S A D - R, cos( 8, - e i ) , as shown in figure 4, and is given by

where is the RMS projected angular spread in the XZ plane due to air scatter between the origin of the electron beam and the secondary collimator, : , u ~ , , ~ is the RMS projected angular spread in the XZ plane due to air scatter between the collimator and S S D ~ ~ and ? , u . ~ , ~ is the RMS projected spatial spread in the XZ plane due to air scatter between the collimator and the S S D , ~ . Evaluation of these quantities (see appendix AS) gives

where SCD is the source to collimator distance, Z D ~ = S S D ~ ~ - SCD is the air gap between the collimator and patient SSD and d82/dZ(ai, is the linear angular scattering power of air. It should be noted that using equations (22a-c) that

-.

The x dependence in equation (20) has the standard error function dependence, with the arguments given by

+ ( ;,".l ] ] SSDy j Z D ,

where

Again using equations (22a-c), equation (25) reduces to

and equation (24) reduces to

326 K R Hogstrom et a1

Substitution of equations (26) and (27) into equation (20) allows evaluation of the Pl l (x i j , y ) , which is required for evaluation of the angular moments of equation (19). In equation (18) the integral over the angular variable y is approximated by numerically integrating between the limits of * 2 . 5 ( ~ , ~ ) ~ around the mean value of y , which equals xi,/SSDij according to equation (20).

2.7. y of-axis factor

The off-axis factor @a( Y)/@, in equation (15) is assumed equal to the y off-axis ratio measured at the depth of maximum dose for the fixed beam. The field size used for this measurement should be the reference width, WXR, by the maximum length of the secondary collimator. The maximum length of the secondary collimator should be great enough so that its penumbra lies outside the maximum off-axis distance I Y / to be used clinically. It would be convenient if this maximum length were also selected as the reference length, WYR, of equation (8) so that the y off-axis ratio data could be measured in conjunction with the depth-dose curve and output of the reference field size. Therefore, the off-axis factor is equated to the measured off-axis ratio, OAR, by the expression

@ d Y ) / @ O = O A R K S ~ W Y R ( O , Y, dm,,), (28)

where dm,, is the depth of maximum dose on central axis for the field size WXR by WYR at isocentre.

2.8. Collimator width correction for conservation of electrons

One weakness of the fixed-beam pencil-beam algorithm using the Fermi-Eyges theory of multiple Coulomb scattering is the inability to predict the off-axis ratios accurately below 20%. This is because the Fermi-Eyges theory is a small-angle scattering theory and it neglects large-angle scattering arising from the electron-electron interaction. For fixed-beam therapy, these inaccuracies are insignificant. However, for arc electron therapy the dose in the tails is important because of its large contribution to dose at a point in the arced beam. This significance is a direct result of (1) a much greater fraction of dose lying in the tails of off-axis dose profile for the narrow arc field than for the broad fixed fields and ( 2 ) the dose at any point being proportional to the integral dose beneath a profile. Therefore, to conserve integral dose in the plane of rotation, the collimator width is modified by a correction factor, C , so that the field widths for all calculations are modified by

WX& = wx( 1 + C). (29)

Illustrated in figure 5, the calculated electron profile is made slightly broader in order that the integral dose of the measured and calculated off-axis profiles are equal. It is recommended that C be determined at the depth of maximum central-axis dose for the reference field width. Expressed mathematically, r X I-, I-, dx O A R ; ; ~ X ~ & ~ ~ ( X , 0, dm,,) = dx O A R G ~ & ~ , W Y ~ ( X , 0, dmaX). (30)

The superscripts indicate whether OAR is measured (mea,) or calculated(ca1c) and whether it is the electron component (e), the gamma-ray component ( y ) , or the total

20 pencil-beam algorithm for electron arc

1OOr

327

I 0 1 2 3 4 5 6 7

Off-axis distance k m )

Figure 5. Comparison of measured (-, W X R = 5 .15 cm) with calculated (- - -, WXR,*= 5.25 cm) off-axis profile for the reference field, 15 MeV beam at a depth of 2.25 cm in a water phantom at an SSD of 85 cm. The width of the calculated beam was increased 2'10, from 5.15 cm to 5.25 cm, in order for the curves to have equal area. The shaded areas represent the differences between the profiles.

(t). The subscripts indicate the field size for the calculation. The calculated OAR is given by

where S C T ~ , , ~ is evaluated at z = dm,,. The value of C in equation (29) is varied until equation (30) holds to within 0.1%. Generally, C should be positive and only a few per cent in magnitude. The value of C obtained in this manner is currently used for other field widths wx and all depths. The measured OAR is for the electron component of the beam only; in other words, the x-ray component must be subtracted from the measured beam profile:

(32)

where fy is the fraction of dose due to the x-ray component. Determination of the x-ray dose components is discussed in the next section.

2.9. X-ray dose component

The x-ray dose component of the total dose must be modelled in a reasonably accurate manner, as the focusing of this component at and near isocentre generates x-ray dose components substantially greater than those of fixed beams. The x-ray dose component of equation (1) is calculated in a conventional manner by summing the fixed-beam

328 K R Hogstrom et a1

dose distribution for a set of discrete angles, i.e. N

D J X , y, 2 ) = c PJx’ , y, 27, (33) j = l

where Py is the dose at the point of calculation arising from the jth arc position and (X‘, z’) are the coordinates in the beam reference system of ( X , 2 ) in the patient system. The x-ray dose at the jth arc position is modelled by

P i (x ’ , Y, z’) = N”D;*(O, 0, z’)oAR,(x’, 0, Z ‘ ) O A R ~ ( O , Y, 2 0 , (34) where the dependence of the off-axis ratios has been assumed to be separable. The central-axis term (labelled CA) is correct for the radiological path length and beam divergence, given by

where Dref(O, 0, zo) is the value of the fixed-beam central-axis percentage depth dose measured in water for the reference field (WXR by WYR) and for the reference SSDR,

at a depth zo, selected to be just beyond the practical range (i.e. no electron dose component present). The term in parentheses corrects for beam divergence to the point of calculation at a distance SAD - R, + E‘ from the source and the exponential term attenuates the x-ray dose component beyond the practical range. The off-axis ratios of equation (34) are assumed to be constant along fan-lines emanating from the virtual electron source so that

OAR,(X’, 0, z’) = SSDR + zo

SAD - Ri + Z ”

where the measured off-axis ratios are measured at depth zo and along the major beam axes.

Note that the arc limits for the x-ray dose are the same as those of equation (12), the actual limits of the arc rotation, not those defined by the skin collimation. The effect of skin collimation is ignored as the collimation is assumed to generate as many x-rays in stopping the electrons as those attenuated by it.

2.10. Beam parameters

The fundamental dosimetry data required at a particular beam energy are (1) a central-axis depth dose, (2) major-axes beam profiles at dm,, and (3) major-axes beam profiles just beyond the practical range for a single-field set-up. These measurements should be made in a water phantom at an SSD typically encountered in arc electron therapy. For example, on a 100 cm SAD machine, the reference SSD (SSDR) of 85 cm is recommended as it corresponds to a typical radius of curvature of 15 cm. These measurements should be made using the reference field size, typically 5 cm wide by 40 cm long (maximum length) at the 100 cm SAD. As discussed by Hogstrom and Leavitt (1987), 5 cm is a typical standard width.

2~ pencil-beam algorithm for electron arc 329

The percentage depth-dose curve is the fundamental measured quantity used in the calculation, similar to that required by the fixed-beam algorithm of Hogstrom et a1 (1981). Measured depth-dose data will be normalised to the maximum dose on central axis. The dose output ( c G ~ / M u ) at dm,, should be carefully measured as it is required in the calculation of treatment monitor units ( M U ) . All doses calculated by the algorithm are normalised relative to the dose at this point, regardless of arc angle, collimator width or radius of curvature.

The initial most probable energy is determined from the depth-dose curve in accordance with ICRU 35 (1984). The depth dose should be measured from the surface to depths at least 11 cm beyond the practical range for determination of the attenuation length of the x-ray beam, which is calculated from a semilogarithmic plot of the relative dose against depth in the bremsstrahlung tail (with the dose fall-off due to beam divergence removed). The attenuation length is extracted from the measured half-value layer (HVL) by p = 0 . 6 9 3 1 ~ ~ ~ .

All measured off-axis profiles are normalised to their value on central axis to form off-axis ratios. The off-axis profile measured at dm,, along the major axis in the plane of rotation is normally measured well outside the penumbra of the beam so as to provide data for an accurate determination of integral dose. Typically 5-10 cm outside the geometric beam edge is sufficient. The off-axis profile measurement at dm,, perpen- dicular to the plane of rotation need only be measured just past the penumbra, as only those data well inside the field edge should ever be required for the off-axis weighting factor.

The off-axis profiles measured along the major axis of the beam for the x-ray dose component are measured at a depth beyond the maximum electron range, R,,, (ICRU 1984). Typically, a depth to the nearest 0.5 cm just beyond R,,, is selected. These profiles should be measured from the central axis to a point corresponding to at least the 10% OAR value in the plane of rotation; this will typically be just outside the projection of the x-ray jaws. In the direction perpendicular to the plane of rotation the profile should be measured to just outside the projection of the x-ray jaws if possible.

The machine geometry required for the calculation comprises (1) the source-axis distance (SAD), (2) the source-collimator distance (SCD) and (3) the air gap between the downstream edge of the collimator and the SAD, denoted by the symbol Z D ~ . These quantities are the same as required by the fixed pencil-beam algorithm (Hogstrom et a1 1981, Hogstrom 1987). It is recommended that the SAD be determined from the same virtual source position as determined for a large field size (> 15 x 15 cm) for the fixed-beam geometry. If the virtual source position is measured, the recommendations of Hogstrom (1987), Meyer et a1 (1984) and Schroder-Babo (1983) should be followed. The value for Z D ~ requires only a physical measurement of that distance, and SCD is given by the difference SAD “ Z D ~ .

As seen above, the algorithm requires only a modest amount of beam data in order to calculate dose distributions for a wide range of conditions. These data and the described theory allow calculation of a patient-dose distribution for an arbitrary arc angle, radius of curvature and width of secondary collimator.

3. Computer implementation

The arc electron-beam algorithm described in this paper has been coded in VAX FORTRAN onto a VAX-based treatment-planning system using standard principles of

330 K R Hogstrom et al

structured programming and modular design. The executable module for this program occupies approximately 90 kbyte of memory. Variables require approximately 100 kbyte, with the largest fraction of this used by the matrices of CT numbers and dose values described below.

Doses are calculated on a Cartesian grid of equally spaced points along both directions in a transverse plane. The grid extends to the maximum patient contour in both directions, with points spaced approximately 3 mm apart. The point spacing is adjusted so that the number of points does not exceed 101 in either direction. For a particular strip beam, the position of each point in the patient-dose grid is transformed into the coordinate system of that strip beam as defined in figure 4. The dose contribution from that strip beam is then calculated at each point. The dose to any point on the dose grid is calculated by accumulating the dose contribution from all strip beams to that point. Dose distributions for each strip beam are calculated on a rectangular grid in the strip beam's coordinate system with the same spacing as the patient-dose grid.

The strip beams are considered to originate from the patient surface with a minimum angular width of 1.5", which corresponds to a strip-beam width of approximately 4 mm for a point on the skin surface 15 cm from isocentre. The angular width is adjusted so that the maximum number of strip beams in a single arced beam is limited to 101. In the presence of skin collimation, the strip beams are positioned between the skin collimators in such a way that the center of a strip beam at the boundary of the arc field, as defined by the skin collimator, is assumed to originate at a point one-half of an angular strip-beam width inside the skin collimation. If skin collimation is absent, the strip beams are calculated 15" beyond the arc for radii of curvature typically greater than 12 cm. For smaller radii (e.g. skull) it is necessary to increase this angle to ensure that contributions from all electrons are included.

Patient CT information in the plane of calculation is required; from these data, CT

numbers at grid points along the central axis of the strip-beam coordinate system are obtained by interpolation. The machine, electron energy, projected width of the secondary electron collimator at the isocentre, the superior and inferior limits of skin collimation, the angular limits of the arc and the angular limits of the skin collimation in the plane of calculation are required as beam input.

The program requires approximately 2 min of calculational time on a MicroVAX I1 (Digital Equipment Corp., Maynard, MA, USA) for a cylindrical water phantom of radius 15 cm, irradiated by a 10 MeV electron beam over an arc angle of 90" and a surface restricted to 60" by the skin collimation. The presence of heterogeneities (e.g. lung) increases by approximately 25% the calculation time. Approximately three- quarters of this time is spent in the actual calculation of the electron arc dose.

The electron arc program calculates doses to points on a Cartesian grid in a set of transverse patient planes. This information is written onto a file for further processing, including summation of doses from multiple arcs, display of dose distributions on a graphics display device and plotting.

4. Results and discussion

In this section selected key results of calculations in the arc electron dose algorithm are presented. Those unique to this new application of pencil-beam theory for arc electron therapy are presented, in particular the normalisation of beam width, strip- beam parameters and dose distributions.

ZD pencil-beam algorithm for electron arc 331

4.1. Collimator width correction

As discussed in Q 2.8 and expressed in equation (29), a correction factor, C, is defined to ensure proper dose to a point by accounting for the underestimate of dose in the tail of the fixed-beam profiles. In the present algorithm, C is determined from an off-axis profile in the X Z plane of the beam using equations (30-32). In figure 5, the measured off-axis profile for the electron dose component has been compared with that calculated with a correction factor. Note that the area under the calculated curve equals that under the measured curve. In this determination, it is imperative that the physical width of the collimator be known to 1'10 or better; otherwise, that error would be accounted for by the correction factor, which would subsequently lead to the incorrect calculation of the arced-beam dose distribution for a field width properly input. The measured data in figure 5 were taken using a brass collimator with machined edges having a width of 5.15 cm at the 100 cm SAD. The value of C for the 15 MeV beam in figure 5 is 2%; typically, the value for C should be less than 5%.

4.2. Pencil-beam parameters

The three parameters used to characterise the distribution of the electron strip beam, i.e. planar fluence, mean direction and the RMS spread about the mean direction, are required for the calculation of the strip-beam dose distribution. For demonstration purposes, these parameters have been calculated for a 10 MeV electron beam with an arc of 60". The hypothetical beam has an SAD of 100 cm and an SCD of 56 cm. The strip beams are defined along a surface with a radius of a curvature of 15 cm about isocentre. The width of the secondary collimator at isocentre is 5 cm. These values are typical of a patient and of the beams used for arc electron therapy. Figure 6 shows

e, ideg)

Figure 6. Plots of strip-beam weight, W,, and mean projected angle, (e,),, as a function of strip-beam angle, l?,, for a 10 MeV beam, 60" arc. The strip beams are defined along a cylindrical phantom with a radius of curvature of 15 cm. The secondary collimator, having a 56 cm SCD, has a field width of 5 cm at the 100 cm SAD. The broken line on the plot of (e,), against 0, represents the mean direction of the ith strip beam if pointing toward isocentre. The broken line on the plot of W, against 8, represents the fall-off in the absence of air scatter.

332 K R Hogstrom et a1

the dependence of the strip-beam weight (see equation (16)) as a function of position along the arc. This quantity represents the relative electron fluence of a strip beam on the surface of the cylinder. The beam weight is observed to be uniform 15" inside the limits of the arc. When the weight is uniform, lateral equilibrium is said to exist; in other words, the arc limits extend sufficiently so that the maximum number of electrons reach the origin of the strip beam. Near the limits of the arc, a penumbra is observed, with the electron fluence becoming insignificant ( < l % ) 15" outside the arc limit. It is interesting to note that the penumbra width, x, as defined by Van Gastern (1984), is about 19". The air scatter was made negligibly small in the code, in which case the fall-off was calculated to be linear (broken line) and x was 16", as predicted solely from geometrical arguments. The difference between this shape and that including air scatter (cf figure 6) demonstrates that both the geometry and the air scatter play significant roles in the formation of the penumbra for this geometry.

In figure 6, the mean angle (e,) of the strip beam, as calculated from equation (19), is plotted as a function of arc angle. If lateral equilibrium exists, then the mean direction of the beam is expected to be opposite that of the location of the strip beam, Bi (see figure 2), indicated by the broken line in figure 6. In the central position of the arc, where the beam weight is uniform, the calculation agrees with that expected by symmetry. Near the arc limits, the mean direction points away from the irradiated region, as equilibrium does not exist. Outside the arc limits the mean angle is nearly constant. The precise dependence, as observed in figure 6, is a complex function described by equation (19). This dependence is a function of both geometry and the position-angle correlation of multiple Coulomb scattering in air.

The final parameter describing the pencil beam is (T~, , the RMS value of the angular distribution in the X2 plane. Plotted as a function of arc angle in figure 7, m8, is observed to be constant (153 mrad) in the central portion of the arc. If there were no multiple Coulomb scattering, the angular distribution for a strip beam would be a result of geometry alone and would be a square pulse distribution with a width of 19.2" and an RMS value of 5.5" (96 mrad). Again this shape has been confirmed by

200

40

0

e, ldeg)

Figure 7. Comparison of plot of the R M S projected angular spread (U@\),, against strip-beam angle, O,, with the strip-beam weight. These values are calculated for the same beam conditions as in figure 6.

Z D pencil-beam algorithm for electron arc 333

performing a calculation after making the air scatter negligibly small. The increase to 8.7" (153 mrad) is attributed to the additional influence of multiple Coulomb scattering in air.

In order to verify the calculated value of go, and to evaluate the validity of the Gaussian approximation to the angular distribution, the angular distribution was measured at mid-arc for a 60" rotation. This was performed by using machined lead plates separated by 3 mm to form a strip beam. A film was then exposed 15 cm beyond the strip. A background film was then exposed with the slit closed to permit background subtraction due to fogging and bremsstrahlung. Repeating the experiment with a 6 mm gap gave almost identical results indicating the gap was narrow enough to be considered a strip beam, but broad enough so that scatter from the edges of the lead plates was insignificant. The resulting angular distribution is plotted in figure 8 and compared with that calculated from

N

It is unclear whether differences below the 10% relative probability are due to the modelling (e.g. large angle scatter) or to measurement artifacts resulting from scatter off the walls of the lead plate defining the strip, although other measured data indicate it is primarily the latter. The calculated curve is almost exactly Gaussian in shape with a (T value of 153 mrad, providing justification for the Gaussian approximation for the current geometry.

-30 -20 -10 0 10 20 30

Projected angle In XZ plane l deg 1

Figure 8. Comparison of the calculated (-1 with the measured (e) angular distribution of a 3 mm wide strip beam located 15 cm from isocentre at mid-arc for a 60" rotation (10 MeV, 56 cm SCD, wx = 5 cm, SAD = 100 cm).

4.3. Patient dose distributions

In the United States, the primary irradiated site using arc electron therapy is the chest wall. In figure 9, the patient anatomy is shown for a typical patient. The treatment aim is normally to treat the internal mammary chain ( I M C ) and the chest wall. Because

334 K R Hogstrom et a1

15 MeV /

Figure 9. Total dose distribution calculated for 15 MeV, I M C arced beam for a typical patient (wx =- 5 cm, SAD = 100 cm).

the IMC normally lies at a greater depth than the thickness of the chest wall, the irradiated region is normally divided into two regions, each irradiated with an arc electron beam of different energy. The dose distribution for the patient's IMC field, irradiated using a 15 MeV electron beam with an arc angle of 48", is plotted in figure 8. Note that the x-ray dose at isocentre is about 7% compared with 4.5% for the fixed beam at 15 MeV. In figure 10, the dose distribution for the chest-wall field, irradiated using a 10 MeV electron beam with an arc angle 69", is plotted. In both figures 9 and 10, the algorithm is able to (1) calculate the penumbral shape at the field edge defined by skin collimation, (2) calculate the penumbral shape at the arc limits that are abutted, (3) calculate the increased penetration of the radiation in lung and (4) calculate areas of increased and decreased dose due to sidescatter. It should be recalled that the algorithm calculates dose relative to the maximum central-axis output of the reference field size ( 5 cm x 35 cm at 100 cm SAD) at 85 cm SSD. In the present example, figures

T l

Figure 10. Total dose distribution calculated for 10 MeV, chest-wall arced beam for a typical patient (wx = 5 cm, SAD = 100 cm).

ZD pencil-beam algorifhm for electron arc 335

T

l

Figure 11. Composite dose distribution (figures 9 and 10) calculated for a typical patient undergoing electron arc therapy (after Hogstrom and Kurup 1987).

9 and 10 were normalised such that 100% equals 52.6% and 36.1% of the reference dose, respectively.

The composite dose plan for this example is shown in figure 11. Note how a simple 1.5 cm slab bolus laid on the skin surface was used, and how the electron beam energies were selected so as to increase the skin dose while sparing as much lung as possible. This example illustrates the usefulness of this algorithm in a clinical environment.

5. Summary

The formalism for a pencil-beam algorithm that calculates the dose distribution for arc electron therapy has been presented. The algorithm can be used to calculate dose for arbitrary location of isocentre, limits of arc rotation, secondary collimator shape, skin collimation and patient geometry. The algorithm can be used to calculate the dose distribution at depth and at the beam edge, both with and without skin collimation. Furthermore, it accounts for heterogeneity corrections in the same manner as the fixed beam algorithm. The algorithm is considered 2~ in that the dose distribution is calculated in a plane perpendicular to isocentre. For the dose calculation in that plane, the algorithm assumes that the patient anatomy and limits of skin collimation have the identical description on other parallel planes (i.e. the patient is a right prism with the cross section of the plane of calculation).

In addition to these fundamental characteristics, the algorithm was designed to have only slightly longer computational time than the fixed pencil-beam, ZD algorithm (Hogstrom er a1 1984) and to require only minimal beam data for implementation. The former is beneficial to the dosimetrist planning treatment and the latter allows the medical physicist to implement this algorithm more quickly into the clinic.

Calculations of strip-beam parameters have been shown to agree with those expected from geometrical arguments alone. Also, the angular distribution calculated using the transport theory presented here has been shown to agree with a measured distribution and to be sufficiently Gaussian in shape to justify the approximation of assuming the angular distribution to be Gaussian with the same first three moments as the actual

336 K R Hogstrom et a1

distribution. The ultimate test of any dose algorithm is in demonstrating its ability to agree with measured dose distributions sufficiently for clinical use. Preliminary analysis showed such agreement between measurement and calculation (Hogstrom and Kurup 1987) and will be presented in greater detail in a subsequent paper.

Acknowledgments

This project was supported in part by National Cancer Institute Grant No CA06294. The coding of the algorithm was primarily supported by a research contract with General Electric Medical Systems, resulting in this algorithm being coded onto their commercial treatment-planning system, as well as the VAX system used for develop- ment. We would like to acknowledge Michael Moyers for his contribution to the measurement of beam data required for both input into the algorithm and its verification. Also, Carl Nyerick assisted in the measurement of the strip-beam angular distributi0.x We thank Dr Marsha McNeese for her support in the clinical applications of this work.

Appendix A. Pencil-beam u s

A.1. Notation

The following convention will be used to define sigma, the root-mean-square ( R M S )

value of the distribution for the electron pencil beam of interest:

pencil S. binual source beam ( c , secondary coliirnator origin S,, ith strip beam orlpin

U secondary collimator d. drift

depth positlon t . total

of interesl parameter

In this convention the origin of the pencil beam appears as a superscript to the left of the symbol (T. After propagating through the intervening medium, the pencil-beam distribution is evaluated in a plane perpendicular to the direction of the pencil beam at its origin. The location of that plane is indicated by the subscript to the left of the symbol (T. All pencil beams considered in this work originate at either the virtual source, the level of the secondary collimator or the skin surface, and are evaluated at either the plane defined by the secondary collimator, the skin surface or depth within the patient. The first subscript to the right denotes which distribution the (T refers to: x, y , 0, or B,,. The second subscript to the right indicates what physically contributed to the (T, e.g. whether it is due to the beam drifting through a vacuum, being multiple- scattered by air, being multiple-scattered by the patient or the RMS total of all physical processes.

20 pencil-beam algorithm for electron arc 337

A.2. Sigmas for pencil beams originating in secondary collimator under arcing conditions (see equations (14, (16) and (21))

;!a),,,, ;,a,,,: RMS spread of projected lateral spatial distribution of electron pencil beam originating at the secondary collimator and propagating to the plane perpendicular to the central axis of the jth arc position and containing the origin of the ith strip beam (see figure 4).

f: ux,a : RMS spread of projected lateral spatial distribution of a monodirectional electron pencil beam propagating from the secondary collimator through air to the plane perpendicular to the central axis of the jth arc position and containing the origin of the ith strip beam (see equation (A.16)).

c , S , ax,d RMS spread of projected lateral spatial distribution due to beam diver-

gence of the electron pencil beam originating at the secondary collimator and subsequent drifting of the electrons to the plane perpendicular to the central axis of the jth arc position and containing the origin of the ith strip beam.

S c(+~,,a : R M S spread of projected angular distribution of electrons in pencil beams

originating at the plane of the secondary collimator due to air scatter between the source and the secondary collimator (see equation (A.12)).

A.3. Sigmas for pencil beams dejiningjxed beam in water phantom (see equations (9) and (15))

S UX,t : R M S spread of projected lateral spatial distribution of electrons in pencil beams originating in the secondary collimator and travelling to depth z in a water phantom at the reference source to surface distance, SSDR.

S ux,d R M S spread of projected lateral spatial distribution of electrons in pencil beams originating in the secondary collimator due to initial angular spread and its subsequent drifting to the plane of calculation at depth z.

:a.,, : R M S spread of projected lateral spatial distribution of electrons in pencil beams originating at the secondary collimator and travelling to depth z due to air scatter between the collimator and the phantom surface (see equation (A.17)).

:av,% : R M S spread of projected lateral spatial distribution of electrons in pencil beams originating at the secondary collimator and travelling to depth z due to scatter in the water between the surface and depth z.

338 K R Hogstrom et a1

A.4. Sigmas for strip beams defining arced field (see equations(7) and ( 1 5 ) )

RMS spread of projected lateral spatial distribution in the plane perpen- dicular to the plane of rotation of electrons propagating from the surface to depth zi for the ith strip beam.

RMS spread of projected lateral spatial distribution of electrons due to patient scattering from the skin surface to depth zi for the ith strip beam (see appendix A.6).

RMS spread of projected spatial distribution in the plane perpendicular to the plane of rotation of electrons drifting from the skin surface to depth zi for the ith strip beam due to beam divergence.

RMS spread of projected angular distribution in the plane perpendicular to the plane of rotation of electrons in the ith strip beam at the skin surface due to scatter (in air) between the j th source position and the skin surface of the ith strip beam which using equation (A.lO) can be approximated by:

RMS spread of projected lateral spatial distribution in the plane of rotation of electrons propagating from the surface to depth zi for the ith strip beam.

RMS spread of projected lateral spatial distribution in the plane of rotation of electrons drifting from the skin surface to depth zi for the ith strip beam due to initial beam divergence

RMS spread of projected angular distribution in the plane of rotation of electrons at the skin surface for the ith strip beam due to arcing and air scatter (see equation (17)).

A.5. In-air sigmas

In the calculations of angular moments in B 2.6, the electrons are propagated from the machine source to the strip beam origins on the patient. Equation (20), which mathematically describes the fundamental propagation, requires three multiple scatter- ing sigmas found in equation (21) and defined in appendix A.4. The first sigma, is the RMs-projected angular spread in the X Z plane due to air scatter between the electron source (e.g. scatter foil or scanning magnet) and the collimator. As pointed out by Jette and Pagnamenta ( 1 9 8 2 ) and Hogstrom and Almond ( 1 9 8 2 ) , this sigma is given as a function of the scattering moments, A , , by

c ~ g , , d S 2 = A;'" 2 A ~ ' ' / s C D + A!"/scD2 (A.lO)

where

(A.11)

ZD pencil-beam algorithm for electron arc 339

giving

(A.12)

The second sigma, ?((+B,,a, is the RMs-projected angular spread in the X Z plane due to air scatter between the collimator and the S S D ~ . In this case, according to the pencil-beam theory (Jette and Pagnamenta 1982, Hogstrom et a1 1981),

where, for this geometry, 1 de2 SSD,,

' 2 d Z air

-

A"" = - __ 1 lScD ( S S D ~ - z'li dz' (A.14)

(A.15)

The final sigma, is ?,U,,, the RMs-projected spatial spread in the XZ plane due to air scatter between the collimator and the S S D ~ . According to pencil-beam theory (Hogstrom er a1 1981)

(A. 16)

In processing the fixed-beam depth-dose data, the RMs-projected spatial spread in the X2 plane at depth Z due to air scatter between the collimator and SSDR, SuX,,,, is required. Similarly to above,

c 7 1 de21 - joSSDR-SCD

= - - 2 d z air

[ Z ' - ( S S D R - S S C D + Z ) ] ~ ~ Z '

(A.17)

In all calculations, it is necessary to input the appropriate value for not only the geometrical parameters, but also for the linear angular scattering power of air (or whatever atmosphere the beam passes through). A reasonable approximation for this quantity is $lair = 4.60p,irE;,b.78 rad' cm" (A.18)

where EO,p is the most probable incident electron beam energy in MeV and pair is the density of air, which at 20 "C and 1.013 x lo5 Pa, is 0.001 205 g cm-3 for dry air (CRC 1986). This expression approximates the ICRU 21 (1972) values for therapeutic energies near 10 MeV.

A.6. Sigmas due to multiple Coulomb scattering in the patient

The pencil-beam sigmas due to multiple Coulomb scattering in the patient, ( : ;u . ,~) , are calculated as a function of depth along the central axis of each strip beam in the plane of calculation, as illustrated in figure 4. These sigmas are calculated by processing data in a CT matrix in the plane of calculation, as originally described by Hogstrom

340 K R Hogstrom et a1

et a1 (1981). The sigmas are calculated using the methodology and recursion relations recently reviewed by Hogstrom (1987).

A.7. Sigma correction factor

In the works of Hogstrom er al (1984) it was reported that calculated dose distributions agreed better in the penumbral region if angular scattering powers in the patient were multiplied by a correction factor of 1.4, particularly for deeper depths and higher electron energies. This correction factor, referred to as FMCS (Hogstrom 1987), partially accounts for the failure of the Eyges small-angle scattering theory to model large-angle scattering. In the present work, linear angular scattering powers in the patient and water phantoms (see A.6) have been increased in value by a factor of 1.4 from that theoretically predicted.

Resume

Algorithme de calcul des distributions de dose en deux dimensions pour I’arc ClectronthCrapie, reposant sur la decomposition en pinceaux 616mentaires.

Les auteurs dkcrivent un algorithme de calcul des distributions de dose en 2 dimensions par dtcomposition en pinceaux elimentaires pour l’arothbrapie par faisceaux d’electrons dans un plan quelconque, perpen- diculaire a l’axe de rotation. Les distributions de dose sont calculkes en modelisant le faisceau mobile comme un faisckau large unique dtfini par la surface irradite du patient. On peut dire que l‘algorithme utilise est B deux dimensions dans la mesure ou la section anatomique transverse du patient et les collimateurs

la peau sont supposes identiques dans des plans paralltles situ& en dehors du plan de calcul. Le faisceau large est modtlise comme une sCrie de faisceaux en bandes, chaque faisceau tltrnentaire etant caracterise par sa fluence planaire, sa direction angulaire moyenne et l’tcart type de la dispersion par rapport B la direction moyenne. La distribution de dose est calculee en utilisant ces parametres a I’aide de la theorie du pinceau eltmentaire. Les auteurs donnent des exemples de parametres pour des faisceaux en bandes et les distributions de dose correspondant a la geometrie d‘un patient. Les caracteristiques de l’algorithme utilise qui comporte: (1) I’intkgration de la thtorie des pinceaux eltmentaires pour le calcul des doses en milieu hittrogtne, (2) des temps de calcul seulement deux fois plus longs que ceux necessaires pour des champs fixes d’klectrons, de dimensions analogues, ( 3 ) des donnees experimentales IimitCes a un seul rendement en profondeur et quatre profils de dose en dehors de l’axe, rendent cet algorithme adapti i son utilisation pour des cas cliniques.

Zusammenfassung

Ein zweidimensionaler Strahlenbiindelalgorithmus zur Berechnung spezieller Elektronendosisverteilungen.

€in zweidimensionaler Strahlenbiindelalgorithmus wird vorgestellt zur Berechnung spezieller Elektronen- dosisverteilungen in beliebigen Ebenen senkrecht zur Rotationsachse. Die Dosisverteilungen wurden berech- net, indem die Elektronenstrahlen als ein breites Strahlenbundel angenomrnen werden, begrenzt durch die bestrahlte Oberflache des Patienten. Der Algorithmus ist zweidimensional insofern, als dap der anatomische Querschnitt des Patienten und der Querschnitt der Hautkollimatoren in parallelen Ebenen auperhalb der Berechnungsebene als identisch angenommen werden. Das breite Strahlenbundel wird simuliert als eine Ansarnmlung von Streifenstrahlen, von denen jeder charakterisiert wird durch seine planare Fluenz, seine rnittleren projizierten Richtungswinkel, sowie eine gewisse Fehlerbreite urn diese Richtung. Mit Hilfe dieser Parameter wird die Dosisverteilung mit Hilfe der Strahlenbundeltheorie berechnet. Beispiele von Streifen- strahlparametern und resultierenden Dosisverteilungen f u r Lerschiedene Patientengeometrien werden vorgestellt. Eigenschaften des Algorithmus, wie ( 1 ) die Beriicksichtigung der Strahlenbiindeltheorie zur Berechnung der Dosis in inhomogenem Gewebe, ( 2 ) Rechenzeiten von nur etwa dem zweifachen ahnlich groper feststehender Elektronenfelder und dap ( 3 ) nur ein einziger Tiefendosiswert und Lier “off axis”- Dosisprofile als Vorgabe erforderlich sind, machen den Algorithmus praktisch nutzbar f u r klinische Anwen- dungen.

20 pencil-beam algorithm for electron arc 34 1

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