COMPUTATIONAL MODELS used for dose distribution calculations done by iSis 3D software

Computational models

Table of contents II-1I-2I-2-1I-2-1-1I-2-1-2I-3I-4I-5I-6I-7I-8-

MODEL FOR PHOTON MODE CALCULATION .............................................. 4 Reference dose and basic dosimetric data...................................................................... 4 Dose at one Point ............................................................................................................... 4 Detailed description........................................................................................... 4 The contribution of primary radiation..................................................................................5 The contribution of scatter radiation....................................................................................5 Corrections for the external surface of the patient (Double cutting out) .................. 6 Influence of the flattening filter ....................................................................................... 7 Entrance dose...................................................................................................................... 8 Penumbra ............................................................................................................................ 8 Wedge filters ....................................................................................................................... 8 Heterogeneities.................................................................................................................... 9

IIII-1II-2II-2-1II-2-2II-2-3II-3II-3-1II-3-2II-4II-4-1II-4-2II-4-3a) b) c) II-5-

Model for Electron mode calculation........................................................... 10 Generalities........................................................................................................................ 10 Primary component ......................................................................................................... 11 Principle........................................................................................................... 11 Experimental determination of the primary ...................................................... 11 Analytical method ............................................................................................ 12 Scatter component............................................................................................................ 12 Principle........................................................................................................... 12 Analytical model .............................................................................................. 13 Penumbra .......................................................................................................................... 13 Penumbra for the total dose d 20-80 ................................................................... 13 Penumbra for the scatter s20-80 ........................................................................ 13 Penumbra due to the primary .......................................................................... 13 Obtaining the values of p 20-80 .............................................................................................13 Analytical representation of p 20-80 ......................................................................................14 Integration of the computational model.............................................................................15 Heterogeneities.................................................................................................................. 15

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INTRODUCTION The models employed by the systems dedicated to the dose calculation for radiotherapy should fulfill the following requirements: -allow calculation for all the situations encountered in clinical practice (SSD, SPD, ARC; coplanar beams or noncoplanar beams; beam modifiers); -ensure correspondence between the calculated doses and the basic dosimetric data acquired by the user (in a water phantom); -ensure compliance of the calculated doses with the real dose distribution for complex cases (involving beam modifiers, heterogeneity's); -safe margins to allow extrapolation beside the limits within which the model is validated; -fast response time, ensuring compatibility with an interactive clinical use (visual “optimization”). The above mentioned requirements have lead us to choose an intermediate solution, in between the “elementary” algorithms, which are not taking into account complex cases, and the “theoretical” ones, requiring detailed knowledge of the spectra and often performing radical simplifications or being hard to control (ICRU 1987, Goitein 1982). Our choice is towards a semiempirical approach, close to the elementary physical phenomena, where the basic data are obtained directly from the beam measurements performed by the user. This approach favors the conformity of the absolute beam calibration to the nationally and internationally recommended protocols. It results in a homogenous methodology for the calculus of the treatment time (or monitor units) as well as for the calculus of dose distribution. In order to achieve a good compromise between precision and computation time, were made available several resolutions of the calculation grid and different levels of sophistication (starting with version v2.0). It is up to the user to choose the options that best fit to the complexity of each study case. The process describe hereafter concerns mainly the calculation of the relatives dose. Concerning the calculation of monitor units, more details are given in the creation of the unit library guide as well as in the calcum software guide.

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I- MODEL FOR PHOTON MODE CALCULATION The algorithm for the calculation of dose in one point is based on the principle of separating primary and scatter radiation, proposed by Cunningham, who starts from an original idea developed by Clarkson [Clarkson 1941, Cunningham 1972]. I-1- Reference dose and basic dosimetric data All the doses are a priori referred to the dose at a reference distance and depth, zref, in a small water column, whose diameter is just as great as to ensure electronic equilibrium. This dose, P(zref), is used as normalization value (=1) and gives the “primary”. By varying the depth and keeping constant the distance source to point, we obtain the transmission curve for primary radiation, in a narrow field geometry or the Tissue - Phantom Ratio (TPR) for a zero field: P(z). The dose D(z, r) at depth z and for a circular field of radius r is given by the sum P(z) + S(z, r), where S(z, r) is the scatter at depth z and for the circular field of radius r. Computational tools make possible the calculation of the basic data P(z) and S(z, r) from Normalized depth dose (NDD) data and the ratio large/ small phantom for square fields [Drouard 1986].

I-2- Dose at one Point for an Irregular Field Shape, a Planar Surface and a Homogenous Medium, Without Beam Modifiers (Simple cutting out) I-2-1- Detailed description In a first approximation the dose D(x, y, z) at a point P of coordinates x, y, z, at a depth p is calculated by integrating over circular sectors as follows: D(p, r) = P(p) + SDS(p, ri) , where DS(p, ri) is the scattering from an angular sector of radius ri (see Figure 1).

y o x p P

z P Figure 1

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I-2-1-1- The contribution of primary radiation The contribution of primary radiation to the dose at a point P can be written as: P(x, y, z) = TAR(p,o) x I x C( x, y) x W(x, y) where: TAR(p, o) is the Tissue - Air Ratio at a depth p for a field of zero section (primary “transmission”); I is the correction factor that accounts for the inverse square (of the distance to the source) law; C(x, y) is the correction factor that accounts for the point position relative to the opening defined by the collimator and the blocks; W(x, y) is a correction factor that accounts for the presence of wedge filters. C(x, y) is calculated imagining the case of a fictive source, of infinite size , whose intensity decays exponentially with the distance from the center to the exterior. It is analyzed the part of the source seen from the calculation point along the open field profile (Wilkinson). C(x,y) is 1 at the field center (x = y = 0) and is practically 0.5 at the geometric boundary of the field and for a collimator transmission enough far from the effective beam. The shape of the dose profile is controlled by a parameter β, called “collimation constant”. β corresponds to the exponent characterizing the exponential variation of the intensity. For a high value of β is obtained a “square” profile. As the value of β decreases the profile becomes more curved. β is important for special treatments depending on the particular cases encountered; for example to account for the geometrical penumbra, the collimator opening, the distance from the collimator to the calculation point, etc. W( x, y) is simply obtained by homothety and interpolation, starting from a wedge transmission profile that is saved in the machine’s library and taking into account the possible rotation of the collimator. The necessary data for characterizing the wedges are the transmission profiles (ratio of doses obtained with and without wedge) measured for an average treatment depth. I-2-1-2- The contribution of scatter radiation The scatter contribution to the dose at a point P is the sum of the elementary scatter DSi(p, ri) corresponding to the contribution at a depth p at a point x, y within an elementary volume DVi exposed to the primary radiation Pi. In practice it is enough to apply the Clarkson method, cutting the irradiation field in circular sectors of angles less or equal with 100 , centered in the calculation point. Each elementary volume is a “piece of pie” of thickness equal with the depth of the calculation point and for which we admit that the produced scatter is proportional with the primary calculated in the field center. The scatter is :

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ai x SAR (p,ri ) 360

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where: SAR (p,ri) scatter air ratio at a depth p for a circular field of radius ri; ri and αi are the radius and the angle respectively of the considered sector. The SAR (p,ri) are calculated by interpolation starting with the tables “primary - scatter” saved in the machine’s library for 20 depths and 20 radius’ values. The SAR values from the tables are obtained from experimental data using the equation: SAR(p,r) = TAR (p,r) - TAR(p,o) To each beam quality is associated a table (identified by FA nn) which contains, apart from SAR values, the TAR vales for a field of zero diameter. Remark : In practice, for high energy photons, the concept of dose “in air” is replaced by the dose “in a Mini phantom ” (SV) because the secondary electrons’ path length cannot be neglected. This leads to replacing the notations TAR and SAR with TMPR and SMPR respectively. This issue will be discussed later when describing the tools for tables’ generation. This simple “cutting out” permits to take into account the field shape and to obtain a correct calculation of the dose, including for the region situated under the blocks and for the region external to the effective field after correcting the primary for the transmission through the corresponding attenuator. The method is accurate (if we neglect the scatter generated under the blocks) in terms that the primary fluence is uniform all over the field surface in a plane perpendicular to the beam axis. In the case of important variations or when a part of the beam is in air the double cutout method gives better results. I-3- Corrections for the external surface of the patient (Double cutting out) The simple cutout method described in the pages before assumes that all the scattering sectors have the same depth as the point P, when applied for an oblique entrance surface or an oblique beam incidence. The same applies for the case where a part of the beam is in air or for a basic implementation, the integration of the scattering accounts only for the field boundaries and not for the medium boundaries. The introduction of the double cutout (over the angle and over the radius), as seen in Figure 2, allows to account for the real scattering volume, in what concerns depth and lateral dimension. More, it is not necessary to assume a homogenous fluence in what concerns the scatter production and it can take into account the changes of the scatter due to the presence of compensators or wedge filters. y o x P

p

z Figure 2

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P

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It is not needed to generate a special S(z, r) table for the double cutting out method. However there is a special manner of using this table. We note that the double cutout allows a dissociation between elementary beams and it is suitable to “pencil beam” modeling for which the basic data are preferentially obtained via a Monte Carlo method. The calculation of the depth p is (starting with version v2.0) realized entirely in 3D for both the simple cutout model and the double cutout model. Each contour is associated with a prism of height equal to the cutout slice thickness. In order to avoid the gaps between the slices, a special algorithm was implemented, ensuring continuity of the anatomical structure between the extreme cutouts. In the calculus of p the contours are merged. It is used the concept of equivalent thickness when the line uniting the source and the point of calculation passes successively through the medium and the air (ears, auxiliary cavity). In the case of double cutout it is equally necessary to calculate the depth for all the scattering elements (for which the reference point is taken at the same distance from the source as the point P). With the aim o gain time we performed a precalculus as illustrated in Figure 3. Reconstructed Slice U

V S

S E

P

U(i), V(i) Figure 3 A grid (u, v) allows to save the SSDs and the depths for each beam [Belshi 1995]. Starting from this data, the depths for each scattering element are calculated from the difference between the sourcepoint distance and the SSD. In the particular case where this “depth” is greater than the thickness at the scatter element level, the thickness is taken into account, and an inverse square of the distances is applied to correct for the fact that the scatter element is brought at the level of the reference point. For the moment there is no correction applied for lack of backscatter [Kappas 1985]. I-4- Influence of the flattening filter In the case of accelerators, the dose profiles are strongly depending on the shape of the flattening filter employed. This is modeled by a radial function, derived from experimental measurements that allows to correct the basic profile (obtained when the radial function is not applied). The variation of the spectrum with the beam axis distance [Zefkili 1994, 1995] is taken into account (staring with version v2.1) through a correction factor obtained from profiles measured at big fields and different depths or (because the accelerators permit) from quality index measurements for asymmetric beams) [Zefkili 1996].

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I-5- Entrance dose The dose in the build- up region is perfectly taken into account by de computational model thanks to the use of the P(z) tables, of significant values for depths between 0 and zmax, where zmax is the depth of maximum dose for a very small field. The variation of the depth of maximum dose, Dmax, and of the surface dose, Ds, with the field size is correctly calculated due to the use of the S(z, r) values taking into account the electronic contamination for small z values [Rosenwald 1991]. The contamination from the accessories interposed in the beam is not actually considered. However is always possible to define directly the machine characteristics in the presence of these accessories, or to define many different machines to consider different possible configurations. I-6- Penumbra The variation of the primary dose at the margins of the collimator or of the block is described by an exponential curve, characterized by an exponent β (“collimation constant”) inversely related to the with of he modeled penumbra. For the cobalt machines, β is calculated via an empirical function based on the geometrical penumbra value at the distance considered t include a correction for the collimator opening and the distance collimator- calculation plane. The validity of this function was verified for most of the cobalt machines currently in use [Rosenwald 1982]. In the case of irregular fields is made a distinction among the different sides of the polygon delimiting the field [starting with version v2.0] such that they correspond to the principal collimator or an additional block. For the accelerators, β is practically independent of the distance to the source. However, one can observe for high energies a rapid increase of the penumbra with the depth and the field size. This increase is too fast to have been taken into account in the increase of the photon scatter contribution. It is due to the phenomena of lack of lateral electronic equilibrium [Simonian 1988, 1990]. It is accounted for through an adequate use of the tabulated values S(z,r) for very small r values, for which the electronic equilibrium is not ensured [Rosenwald 1987, Woo 1990]. Computational tools allow to achieve a good agreement with the experimental data of β and the scatter tables for small fields. I-7- Wedge filters The introduction of a wedge filter modifies mainly the primary radiation component and less the scatter component. The present model corrects globally the primary and the scatter through an effective transmission factor, calculated from experimental data. It is recommended to provide in the basic data the variation of the transmission perpendicular to the back of the wedge, measured at the reference depth. This guaranties a good representation of the dose in the filter’s presence at this depth, small differences may arise at different depths. An improvement of the method is about to be integrated (covering the general case of the compensators), taking truly into account the change of scatter (with the double cutting out) as well as the spectrum modification (filtration effect) [Castellanos 1996]. In the case of motorized or dynamic wedge, the method consist to think like "equivalent physical wedge filter" [Papathéodorou 1999]. About dynamic wedge a specific problem remain for the calculation of monitor unit that are very sensitive to the field size (according to the jaws movement way) as well as to possible asymmetries. So a special process has been implemented for distinctive case of "Varian" enhanced dynamic wedge filter [Papathéodourou 1998, Papathéodourou 1999]. March 2000

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I-8- Heterogeneities The corrections for heterogeneities are applied to the dose calculated for a water equivalent medium. They are based on the beam subtraction method [Kappas 1985, 1986] and they use as basic correction, for a heterogeneity equal or greater than the field size, the Batho generalized method [Batho 1964, ElKhatib 1986, Kappas 1986]. The beam subtraction method allows to account not only for the modifications introduced by the heterogeneities along the line connecting the source with the calculation point but also for the modifications of the scatter due to heterogeneities situated laterally relative to this point (for example mediastin between the two lungs [Kappas 1985, 1995]). The method being initially developed for paralelipipedic heterogeneities located in rectangular fields, the following approximations are made: - The modification of scatter at a point P is supposed to be the same as for a paralelipipedic heterogeneity of thickness equal to the thickness crossed by the SP line and of lateral dimensions equal to the heterogeneity lateral dimensions, and the longitudinal dimension delimited by the longitudinal dimension of the field. If the SP line does not cross the heterogeneity, its thickness is considered to be equal to the maximum thickness of the heterogeneity. A special study, “by slices” was done when the right SP intersects for more than two times the heterogeneity’ s contour. - For an irregular field, the modification of the scatter is supposed to be the same as for a rectangular field having the same dimensions as those obtained along he principal axes passing through P. - No heterogeneities correction are performed outside the beam geometrical limits (particulary in case of blocks). For the moment the beam subtraction method is applied only for “coplanar” beams, whose axes are parallel to the slices’ plane. For noncoplanar beams, the generalized Batho method is applied as if the heterogeneity is completely covered by corresponding fields. The possibility to make a “voxel by voxel” 3D correction is also available (starting with version v2.1) [Chiotti 1995]. It can be used only when the dosimetric study is based on a set of scanographic (preferably jointed) slices. The principle of this method consists in calculating the “radiological” thickness (geometrical thickness weighted by the electronic density) between the entrance surface and the calculation point. For this determination is done first a reconstruction of the voxels’ volume corresponding to the sliced volume and then the line segment uniting the entrance point with the calculation point is partitioned looking for each partition point the density of the corresponding voxel. The correction factor CF applied to the dose without heterogeneity is obtained calculating the ratios of TARs (or TPRs), recalculated from the tables of scatter for an equivalent square field and for the radiological depth and geometrical depth. CF =

(å r Z ) TAR ( å Z )

TAR

i

i

i

i

i

This method permits to avoid the case of complex heterogeneities, but it cannot really account for the modifications of the scatter beside the line uniting the source with the calculation point.

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II- MODEL FOR ELECTRON MODE CALCULATION II-1- Generalities The starting point of the method is based on the idea of A. Dutreix and J. Van de Geijn. It consists in following exactly the methodology developed by for the high energy photons (separately for primary and scatter) and adding modifications to it to account for the specificity of electrons. We cannot speak rigorously about “primary” and “scatter” in the case of electrons. In fact all electrons might be scattered via coulombian interactions. We call “primary” the electrons whose direction is practically unchanged relative to the direction of the incident electrons (assumed to come along a straight line from a point source), and “scatter” those whose direction is significantly changed. We can consider, under these conditions, that for a non zero field dimension, the dose is the sum of the contributions of primary and scattered electrons. The dose due to the primary electrons is obtained from the dose in air at a given distance (at the surface for example) applying a correction for the increasing distance (the inverse square of the distance to a fictive position of the source) and mainly for the interactions with the medium (the “primary” electrons are progressively transformed in “scattered” electrons). Hence the primary dose does not depend on the distance to the source and on the depth z of the calculation point. It is not dependent on the filed size. The dose due to the scattered “electrons” is obtained according to the Clarkson method, integrating over the field surface the contributions of the elementary sectors. For calculating it is thus necessary to use the “scatter” tables (equivalent of the scatter - air ratio , SAR, of the photons) calculated for circular fields, and which take into account the depth z of the calculation point and the radius r of the field. These tables are obtained experimentally from the difference between the total and the primary dose [Rosenwald 19]. Last, it is necessary to give a particular attention to the calculation of the penumbra [Rosenwald], which in contrast to he case of the photons, increases rapidly with the depth. The penumbra is strongly dependent on the distance from the collimator base to the skin.

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II-2- Primary component II-2-1- Principle In the following we consider that the collimator remains completely open and that the field size variation is obtained by varying the volume of the scattering medium (for example recolimating using the lead blocks situated at the surface). Dose 100 DS Increasing field size field 0 x 0 (extrapolated) DX RP

depth

Figure 4: Normalized depth dose at SPD ¥ for different field dimensions Starting from the NDD curves for a given energy and for different field dimensions, we want to obtain the extrapolated curve far a zero field dimension. The curves are corrected with the inverse square of the distance (SPD¥) and normalized at the depth of maximum for a field big enough to ensure maximum scatter on the axis (in practice it should be >15 x 15 cm 2 for any energy). This result is predicted by the theory since the dose at the surface Ds remains unchanged when the field dimension is varied (Figure 4). The extrapolated curve should decrease constantly starting from Ds (for z=0), and until Dx, corresponding to the braking radiation (for z = R p). II-2-2- Experimental determination of the primary The experimental determination is not at hand if we consider the need of very small detectors and the perturbations introduced by the collimating system. It is not very precise because of the rapid variation of the curve shape with the field dimension for field diameters in the order of the centimeters. On the other hand, it is not needed to know with a high accuracy the values used since they are going to be anyway recombined with the values of the “scatter”, reproducing the normalized depth dose for fields of finite dimensions. It is therefore acceptable to generalize the experimental data obtained by other authors in the idea to propose a general methodology, applicable to any case.

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II-2-3- Analytical method The analyze of the experimental results obtained in the Institute Gustave Roussy [G. Puel, O. Villeret] shows that for low energies ( R p

SPD is the distance for which the NDDs (Percent Depth Dose) have been determined. P(z) corresponds to the NDD for the zero field. a is a parameter who’s value is between 0 and 1. For a = 0 is found a linear relation, similar with the one obtained for low energies. For a = 1 is found a quadratic relation, with a horizontal tangent in the point z = R p. We adopted the following expressions: a=0

for Rp ≤ 3 cm

a = 0,25(Rp-3)

for 3 < Rp≤ 7 cm

a =1

for Rp>7 cm

II-3- Scatter component II-3-1- Principle As we have seen in the case of the determination of the primary component, one needs the NDD curves corrected with the inverse square of the distances for a given energy and different square or circular field dimensions. These curves are part of the basic experimental data measured when installing an accelerator1, apart from what concerns the field dimensions clinically used (for example bigger than 3x3 cm 2 ). At the beginning they are all normalized to the maximum dose for the corresponding field. Let Ds be the dose relative to the surface for the biggest field. We use the value of Ds to renormalize to 1 at the surface all the curves corresponding to a given energy. The D curves are theoretically ordered, as indicated in Figure 4, so that they do not intersect again 2. During the experimental determination of the NDDs for small fields, it should be avoided the perturbation from the penumbra region. It should be given particular attention to the beam collimator proximal to the skin 1

If the curves obtained do intersect again this may be due to the superficial contamination which increases with the field size. This phenomena will be taken into account by looking for the field for which Ds is minimum (see Chapter 4, The creation of the basic dosimetric files for the electron beams, page II-33). 2

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II-3-2- Analytical model The model proposed for the analytical representation of S is: S (z,r) = S(z,rmax) x (1 - g-r) In reality, when r increases is rapidly reached a plateau region. In practice, rmax is the radius from which the NDD is not varying any more. It is greater for a greater energy and it is always considerably smaller than the practical (extrapolated) range R p. g is a parameter which depends on the energy as well as on the depth and which permits to adjust the speed with which varies S as a function of r. The higher the g the faster the NDD is reaching the plateau. At shallow depths, there is practically no scatter and g is very high. At greater depths, g decreases to reach values between 1.5 - 2 in the vicinity of the extrapolated path. The values of g should be adjusted so that the NDD recalculated as D(z, r) = P(z, r) + S(z, r) for the fields effectively measured will correspond to the experimental values of the NDD. As an indication, the values found for the Saturne 25 at the Institute Curie are given at the pages 87-90 of this document. We can observe that in a first approximation, g depends mainly on the average energy Ez at the depth z of the considered point. g remains smaller than 5 for Ez < 12 MeV. For Ez > 12 MeV, g increases rapidly with the energy. The choice of g affects as well the dose representation in the penumbra region II-4- Penumbra Applying the Clarkson method for a square field, the basis of the scatter tables obtained from the analytical form of S( z, r), it is possible to calculate the dose profiles for each of the primary, scatter contributions or the total dose. On each of the profiles obtained this way we can measure the 20%-80% penumbra region. II-4-1- Penumbra for the total dose d20-80 The penumbra for the total dose d20-80 is experimentally determinable with the condition to have available profiles at different depths, for different field sizes. Anyhow, it varies slightly with the field size and we consider for the moment that it depends only on the depth, choosing as reference field a 10 x 10 cm 2. II-4-2- Penumbra for the scatter s20-80 The penumbra for the scatter is a function of the chosen value g, independent of other parameters. Higher is the g closer is the conformation of the S20-80 penumbra according to the values listed below. g

1,5

S20-80 (cm) 2.14

2

3

4

5

8

10

15

30

50

1.50

0.98

0.79

0.70

0.60

0.55

0.50

0.45

0.21

II-4-3- Penumbra due to the primary a) Obtaining the values of p20-80 March 2000

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The penumbra due to the primary should be such that, recombined with s 20-80, will give d20-80. We proved that the recombination of the penumbras satisfies the law: d20-80 = (p20-80)Cp . (s 20-80) Cs where Cp and Cs represent the contributions to the dose of the primary and scatter, respectively. Cp =

P( z) S( z, r ) and Cs = D( z, r ) D( z, r )

With Cp + Cs = 1. Hence we can start from d 20-80 and s20-80 and deduce the primary penumbra p 20-80, knowing Cp and Cs. From our experience it is possible to propose a satisfactory analytical model for the representation of p20-80. b) Analytical representation of p20-80

For a given collimator-skin distance, the representation of p 20-80 as a function of z results in a curve that can be modeled by to linear segments (Figure 5: Analytical representation of the primary).

Primary penumbra P20-80

m2 P0

m1 Depth Z Zl

Figure 5: Analytical representation of the primary The first linear part represents the region between z = 0 and z = z1, where z1 is close to the depth of 100% (R100). In this region, the electrons are not too much scattered and the penumbra is slowly increasing. Let m denote the inclination corresponding to this increase. The second segment, for z > z1, corresponds to a faster increase (an inclination m2), because of the more important amount of scatter at the end of the electronic path. When the collimator- skin distance varies, we observe mainly a global translation of the curve, parallel to the axis of coordinates. It is thus enough to know the penumbra width at the surface p 0 to be able to derive the curve’s shape, whatever is the collimator - skin distance. This penumbra width can be considered as linearly varying with the distance to the collimator CSD. More, it can be easily measured since at the surface S=0, where from it comes that d 20-80 = p20-80. Finally, for CSD = 0, we can neglect the scatter through the collimator hole and d20-80 = 0. It is thus enough to measure d 0 from d 20-80 to CSD = CSD0. In practice, we will choose CSD0 close to clinically useful values (in general in the order of 10 cm). Finally the primary penumbra is represented by the following function. March 2000

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p20-80 = k . CSD + m1. z p20-80 = k . CSD + m1. z1 + m2 . (z-z1)

for z ≤ z1 for z > z1

It is thus a function of four parameters: k (dimensionless), z1 (cm), m 1 and m 2 (dimensionless) . k is easily determined experimentally. z1, m1 and m2 are determined from the curve p20-80 as a function of z, at the normally used collimatorskin distance. As an indication, the values of z1, m1 and m2 found for the Saturn 25 of Institute Curie are given at the page 90 of this document. We note that m1 and m2 are constant while k and z1 can be obtained approximately by the following analytical expressions: k = 0,015 +0,25/ Rp z1 = 1 + 0,25 . Rp Using these values or these formulas can serve as a starting point for the adjustment of the parameters values for a different machine or for different energies. c) Integration of the computational model

The computational model used is that proposed by J. R. CUNNINGHAM: it consists in calculating the primary, integrating the fraction from an imaginary source of exponentially decreasing intensity, from the enter to the periphery, so as it is seen from the calculation point through the collimator opening. The coefficient b of the exponential function characterizing the source allows to “play” with the primary penumbra width. The relation between b and p 20-80 is: b x p20-80 = 2,5 cm. The calculation of the primary dose is done by the program using the value of b deduced from the expression above. II-5- Heterogeneities The method used for heterogeneities' corrections, consist simply to calculate a equivalent depth based on the summary of corrected geometrical thickness gone right through of the corresponding density and to use this depth for "primary" calculation "RTA"(z,O) and for "scatter" calculation "RDA(z,ri). This equivalent depth is obtained either by calculation of the intersections with heterogeneous structures, or voxel by voxel. We will note that, this solution is incompatible with double cutting out. Therefor the double cutting out calculation option is only authorize without heterogeneities.

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