The Dose Kernels of Pencil and Differential Pencil ... - Springer Link

ISSN 0027-1349, Moscow University Physics Bulletin, 2016, Vol. 71, No. 4, pp. 431–439. © Allerton Press, Inc., 2016. Original Russian Text © V.A. Klimanov, A.N. Moiseev, M.A. Kolyvanova, V.L. Romodanov, A.P. Chernyaev, 2016, published in Vestnik Moskovskogo Universiteta, Seriya 3: Fizika, Astronomiya, 2016, No. 4, pp. 83–91.

BIOPHYSICS AND MEDICAL PHYSICS

The Dose Kernels of Pencil and Differential Pencil Photon Beams with the Spectrum of Treatment Machines with a 60Co Source in Water and Their Analytical Approximation V. A. Klimanova, b, c*, A. N. Moiseevd, M. A. Kolyvanovab, c**, V. L. Romodanove, and A. P. Chernyaevb*** a

Department of Medical Physics, National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe sh. 31, Moscow, 115409 Russia b Department of Physics, Moscow State University, Moscow, 119991 Russia c State Research Center Burnasyan Federal Medical Biophysical Center of the Federal Medical Biological Agency, ul. Zhivopisnaya 46, Moscow, 123182 Russia d Medical Rehabilitation Center, Ministry of Health of the Russian Federation, Ivan’kovskoe sh. 3, Moscow, 125367 Russia e Department of Experimental and Theoretical Physics of Nuclear Reactors, National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe sh. 31, Moscow, 115409 Russia e-mail: *[email protected], **[email protected], ***[email protected] Received October 16, 2015; in final form, May 13, 2015

Abstract—This article presents the results of calculations by the Monte-Carlo method in the EGSnrc toolkit of the spatial distributions of absorbed energy or dose kernels in water for a pencil beam or differential pencil beam (point spread function) with the spectrum of ROKUS-M treatment machines. The photon spectrum of the ROKUS-M machines was also calculated by the Monte-Carlo method. The calculated dose-kernel results were approximated separately for the radial distribution of the primary and the scattered component of dose kernels by sums of exponential functions divided by the squared radius for a differential pencil beam and by the radius for a pencil beam. This approximation makes direct implementation possible for wellknown model-based techniques for finding 3D dose distributions in external radiation therapy. A simple analytical procedure to verify approximation formulas is proposed. It is also applicable in independent checks of dose distributions along the treat beam axis, which is an important guideline of the Radiation Therapy Quality Assurance Program. Keywords: external radiation therapy, photons, ROKUS treatment machine, pencil beam, differential pencil beam, 3D treatment planning, collapsed cone convolution, pencil beam algorithm. DOI: 10.3103/S0027134916040111

INTRODUCTION Model-based techniques [3] for using dose kernels for basic photon sources have found extensive application in 3D dosimetry planning systems (DPSs) of external radiation therapy with photon beam. These kernels are the relative spatial distributions of the energy absorbed in a unit of water volume near a random point in space; they are usually calculated in advance by the Monte-Carlo method. The most popular models are the differential-pencil-beam (DPB) or point-spread function model and the pencil-beam model [2]. After [1, 2] were published kernel techniques have gradually become the most popular techniques in DPS radiation. They became especially popular with the advent of dosimetry planning and the development of new radiation procedures and technologies [3].

Their undoubted strong points are good suitability with dose calculations in a complex non-homogeneous 3D geometry and high speed of calculation. Notwithstanding the intensive development of computer technology, which sometimes makes it possible to apply stricter techniques, these methods still remain topical and interesting for specialists due to the very high volume of calculations for radiation-therapy planning [3]. The upgrade of RT equipment and technologies and the raised requirements to calculation accuracy (cumulative error for the released dose must be more than 5% [3]) have triggered the development of specialized dose kernels for use in particular RT technologies. As an example, the development of stereotactic radiation exposure and radiosurgery encouraged a number of studies that propose dose kernels for narrow beams with a circular cross section; let us take, for

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example, the phenomenological model in [4]. The emergence of the Intensity Modulated Radiation Technology (IMRT) encouraged the development of dose kernels for so-called Finite-Size Pencil Beams (FSPBs) (diverging photon beam with a small square cross section). As an example, these kernels were obtained in [5]. When it appeared that the dose-kernel methods of that time sometimes failed to provide the required accuracy, particularly, in calculations of the dose in heterogeneous media and off-normal incidence of radiation on the patient, a new algorithm called AAA (anisotropic analytical algorithm) was developed on the basis of PB model. The theoretical framework for this algorithm is found in [6, 7] and it was implemented in TPS Eclipse in 2009 [8]. The interest in improving dose-kernel techniques still remains quite strong. Advances in computer technologies and development of new RT technologies and new programs intended to solve radiation transfer problems have sometimes stimulated the reconsideration of available data on dose kernels and their identification with smaller statistical and system errors, more detailed results, and link with particular types of treatment machines, for instance, [9–11]. The dosekernel techniques described in some papers [8, 12] are not based on calculations but rather on test measurements of dose distributions in the water phantom. The subject of research in most papers was dose kernels for photons with a bremsstrahlung spectrum. In some published works, however, dose kernels for single-energy photon sources were studied. For the most detailed information in this field see [13, 14]. In these papers the Monte-Carlo method was used to calculate dose kernels for a large set of single-energy photon sources with DPB and PB geometry in an energy range of 0.1–25.0 MeV. It is somewhat difficult to make direct use of dose kernels data as numerical arrays in RT dosimetry planning. The reason is that rigorous calculation of dose values at particular points is linked with numerical calculation of multidimensional integrals (for nD space, incidence directions, and the photon-beam spectrum) [1–3]. At the same time, the dose distribution for preliminary RT dosimetry planning is calculated in several hundreds of thousands of points inside the patient in a complex nD heterogeneous geometry. These calculations are repeated by the planner several dozens of times in the TPS and hundreds and thousands of times in the optimization system by changing the number, size, and shape of radiation exposure fields, directions of beams, etc. for the purpose of defining the best radiation exposure plan. The second reason is that spatial relations of dose kernels have very high gradients [3, 15] that make it difficult to interpolate numerical data. That is the reason that dose-kernel techniques must be applied in practice with regard to such important factors as form, correctness, and conciseness of methods of representing and interpolating dose-kernel values. These factors largely determine the calculation time

and accuracy. In this regard analytical approximation equations proposed for dose kernels of DPB and PB of braking photons in [1, 2] were very useful in calculating the dose from treatment beams. In elaboration of the differential pencil beam model in [1] a convenient analytical approximation was found for the dose kernel of photon DPB and generalized for heterogeneous media (only to differentiate them by density). This form contains four empirical coefficients with their values have been chosen in [1] by the nonlinear least-squares method using the results of DPB dose-kernel calculations by the MonteCarlo method. A similar procedure was conducted in [2] for the photon pencil-beam (PB) model. In both works the empirical coefficient values are given for continuous spectra of braking radiation with energy peaks of 4, 6, 10, 15, and 24 MeV. As asserted by the authors of [1, 2], the results of dose-kernel calculations using the proposed approximation formulas satisfactorily match the original data found by the Monte-Carlo method (the errors are not cited) but for the nearest and the remote zone relative to the sources. We can conclude from graphical comparisons in [1, 2] that the discrepancies in these zones are 20–25%. These analytical approximations provided the basis for two rapid classical algorithms for calculating nD dose distributions, as elaborated by the authors of [1, 2] and referred to as collapsed cone convolution and the pencil-beam model. In these algorithms calculations are accelerated by analytical integration through the space variable. This was made possible using convenient functional forms of approximation equations for dose kernels, discretization by the angular variable in the DPB technique, radiation exposure field triangulation in the PB technique, and averaging dose kernels for the beam spectrum. The approximation formulas for single-energy photon sources in PB geometry were proposed in [13, 14] separately for the primary and the scattered component of dose kernel. Those formulas were more accurate (with a mean square error of more than 5%) than the approximation formulas from [2] but not as convenient for use in finding the dose in RT through integration by irradiated volume. The late 20th century was marked by the beginning of the broad adoption of medicinal linear electron accelerators (LEAs) as substitutes for 60Co treatment machines. At first the substitution process was quite intensive. As an example, while in the 1980s there were more than a thousand gamma machines in the United States, by 2007 their number fell to 100 in the United States and 60 in Japan. By that year the largest numbers of those machines were used in China, India, and Russia (500, 256, and 250 units, respectively) [16, 17]. However, the replacement of gamma machines has currently become less intensive: notwithstanding their weak points, they do have some advantages compared to LEAs, such as lower cost, better reliability, longer

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THE DOSE KERNELS OF PENCIL AND DIFFERENTIAL PENCIL PHOTON BEAMS

service life, and lower maintenance costs. Upgraded versions of some new models of 60Co gamma machines with specifications similar to LEAs [17] are still being manufactured. First of all, we will take note of Theratron-Equinox made by MDS Nordion (Canada) with a multileaf collimator and Teragram made by Canberra (Canada). For the necessary information on these machines see [18, 19]. In the Soviet Union the first treatment gamma machines were manufactured in the late 1960s [20]. Those were models such as AGAT-S, AGAT-R (from the works in Narva) and ROKUS (the former Ravenstvo works in St. Petersburg) [20]. After the collapse of the Soviet Union the manufacture of AGAT machines ceased and the manufacture of ROKUS (Russian for rotation converging plant) models was suspended for some time. In 2007, however, OAO Ravenstvo declared that it was ready to manufacture the ROKUS-AM computerized set with a new control system for individual orders and offered all owners of ROKUS units with an expired lifespan an upgrade of their equipment in cooperation with OOO Agat– Chernobyl’ [21]. At the same time, an equipment upgrade alone will not be enough. ROKUS sets are currently operated in conjunction with outdated TPS that do not meet contemporary requirements on dose-calculation accuracy [3]. Unfortunately, existing commercial TPS are closed and cannot be used in other systems. In this regard there is a serious need to make open investigations and developments in this area. ROKUS-M is a cobalt machine with a typical source design and beam collimation system. The sources in these machines (cylinders with D = ~20 mm and H = ~20 mm) have dimensions that are comparable with the length of the free flight of emitted photons; this is the reason that the spectrum of the radiation coming from these sources does not consist of two lines with a mean energy of 1.25 MeV (the energy values of emitted photons for a 60Co radioactive nuclide are 1.17 and 1.33 MeV) but is modified by the absorption and scattering of photons in the machine source and head. Conversely, specialized references do not provide any dose-kernel data for elementary sources of photon DPB and PB with the ROKUS-M beam spectrum. Therefore, it becomes, necessary to obtain DPB and PB photon dose kernels with a spectrum typical of ROKUS-M-type gamma machines, including their analytical approximation in a shape that is convenient for analytical integration through the space variable, while calculating the treatment beam dose. METHODS OF OBTAINING DOSE KERNELS AND THEIR ANALYTICAL APPROXIMATION ROKUS-M treatment machines are equipped with grid-type sources of fine discs or grit [22]. They are MOSCOW UNIVERSITY PHYSICS BULLETIN

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20 ROKUS head 20 Grid-type cobalt source Stainless steel capsule 750

244 304 Collimation blinds

100 × 100 Water phantom

Fig. 1. The configuration of the head of the ROKUS gamma treatment machine, modeled in MCNP4C2 (dimensions are given in mm).

placed in stainless steel cylindrical capsules with double sealing, D = 23 mm, and H = 22.5 mm. The capsule end wall serves as the work surface; its thickness is 0.6–1.2 mm. The volume density of the active part is approximately 5.0–5.9 g/cm3 [22]. The radiation that is emitted through the work surface is collimated by the primary conical collimator and then penetrates the aperture, which makes it possible to produce 2 × 2 to 20 × 20 cm2 fields at a distance of 75 mm from the source. This irradiation geometry was simulated in geometric module MCNP4C2 [23] and is shown in Fig. 1. The energy distribution of photons averaged for the 20 × 20 cm2 field was also found in MCNP4C2 by the Monte-Carlo method. The resulting spectrum is shown in Fig. 2. The photons of this spectrum have a mean energy of 1.058 MeV. The energy spectrum of the ROKUS-M beam was recorded in the EGSnrc spectra library [24]. For this spectrum the DPB dose kernel in a spherical geometry was found in the edknrc subprogram by the MonteCarlo method and the PB dose kernel in cylindrical geometry was found in the dosrznrc subprogram by the Monte-Carlo method as well. The geometry of these elementary sources and the agreed notation of the coordinates on which dose kernels depend are shown in Fig. 3. The dose-kernel calculations in [1, 2, 13, 14] were made by means of EGS4 [25] or GEWATER, which is a subprogram of the ELIZA software [26]. In this paper the calculations are made using EGSnrc, which No. 4

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f(E), group photon/source photon 0.35

(a)

(b) Ω

Ey

0.30

Impact point

0.25

Ω

Ey z

0.20

r

θ

θ

0.15 0.10

Detectors

0.05 0

0.2

0.4

0.6

0.8

1.0

1.2 1.4 E, MeV

Fig. 2. The energy distribution of the photon beam of the ROKUS-M gamma treatment machine, averaged for a 20 × 20 cm2 field. The group width in the spectrum is 40 keV.

is an improved version of EGS4. In particular, the simulation of the transport of low-energy photons and electrons has been seriously modified and the accounting of electron bonds in the atom and relaxation of atoms after Compton scattering and photoelectric absorption of photons is included. The EGSnrc kit has been tested multiple times and verified using experimental data (for instance, in [27–29]). It is currently a sort of standard reference that is used in calculations of electron and photon transport in lowenergy and medium-energy areas and with which results of calculations in other programs (MNCP, PENELOPE, GEANT4, etc.) by the Monte-Carlo method [28, 30] are compared. At the same time, EGSnrc has a user-friendly interface and is characterized by a rapid response time. To round out the picture, we will give the definition of a dose kernel for the DPB of photons according to [1]. We assume that photons with a given energy spectrum and line of motion Ω have an initial contact with the medium in point r'. Thus, the dose kernel for DPB will be defined as the relative share of photon energy absorbed by a unit of volume near random point r and referred to as KDPB (E, Ω, r', r). Because of azimuthal symmetry in the infinitely homogeneous medium this kernel depends not on energy alone but also on the distance between the contact point and calculation point r and polar angle θ measured from the photon travel direction. Since this work investigates DPB with the continuous spectrum of the ROKUS-M machine, we will denote its dose kernel as KDPB (r, θ). A dose kernel as defined in PB geometry [2] is the relative share of the energy of a unidirectional point photon (PB) source with normal incidence on the sur-

Fig. 3. The geometries of the basic photon sources and their geometric variables: (a) pencil beam; (b) differential pencil beam.

face of the semi-infinite water medium that is absorbed by a unit of volume near a point with coordinates (z, r). We will refer to this source as KPB (z, r). The calculation of KDPB (r, θ) was made in the spherical coordinate system for all the cells formed by the intersection of spherical surfaces for r values of 0.025–50.0 cm and cones for θ angle values identical to the ones found in [2] (48 equidistant values in the range of 0–180°). The calculation of KPB (z, r) was made in the cylindrical system of coordinates for the cells formed by the intersection of cylindrical surfaces for r values of 0.025–45 cm and planes for z values of 0.2–40 cm. In both geometries the statistical calculation error was ≤2%; in certain cells it was as high as 5% at large θ angles and high r values. As noted above, the authors of [1, 2] proposed to use approximation equations for dose kernels of DPB and PB photons for a number of braking radiation spectra in the range of 6–18 MV. These equations are conveniently subjected to analytical integration by irradiated volume in 3D calculations of the dose in RT. They are recorded as follows:

K DPB = ( Aθe −aθr + BΘe −bθr )/ r 2,

f or

DPB ; (1)

K PB = ( Aθe −aθr + BΘe −bθr )/ r,

for

PB ,

(2)

where A, a, B, and b are the empirical coefficients for the given photon spectrum that depend on θ for DPB and calculation point depth for PB. The exponential terms and denominators in the formulas give an approximate description of radiation attenuation due to interaction and geometry, respectively. The values of the empirical coefficients were determined by fitting with the results of Monte-Carlo calculations using the nonlinear least-squares method. Another important characteristic of the proposed equations is that the first term in the formulas approximately describes the contribution to the dose kernel by the primary component (the dose of electrons gen-


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erated by the interaction between primary radiation and water) and the second term describes the contribution of the scattered component (the dose of photons scattered in water) [1, 2]. However, the values of the errors for this representation are not given. In [1, 2] photons with the spectrum of gamma machines with 60Co and other radio nuclides were not considered. However, we developed a similar conceptual approach for photons with the beam spectrum of ROKUS-M-type machines in [31, 32]. Separate dosekernel calculations for water were made for the primary and the scattered component. When we carried out the approximation, we retained the functional form of certain terms, as in (1) and (2), increasing, however, the number of terms in the approximation equations. The comparison of the calculation by approximation formulas with the original data found by the Monte-Carlo method showed that the approximation error for both components of the DPM dose kernel decreased, when the number of terms increased to three, but then remained almost unchanged. As to the PB dose kernel, the optimal number of terms for both components is four. Our approximation formulas for water (the generalization for other light media will be published later) are:

K = K p + K s; 3

K DPB, j (r,θ ) =

∑ C (θ)e

−ki (θ)r

i

2

/r ,

for

DP B ; (3)

i =1

4

K PB, j (z, r ) =

∑ C (z)e i

−ki ( z )r

/ r,

for

PB;

i =1

where r, z, θ are the coordinates of the detection points (see Fig. 3); KDPB is the dose kernel for differential pencil beam; KPB is the dose kernel for the pencil beam; j = p (primary component or j = s (scattered component); Ci(θ), and ki(θ) and Ci(z), ki(z) are the empirical components for DPB and PB, respectively. The values of those components were found by nonlinear regression combined with random search via fitting the results of the calculations by formulas (3) with the results of the calculations by the Monte-Carlo method. The procedure that was applied to find the empirical coefficient was tested in [33]. For a more detailed description of the procedure see [31]. Of course, the increased number of terms in approximation formulas (3) compared with (1) and (2) somewhat extends the time to find dose distributions in the collapsed-cone convolution [1] and pencil-beam techniques [2]. However, this increase is insignificant (approximately 10%) because most of the calculation time in these methods is spent on the beam analysis of the object’s nD geometry. MOSCOW UNIVERSITY PHYSICS BULLETIN

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EXPERIMENTAL RESULTS The results of our dose-kernel calculations for ROKUS-M gamma treatment machines, the approximation results, and the coefficients included in the approximation formulas were described in [21, 32]. The approximation results are compared with the original data for some common cases in Fig. 4. We see that the approximation results for the primary components of both elementary sources match the MonteCarlo calculations in the significant region with a discrepancy of ±5% but for r values ≥ 0.5 cm, where the concurrence gets worse (the discrepancy increases to 20%). However, the cumulative dose-kernel value in this region is determined by the scattered component. The situation for the scattered component is opposite. The discrepancy between the approximation results and the Monte-Carlo calculation results considerably increases at low r ≤ 0.02 cm but remains acceptable over the residual range of ≤5%. The discrepancy for the scattered DPB component at low r and θ ≤ 15° is especially significant; the approximation results are two-to-three times as high as the Monte-Carlo calculation results. In this region, however, the primary component values exceed the dissipated component values by more than two orders of magnitude. Thus, the approximation error in the cumulative dose-kernel values remains within ±5%. The rigid requirements on error posed in the calculations of dose distributions from treatment beams apply to the definition of the full dose value in individual voxels. However, the design equations include dose kernels in their integral shape (definite integral by the radial variable). Thus, it will be interesting to compare integrals of DPB dose-kernel values found by the Monte-Carlo method and by approximation formula (3) for radii of different directions. The comparison is given in Fig. 5. In this case the discrepancy is below 5%. The approximation formulas and their coefficients were additionally verified by calculations in conditions typical of PB. To that end the central-axial distributions of the absorbed dose and its components were found in EGSnrc by the Monte-Carlo method in the water phantom for disc-shaped unidirectional sources with the beam spectrum of ROKUS-M gamma treatment machines with normal incidence of photons on the water phantom surface (Fig. 6). The mean divergence of treatment beams is within 3°–4°. However, this divergence is insignificant for the purpose of defining the adequacy of the approximation formulas for dose kernels. These dose distributions were found in the same geometry using the approximation formulas derived in the paper. If we apply the reciprocity principle, we will easily determine that the radial variable integral of 0–R that is derived from the product of the radial variable by the PB dose-kernel value at the specific depth in water z and multiplied by 2π is equivalent to the dose proNo. 4

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10−1

10−1

z = 0.1 cm

z = 0.1 cm

r × KPB(z, r), cm/cm3

(a) 10−2

12.0 cm

23.0 cm

23.0 cm 10

10

−3

10

−4

−3

12.0 cm

23.0 cm

10−4

10−5

z = 0.1 cm

10−5

10−2

10−1

10−6 10−2

100

10−1

Δθ = 0 − 3.75° r × KPB(z, r), cm2/cm3

56.25 − 60° 10−1

86.25 − 90° 116 − 120°

10−2

10−3

100

101

Δθ = 0 − 3.75°

10−1

(c)

100 r × KPB(z, r), cm2/cm3

(b)

12.0 cm 10−2

102 (d)

56.25 − 60° 10−2

86.25 − 90°

116 − 120° 10−3

10−4

10−4 2 10−

10−1 r, cm

100

10−2

10−1

100 r, cm

101

102

Fig. 4. The results of the approximation (curves) of individual components and the cumulative dose kernel of (a, b) pencil beam (PB) and (c, d) differential pencil beam (DPB) compared with the results of calculations by the Monte-Carlo method (mark points) for the primary (diagrams on the left) and the scattered component and the cumulative dose-kernel value (diagrams on the right); o, the markers for the primary component and the cumulative value; *, the scattered component markers.

duced by the disc-shaped unidirectional source with radius R at the same z in the semi-infinite medium given the normal incidence of photons on the medium and the identical spectrum (see [34], for example). It is exactly this characteristic of the PB dose kernel that was used to additionally verify the integral accuracy of approximation of the given kernel (in terms of the error in finding the dose of treatment-like photon beams). After the dose kernel is integrated this way, the dose in the phantom along the axis of the discshaped unidirectional source will be: R

∫

D j (z, R) = 2π rK PB, j (z, r )dr = 2π 0

4

C j,i (z ) −k ( z )R [1 − e ji ], × k j,i (z) i =1

∑

(4)

where Cj, i and kj, i are the coefficients of the approximation formula for the PB dose-kernel component at depth z. As an example, some results of comparing the values of the indicated integrals with the direct dose calculations are given in Fig. 6. We see that the results of calculating by equation (4) slightly exceed the results of the direct Monte-Carlo calculations only at z = 0.5–1.5 cm (the peak error is 5% at z = 0.5 cm). Let us note that the dose generated by photon beams at this depth is determined not by the photons but by the electrons that pollute the beam [3]. That is reason that the dose in this region is calculated on the basis of experimental data [3]. In the strict sense, the integral accuracy of the dose calculation using the approximation formula of the dose kernel for DPB must be verified as part of the program for applying the collapsed-cone convolution


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∫

θ

∫

2 ρ θ jj −1 sin θ d θ 0.040

R 0

r 2K DPB( r , θ ) dr

D, (MeV/g)/(MeV/cm2) 0.035

0.035

0.030

0.030

R = 2.26 cm

0.020

0.020

3.75 − 7.5°

0.015

0.015 0.010

0.010

7.5 − 11.75° 0.005

0.005 0 10−2

R = 5.64 cm

0.025

Δθ = 0 − 3.75°

0.025

437

10−1

100

101

102 R, cm

Fig. 5. Comparison of the integrals of DPB dose-kernel values found by the approximation formula (–) and the Monte-Carlo method (○) for different values of Δθ with respect to R.

technique. However, the commercial dosimetry planning systems that make use of this method do not permit one to enter these data in themselves, whereas the development of in-house general-purpose software is very labor intensive. On the other hand, if there is azimuthal symmetry, the dose-calculation algorithm can be made much simpler to solve tasks with more limited goals. The algorithm we propose in this paper to use for finding the central-axial dose distribution generated by disc-shaped unidirectional sources is based on approximation formula (3) for the DPB dose kernel and described below. First of all, let us distinguish the scattering volume element dV in the water phantom (Fig. 7). This element is formed by the intersection of conical and spherical surfaces; it is found in the phantom at depth z and distance t from the calculation point and is oriented towards this point at angle θn. The share in the dose of the scattered radiation of photons exposed to the first interaction in this volume is: 3

dD s (z d , θ n ) = μ e

−μ z

dν

∑ C (θ ) e i

i =1

n

0

5

10

15

20

25

30

35

40 45 z, cm

Fig. 6. Dose distributions in water along the axis of disclike unidirectional photon sources compared with the spectrum of the ROKUS-M treatment machine with R = 5.64 cm (equivalent in area to a standard beam of 10 × 10 cm2) and R = 2.26 cm (equivalent in area to a standard beam of 4 × 4 cm2). These distributions were obtained directly by the Monte-Carlo method (*) and calculated using formula (4) (–). The dose is measured in MeV /g and normalized per unit value of energy fluence uniformly distributed across the radiated surface area, i.e., by 1.0 MeV/cm2.

Having integrated equation (5) by variable t, we obtain the contribution to the target component from the space between two conical surfaces with spanning angle θ j – 1 and θ j and sphere with R = tm:

D s (z d , θ n ) = 2π(cos θ j −1 − cos θ j )μ e 3

×

−{[ki (θ n )−μ cos θ n ]t m }

C i (θ n )1 − e , k i (θ n ) − μ cos θ n i =1

∑

−μ z d

(6)

where θn = (θ j – 1 + θ j)/2. The full value of the component is found by summing contribution (6) for all the discrete values of θn for which the DPB dose-kernel values were calculated, i.e.,

−ki (θ n )t

t2

,

(5)

where μ is the effective linear attenuation coefficient of photons with the beam spectrum of ROKUS-M; Ci(θ) and ki(θ) are the θ-dependent coefficients of approximation formula (3) for the scattered component of the DPB dose kernel; t, z, and zd are the coordinates found as shown in Fig. 7; dν = 2π · t2 · sinθ · dθ · dt. In the case of water the absorbed energy in a unit of volume is numerically equal to the absorbed dose. The contribution of the primary component to the dose is considered in a similar manner. MOSCOW UNIVERSITY PHYSICS BULLETIN

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Ds (z) =

∑ D (z, θ ). s

n

(7)

n =1

In Fig. 8 the results of finding the central-axial distribution in water for the full dose and the scattered dose component by the above-described procedure are compared with the results of calculating these distributions directly by the Monte-Carlo method. The discrepancy between the data does not go beyond the limits of the error in their calculation. The mean square error is below 5.0% and the maximum error is observed at a depth of 0.25 cm; it equals 19%. As noted No. 4

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Beam

D, (MeV/g)/(MeV/cm2) 0.035 R 0.030 0.025

dV zd

z

t

θ

0.015

tm

d

Cumulative dose

0.020

Point of calculation Water phantom

0.010

Scattered radiation dose

0.005 0

Fig. 7. The geometry of calculating the central-axial distribution of the absorbed dose in the water phantom on the basis of the DPB dose kernel for the disc-shaped unidirectional source.

above, however, the dose near the surface is found by processing experimental data for particular beams. CONCLUSIONS In this study we used the Monte-Carlo method to find the values of the primary, the scattered, and the cumulative component of the dose kernels of PB and DPB of photons with the energy spectrum of the ROKUS-M treatment machine for water and proposed the approximation formulas for these quantities. Since the proposed formulas are analytical in form, they make a far more rapid multidimensional integration of dose kernels by irradiated volume possible; this is needed in calculations of nD dose distributions produced by treatment beams in RT. The derived formulas and their coefficients can be verified by the simple analytical procedures proposed in this article. As shown by the analysis, the mean error in the integral approximation of the dose kernel is less than 5%. It should be noted that, combined with the allowance for beam divergence (geometric attenuation by the inverse square law with regard to the distance from the source), the procedure for finding dose distributions on the basis of the approximation formulas for PB and DPB dose kernels can be efficiently used in operative control dose calculations in remote radiation therapy, especially in such technologies as stereotaxic and radiation surgery, where rotationally symmetric photon beams with circular cross section are frequently used. One can make independent operative dose calculations along the beam axis with a rectangular cross section using the formulas of transfer to the equivalent beam with a square section and then to the equivalent beam with a circular section [3]. These independent control calculations are strongly encour-

5

10

15

20

25

30 z, cm

Fig. 8. A comparison of the central-axial distribution of the cumulative dose and the scattered radiation dose in water for a disc-shaped unidirectional photon source with R = 5.64 cm with the spectrum of the ROKUS-M treatment machine. These distributions were found by the procedure considered in this paper (curve) and calculated by the Monte-Carlo method in EGSnrc (o, +). The dose is measured in MeV/g and normalized per unit value of energy fluence uniformly distributed across the irradiated surface area, i.e., by 1.0 MeV/cm2.

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Translated by S. Kuznetsov

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