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Comments on the primary and scatter dose-spread kernels used for convolution methods
Radiation Physics and Chemistry 65 (2002) 595–597
Comments on the primary and scatter dose-spread kernels used for convolution methods Akira Iwasaki* School of Health Sciences, Hirosaki University, 66-1 Hon-cho, Hirosaki, Aomori 036-8564, Japan Received 21 February 2002; accepted 6 June 2002
Abstract The basic primary and scatter dose-spread kernels used for convolution methods are usually produced by Monte Carlo simulations with the interaction point forced to the center of a large water phantom. However, it is still not clear whether such Monte Carlo based kernels allow accurate dose calculations with a wide range of field sizes and depths, especially in thorax phantoms. Using the differential primary and scatter concept, this paper proposes another type of basic kernel, with which perfectly accurate primary and scatter absorbed dose calculations can be performed under conditions that the beam is parallel, the incident beam intensity is uniform within and zero outside the field, and the primary beam attenuation coefficient along raylines is not a function of depth and off-axis distance. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Convolution method; Primary dose-spread kernel; Scatter dose-spread kernel
1. Introduction Calculation of the absorbed dose at each point in a medium irradiated by high-energy photons has recently also been carried out using convolution methods. The technique spatially convolves the primary fluence with a kernel that describes the transport and energy deposition by secondary particles. Primary and scatter dosespread kernels are used for calculation of primary and scatter absorbed dose components, respectively. The basic primary and scatter dose-spread kernels are usually produced by Monte Carlo simulations with the interaction point forced to the center of a large water phantom (Mackie et al., 1985; Mohan et al., 1986; . 1989). Ahnesjo. et al., 1987; Mackie et al., 1988; Ahnesjo, It is a prerequisite for dose calculations using convolution that the applied kernels be relatively spatially invariant. Primary and scatter dose-spread kernels in materials different from water are usually converted
*Tel.: +81-172-39-5957; fax: +81-172-39-5912. E-mail address: [email protected] (A. Iwasaki).
from the basic ones using the density scaling theorem (O’Connor, 1957). However, it is still not clear whether such Monte Carlo based kernels allow accurate dose calculations with a wide range of field sizes and depths, especially in thorax phantoms. The author proposes another type of basic kernel, with which perfectly accurate primary and scatter absorbed dose calculations can be performed under certain irradiation conditions.
2. Theoretical considerations To achieve a stage of being able to calculate the absorbed dose with sufficient accuracy at points in heterogeneous media using a convolution method, we need to be capable of performing accurate calculations of absorbed doses, at least at points on the beam axis in water for a single beam irradiation using the convolution method. Here we assume that the incident beam intensity is uniform within and zero outside the field and that the primary beam attenuation coefficient (m) along raylines is not a function of depth and off-axis distance.
0969-806X/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 9 - 8 0 6 X ( 0 2 ) 0 0 3 0 9 - 2
596
A. Iwasaki / Radiation Physics and Chemistry 65 (2002) 595–597
For the primary fluence to be convolved with a kernel, we take the primary water collision kerma (Attix, 1979a, b, 1983, 1986). Let K0 denote an incident water collision kerma, then the primary water collision kerma at a depth X within the field can be expressed as KðX Þ ¼ K0 expðmX Þ:
ð1Þ
As illustrated in Fig. 1, we take a semi-infinite water phantom and a parallel beam circular field with a radius of R: The parallel beam is normally incident on the surface of the phantom. We set a point O on the beam axis, at which the primary or scatter absorbed dose is evaluated. We define FðZ; RÞ as the primary or scatter absorbed dose at the point O; which is at a depth Z for the field radius R: Fig. 1 also shows the case of acquiring F ðZ þ dZ; RÞ at the point O; which is at a depth Z þ dZ: Mathematically, we can expand this as F ðZ þ dZ; RÞ ¼ F ðZ; RÞ þ qF ðZ; RÞ=qZ dZ:
ð2Þ
Here we introduce a forward primary or scatter dosespread function S1 ðZ; rÞ; expressing the primary or scatter absorbed dose at the point O; arising from a point (Z; r) on the incident surface per unit primary water collision kerma and per unit volume at (Z; r). If the S1 ðZ; rÞ function is taken for the absorbed dose calculation, it can be understood that F ðZ þ dZ; RÞ is composed of two components when neglecting the
primary or scatter absorbed dose caused by the interaction between the dZ layer and the phantom yielding FðZ; RÞ: One is the primary or scatter absorbed dose produced by the incident primary water collision kerma of K0 expðm dZÞ for the phantom yielding F ðZ; RÞexpðm dZÞ: The other is the primary or scatter absorbed dose arising from the dZ layer. Namely, we have F ðZ þ dZ; RÞ ¼ F ðZ; RÞ expðm dZÞ Z R þ 2pK0 dZ S1 ðZ; rÞr dr:
ð3Þ
0
In the equation constructed by letting the right-hand sides of Eqs. (2) and (3) be equal to each other, we make expðm dZÞ ¼ 1 m dZ and then differentiate both sides with respect to R; to obtain S1 ðZ; RÞ ¼ ½q2 F ðZ; RÞ=qZqR þ mqF ðZ; RÞ=qR=ð2pRK0 Þ:
ð4Þ
It should be noted that S1 ðZ; RÞ is evaluated for the point situated at the margin of the circular beam field on the phantom surface. Next, we summarize the forward primary or scatter dose-spread function with a backward primary or scatter dose-spread function (almost the same description as given by Iwasaki (1992, 2002)). With respect to the two semi-infinite water phantoms illustrated in Fig. 2, (a) S1 ðx; rÞ expresses the forward primary or scatter absorbed dose at a point (x; r), arising from the pencil-beam interaction point O (x ¼ 0; r ¼ 0) situated on the phantom surface, per unit primary water collision kerma per unit volume at point O; and (b) S2 ðZ; rÞ expresses the backward primary or scatter absorbed dose at a point (Z; r) on the phantom surface, arising from the pencil-beam interaction point O (Z ¼ 0; r ¼ 0) situated at a depth Z below the phantom surface, per unit primary water collision kerma per unit volume at point O: With respect to the S1 ðx; rÞ and S2 ðZ; rÞ functions, we can obtain S1 ð0; rÞ ¼ S2 ð0; rÞ for x-0 and Z-0: Using the above S1 ðx; rÞ and S2 ðZ; rÞ functions, we can derive FðZ; RÞ of Fig. 1 as Z RZ Z F ðZ; RÞ ¼ 2pK0 S1 ðx; rÞ exp½mðZ xÞrd rdx 0 0 Z RZ N þ2pK0 S2 ðZ; rÞ exp½mðZ þ ZÞrd rdZ: 0
0
ð5Þ
Fig. 1. Using a semi-infinite water phantom, F ðZ; RÞ and F ðZ þ dZ; rÞ are defined as the primary or scatter absorbed doses at the point O at depths Z and Z þ dZ; respectively, on the beam axis for a parallel photon beam irradiation with a field radius of R and an incident water collision kerma of K0 :
The right-hand side of Eq. (5) describes a convolution expression. The first and second terms express the forward and backward absorbed dose components, respectively, at the point O: It can also be proved using Eq. (5) itself that the second term is exactly equal to F ð0; RÞexpðmZÞ:
A. Iwasaki / Radiation Physics and Chemistry 65 (2002) 595–597
597
Fig. 2. Diagrams showing how (a) the S1 ðx; rÞ function and (b) the S2 ðZ; rÞ function should be used in water for a given pencil–beam, where the interaction point is situated at point O:
From the above-described theoretical considerations, it may be suggested that for performing more accurate dose calculations, the basic primary and scatter dosespread kernels used for convolution methods should not be obtained with the interaction point forced to the center of a large water phantom. The approach employing the S1 ðx; rÞ function for scatter absorbed dose calculations is essentially the same as the differential scatter method (Cunningham, 1972; Iwasaki and Ishito, 1984). The method of using the S1 ðx; rÞ and S2 ðZ; rÞ functions for calculation of 10 MV X-ray primary and scatter absorbed doses has already been developed by Iwasaki (1992), where the primary dose-spread kernel is constructed based on the zero-area TMR (tissue-maximum ratio) and the LSD (laterally spread primary dose) function, and the scatter dosespread kernel is constructed based on the SMR (scattermaximum ratio).
References Ahnesj.o, A., 1989. Collapsed cone convolution of radiant energy for photon dose calculation in heterogeneous media. Med. Phys. 16, 577–592. Ahnesj.o, A., Andreo, P., Brahme, A., 1987. Calculation and application of point spread functions for treatment planning with high energy photon beams. Acta Oncol. 26, 49–56. Attix, F.H., 1979a. The partition of kerma to account for bremsstrahlung. Health Phys. 36, 347–354.
Attix, F.H., 1979b. Addendum to ‘The partition of kerma to account for bremsstrahlung. Health Phys. 36, 536. Attix, F.H., 1983. Energy imparted, energy transferred and net energy transferred. Phys. Med. Biol. 28, 1385–1390. Attix, F.H., 1986. Introduction to Radiological Physics and Radiation Dosimetry. Wiley-Interscience, New York, pp. 24–26. Cunningham, J.R., 1972. Scatter-air ratios. Phys. Med. Biol. 17, 42–51. Iwasaki, A., 1992. A convolution method for calculating 10-MV X-ray primary and scatter dose including electron contamination dose. Med. Phys. 19, 907–915. Iwasaki, A., 2002. A method for 10 MV X-ray scatter dosespread kernel construction based on the Bj.arngard scatter factor formulation. Radiat. Phys. Chem. 64, 89–98. Iwasaki, A., Ishito, T., 1984. The differential scatter-air ratio and differential backscatter factor method combined with the density scaling theorem. Med. Phys. 11, 755–763. Mackie, T.R., Bielajew, A.F., Rogers, D.W.O., Battista, J.J., 1988. Generation of photon energy deposition kernels using the EGS Monte Carlo code. Phys. Med. Biol. 33, 1–20. Mackie, T.R., Scrimger, J.W., Battista, J.J., 1985. A convolution method of calculating dose for 15-MV X-rays. Med. Phys. 12, 188–196. Mohan, R., Chui, C., Lidofsky, L., 1986. Differential pencil beam dose computation model for photons. Med. Phys. 13, 64–73. O’Connor, J.E., 1957. The variation of scattered X-rays with density in an irradiated body. Phys. Med. Biol. 1, 352–369.
Comments on the primary and scatter dose-spread kernels used for convolution methods Akira Iwasaki* School of Health Sciences, Hirosaki University, 66-1 Hon-cho, Hirosaki, Aomori 036-8564, Japan Received 21 February 2002; accepted 6 June 2002
Abstract The basic primary and scatter dose-spread kernels used for convolution methods are usually produced by Monte Carlo simulations with the interaction point forced to the center of a large water phantom. However, it is still not clear whether such Monte Carlo based kernels allow accurate dose calculations with a wide range of field sizes and depths, especially in thorax phantoms. Using the differential primary and scatter concept, this paper proposes another type of basic kernel, with which perfectly accurate primary and scatter absorbed dose calculations can be performed under conditions that the beam is parallel, the incident beam intensity is uniform within and zero outside the field, and the primary beam attenuation coefficient along raylines is not a function of depth and off-axis distance. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Convolution method; Primary dose-spread kernel; Scatter dose-spread kernel
1. Introduction Calculation of the absorbed dose at each point in a medium irradiated by high-energy photons has recently also been carried out using convolution methods. The technique spatially convolves the primary fluence with a kernel that describes the transport and energy deposition by secondary particles. Primary and scatter dosespread kernels are used for calculation of primary and scatter absorbed dose components, respectively. The basic primary and scatter dose-spread kernels are usually produced by Monte Carlo simulations with the interaction point forced to the center of a large water phantom (Mackie et al., 1985; Mohan et al., 1986; . 1989). Ahnesjo. et al., 1987; Mackie et al., 1988; Ahnesjo, It is a prerequisite for dose calculations using convolution that the applied kernels be relatively spatially invariant. Primary and scatter dose-spread kernels in materials different from water are usually converted
*Tel.: +81-172-39-5957; fax: +81-172-39-5912. E-mail address: [email protected] (A. Iwasaki).
from the basic ones using the density scaling theorem (O’Connor, 1957). However, it is still not clear whether such Monte Carlo based kernels allow accurate dose calculations with a wide range of field sizes and depths, especially in thorax phantoms. The author proposes another type of basic kernel, with which perfectly accurate primary and scatter absorbed dose calculations can be performed under certain irradiation conditions.
2. Theoretical considerations To achieve a stage of being able to calculate the absorbed dose with sufficient accuracy at points in heterogeneous media using a convolution method, we need to be capable of performing accurate calculations of absorbed doses, at least at points on the beam axis in water for a single beam irradiation using the convolution method. Here we assume that the incident beam intensity is uniform within and zero outside the field and that the primary beam attenuation coefficient (m) along raylines is not a function of depth and off-axis distance.
0969-806X/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 9 - 8 0 6 X ( 0 2 ) 0 0 3 0 9 - 2
596
A. Iwasaki / Radiation Physics and Chemistry 65 (2002) 595–597
For the primary fluence to be convolved with a kernel, we take the primary water collision kerma (Attix, 1979a, b, 1983, 1986). Let K0 denote an incident water collision kerma, then the primary water collision kerma at a depth X within the field can be expressed as KðX Þ ¼ K0 expðmX Þ:
ð1Þ
As illustrated in Fig. 1, we take a semi-infinite water phantom and a parallel beam circular field with a radius of R: The parallel beam is normally incident on the surface of the phantom. We set a point O on the beam axis, at which the primary or scatter absorbed dose is evaluated. We define FðZ; RÞ as the primary or scatter absorbed dose at the point O; which is at a depth Z for the field radius R: Fig. 1 also shows the case of acquiring F ðZ þ dZ; RÞ at the point O; which is at a depth Z þ dZ: Mathematically, we can expand this as F ðZ þ dZ; RÞ ¼ F ðZ; RÞ þ qF ðZ; RÞ=qZ dZ:
ð2Þ
Here we introduce a forward primary or scatter dosespread function S1 ðZ; rÞ; expressing the primary or scatter absorbed dose at the point O; arising from a point (Z; r) on the incident surface per unit primary water collision kerma and per unit volume at (Z; r). If the S1 ðZ; rÞ function is taken for the absorbed dose calculation, it can be understood that F ðZ þ dZ; RÞ is composed of two components when neglecting the
primary or scatter absorbed dose caused by the interaction between the dZ layer and the phantom yielding FðZ; RÞ: One is the primary or scatter absorbed dose produced by the incident primary water collision kerma of K0 expðm dZÞ for the phantom yielding F ðZ; RÞexpðm dZÞ: The other is the primary or scatter absorbed dose arising from the dZ layer. Namely, we have F ðZ þ dZ; RÞ ¼ F ðZ; RÞ expðm dZÞ Z R þ 2pK0 dZ S1 ðZ; rÞr dr:
ð3Þ
0
In the equation constructed by letting the right-hand sides of Eqs. (2) and (3) be equal to each other, we make expðm dZÞ ¼ 1 m dZ and then differentiate both sides with respect to R; to obtain S1 ðZ; RÞ ¼ ½q2 F ðZ; RÞ=qZqR þ mqF ðZ; RÞ=qR=ð2pRK0 Þ:
ð4Þ
It should be noted that S1 ðZ; RÞ is evaluated for the point situated at the margin of the circular beam field on the phantom surface. Next, we summarize the forward primary or scatter dose-spread function with a backward primary or scatter dose-spread function (almost the same description as given by Iwasaki (1992, 2002)). With respect to the two semi-infinite water phantoms illustrated in Fig. 2, (a) S1 ðx; rÞ expresses the forward primary or scatter absorbed dose at a point (x; r), arising from the pencil-beam interaction point O (x ¼ 0; r ¼ 0) situated on the phantom surface, per unit primary water collision kerma per unit volume at point O; and (b) S2 ðZ; rÞ expresses the backward primary or scatter absorbed dose at a point (Z; r) on the phantom surface, arising from the pencil-beam interaction point O (Z ¼ 0; r ¼ 0) situated at a depth Z below the phantom surface, per unit primary water collision kerma per unit volume at point O: With respect to the S1 ðx; rÞ and S2 ðZ; rÞ functions, we can obtain S1 ð0; rÞ ¼ S2 ð0; rÞ for x-0 and Z-0: Using the above S1 ðx; rÞ and S2 ðZ; rÞ functions, we can derive FðZ; RÞ of Fig. 1 as Z RZ Z F ðZ; RÞ ¼ 2pK0 S1 ðx; rÞ exp½mðZ xÞrd rdx 0 0 Z RZ N þ2pK0 S2 ðZ; rÞ exp½mðZ þ ZÞrd rdZ: 0
0
ð5Þ
Fig. 1. Using a semi-infinite water phantom, F ðZ; RÞ and F ðZ þ dZ; rÞ are defined as the primary or scatter absorbed doses at the point O at depths Z and Z þ dZ; respectively, on the beam axis for a parallel photon beam irradiation with a field radius of R and an incident water collision kerma of K0 :
The right-hand side of Eq. (5) describes a convolution expression. The first and second terms express the forward and backward absorbed dose components, respectively, at the point O: It can also be proved using Eq. (5) itself that the second term is exactly equal to F ð0; RÞexpðmZÞ:
A. Iwasaki / Radiation Physics and Chemistry 65 (2002) 595–597
597
Fig. 2. Diagrams showing how (a) the S1 ðx; rÞ function and (b) the S2 ðZ; rÞ function should be used in water for a given pencil–beam, where the interaction point is situated at point O:
From the above-described theoretical considerations, it may be suggested that for performing more accurate dose calculations, the basic primary and scatter dosespread kernels used for convolution methods should not be obtained with the interaction point forced to the center of a large water phantom. The approach employing the S1 ðx; rÞ function for scatter absorbed dose calculations is essentially the same as the differential scatter method (Cunningham, 1972; Iwasaki and Ishito, 1984). The method of using the S1 ðx; rÞ and S2 ðZ; rÞ functions for calculation of 10 MV X-ray primary and scatter absorbed doses has already been developed by Iwasaki (1992), where the primary dose-spread kernel is constructed based on the zero-area TMR (tissue-maximum ratio) and the LSD (laterally spread primary dose) function, and the scatter dosespread kernel is constructed based on the SMR (scattermaximum ratio).
References Ahnesj.o, A., 1989. Collapsed cone convolution of radiant energy for photon dose calculation in heterogeneous media. Med. Phys. 16, 577–592. Ahnesj.o, A., Andreo, P., Brahme, A., 1987. Calculation and application of point spread functions for treatment planning with high energy photon beams. Acta Oncol. 26, 49–56. Attix, F.H., 1979a. The partition of kerma to account for bremsstrahlung. Health Phys. 36, 347–354.
Attix, F.H., 1979b. Addendum to ‘The partition of kerma to account for bremsstrahlung. Health Phys. 36, 536. Attix, F.H., 1983. Energy imparted, energy transferred and net energy transferred. Phys. Med. Biol. 28, 1385–1390. Attix, F.H., 1986. Introduction to Radiological Physics and Radiation Dosimetry. Wiley-Interscience, New York, pp. 24–26. Cunningham, J.R., 1972. Scatter-air ratios. Phys. Med. Biol. 17, 42–51. Iwasaki, A., 1992. A convolution method for calculating 10-MV X-ray primary and scatter dose including electron contamination dose. Med. Phys. 19, 907–915. Iwasaki, A., 2002. A method for 10 MV X-ray scatter dosespread kernel construction based on the Bj.arngard scatter factor formulation. Radiat. Phys. Chem. 64, 89–98. Iwasaki, A., Ishito, T., 1984. The differential scatter-air ratio and differential backscatter factor method combined with the density scaling theorem. Med. Phys. 11, 755–763. Mackie, T.R., Bielajew, A.F., Rogers, D.W.O., Battista, J.J., 1988. Generation of photon energy deposition kernels using the EGS Monte Carlo code. Phys. Med. Biol. 33, 1–20. Mackie, T.R., Scrimger, J.W., Battista, J.J., 1985. A convolution method of calculating dose for 15-MV X-rays. Med. Phys. 12, 188–196. Mohan, R., Chui, C., Lidofsky, L., 1986. Differential pencil beam dose computation model for photons. Med. Phys. 13, 64–73. O’Connor, J.E., 1957. The variation of scattered X-rays with density in an irradiated body. Phys. Med. Biol. 1, 352–369.