Home
Search
Collections
Journals
About
Contact us
My IOPscience
Application of an adapted Fano cavity test for Monte Carlo simulations in the presence of Bfields
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 Phys. Med. Biol. 60 9313 (http://iopscience.iop.org/0031-9155/60/24/9313) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 128.233.210.97 This content was downloaded on 22/02/2016 at 07:00
Please note that terms and conditions apply.
Institute of Physics and Engineering in Medicine Phys. Med. Biol. 60 (2015) 9313–9327
Physics in Medicine & Biology doi:10.1088/0031-9155/60/24/9313
Application of an adapted Fano cavity test for Monte Carlo simulations in the presence of B-fields J A de Pooter1, L A de Prez1 and H Bouchard2 1
VSL, Thijsseweg 11, 2629 JA Delft, The Netherlands Acoustics and Ionising Radiation Team, National Physical Laboratory, Hampton Road, Teddington TW11 0LW, UK 2
E-mail: [email protected] Received 27 February 2015, revised 30 September 2015 Accepted for publication 15 October 2015 Published 18 November 2015 Abstract
With the advent of MR guided radiotherapy the relevance of Monte Carlo radiation transport simulations in the presence of strong magnetic fields (B-fields) is increasing. While new tests are available to benchmark these simulation algorithms for internal consistency, their application to known codes such as EGSnrc, PENELOPE, and GEANT4 is yet to be provided. In this paper a method is provided to apply the Fano cavity test as a benchmark for a generic implementation of B-field effects in PENELOPE. In addition, it is investigated whether violation of the conditions for the Fano test can partially explain the change in the response of ionization chambers in the presence of strong B-fields. In the present paper it is shown that the condition of isotropy of the secondary particle field (Charged Particle Isotropy, CPI) is an essential requirement to apply the Fano test in the presence of B-fields. Simulations in PENELOPE are performed with (B = 0.0 T) and (B = 1.5 T) for cylindrical cavity geometry. The secondary particle field consists of electrons generated from a mono-energetic source (E = 0.5–4.0 MeV) with a uniform source density and different angular distributions; isotropic, mono-directional, and Compton. In realistic photon fields the secondary radiation field has a non-isotropic angular distribution due to the Compton process. Based on the simulations for the Compton angular distribution (non-CPI), the response change of the cavity model in a uniform radiation field in the presence of B-fields is investigated. For the angular distributions that violate the CPI condition and B = 1.5 T, the deviations from 1 are considerable, which emphasizes the requirement of CPI. For the isotropic angular distributions obeying this requirement, both the results for B = 0.0 T and B = 1.5 T shows deviations from the predictions for E ⩾ 1.5 MeV with values up to 1.0% for E = 4.0 MeV. Nevertheless, 0031-9155/15/249313+15$33.00 © 2015 Institute of Physics and Engineering in Medicine Printed in the UK
9313
J A de Pooter et al
Phys. Med. Biol. 60 (2015) 9313
due to the high correlation in the deviation for B = 0.0 T and B = 1.5 T, the accuracy of the PENELOPE code for the simulation of the change in detector response in the presence of B-fields is within 0.3%. The effect of the B-field on the detector response for non-isotropic angular distributions suggests that violation of CPI is a major contribution to the response change of ionization chambers in the presence of B-fields. Keywords: Fano cavity test, magnetic fields, Monte Carlo radiation transport, dosimetry, reference dosimetry, MRI-guided radiotherapy, ionization chamber response (Some figures may appear in colour only in the online journal) Introduction With the advent of MR guided radiotherapy (Raaymakers et al 2004, Raaijmakers et al 2005, Fallone et al 2009) the interest in and the relevance of dosimetry in the presence of strong magnetic fields (B-fields) is increasing. Since no reference dosimetry standards are available for such conditions that are able to measure the dose to water in an absolute way in the presence of strong B-fields, attempts to adapt reference dosimetry is largely based on Monte Carlo simulations (Meijsing et al 2009, Reynolds et al 2013). A few investigations calculated response variations for Farmer-type chambers in clinical beams and compared them with relative measurements, leading to the conclusion that the response can be simulated within reasonable agreement with measurements. A number of Monte Carlo codes exist that include the influence of B-fields on the transport of the charged particles (Agostinelli et al 2003, Salvat et al 2011). Most Monte Carlo codes use an approximation to simulate the trajectories of the charged particles in the presence of a B-field in between two interactions (Salvat et al 2011). Several parameters can be used to set the accuracy of the simulated trajectories. In addition, the crossing of boundaries between volumes for curved trajectories is different than for straight trajectories. These two examples indicate that accurate Monte Carlo simulations in the presence of B-fields are not trivial and tests to benchmark the codes are needed. The Fano theorem (Fano 1954) plays an important role in conventional radiation dosimetry (i.e. without a B-field). It is based on an analytical solution to the Boltzmann transport equation for the secondary particle fluence of a uniform field in a phantom of uniform atomic properties with varying densities (Smyth 1986). Based on the analytical solution, the internal consistency of radiation transport simulations for a particular geometry can be assessed. Two major fields of applications are: in the development of cavity standards (La Russa and Rogers 2009, Yi et al 2006) and in the validation of Monte Carlo radiation transport codes for the simulation of ionization chambers and other detectors (Smyth 1986, Kawrakow 2000, Poon et al 2005, Sempau and Andreo 2006, Sterpin et al 2014). In the case of Sempau and Andreo (2006), it is used as a benchmark to estimate the influence of the set of simulation parameters, such as the maximum step size and the parameters for elastic scattering (Salvat et al 2011) on the transport algorithm self-consistency. Lately, it was shown that the classical Fano test is not applicable for radiation transport simulations in the presence of a B-field (Bouchard and Bielajew 2015). Based on the coupled Boltzmann transport equation, two general benchmarking tests have been proposed and their theoretical basis was mathematically proven (Bouchard et al 2015). 9314
J A de Pooter et al
Phys. Med. Biol. 60 (2015) 9313
In this paper, a method is provided to apply the first general Fano test of Bouchard et al (2015) to the case of a photon radiation field in the presence of an external B-field. This is done by Monte Carlo calculations in which only the secondary electron field is considered using the PENELOPE code. An alternative more intuitive proof is given for the required conditions for the Fano test in the presence of an external B-field. In addition, it is investigated whether violation of the conditions of the Fano test can partially explain the previously observed response changes of ionization chambers in the presence of strong B-fields (Meijsing et al 2009, Reynolds et al 2013). Materials and methods Extension of the Fano test conditions for the influence of B-fields
The classical Fano theorem was formulated as follows (Fano 1954): ‘In a phantom of given composition exposed to a uniform fluence of primary radiation, the fluence of the secondary radiation field is also uniform and independent of the density of the phantom as well as of the density variations from point to point.’ The Fano theorem for applications in the presence of B-fields can be formulated in the following way (Fano 1954, Bouchard et al 2015): ‘In a phantom of given composition exposed to a uniform and isotropic fluence of primary radiation in the presence of a B-field, the fluence of the secondary radiation is also uniform and isotropic, and independent of the density of the phantom and the B-field strength as well as of the density and B-field variations from point to point.’ The main difference with the classical Fano theorem is the extension with the condition for isotropy of the primary and secondary radiation fields, which we call charged particle isotropy (CPI). The proof for the extension of the conditions uses to a large extent the same arguments as for the classical formulation and starts from the general transport equation for secondary particle fields as given by (Fano 1954, Bouchard 2010) and illustrated in figure 1. →
→ → dϕe(r→, E , Ω) = ρ(r→)Se(r→, E , Ω) − ρ(r→)ϕe(r→, E , Ω) dl
+ρ(r→)
E0
∫E
dE′
∫0
E
dE′
∬4π dΩ′k(E′, E , Ω′ ⋅ Ω) →
→
∬4π dΩ′k(E , E′, Ω′ ⋅ Ω)ϕe(r , E′, Ω′) →
→
→
→
→
→
→
→ + → = ρ(r→) [Se(r→, E , Ω) − I − (1) e (r , E , Ω) + I e (r , E , Ω)] Please note, for the remainder only the electrons of the secondary particle field will be considered. The formulation for positrons follows an analogous way. With:
•
→
dϕe(→ r , E , Ω) dl
the left-hand side of equation (1) being the change in fluence for an infinitesi→
r and energy E, mally small step in the direction of motion Ω for electrons with position → → which is identical to ∇r ϕe in the classical formulation (13), i.e. without the presence of the B-field (see figure 1). energy of the secondary particles in the system. • E0 the maximum → • Se(→ r , E, Ω): the number of secondary electrons generated per unit of mass due to interac→ tions of the primary photons at position → r with energy E and direction Ω = (u, v, w ) (see figure 1). 9315
J A de Pooter et al
Phys. Med. Biol. 60 (2015) 9313
Figure 1. Illustration of the transport of secondary particles as given by equation (1) → for a fluence on position →r with direction Ω and energy E, perpendicular to a slab with thickness dl. The derivative of the fluence with respect to l is the change in fluence for an infinitesimally small step in the direction of motion, which equals the net effect of → → → the source term Se and the two interaction terms, I−e (→r , E, Ω) and I+ e ( r , E, Ω). Se are the secondary particles generated in the slab by primary particles. →
• ϕe(→ r , E, Ω): the fluence of the secondary electrons written as ϕe in the remainder for presentation purposes. → → • k (E ′, E , Ω′ ⋅ Ω)dE′dΩ′: the probability per unit of mass traversed that the electrons with → energy E and direction of motion Ω have an inelastic interaction with a resulting energy → E′ and a direction of motion Ω′ E → → → → → ′ ′ ′ → • I − e (r , E , Ω) = ϕ (r , E , Ω) ∫ dE ′ ∬ dΩ k (E , E , Ω ⋅ Ω), describing the loss of electron e
→
0
4π
fluence with direction Ω and energy E per unit of mass due to electron interactions (see figure 1). E0 → → → → → ′ ′ ′ ′ → ′ • I + e (r , E , Ω) = ∫E dE ′ ∬ dΩ k (E , E , Ω ⋅ Ω)ϕe(r , E , Ω ), describing the increase of elec4π
→
tron fluence with direction Ω and energy E per unit of mass due to electron interactions (see figure 1). The main difference with the proof of the classical theorem lies in the expression of the left-hand side of equation (1), which changes due to the presence of the B-field. The left-hand side of equation (1) can be rewritten as (according to the derivation in the appendix with as the end result equation (A.9)): ⎡→ → dϕe → → e ⎤ = Ω ⋅ ⎢∇r ϕe − ∇Ω ϕe × B(r→) ⎥ (2) dl meγ v ⎦ ⎣
→
With ∇r ϕe =
(
∂ϕe ∂ϕe ∂ϕe , , ∂x ∂y ∂z
) and ∇ ϕ = →
Ω e
∂ϕe Θ ∂Θ
+
1 ∂ϕe φ. sin Θ ∂φ
The proof consists of two steps.
In the first step, it is shown that for a uniform and isotropic primary radiation field in a phantom of uniform composition and density in the presence of a uniform B-field, the fluence of the secondary field is uniform and isotropic as well. In the second step, it is shown that the solution of the first step is the same as the solution for a phantom with non-uniform density and B-field, with the same properties as the primary radiation field. Step 1. In a uniform and isotropic primary radiation field the production of the second→
→
ary electron, Se(→ r , E, Ω), is uniform and isotropic as well, yielding Se(→ r , E, Ω) = Se(E). 9316
J A de Pooter et al
Phys. Med. Biol. 60 (2015) 9313
In a homogenous phantom with a homogenous B-field for any straight line l,
dϕe dl
= 0. There-
fore, from equation (1) it follows that, →
→
+ [(3) Se(E ) − I − e (E , Ω) + I e (E , Ω)] = 0 →
→
[Se(E ) − ϕ(E , Ω)µ(E ) + I + e (E , Ω)] = 0
which can be written as, →
→ I +(E , Ω) S (E ) ϕ (E , Ω) = e + e (4) µ (E ) µ (E ) →
From equation (1) it can be seen that, for the maximum energy in the system, E0, I + e→(E 0, Ω) = 0 and since the left term on the right-hand side of equation (4) does not depend on Ω →
(5) ϕ (E0, Ω) = ϕ(E0) In addition, by definition, →
→
+ I+ (6) e (E , Ω) = I e (E ) if ϕ(E ′, Ω) = ϕ(E ′) for all E ′ > E
Therefore, from equations (4), and (6), →
→
ϕ(E0 − dE , Ω) =
I +(E0 − dE , Ω) I +(E0 − dE ) Se(E0 − dE ) S (E − dE ) = e 0 + e + e = ϕ(E0 − dE ) µ(E0 − dE ) µ(E0 − dE ) µ(E0 − dE ) µ ( E 0 − dE )
(7) for a choice of dE → 0. The steps made in equations (4), (6), and (7) can be repeated for i = 1 ... n start→ ing with ϕ(E0 − (i − 1)dE , Ω) in equation (4). From this, it is concluded that → ϕ(E0 − (i − 1)dE , Ω) = ϕ(E0 − (i − 1)dE ) for any i 1, and therefore, →
(8) ϕ (E , Ω) = ϕ(E ) the fluence of the secondary particle field is isotropic as well. Step 2. To prove the extension of the conditions for a non-uniform B-field and density, it
has to be proven that the solution, ϕ(E ) (equation (8)) for the situation of a uniform isotropic primary radiation field in a phantom with a uniform composition and density is equal to the solution for equation (1) (i.e. for a phantom with a uniform composition with density variations in the presence of a non-uniform B-field). The left-hand side of equation (1) as given by equation (2) is only zero when the secondary electron field is uniform and isotropic. In that case equation (1) reduces to →
→
+ 0(9) = [Se(E ) − I − e (E , Ω) + I e (E , Ω)] ,
which is identical to equation (3). Hence, under the conditions of a uniform isotropic primary radiation field the solution for the homogeneous phantom, equation (8) is the solution for the phantom in which the B-field and the density varies as a function of the position (equations(1) and (2). Therefore the solution is unique. Monte Carlo model
The Fano test is applied to a Monte Carlo model using the Penmain routine from the PENELOPE 2011 code (Salvat et al 2011). This routine is modified to allow for simulation 9317
J A de Pooter et al
Phys. Med. Biol. 60 (2015) 9313
of the particle transport in the presence of a uniform B-field using the prescription in the Penfield package. Parameters are added to the input file to define the uniform B-field vector, and to set tolerances on the simulation of trajectories of charged particles between two interactions. Three parameters for the tolerances are used to put a maximum on the change in B-field, energy, and speed of the charged particle along the trajectory between two hard collisions. Note that the first tolerance parameter is not applicable in the case of a uniform B-field. If in a step movement of a single particle in the simulation the deviation on one of the abovementioned quantities is higher than its tolerance parameter, the step length of the particle is reduced to a value for which the deviations obey the tolerance parameters. In case a maximum allowed step length is defined in the input file by the parameter DSMAX, the minimum of the two limitations on the step length is applied. In the study of Bouchard et al (2015) it is shown that the energy deposited under Fano conditions in the presence of an arbitrary B-field is the same as in the absence of the B-field, which implies that the validity of the Fano test can be investigated with a similar approach as used by Sempau and Andreo (2006), in which only the secondary electron field is simulated. Geometry. The simulations are performed for a simplified geometry of a Farmer-type
i onization chamber. The geometry consists of three bodies: the cavity, the wall, and the phantom (see figure 2). The dimensions of the cavity are the same as for a Farmer-type ionization chamber with a cavity diameter of 3.15 mm and a cavity length of 24.1 mm. The thickness of the wall is 0.35 mm. To avoid boundary effects, the size of the phantom is such that the minimum distance between the cavity and edges of the phantom is larger than 1.2 times the CSDA range of the electrons. The material for the phantom is set to water. The material for the cavity is set to water vapour. To avoid influence of the density effect parameter, the same material data file as for water is used with the density modified to the density of water vapour, in this way keeping the density effect parameter constant. The orientation of the cavity with respect to the B-field is shown in figure 2. Implementation of the adapted Fano test and modelling of the secondary particle field. For B = 0 several approaches exist to implement the Fano test: the regeneration technique (Kawrakow 2000), the re-entrance technique (Yi et al 2006), and the technique in which only the secondary particle field is modelled as used by Sempau and Andreo (2006). Here the Fano test adapted for application in the presence of B-fields is implemented based on the Sempau approach by modelling only the secondary particle field. In the section ‘Extension of the Fano test conditions for the influence of B-fields’ it is shown that a uniform and isotropic production per unit mass of secondary particles yields a uniform and isotropic secondary particle fluence, ϕe, regardless of spatial variations in density and magnetic field (strength and direction). This is a direct consequence of the cancelation of → ∇Ω ϕe in equation (2). As a result of the uniform ϕe, the absorbed dose in an infinite phantom of uniform atomic composition is uniform and equal to the energy of the secondary particles produced per unit mass, K. In photon fields the produced secondary electrons have in general a non-isotropic directional distribution due to the Compton process. This distribution varies as a function of the photon energy. Non-isotropic secondary particle production will not cancel → the ∇Ω ϕe term in equation (2). Therefore, it is expected that the spatial distribution of ϕe will depend on spatial variations in density and magnetic field strength. The extent of these spatial variations in ϕe for non-isotropic distributions is investigated by simulating two additional non-isotropic angular distributions. The three different angular distributions for the starting direction of the electrons that have been simulated are: 9318
J A de Pooter et al
Phys. Med. Biol. 60 (2015) 9313
Figure 2. Geometry of the cavity, wall, and the phantom as well as their orientation with respect to the beam axis (z-axis) and the B-field.
• Isotropic distribution, which obeys the CPI condition and which corresponds to an isotropic photon field with an artificial production process in which the total photon energy and momentum is transferred to the secondary electron. • Compton distribution, which corresponds to the production of electrons due to the Compton interactions of a mono-directional uniform photon field. • Mono-directional, which corresponds to a mono-directional photon field with an artificial production process in which the total photon energy and momentum is transferred to the secondary electron. The angular distributions of the secondary electrons produced for the Compton distribution and the isotropic distribution are given in figure 3 as a function of the angle θ between the direction of the primary photon and the secondary electron. θ is the angle with the z-axis (figure 2). The Compton distribution was implemented in Penmain using the Source routine. The other two angular distributions can be generated in a straightforward way using the Penmain input file. Dose calculation. The secondary electron field is modelled as a uniform source over the →
geometry scaled with the local mass density, in such a way that Se(Ω, E ) is constant. The result of the calculation is the ratio between the dose absorbed in the cavity, Dcav and the energy of the secondary particles produced per unit mass, K, in the source, which is given by, K = E Se (10)
The average energy of the produced secondary particles is calculated from µ tr
ρ
E = hν µ (11) ρ
To keep the model consistent the ratio between µtr ρ and µ ρ was separately calculated for the Compton angular distribution with the same Monte Carlo code for each energy. For the other angular distributions (isotropic, mono-directional) this ratio is 1. 9319
J A de Pooter et al
Phys. Med. Biol. 60 (2015) 9313
Figure 3. Angular distributions for the uniform source of secondary electrons for the Compton distribution and the uniform distribution.
Since a source with a uniform number of particles per unit mass emitted cannot be modelled in Penmain using the input file, three separate simulations have been performed, one for each body. The final result is calculated from a summation over all the three simulations, Dcav, i Dcav = (12) K Ki i
∑
Ki is calculated analytically from the number of simulated histories and the mass of the body. In the case where the conditions for the (reformulated) Fano theorem are violated, the ratios will deviate from 1. The Monte Carlo calculations have been performed for photon energies between 0.5 MeV and 4.0 MeV with steps of 0.5 MeV and for a B-field strength B = 1.5 T and B = 0.0 T. The simulations have been repeated for a geometry in which the cavity consists of normal density water, i.e. the whole geometry has a uniform density. Previous studies (Meijsing et al 2009, Reynolds et al 2013) on the response change of ionization chambers in the presence of a magnetic field have calculated the response change as a ratio between dose to the cavity with and without a B-field. A more solid approach is to calculate the ratios of calibration coefficients. Here the ratio, RB, of relative calibration coefficients Dcav(ρ = ρwater) / Dcav(ρ = ρvapour) with and without a B-field is calculated as an artificial response change of the cavity model (figure 2) in B-fields for a uniform radiation field. RB is calculated from the Dcav/K results for the simulations with B = 0.0 T and B = 1.5 T, and with two cavity densities (vapour and water) by, Dcav (B = 1.5 T, ρ Dcav (B = 0.0 T, ρ K cav = ρvapour) K cav = ρwater) RB = D ⋅ D (13) cav (B = 0.0 T, ρ cav (B = 1.5 T, ρ K K cav = ρvapour) cav = ρwater)
Input parameters. The other parameters that influence the transport of the electrons are set
in agreement with the criteria defined by Sempau and Andreo (2006). The absorption energy for electrons is set to 1 keV. No bremsstrahlung interactions have been simulated by setting the absorption energy of photons to the maximum energy in the system. The scattering parameters 9320
J A de Pooter et al
Phys. Med. Biol. 60 (2015) 9313
(C1 and C2) are set to 0.01 and the cut-off value for inelastic interactions to 1 keV. The maximum allowed step length in the cavity (see figure 2) is set to 0.1 mm. For each body the same material data file is used, with the exception that for the cavity the density is set to the density of air (for the simulations with density variation in the geometry). In this way PENELOPE treats only boundaries between two volumes with different materials (i.e. densities here) as a real boundary. Results The results for the Dcav/K ratios are presented in figure 4. In the remainder statistical uncertainties are presented as error bars with a coverage factor k = 2. For the results of the geometry with a homogenous density (figures 4(a) and (c)) the ratio is 1 within their statistical uncertainty, which shows the consistency of the Monte Carlo model both for the simulations without (a) and with (c) a B-field. In case of density variation and a non-isotropic angular distribution of the generated secondary electrons, the modified Fano theorem predicts that the fluence is not uniform over the geometry and that ratios Dcav/K different from unity are to be expected. This is confirmed by the results in figure 4(d) (Compton distribution and mono-directional distribution). Since only the simulations for the isotropic angular distribution will give information on the accuracy of PENELOPE for Monte Carlo simulations of radiation transport in the presence of magnetic fields, the results for these simulations have been calculated with a lower statistical uncertainty (
Search
Collections
Journals
About
Contact us
My IOPscience
Application of an adapted Fano cavity test for Monte Carlo simulations in the presence of Bfields
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 Phys. Med. Biol. 60 9313 (http://iopscience.iop.org/0031-9155/60/24/9313) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 128.233.210.97 This content was downloaded on 22/02/2016 at 07:00
Please note that terms and conditions apply.
Institute of Physics and Engineering in Medicine Phys. Med. Biol. 60 (2015) 9313–9327
Physics in Medicine & Biology doi:10.1088/0031-9155/60/24/9313
Application of an adapted Fano cavity test for Monte Carlo simulations in the presence of B-fields J A de Pooter1, L A de Prez1 and H Bouchard2 1
VSL, Thijsseweg 11, 2629 JA Delft, The Netherlands Acoustics and Ionising Radiation Team, National Physical Laboratory, Hampton Road, Teddington TW11 0LW, UK 2
E-mail: [email protected] Received 27 February 2015, revised 30 September 2015 Accepted for publication 15 October 2015 Published 18 November 2015 Abstract
With the advent of MR guided radiotherapy the relevance of Monte Carlo radiation transport simulations in the presence of strong magnetic fields (B-fields) is increasing. While new tests are available to benchmark these simulation algorithms for internal consistency, their application to known codes such as EGSnrc, PENELOPE, and GEANT4 is yet to be provided. In this paper a method is provided to apply the Fano cavity test as a benchmark for a generic implementation of B-field effects in PENELOPE. In addition, it is investigated whether violation of the conditions for the Fano test can partially explain the change in the response of ionization chambers in the presence of strong B-fields. In the present paper it is shown that the condition of isotropy of the secondary particle field (Charged Particle Isotropy, CPI) is an essential requirement to apply the Fano test in the presence of B-fields. Simulations in PENELOPE are performed with (B = 0.0 T) and (B = 1.5 T) for cylindrical cavity geometry. The secondary particle field consists of electrons generated from a mono-energetic source (E = 0.5–4.0 MeV) with a uniform source density and different angular distributions; isotropic, mono-directional, and Compton. In realistic photon fields the secondary radiation field has a non-isotropic angular distribution due to the Compton process. Based on the simulations for the Compton angular distribution (non-CPI), the response change of the cavity model in a uniform radiation field in the presence of B-fields is investigated. For the angular distributions that violate the CPI condition and B = 1.5 T, the deviations from 1 are considerable, which emphasizes the requirement of CPI. For the isotropic angular distributions obeying this requirement, both the results for B = 0.0 T and B = 1.5 T shows deviations from the predictions for E ⩾ 1.5 MeV with values up to 1.0% for E = 4.0 MeV. Nevertheless, 0031-9155/15/249313+15$33.00 © 2015 Institute of Physics and Engineering in Medicine Printed in the UK
9313
J A de Pooter et al
Phys. Med. Biol. 60 (2015) 9313
due to the high correlation in the deviation for B = 0.0 T and B = 1.5 T, the accuracy of the PENELOPE code for the simulation of the change in detector response in the presence of B-fields is within 0.3%. The effect of the B-field on the detector response for non-isotropic angular distributions suggests that violation of CPI is a major contribution to the response change of ionization chambers in the presence of B-fields. Keywords: Fano cavity test, magnetic fields, Monte Carlo radiation transport, dosimetry, reference dosimetry, MRI-guided radiotherapy, ionization chamber response (Some figures may appear in colour only in the online journal) Introduction With the advent of MR guided radiotherapy (Raaymakers et al 2004, Raaijmakers et al 2005, Fallone et al 2009) the interest in and the relevance of dosimetry in the presence of strong magnetic fields (B-fields) is increasing. Since no reference dosimetry standards are available for such conditions that are able to measure the dose to water in an absolute way in the presence of strong B-fields, attempts to adapt reference dosimetry is largely based on Monte Carlo simulations (Meijsing et al 2009, Reynolds et al 2013). A few investigations calculated response variations for Farmer-type chambers in clinical beams and compared them with relative measurements, leading to the conclusion that the response can be simulated within reasonable agreement with measurements. A number of Monte Carlo codes exist that include the influence of B-fields on the transport of the charged particles (Agostinelli et al 2003, Salvat et al 2011). Most Monte Carlo codes use an approximation to simulate the trajectories of the charged particles in the presence of a B-field in between two interactions (Salvat et al 2011). Several parameters can be used to set the accuracy of the simulated trajectories. In addition, the crossing of boundaries between volumes for curved trajectories is different than for straight trajectories. These two examples indicate that accurate Monte Carlo simulations in the presence of B-fields are not trivial and tests to benchmark the codes are needed. The Fano theorem (Fano 1954) plays an important role in conventional radiation dosimetry (i.e. without a B-field). It is based on an analytical solution to the Boltzmann transport equation for the secondary particle fluence of a uniform field in a phantom of uniform atomic properties with varying densities (Smyth 1986). Based on the analytical solution, the internal consistency of radiation transport simulations for a particular geometry can be assessed. Two major fields of applications are: in the development of cavity standards (La Russa and Rogers 2009, Yi et al 2006) and in the validation of Monte Carlo radiation transport codes for the simulation of ionization chambers and other detectors (Smyth 1986, Kawrakow 2000, Poon et al 2005, Sempau and Andreo 2006, Sterpin et al 2014). In the case of Sempau and Andreo (2006), it is used as a benchmark to estimate the influence of the set of simulation parameters, such as the maximum step size and the parameters for elastic scattering (Salvat et al 2011) on the transport algorithm self-consistency. Lately, it was shown that the classical Fano test is not applicable for radiation transport simulations in the presence of a B-field (Bouchard and Bielajew 2015). Based on the coupled Boltzmann transport equation, two general benchmarking tests have been proposed and their theoretical basis was mathematically proven (Bouchard et al 2015). 9314
J A de Pooter et al
Phys. Med. Biol. 60 (2015) 9313
In this paper, a method is provided to apply the first general Fano test of Bouchard et al (2015) to the case of a photon radiation field in the presence of an external B-field. This is done by Monte Carlo calculations in which only the secondary electron field is considered using the PENELOPE code. An alternative more intuitive proof is given for the required conditions for the Fano test in the presence of an external B-field. In addition, it is investigated whether violation of the conditions of the Fano test can partially explain the previously observed response changes of ionization chambers in the presence of strong B-fields (Meijsing et al 2009, Reynolds et al 2013). Materials and methods Extension of the Fano test conditions for the influence of B-fields
The classical Fano theorem was formulated as follows (Fano 1954): ‘In a phantom of given composition exposed to a uniform fluence of primary radiation, the fluence of the secondary radiation field is also uniform and independent of the density of the phantom as well as of the density variations from point to point.’ The Fano theorem for applications in the presence of B-fields can be formulated in the following way (Fano 1954, Bouchard et al 2015): ‘In a phantom of given composition exposed to a uniform and isotropic fluence of primary radiation in the presence of a B-field, the fluence of the secondary radiation is also uniform and isotropic, and independent of the density of the phantom and the B-field strength as well as of the density and B-field variations from point to point.’ The main difference with the classical Fano theorem is the extension with the condition for isotropy of the primary and secondary radiation fields, which we call charged particle isotropy (CPI). The proof for the extension of the conditions uses to a large extent the same arguments as for the classical formulation and starts from the general transport equation for secondary particle fields as given by (Fano 1954, Bouchard 2010) and illustrated in figure 1. →
→ → dϕe(r→, E , Ω) = ρ(r→)Se(r→, E , Ω) − ρ(r→)ϕe(r→, E , Ω) dl
+ρ(r→)
E0
∫E
dE′
∫0
E
dE′
∬4π dΩ′k(E′, E , Ω′ ⋅ Ω) →
→
∬4π dΩ′k(E , E′, Ω′ ⋅ Ω)ϕe(r , E′, Ω′) →
→
→
→
→
→
→
→ + → = ρ(r→) [Se(r→, E , Ω) − I − (1) e (r , E , Ω) + I e (r , E , Ω)] Please note, for the remainder only the electrons of the secondary particle field will be considered. The formulation for positrons follows an analogous way. With:
•
→
dϕe(→ r , E , Ω) dl
the left-hand side of equation (1) being the change in fluence for an infinitesi→
r and energy E, mally small step in the direction of motion Ω for electrons with position → → which is identical to ∇r ϕe in the classical formulation (13), i.e. without the presence of the B-field (see figure 1). energy of the secondary particles in the system. • E0 the maximum → • Se(→ r , E, Ω): the number of secondary electrons generated per unit of mass due to interac→ tions of the primary photons at position → r with energy E and direction Ω = (u, v, w ) (see figure 1). 9315
J A de Pooter et al
Phys. Med. Biol. 60 (2015) 9313
Figure 1. Illustration of the transport of secondary particles as given by equation (1) → for a fluence on position →r with direction Ω and energy E, perpendicular to a slab with thickness dl. The derivative of the fluence with respect to l is the change in fluence for an infinitesimally small step in the direction of motion, which equals the net effect of → → → the source term Se and the two interaction terms, I−e (→r , E, Ω) and I+ e ( r , E, Ω). Se are the secondary particles generated in the slab by primary particles. →
• ϕe(→ r , E, Ω): the fluence of the secondary electrons written as ϕe in the remainder for presentation purposes. → → • k (E ′, E , Ω′ ⋅ Ω)dE′dΩ′: the probability per unit of mass traversed that the electrons with → energy E and direction of motion Ω have an inelastic interaction with a resulting energy → E′ and a direction of motion Ω′ E → → → → → ′ ′ ′ → • I − e (r , E , Ω) = ϕ (r , E , Ω) ∫ dE ′ ∬ dΩ k (E , E , Ω ⋅ Ω), describing the loss of electron e
→
0
4π
fluence with direction Ω and energy E per unit of mass due to electron interactions (see figure 1). E0 → → → → → ′ ′ ′ ′ → ′ • I + e (r , E , Ω) = ∫E dE ′ ∬ dΩ k (E , E , Ω ⋅ Ω)ϕe(r , E , Ω ), describing the increase of elec4π
→
tron fluence with direction Ω and energy E per unit of mass due to electron interactions (see figure 1). The main difference with the proof of the classical theorem lies in the expression of the left-hand side of equation (1), which changes due to the presence of the B-field. The left-hand side of equation (1) can be rewritten as (according to the derivation in the appendix with as the end result equation (A.9)): ⎡→ → dϕe → → e ⎤ = Ω ⋅ ⎢∇r ϕe − ∇Ω ϕe × B(r→) ⎥ (2) dl meγ v ⎦ ⎣
→
With ∇r ϕe =
(
∂ϕe ∂ϕe ∂ϕe , , ∂x ∂y ∂z
) and ∇ ϕ = →
Ω e
∂ϕe Θ ∂Θ
+
1 ∂ϕe φ. sin Θ ∂φ
The proof consists of two steps.
In the first step, it is shown that for a uniform and isotropic primary radiation field in a phantom of uniform composition and density in the presence of a uniform B-field, the fluence of the secondary field is uniform and isotropic as well. In the second step, it is shown that the solution of the first step is the same as the solution for a phantom with non-uniform density and B-field, with the same properties as the primary radiation field. Step 1. In a uniform and isotropic primary radiation field the production of the second→
→
ary electron, Se(→ r , E, Ω), is uniform and isotropic as well, yielding Se(→ r , E, Ω) = Se(E). 9316
J A de Pooter et al
Phys. Med. Biol. 60 (2015) 9313
In a homogenous phantom with a homogenous B-field for any straight line l,
dϕe dl
= 0. There-
fore, from equation (1) it follows that, →
→
+ [(3) Se(E ) − I − e (E , Ω) + I e (E , Ω)] = 0 →
→
[Se(E ) − ϕ(E , Ω)µ(E ) + I + e (E , Ω)] = 0
which can be written as, →
→ I +(E , Ω) S (E ) ϕ (E , Ω) = e + e (4) µ (E ) µ (E ) →
From equation (1) it can be seen that, for the maximum energy in the system, E0, I + e→(E 0, Ω) = 0 and since the left term on the right-hand side of equation (4) does not depend on Ω →
(5) ϕ (E0, Ω) = ϕ(E0) In addition, by definition, →
→
+ I+ (6) e (E , Ω) = I e (E ) if ϕ(E ′, Ω) = ϕ(E ′) for all E ′ > E
Therefore, from equations (4), and (6), →
→
ϕ(E0 − dE , Ω) =
I +(E0 − dE , Ω) I +(E0 − dE ) Se(E0 − dE ) S (E − dE ) = e 0 + e + e = ϕ(E0 − dE ) µ(E0 − dE ) µ(E0 − dE ) µ(E0 − dE ) µ ( E 0 − dE )
(7) for a choice of dE → 0. The steps made in equations (4), (6), and (7) can be repeated for i = 1 ... n start→ ing with ϕ(E0 − (i − 1)dE , Ω) in equation (4). From this, it is concluded that → ϕ(E0 − (i − 1)dE , Ω) = ϕ(E0 − (i − 1)dE ) for any i 1, and therefore, →
(8) ϕ (E , Ω) = ϕ(E ) the fluence of the secondary particle field is isotropic as well. Step 2. To prove the extension of the conditions for a non-uniform B-field and density, it
has to be proven that the solution, ϕ(E ) (equation (8)) for the situation of a uniform isotropic primary radiation field in a phantom with a uniform composition and density is equal to the solution for equation (1) (i.e. for a phantom with a uniform composition with density variations in the presence of a non-uniform B-field). The left-hand side of equation (1) as given by equation (2) is only zero when the secondary electron field is uniform and isotropic. In that case equation (1) reduces to →
→
+ 0(9) = [Se(E ) − I − e (E , Ω) + I e (E , Ω)] ,
which is identical to equation (3). Hence, under the conditions of a uniform isotropic primary radiation field the solution for the homogeneous phantom, equation (8) is the solution for the phantom in which the B-field and the density varies as a function of the position (equations(1) and (2). Therefore the solution is unique. Monte Carlo model
The Fano test is applied to a Monte Carlo model using the Penmain routine from the PENELOPE 2011 code (Salvat et al 2011). This routine is modified to allow for simulation 9317
J A de Pooter et al
Phys. Med. Biol. 60 (2015) 9313
of the particle transport in the presence of a uniform B-field using the prescription in the Penfield package. Parameters are added to the input file to define the uniform B-field vector, and to set tolerances on the simulation of trajectories of charged particles between two interactions. Three parameters for the tolerances are used to put a maximum on the change in B-field, energy, and speed of the charged particle along the trajectory between two hard collisions. Note that the first tolerance parameter is not applicable in the case of a uniform B-field. If in a step movement of a single particle in the simulation the deviation on one of the abovementioned quantities is higher than its tolerance parameter, the step length of the particle is reduced to a value for which the deviations obey the tolerance parameters. In case a maximum allowed step length is defined in the input file by the parameter DSMAX, the minimum of the two limitations on the step length is applied. In the study of Bouchard et al (2015) it is shown that the energy deposited under Fano conditions in the presence of an arbitrary B-field is the same as in the absence of the B-field, which implies that the validity of the Fano test can be investigated with a similar approach as used by Sempau and Andreo (2006), in which only the secondary electron field is simulated. Geometry. The simulations are performed for a simplified geometry of a Farmer-type
i onization chamber. The geometry consists of three bodies: the cavity, the wall, and the phantom (see figure 2). The dimensions of the cavity are the same as for a Farmer-type ionization chamber with a cavity diameter of 3.15 mm and a cavity length of 24.1 mm. The thickness of the wall is 0.35 mm. To avoid boundary effects, the size of the phantom is such that the minimum distance between the cavity and edges of the phantom is larger than 1.2 times the CSDA range of the electrons. The material for the phantom is set to water. The material for the cavity is set to water vapour. To avoid influence of the density effect parameter, the same material data file as for water is used with the density modified to the density of water vapour, in this way keeping the density effect parameter constant. The orientation of the cavity with respect to the B-field is shown in figure 2. Implementation of the adapted Fano test and modelling of the secondary particle field. For B = 0 several approaches exist to implement the Fano test: the regeneration technique (Kawrakow 2000), the re-entrance technique (Yi et al 2006), and the technique in which only the secondary particle field is modelled as used by Sempau and Andreo (2006). Here the Fano test adapted for application in the presence of B-fields is implemented based on the Sempau approach by modelling only the secondary particle field. In the section ‘Extension of the Fano test conditions for the influence of B-fields’ it is shown that a uniform and isotropic production per unit mass of secondary particles yields a uniform and isotropic secondary particle fluence, ϕe, regardless of spatial variations in density and magnetic field (strength and direction). This is a direct consequence of the cancelation of → ∇Ω ϕe in equation (2). As a result of the uniform ϕe, the absorbed dose in an infinite phantom of uniform atomic composition is uniform and equal to the energy of the secondary particles produced per unit mass, K. In photon fields the produced secondary electrons have in general a non-isotropic directional distribution due to the Compton process. This distribution varies as a function of the photon energy. Non-isotropic secondary particle production will not cancel → the ∇Ω ϕe term in equation (2). Therefore, it is expected that the spatial distribution of ϕe will depend on spatial variations in density and magnetic field strength. The extent of these spatial variations in ϕe for non-isotropic distributions is investigated by simulating two additional non-isotropic angular distributions. The three different angular distributions for the starting direction of the electrons that have been simulated are: 9318
J A de Pooter et al
Phys. Med. Biol. 60 (2015) 9313
Figure 2. Geometry of the cavity, wall, and the phantom as well as their orientation with respect to the beam axis (z-axis) and the B-field.
• Isotropic distribution, which obeys the CPI condition and which corresponds to an isotropic photon field with an artificial production process in which the total photon energy and momentum is transferred to the secondary electron. • Compton distribution, which corresponds to the production of electrons due to the Compton interactions of a mono-directional uniform photon field. • Mono-directional, which corresponds to a mono-directional photon field with an artificial production process in which the total photon energy and momentum is transferred to the secondary electron. The angular distributions of the secondary electrons produced for the Compton distribution and the isotropic distribution are given in figure 3 as a function of the angle θ between the direction of the primary photon and the secondary electron. θ is the angle with the z-axis (figure 2). The Compton distribution was implemented in Penmain using the Source routine. The other two angular distributions can be generated in a straightforward way using the Penmain input file. Dose calculation. The secondary electron field is modelled as a uniform source over the →
geometry scaled with the local mass density, in such a way that Se(Ω, E ) is constant. The result of the calculation is the ratio between the dose absorbed in the cavity, Dcav and the energy of the secondary particles produced per unit mass, K, in the source, which is given by, K = E Se (10)
The average energy of the produced secondary particles is calculated from µ tr
ρ
E = hν µ (11) ρ
To keep the model consistent the ratio between µtr ρ and µ ρ was separately calculated for the Compton angular distribution with the same Monte Carlo code for each energy. For the other angular distributions (isotropic, mono-directional) this ratio is 1. 9319
J A de Pooter et al
Phys. Med. Biol. 60 (2015) 9313
Figure 3. Angular distributions for the uniform source of secondary electrons for the Compton distribution and the uniform distribution.
Since a source with a uniform number of particles per unit mass emitted cannot be modelled in Penmain using the input file, three separate simulations have been performed, one for each body. The final result is calculated from a summation over all the three simulations, Dcav, i Dcav = (12) K Ki i
∑
Ki is calculated analytically from the number of simulated histories and the mass of the body. In the case where the conditions for the (reformulated) Fano theorem are violated, the ratios will deviate from 1. The Monte Carlo calculations have been performed for photon energies between 0.5 MeV and 4.0 MeV with steps of 0.5 MeV and for a B-field strength B = 1.5 T and B = 0.0 T. The simulations have been repeated for a geometry in which the cavity consists of normal density water, i.e. the whole geometry has a uniform density. Previous studies (Meijsing et al 2009, Reynolds et al 2013) on the response change of ionization chambers in the presence of a magnetic field have calculated the response change as a ratio between dose to the cavity with and without a B-field. A more solid approach is to calculate the ratios of calibration coefficients. Here the ratio, RB, of relative calibration coefficients Dcav(ρ = ρwater) / Dcav(ρ = ρvapour) with and without a B-field is calculated as an artificial response change of the cavity model (figure 2) in B-fields for a uniform radiation field. RB is calculated from the Dcav/K results for the simulations with B = 0.0 T and B = 1.5 T, and with two cavity densities (vapour and water) by, Dcav (B = 1.5 T, ρ Dcav (B = 0.0 T, ρ K cav = ρvapour) K cav = ρwater) RB = D ⋅ D (13) cav (B = 0.0 T, ρ cav (B = 1.5 T, ρ K K cav = ρvapour) cav = ρwater)
Input parameters. The other parameters that influence the transport of the electrons are set
in agreement with the criteria defined by Sempau and Andreo (2006). The absorption energy for electrons is set to 1 keV. No bremsstrahlung interactions have been simulated by setting the absorption energy of photons to the maximum energy in the system. The scattering parameters 9320
J A de Pooter et al
Phys. Med. Biol. 60 (2015) 9313
(C1 and C2) are set to 0.01 and the cut-off value for inelastic interactions to 1 keV. The maximum allowed step length in the cavity (see figure 2) is set to 0.1 mm. For each body the same material data file is used, with the exception that for the cavity the density is set to the density of air (for the simulations with density variation in the geometry). In this way PENELOPE treats only boundaries between two volumes with different materials (i.e. densities here) as a real boundary. Results The results for the Dcav/K ratios are presented in figure 4. In the remainder statistical uncertainties are presented as error bars with a coverage factor k = 2. For the results of the geometry with a homogenous density (figures 4(a) and (c)) the ratio is 1 within their statistical uncertainty, which shows the consistency of the Monte Carlo model both for the simulations without (a) and with (c) a B-field. In case of density variation and a non-isotropic angular distribution of the generated secondary electrons, the modified Fano theorem predicts that the fluence is not uniform over the geometry and that ratios Dcav/K different from unity are to be expected. This is confirmed by the results in figure 4(d) (Compton distribution and mono-directional distribution). Since only the simulations for the isotropic angular distribution will give information on the accuracy of PENELOPE for Monte Carlo simulations of radiation transport in the presence of magnetic fields, the results for these simulations have been calculated with a lower statistical uncertainty (
