Inverse square corrections for FACs and WAFACs

Applied Radiation and Isotopes 153 (2019) 108638

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Inverse square corrections for FACs and WAFACs

T

D.W.O. Rogers Carleton Laboratory for Radiotherapy Physics, Physics Dept, Carleton University, Ottawa, K1S 5B6, Canada

A R T I C LE I N FO

A B S T R A C T

Keywords: Air-kerma standards Free-air chambers Monte Carlo EGSnrc egs_brachy

Inverse square correction factors for wide-angle free-air chambers (WAFACs) and free-air chambers (FACs) for cylindrical, conical and square-prism detectors are required for determining the on-axis air kerma from measurements or Monte Carlo calculations made with these different shaped detectors. Values of air kerma measured with these detectors use an effective volume technique related to the inverse square correction factors. This paper presents these factors in a consistent framework and the relationships between them are made clear. Using Monte Carlo simulations, the various corrections and techniques are shown to be accurate within a statistical precision of about 0.04% or better with the exception of the published correction for square prism detectors which is shown to hold only for thin detectors which have an opening angle corresponding to the NIST and NRCC WAFAC primary standards. A more accurate correction for square prism detectors is presented which properly averages 1/ d 2 rather than d 2 where d is the distance away from the source.

1. Introduction

Carlo verification clarifies the techniques and uncovers one incorrect previous approach. For a measurement standard the correction factors studied here are only a subset of the corrections needed. Other corrections needed include those for photon attenuation, photon scatter, electron loss, aperture penetration etc (see, e.g., Mainegra-Hing et al. (2008) and references therein). None of these are relevant to the present study which is for an idealized model which avoids the need for all these other corrections.

When doing Monte Carlo calculations related to, or measurements with a Free-Air Chamber (FAC) or Wide-Angle Free-Air Chamber (WAFAC) it is necessary to correct the results to give the air kerma on the axis at the front of the detector's aperture rather than averaged over the sensitive region of the detector. There are two effects to take into account. The first is the longitudinal 1/ r 2 drop-off of both the photon fluence and air kerma. Experimentally this is accounted for using an effective volume rather than the actual volume of the irradiated air. As will be shown below, assuming a point source of radiation, using the effective volume exactly cancels the longitudinal 1/ r 2 effects to give the air kerma at the front of the detector. The second effect, which is only significant for WAFACs, is the fact that the photon fluence and hence air kerma are not uniform over the front face of the detector, again because of the 1/ r 2 effects. A correction factor must be used to convert the average air kerma over the detector face to that on the detector's axis. The purposes of this paper are: i) to rigorously demonstrate that the effective volume technique is valid; ii) to derive the various correction factors for use with Monte Carlo studies which calculate the average air kerma in the sensitive volume of different shaped detectors rather than using the effective volume technique; iii) to establish the required correction factor to give the on-axis air kerma from the aperture-averaged air kerma; and iv) to verify the specific formulae developed are accurate by using very-high precision simulations of simplified FAC and WAFAC geometries. With one exception, the specific formulae developed are not new but a systematic derivation together with the Monte

2. Theoretical framework 2.1. Effective volume technique for FACs Since the 1920s, authors have shown that measuring the charge released in the sensitive volume of an FAC is related to the air kerma at the front of the FAC's aperture by considering the charge released to be from a cylinder with the radius of the aperture and length of the actual sensitive region (see, e.g., Taylor (1930) and references therein and Attix (1986)). The standard derivation ignores attenuation in the air, assumes a perfect point source of radiation and ignores minor effects such as photon scattering, electron loss etc, all of which are corrected for separately. ¯ 0 , entering an aperture of Consider an average photon fluence, Φ ¯ 0 A0 radius R 0 and area A0 at distance s0 from the point source with Φ photons entering the aperture. Ignoring any off-axis variation, the ¯ , decreases as 1/s 2 where s is the distance to the point photon fluence, Φ

E-mail address: [email protected]. https://doi.org/10.1016/j.apradiso.2019.03.008 Received 14 December 2018; Received in revised form 6 February 2019; Accepted 4 March 2019 Available online 09 March 2019 0969-8043/ © 2019 Elsevier Ltd. All rights reserved.

Applied Radiation and Isotopes 153 (2019) 108638

D.W.O. Rogers

source. At the same time, the area irradiated, A (s ) , increases as s 2 . At any given distance s, 2

s A (s ) = A0 ⎛ ⎞ ⎝ s0 ⎠ ⎜

2

¯ (s ) = Φ ¯ 0 ⎛ s0 ⎞ . Φ ⎝s⎠

and

⎟

(1)

Hence: 2

2 ¯ (s ) A (s ) = Φ ¯ 0 ⎛ s0 ⎞ A 0 ⎛ s ⎞ = Φ ¯ 0 A0 , Φ ⎝s⎠ ⎝ s0 ⎠ ⎜

⎟

(2)

i.e., attenuation and scatter being ignored, the total number of photons is a constant which is how it is known the fluence decreases as 1/ s 2 in the first place. The exposure, X, is dQ/ dm where dQ is the charge of one sign released in a mass of dry air, dm . It is related to the air kerma, K, by X = K (1 − g¯)(e / W )air where (e / W )air is the charge released in air per energy deposited (units C/J) and K (1 − g¯) = Ψ(μen / ρ)air with the energy fluence Ψ = E Φ . For x rays, g¯ , the fraction of kinetic energy lost to bremsstrahlung by electrons set in motion is less than 3 × 10−4 and is ignored here, consistent with ignoring other corrections of this magnitude since they are much less than experimental uncertainties, and so K = Ψ(μen / ρ)air . Consider a small element of charge, dQ0 = dmX , released in a thickness ds in the aperture:

μ e μ e dQ0 = dm 0 Ψ0 ⎜⎛ en ⎟⎞ = ρA 0 ds Ψ0 ⎜⎛ en ⎟⎞. ⎝ ρ W⎠ ⎝ ρ W⎠

Fig. 1. Schematic defining various angles and dimensions for a cylindrical sensitive region. For a conical geometry, θ ≤ θm in the upper panel and R would d refer to the radius at the front of the detector (= d + L R in this figure).

detector means that the lateral variation due to 1/ r 2 effects must be accounted for. The following derivation is based on that of Culberson (2006) which in turn was based on that of Seltzer et al. (2003). WAFAC detectors differ from FACs in that the charge is collected with an electric field parallel to the beam axis whereas in an FAC the electric field is perpendicular to the axis. In the WAFACs at NIST and NRCC, the collecting volume includes the entire conical region irradiated by the photon beam whereas in the PTB's GROVEX detector (Selbach et al., 2008) the collecting region is a cylinder on the beam axis which is entirely contained within the conical region being irradiated. In both cases it is important to account for the longitudinal and lateral 1/ r 2 effects. As seen above, the charge released and the air kerma are proportional to the average photon fluence in a region. Concentrating on the collecting region of the detector, and ignoring for the moment the aperture collimating the beam, then for a cylindrical collecting region, y (θ) , the pathlength in the collecting volume for a photon at angle θ with respect to the central axis is in two parts, depending on whether the photon hits the side or back of the collecting volume (see Fig. 1). For θ < θm , the photon hits the back of the collecting volume and y (θ) = L/cos θ as seen in Fig. 1's lower panel. For θm < θ < θc , the photon hits the side of the cylinder and y (θ) = R/sin θ − d/cos θ . For a conical collecting volume, only the first component (θ < θm ) contributes to the pathlength. For an isotropic point source, the probability of being within the solid angle dΩ is 1/4π where d Ω= sin θdθdϕ = −d (cos θ) dϕ . Integrating over the available angles, the total pathlength in the detector's collecting volume is

(3)

Further, consider Ψ(s ) and dm (s ) , the values of the quantities at the distance s on the axis from the point source and ignore the increased distance when the point is off axis (it is actually s 2 + r 2 where r is the distance off-axis but in a typical FAC this is s within 0.005%).

s 2 Ψ(s ) = Ψ0 ⎛ 0 ⎞ ⎝s⎠

2

s dm (s ) = ρA (s ) ds = ρA 0 ds ⎛ ⎞ . ⎝ s0 ⎠ ⎜

⎟

(4)

Hence the charge released in a thin slab at s is 2

μ e s s 2 μ e dQ (s ) = dm (s )Ψ(s ) ⎜⎛ en ⎟⎞ = ρA 0 ds ⎛ ⎞ Ψ0 ⎛ 0 ⎞ ⎜⎛ en ⎞⎟ ρ W s s ⎠ ⎝ ρ W⎠ ⎝ 0 ⎝ ⎠ ⎝ ⎠

(5)

μ e = ρA 0 ds Ψ0 ⎛⎜ en ⎞⎟, ⎝ ρ W⎠

(6)

⎜

⎟

and the total charge in the sensitive region from s1 to s2 = s1 + L is

Q=

∫s

s1+ L

1

μ e μ e ρA0 Ψ0 ⎜⎛ en ⎟⎞ ds = ρA0 Ψ0L ⎜⎛ en ⎟⎞. ρ W ⎝ ⎠ ⎝ ρ W⎠

(7)

By treating this charge as coming from a cylinder of air with the radius of the aperture and length of the sensitive volume, i.e., with an effective volume given by

Veff,cyl = πR 02 L = A0 L,

(8)

the mass of that air is ρAo L and hence:

Q = m

ρA0 Ψ0L

(

μen e ρ W

ρA0 L

) = Ψ ⎛μ

en e ⎞ ⎟, ⎝ ρ W⎠

0⎜

y¯ =

(9)

−1 ⎡ ⎢ 4π ⎣

2π

∫0 ∫0

cos θm

L d (cos θ) dϕ cos θ

which is the exposure at the front aperture. This is the standard approach used with FACs and the use of this effective volume rather than the true volume of the sensitive region cancels the 1/ r 2 drop-off in the fluence. Issues of the kerma on-axis vs the average over the area of the aperture still occur but is a negligible effect for FACs.

where the right hand term is not needed for a conical collecting volume. Consider the first term:

2.2. A rigorous derivation for WAFACs

y¯1 =

+

2π

cos θc

∫0 ∫cos θ

m

⎛ R − d ⎞ d (cos θ) dϕ ⎤, ⎥ cos θ ⎠ ⎝ sin θ ⎦

2π cos θm −L 1 d (cos θ) dϕ 0 0 4π cos θ −2πL = (ln(cos θm) − ln(cos(0))) 4π

The above derivation ignores the lateral variation in photon fluence and thus does not apply exactly to WAFACs where the diameter of the 2

(10)

∫ ∫

(11)

Applied Radiation and Isotopes 153 (2019) 108638

D.W.O. Rogers

=

−L ln(cos θm) 2

=

−L ⎛ 1 ⎞ ln ⎜ 2 ⎟ 2 ⎝ 1+β ⎠

=

L ln(1 + β 2), 4

Vcone = (12)

=

−2π 4π

−1 4π

2π cos θ ∫0 ∫cos θmc ( sinR θ −

d cos θ

) d (cos θ) dϕ

[R arcsin(cos θ) − d ln(cos

cone krsq , R, L =

θ)]θθcm

(13)

−1 ⎡ ⎛ π π ⎛ 1 + α2 ⎞ ⎤ − θc − ⎛ − θm ⎞ ⎞ + d ln ⎜ = ⎢R 2 ⎟⎥ 2 2 ⎝2 ⎠⎠ ⎝ 1 + β ⎠⎦ ⎣ ⎝

=

R d 1 + α2 ⎞ ⎟, (θc − θm) − ln ⎛⎜ 2 4 ⎝ 1 + β2 ⎠

⎜

⎜

(14)

krsq, R,0 ≡ k r 2 = 2.2.1. Corrections for cylindrical collecting volumes The average photon fluence in the detector's collecting volume (assuming no attenuation) is

y¯1 + y¯2 1 L R = ⎛ 2 ⎞ ⎡ ln(1 + β 2) + (θc − θm) 4 2 Vcyl ⎝ πR L ⎠ ⎢ ⎣

NIST krsq , R, L =

α2

1 . 4πd 2

NIST krsq , R, L =

ϕ (d) ¯ cyl Φ R, L

= Lα 2

=

⎜

=

⎡ln(1 + β2) + 2R (θ − θ ) − d ln ⎛ 1 + α2 ⎞ ⎤ c m ⎢ L L ⎝ 1 + β2 ⎠ ⎥ ⎣ ⎦ ⎜

⎟

⎟

,

α2 . ln(1 + α 2)

α2 , ln(1 + α 2)

α = R/d, (23)

α2 , ln(1 + α 2)

(24)

α2 L 2 ⎛1 + ⎞ . ln(1 + α 2) ⎝ d⎠

(25)

(26)

2.3. An alternative and more exact derivation for FACs (19) Section 2.1 showed that use of an effective volume in an FAC led to determining the air kerma at the front of the aperture. This derivation ignored the off-axis variation of the fluence and the fact that the photon pathlength in the sensitive region was L/cos θ rather than just L. Given the general results for conical chambers derived above, a somewhat more complete derivation for FACs is possible. Using Fig. 2’s notation, the FAC's effective volume, as defined in eqn (8), is

where α = R/ d , β = R/(L + d ) , θc = arctan α and θm = arctan β (see Appendix). In the limit L → 0 , the correction is just for the variation across the front face of the detector. In this limit, α = β and hence θc = θm and by going back to eqn (12) and (15) it can be shown that the limit leads to the correction factor: cyl krsq , R,0 =

∫

=

If this approach is carried through in the measurement's analysis, the actual volume used cancels out, but using the larger volume means that eqn (22) cannot be used to correct Monte Carlo calculations as done below unless the Monte Carlo calculations also use the larger volume.

⎟

⎡L ln(1 + β2) + 2R (θ − θ ) − d ln ⎛ 1 + α2 ⎞ ⎤ c m 2 ⎥ ⎢ ⎝ 1 + β ⎠⎦ ⎣ α2

1 d2 R 2πrdr 1 πR2 0 (r 2 + d2)

R2 = R (1 + L/ d ).

πR2L 1 2 4πd2 ⎡ L 2) + R (θ − θ ) − d ln ⎛ 1 + α ⎞ ⎤ + β ln(1 c m ⎢4 2 4 ⎝ 1 + β2 ⎠ ⎥ ⎦ ⎣ 1 ⎜

(22)

This differs from eqn (22) above, which notionally should be equivalent. This is because the NIST derivation does not use the actual volume of the conical collecting region in the WAFAC but uses the larger cylindrical volume from which charge is collected, i.e., πR22 L , where R2 is the radius at the back of the detector so

(18)

cyl Hence, krsq , R, L , the correction factor from the average fluence in the detector's cylindrical collecting volume to that on-axis at the front face is cyl krsq , R, L =

α = R/d.

i.e., all that is being corrected for is the variation in the fluence across the front face of the detector as in eqn (20). In Seltzer et al. (2003), their eqns (15) and (16) can be multiplied together to give:

(17)

The quantity of interest is the fluence on the axis of the detector at its front face. For an isotropic point source, the fluence per initial particle at a distance d is

ϕ (d ) =

⎟

2.2.2.1. NIST's approach. Rather than use the actual volume in the WAFAC that is irradiated by the source, i.e., eqn (21), NIST makes use of an effective volume, similar to the FAC case (Seltzer et al., 2003). Here Veff,WAFAC = πR2L (with R the radius of the aperture) and using this to establish the average fluence and hence air kerma in the detector leads to:

where V is the collecting region's volume. For a cylinder, Vcyl = πR2L . The average fluence in the cylindrical collecting volume is given by combining eqns (12), (15) and (16) to give

d ⎛ 1 + ⎞⎤ ln ⎜ . ⎟ 4 ⎝ 1 + β2 ⎠ ⎥ ⎦

(21)

which, as required, is the L = 0 limiting case of eqns (19) and (22).

(16)

−

⎟

2 ϕ (d ) α2 ⎛1 + L + L ⎞, = 2) 2 ¯ cone Φ ln(1 3 α d d + R, L ⎝ ⎠

where α = R/ d and θc = arctan α and θm = arctan β .

¯ cyl Φ R, L =

⎜

As expected, if the collecting volume has 0 thickness, the limit L → 0 leads to the same result as eqn (20) for krsq, R,0 in the cylindrical case in the same limit. If the sole purpose is to account for the variation across the front face/aperture of the conical or cylindrical detector, this can be done directly by accounting for the average value of the distance to the point source (Daskalov and Williamson, 2001), i.e.,

(15)

¯ = y¯ , Φ V

⎟

where R is now the radius of the front face of the conical collecting ¯ cone volume. The average fluence in the conical region is Φ R, L = y¯1 / V (with y¯1 from eqn (12)), from which the correction to the fluence on-axis at the front face of the detector is

where β = R/(L + d ) and some basic mathematical relationships are used from the Appendix. Consider the 2nd term in eqn (10):

y¯2 =

π ⎛ R2 ((d + L)3 − d3) ⎞ L L2 ⎞ = πR2L ⎛1 + , + 2 3⎝ d d 3d 2 ⎠ ⎝ ⎠

(20)

x 2 Veff,cyl = πR 02 L = πR2L ⎛1 − ⎞ , d⎠ ⎝

2.2.2. Corrections for conical collecting volumes For a conical detector's collecting volume, θc = θm and α = β and hence y¯2 (eqn (15)) is identically zero. The volume of the detector's conical region is

(27)

where R 0 is the radius of the aperture and R is the radius at the front of cone the FAC's actual sensitive region. Eqn (22) for krsq , R, L corrects the average fluence in the conical region of the FAC (determined using the 3

Applied Radiation and Isotopes 153 (2019) 108638

D.W.O. Rogers

extending for the length of the sensitive region of the FAC so that

Veff,cyl = πR 02 L = πR2L ⎛ ⎝ =

d − x ⎞2 d − x ⎞2 = Vcyl ⎛ , d ⎠ ⎝ d ⎠

i. e. ,

1 Vcyl

2 ⎛d − x ⎞ , Veff,cyl ⎝ d ⎠

1

(36)

Substituting into eqn (34);

K ap =

Fig. 2. Exaggerated schematic of a standard FAC geometry. The FAC's aperture has a radius R 0 at a distance d − x from the source. The aperture has a thickness of t cm. The FAC's sensitive region is a distance x past the front of the aperture and hence at a distance d from the point source. The FAC's sensitive region has a length L.

=

⎜

For air-kerma strength calculations with the BrachyDose code (Taylor et al., 2007; Rodriguez and Rogers, 2013) a squared-faced rectangular prism was used as the detector/scoring region with roughly the opening angle of the NIST WAFAC (2.66 × 2.66 cm square, 0.05 cm thick at 10 cm from the source). This was because the BrachyDose code did not have the ability to model a conical detector and modelling a squared-faced rectangular prism was straightforward. To correct to the on-axis value of the kerma, the correction used was:

Dividing the conical region's true volume (eqn (21)) by the volume of a cylinder with the same radius as the front of the conical sensitive region, πR2L , gives:

Vcone L L2 ⎞ = ⎛1 + + . Vcyl d 3 d2 ⎠ ⎝ ⎜

⎟

(29)

Hence

kr2 = α2 Vcone ⎛ d ⎞2 . ln(1 + α 2) Vcyl ⎝ d − x ⎠

kFAC =

Edep ρVcone

,

K ap =

(31)

cone,

k sq2 − prism = r

(32)

=

α2 Vcone ⎛ d ⎞2 Edep 2 ln(1 + α ) Vcyl ⎝ d − x ⎠ ρVcone

2 α2 ⎛ d ⎞ , ρVcyl ln(1 + α 2) ⎝ d − x ⎠

w /2

. (39)

2.5. Off-axis considerations Except for eqn (38), all the derivations above have assumed a point source on the axis of the various detectors. In practice, brachytherapy seeds are 3–4 mm long and x-ray tubes can have beam spots of the order of a few mm in diameter. Both Taylor (1930) and Aitken (1958) have investigated the effect of these off-axis sources of radiation and shown them to have a negligible impact on the analysis. Similarly the k r 2 correction of Rodriguez and Rogers (2013) included the effect of the length of the brachytherapy seed's active length but this was only 0.03% for a 6 mm long seed (about twice the length of most seeds).

(34)

πR2L

where recall that Vcyl = corresponds to a cylinder with the same length as the sensitive region and same radius as the front of the sensitive conical region so that

d R0, d−x

w /2

∫−w /2 ∫−w /2 ∫

(33)

Edep

R=

1 w 2L

1 d2 d+L 1 dxdydz d (x 2 + y 2 + z 2)

This does not appear to have a closed form solution, even for the case of L = 0 but can be numerically determined. As will be shown below, for situations in which d ≫ w/2 and d ≫ L , the distinction between averaging d 2 or 1/ d 2 is quite small and eqn (38) is still sufficiently accurate.

Hence, using eqn (30):

K ap =

(38) 2

where kr2 represents the ratio to d of the average distance r between a vertical line source of length h centered on the origin and the scoring volume with its front face at a distance d from the origin, and with w and L the width and thickness of the scoring volume, respectively. This was thought to be comparable to krsq,R,Lcone in eqn (22) except applying to a detector with a square face. Unfortunately this equation is incorrect because it was determining the mean value of d 2 whereas what is actually needed is the mean value of 1/ d 2 . For a point source this is

(30)

where Edep is the energy deposited in the sensitive conical region. Even in a worst case scenario for an FAC with a 2 cm diameter aperture as close as 50 cm from the source, the correction between the average kerma over the front of the aperture vs on the axis is 0.02% (eqn (20)) so when using practical FACs with an even smaller correction, it is assumed that the kerma on the axis is equal to the average kerma in the aperture region. By definition of kFAC

kFACK

1 ⎡ h2 2w 2 L2 ⎤ , + + d 2 + dL + d2 ⎢ 12 12 3⎥ ⎣ ⎦ 2

The average air kerma in the cone is

K cone =

(37)

⎟

(28)

= R/d.

α = R/d.

2.4. The BrachyDose approach

2 2 d ⎞2 α2 ⎛1 + L + L ⎞ ⎛ d ⎞ , α = 2) 2 d − x ln(1 + α d 3 d d − x ⎝ ⎠ ⎠ ⎝ ⎠⎝

cone ⎛ krsq , R, L

α2 ρVeff,cyl ln(1 + α 2)

Eqn (37) states that the air kerma at the front of the aperture is given by the energy deposited in the conical sensitive region of the FAC divided by the mass of air in a cylinder of the same sensitive length but with the radius of the aperture along with an additional correction factor of α 2/ln(1 + α 2) . Except for the α 2 factor, this is the standard result used with FACs (derived in section 2.1). In the limit of α = R/ d → 0 , this α 2 factor is 1. In a typical FAC's worst case (i.e., when it is closest to the source, say 50 cm) and for the sensitive region starting 24 cm past a 10 mm aperture, the α 2 factor differs from 1 by 18 ppm.

actual volume) to give the fluence on the axis at distance d from the source. This then can be corrected by a simple 1/ r 2 correction to give the fluence at the front of the aperture, i.e., at distance d − x from the source. Thus the correction from the average air kerma in the FAC's sensitive volume to the on-axis air kerma at the front aperture is given by kFAC where:

kFAC

Edep

(35)

where R 0 is the radius of the aperture (Fig. 2), d − x is the distance from the source to the front of the aperture and d is the distance to the front of the sensitive region from the source. As in eqn (8), define an effective cylinder of air starting at the position of the front of the aperture and

3. Methods To verify the accuracy of the expressions derived above, a series of 4

Applied Radiation and Isotopes 153 (2019) 108638

D.W.O. Rogers

and (22). These corrected values (col 5) should give the air kerma onaxis at the front face of the detector. To the extent that the correction factors are accurate, the values in column 5 for a given distance should all be the same. As can be seen by the normalized values in column 6, the k rsq,R,L values for the cylindrical and conical detectors are accurate within the statistics of the order of 0.02%–0.06%. Furthermore, scaling the on-axis air kerma by a simple 1/ r 2 factor of 9, the estimated on-axis air kerma values from detectors at 30 cm agree within 0.04% with those at 10 cm. In addition, the absolute values agree within 0.11% with a simple analytic calculation of the air kerma per initial photon obtained by multiplying the energy fluence per initial photon by the air mass energy absorption coefficient for the initial photon spectrum as calculated by EGSnrc. For detectors at 10 cm distance from the source, the difference between the correction factors for a 0.05 cm thick detector and a 0.0001 cm thick detector are initially surprisingly large at 0.5% but this is just the longitudinal 1/ r 2 effect from (10.025/10.0)2 = 1.005 where the mid-point has been used in the estimate. The last entry in Table 1 is an extreme case of a very broad but thin detector which demonstrates the accuracy of eqn (20) and (23) for the correction for the 0 length detector.

Monte Carlo calculations of the WAFAC simplified geometry were done with the egs_brachy code (Chamberland et al., 2016). An isotropic point source of bare 125I is at distances of 10 or 30 cm from the front face of detectors that are a cylinder, a cone or a square prism. The geometry is all vacuum except for the detector's sensitive region which has a very low-density air in it (10−13 g/cm3). This means there is negligible photon attenuation in the simulation. The air kerma in the actual detector volume was scored using egs_brachy's tracklength scoring option based on mass energy absorption coefficients calculated based on the XCOM photon cross sections (Berger et al., 2010). The appropriate correction from eqns (19) and (22), (38) or (39) for the cylindrical, conical or (2) square prism cases respectively were applied to give the air-kerma values on the axis at the front face of each detector. Several calculations for WAFAC chambers were done with the point source deliberately off-axis by 2–10 mm. The calculated average air kerma in the detector volumes were compared. Similar calculations were done for an FAC geometry with distances to the aperture of 200 cm or 20 cm, an aperture radius of 0.5 cm (relatively large for FACs) or 0.333 cm and the sensitive region starting 17, 24 or 34 cm past the aperture's front face. The air kerma was scored in the FAC's sensitive region, K cone (based on the actual volume), and in the aperture region, K ap . To verify eqn (37), use Edep = K cone ρVcone and rework the equation to give:

4.1.2. Square prism detectors Fig. 3 presents the values of k r 2 for a square prism detector as calculated using eqn (38) based on averaging d2 throughout the detectors or the correct eqn (39) which averages 1/ d 2 . When the values are plotted as a function of detector width for the standard thickness used for BrachyDose and egs_brachy calculations of air-kerma strength, i.e., 0.05 cm, it is seen that the error in eqn (38) is less that 0.03% for detectors as wide as 4 cm. Conversely, if the standard width for these same calculations is used, i.e., 2.66 cm (which has the same opening angle when at 10 cm from the source as the NIST WAFAC), the ratio of correct to incorrect equation values is within 0.06% of unity for thicknesses up to 0.4 cm. Consistent with the results in Fig. 3, the results in Table 2 for the square prism detectors show that in general the correction factors given by eqn (38) break down for thicker or wider detectors. However, for the detectors with thickness ≤0.05 cm, the predicted on-axis kerma agrees with the comparable values from the cylindrical and conical detectors

K ap

V α2 / cone = ≈ 1.000. K cone Veff,cyl ln(1 + α 2)

(40)

Since all the quantities on the left hand side of eqn (40) are known from the simulation, eqn (37) is verified if the left hand side equals 1.0 since the right hand side is 1.0000 at the 0.02% level in a worst case FAC geometry. 4. Results 4.1. WAFAC detectors 4.1.1. Cylindrical and conical detectors Table 1 presents the results of the egs_brachy calculations of the air kerma in cylindrical and conical WAFAC detectors and their corrections by the 2 different k rsq,R,L factors for the respective shapes, i.e., eqn (19)

Table 1 Monte Carlo calculated average air kerma in cylindrical or conical WAFAC detector shapes irradiated by a bare point source of 125I photons. There is no photon attenuation. The raw results are multiplied by the appropriate correction factors from eqn (19) and 22 to give the air kerma on-axis at the front of the detector. The equality of the normalized values show the accuracy of the conical and cylindrical correction factors. Statistical uncertainties in the last digit are shown in brackets. K det krsq,R,L × 1016

Normalized

d, R, L

K det × 1016

Vdet

cm

Gy/photon

cm3

Cylindrical detector 10, 0.0564, 0.0001 10, 0.0564, 0.05 10, 0.0564, 5.0 10, 1.5, 0.0001 10, 1.5, 0.05 10, 1.5, 5.0

6.676 6.643 4.453 6.601 6.568 4.416

9.9934E-7 4.9966E-4 4.9966E-2 7.06858E-4 0.353429 35.3429

1.00003 1.00502 1.50002 1.01122 1.01621 1.51184 Average:

6.6759 6.6764 6.6796 6.6752 6.6745 6.6763 6.6763

[1.0000] 1.0001 (2) 1.0006 (6) 0.9999 (2) 0.9998 (3) 1.0001 (3)

Conical detector 30, 4., 0.0001 30, 4., 5.0 30, 4., 12.0 30, 0.4, 0.0001

0.7349 0.6250 0.5057 0.7416

5.0267E-3 295.542 876.630 5.0266E-5

6.617 (1.3) 2.836 (1) 5.0344 (5)

0.74144 (10) 0.74149 (10) 0.74148 (10) 0.74167 (8) 0.74152† 6.676 (1.3) 6.676 (1) 6.676 (1)

[1.0000] 1.0001 (2) 1.0001 (2) 1.0003 (1)

10, 1.3333, 0.0001 10, 1.3333, 10.0 10, 8.4628, 0.0001

1.0089* 1.18635 1.46621 1.00009 Average: 1.0089 2.3540 1.32603

(1) (1) (3) (1) (1) (1)

(1) (1) (1) (1)

krsq,R,L

Gy/photon

5.58483E-4 130.311 2.25E-2

*This is the geometry and the numerical correction used by NIST (eqn (24)). †Correcting for 1/d2 effects implies a value 9 × 0.74152 = 6.6737 at d = 10 cm which differs by 0.04% from the average at 10 cm. 5

[1.0000] 1.0000 1.0001 (1)

Applied Radiation and Isotopes 153 (2019) 108638

D.W.O. Rogers

Fig. 3. Differences in calculated k r 2 values using eqns (38) and (39) which average over d 2 and 1/ d 2 respectively, the latter being the correct equation.

4.2. FAC detectors

within 0.04% for widths corresponding to or narrower than a WAFAC geometry. When kr2 values based on numerical evaluation of eqn (39) are used, the predicted on-axis air kerma is consistent with what is expected.

Table 3 compares the Monte Carlo calculated values of the kerma in the FAC's actual sensitive conical region, K cone , to the air kerma in the

Table 2 Monte Carlo calculated average air kerma in a square prism WAFAC detector irradiated by a bare point source of 125I photons. There is no photon attenuation. The raw results are multiplied by the k r 2 correction factor from eqn (38) to attempt to give the air kerma on-axis at the front of the detector. The results show the failure of eqn (38) for the square prism for detectors of larger thicknesses or width but demonstrate the eqn is accurate for thin detectors with small or moderate width. Values of k r 2 derived numerically using eqn (39) are shown as bold values with a † and these accurately estimate the on-axis kerma for a variety of extreme detector shapes. K det k r 2 × 1016

Normalized

d, w, L

K det × 1016

Vdet

cm

Gy/photon

cm3

Square prism detector 10, 2.66, 0.0001 10, 2.66, 0.05 10, 2.66, 5.0 10, 2.66, 10.0 10, 2.66, 10.0

6.5968 6.5645 4.4132 3.3147 3.3147

(9) (9) (7) (5) (5)

7.076E-4 0.35378 35.378 70.756 70.756

1.0118 1.0168 1.5951 2.3451 2.0137†

6.6746 6.6748 7.0395 7.7733 6.6748

10, 10, 10, 10,

0.1, 0.05 0.1, 5.0 15, 0.0001 15, 0.0001

6.644 (5) 4.451 (4) 4.9981 (2) 4.9981 (2)

5.000E-4 5.000E-2 2.250E-2 2.250E-2

1.00503 1.58335 1.3750 1.33569†

6.677 (5)‡ 7.047 (6) 6.8724 (2) 6.6759 (1)†

[1.0000] 1.0553 (11) 1.0294 (3) 1.00006 (3)

30, 30, 30, 30,

7.09, 7.09, 7.09, 7.09,

0.73356 0.63056 0.55225 0.55225

2.5134 251.34 502.68 502.68

1.01098 1.18523 1.3797 1.34287†

0.74161 0.74736 0.76187 0.74160

[1.0000] 1.0078 (3) 1.0274 (3) 1.0000 (3)

0.05 5.0 10.0 10.0

(12) (10) (9) (9)

kr2

Gy/photon

(9)* (11) (11) (12) (12)

(13)†† (12) (13) (13)

[1.0000] 1.0000 (2) 1.0547 (2) 1.1647 (3) 1.0000 (3)

* Within 0.03% of the average value for the cylindrical detector in Table 1. † The kr2 value is from the numerical integration of eqn (39) and leads to an on-axis value within 0.01% of the average on-axis value for the cylindrical detectors. ‡ Within 0.04% of value for 10, 2.66, 0.05. †† Correcting for 1/ d 2 effects ( × 9) gives 6.6745, within 0.04% of the prism value at 10 cm and within 0.03% of the average value for the conical and cylindrical averages. 6

Applied Radiation and Isotopes 153 (2019) 108638

D.W.O. Rogers

Table 3 Monte Carlo results for ideal FAC simulations with no attenuation. d − x is the distance from source to aperture, d is the source to front of sensitive region distance, x is the distance from the front of the aperture to the front of the sensitive region, R 0 the radius of the aperture and t is its thickness. K ap etc are the MC kerma per initial photon values × 1017 at 20 cm and × 1019 at 200 cm. The Vap etc are the real volumes of the regions except for Veff,cyl = πR 02 L which is the effective volume. K ′ratio is the ratio K cone/ K ap with the detector's volume taken to be Veff,cyl which is just column 6 divided by column 7 (see eqn (40)). The high correlation between K cone and K ap (caused by the same photons) is not accounted for in the statistical uncertainties which are therefore smaller than shown. Kcone x (1017 or 1019)

Vcone

Kap x (1017 or 1019)

Vap

Gy/photon

cm3

Gy/photon

cm3

200,0.5,17,0.1, 10 200,0.5,17,0.01,10

5.302 (3) 5.304 (3)

9.6785 9.6785

6.531 (3) 6.536 (3)

200,0.5,34,0.1, 10 200,0.5,34,0.01,10

4.576 (2) 4.576 (2)

11.217 11.217

200,0.5,24,0.01,10

4.987 (2)

20,0.333,10,0.1, 10 20,0.333,10,0.01,10

2.119 (1) 2.119 (1)

d − x , R 0 , x , t , L cm

a

K cone/ Kap

Veff,cyl/ Vcone

K ′ratio

0.07854 0.007854

0.8118 (5) 0.8115 (5)

0.81149 0.81149

1.0004 (5)a 1.0001 (5)

6.532 (3) 6.535 (3)

0.07854 0.007854

0.7006 (5) 0.7002 (4)

0.70017 0.70017

1.0005 (7)a 1.0001 (7)

10.298

6.537 (3)

0.007854

0.7628 (4)

0.76264

1.0002 (6)

10.763 10.763

6.502 (3) 6.531 (3)

0.03491 0.003491

0.3259 (3) 0.3245 (3)

0.32432 0.32432

1.0049 (6)a 1.0001 (6)

For thicker apertures, there is a small 1/r2 effect. For aperture at 200 cm it is roughly (200.05/200)2 = 1.0005 whereas at 20 cm it is roughly (20.05/20)2 = 1.0050 .

volume of the aperture, K ap . By taking the ratio of these two values (column 6) to the ratio (column 7) of the conical region's actual volume, Vcone , to the effective FAC volume (Veff,cyl , eqn (27)) the comparison of eqn (40) is obtained. The fact that this ratio is very close to 1.0000 implies that the approach of using the effective volume is accurate. The one exception is for the relatively thick aperture scoring region at 20 cm distance where the ratio is 0.5% greater than unity, but this is on account of the 1/ r 2 effect noted above and caused by the thickness of the simulated aperture. It does not imply that the Veff,cyl technique is incorrect.

effects and are not directly used experimentally because of the use of an effective volume. However, the correction for lateral 1/r2 effects is used experimentally and the calculations related to very thin detectors indicate the excellent accuracy of these corrections, i.e., krsq, R,0 . However, the complete krsq, R, L correction factors are needed when calculating airkerma strengths using Monte Carlo calculations with detectors of finite thickness and these are also very accurate. The results for square prism detectors in Table 2 demonstrate that eqn (38), the krsq correction used by Taylor et al. (2007) and Rodriguez and Rogers (2013) is, in general, incorrect since it fails by several percent for thick detectors and for very wide detectors. This is because the original derivation averaged d 2 over the detector volume rather that the required 1/ d 2 . However, the results also demonstrate that for detectors roughly corresponding to the NIST WAFAC's solid angle and for thin detectors, i.e., ≤0.05 cm thick, eqn (38) is accurate to better than 0.04%. This is because, for small corrections, the distinction between averaging d 2 rather than 1/ d 2 is negligible. This is important since all of their calculated air-kerma strengths and hence calculated dose-rate constants are based on corrections for square prism detectors which are 0.05 cm thick and many of these contribute to the values recommended for clinical use (Rivard et al., 2017). The bolded entries in Table 2 demonstrate that using numerically evaluated values based on the proper eqn (39) accurately yields the on-axis air kerma for square prism detectors which are thick and/or very wide. The results in Table 3 demonstrate that the standard technique of using an effective volume rather than the actual volume in a free-air chamber is accurate within a precision of 0.02%. However, to achieve this accuracy it is necessary to calculate the average air kerma at the very front of the aperture rather than average over even a 1 mm thick aperture. The results also demonstrate that the lateral 1/ r 2 effects (the α 2 term in eqn (37)) are completely negligible as was already well known. The results in Table 4 demonstrate that the effect of the finite size of the photon source is negligible for a typical WAFAC geometry and hence is not an issue. All of the calculations reported here are for an isotropic point source and present the formulae used when correcting WAFAC and FAC calculations. However, some brachytherapy sources are known to be anisotropic and for certain designs there is a significant difference between the average air kerma calculated over the volume of a WAFAC and that present on the axis due to self-shielding in the seed (see Taylor et al. (2007) and references therein). Measured air-kerma strengths are based on WAFAC measurements which are corrected to give the on-axis value assuming an isotropic point source. Therefore it is essential to do Monte Carlo calculations for something approaching the WAFAC geometry

4.3. Off-axis effects Table 4 presents the variation in the calculated average air kerma in a conical detector of roughly the NIST WAFAC opening angle as a function of the distance of the point source vertically off axis. Even for the extreme case of the WAFAC very close to the seed and the isolated point source 3 mm off axis (seeds are typically 3 mm in length and thus radioactive material is less than 2 mm off axis) the change in the result is < 0.04%, thus confirming the earlier results that this is a negligible effect for realistic geometries although if off-axis by 10 mm there is a significant effect (0.5%). 5. Summary and conclusions The results for conical and cylindrical detectors in Table 1 demoncyl cone strate clearly that eqn (19) and (22) for krsq , R, L and k rsq, R, L are accurate at the level of 0.02% for geometries which require correction by up to a factor of 2.35. Most of these corrections are from longitudinal 1/r2 Table 4 Monte Carlo calculated average air kerma per initial photon in a conical WAFAC detector 10 cm away on the axis from isotropic point sources which are vertically offset from the origin. The detector's radius is 1.333 cm and length is 10 cm. offset mm

K cone × 1016

K cone (0)/ K cone (offset)

Gy/photon 0 2 2.5 3 4.0 5 10

2.8356 2.8356 2.8352 2.8346 2.8339 2.8326 2.8226

(1.4) (1.4) (1.4) (1.4) (1.4) (4) (4)

[1.0000] 1.00000 (5) 1.00014 (5) 1.00035 (5) 1.00060 (6) 1.00106 (15) 1.00461 (15)

7

Applied Radiation and Isotopes 153 (2019) 108638

D.W.O. Rogers

Acknowledgements

and correct this calculated value to the on-axis air-kerma strength using the same assumption of an isotropic point source based on the kr2 corrections presented here. This is despite the fact that the Monte Carlo could directly calculate the air-kerma strength on the axis as per its definition. But doing that would mean the calculated value would not correspond to the measured value and lead to incorrect dose-rate constants.

I want to thank Randy Taylor, of Multi Leaf Consulting, who extended the egs_brachy code to handle the case of conical detectors. Thanks also to Marc Chamberland and Randy Taylor for answering various questions about input file formats for egs_brachy. This research supported by an NSERC Discovery Grant.

Declarations of interest None. Appendix The following standard relationships are used in the derivations and are here as an aide-mémoire. Symbols are those defined in Fig. 1.

α = R/d

β = R/(L + d )

∫ dxx = ln(x )

∫

ln

(

dx 1 − x2

cos θm =

cos θc =

R2 + (d + L)2

d R2 + d 2

=

=

1 2

2

= arcsin(x )

∫ x (1 +dxd2 / x 2) dx = 12 ln(d 2 + x 2) d+L

) = − ln(1 − x )

1 1 − x2

1+

d+L 2 R

( )

=

=1

arcsin(cos x ) =

θc = arctan

(d + L)/ R

x2 2 x → 0 ln(1 + x )

lim

( ) = arctan α, R d

1 β

1+

1 2 β

()

=

1 β2 + 1

π 2

θm = arctan β

, (41)

1 1 + α2 ,

(42)

calculation of free-air chamber correction factors. Med. Phys. 35, 3650–3660. Rivard, M.J., et al., 2017. Supplement 2 for the 2004 update of the AAPM task group No. 43 report: joint recommendations by the AAPM and GEC-ESTRO. Med. Phys. 44 (9), e297–e338. Rodriguez, M., Rogers, D.W.O., 2013. On determining dose rate constants spectroscopically. Med. Phys. 40, 011713 (10pp). Selbach, H.J., Kramer, H.M., Culberson, W.S., 2008. Realization of reference air-kerma rate for low-energy photon sources. Metrologia 45 (4), 422–428. Seltzer, S.M., Lamperti, P.J., Loevinger, R., Mitch, M.G., Weaver, J.T., Coursey, B.M., 2003. New national air-kerma-strength standards for 125I and 103Pd brachytherapy seeds. J. Res. Natl. Inst. Stand. Technol. 108, 337–358. Taylor, L.S., 1930. Analysis of diaphragm system for the x-ray standard ionization chamber. Radiology 15, 49–65. Taylor, R.E.P., Yegin, G., Rogers, D.W.O., 2007. Benchmarking BrachyDose: voxel-based EGSnrc Monte Carlo calculations of TG–43 dosimetry parameters. Med. Phys. 34, 445–457.

References Aitken, J.H., 1958. An analysis of the free-air ionization chamber for extended sources of radiation. Phys. Med. Biol. 3, 27–36. Attix, F.H., 1986. Introduction to Radiological Physics and Radiation Dosimetry. Wiley, New York. Berger, M.J., Hubbell, J.H., Seltzer, S.M., Chang, J., Coursey, J., Sukumar, R., Zucker, D., Olsen, K., 2010. XCOM: Photon Cross Section Database (Version 1.5) Technical Report NBSIR87-3597. NIST, Gaithersburg, MD. http://physics.nist.gov/xcom. Chamberland, M., Taylor, R.E.P., Rogers, D.W.O., Thomson, R.M., 2016. egs_brachy: a versatile and fast Monte Carlo code for brachytherapy. Phys. Med. Biol. 61, 8214–8231. Culberson, W.S., 2006. Large-Angle Ionization Chambers for Brachytherapy Air-KermaStrength Measurements. PhD Thesis. University of Wisconsin-Madison. Daskalov, G.M., Williamson, J.F., 2001. Monte Carlo-aided dosimetry of the new Bebig IsoSeed 103Pd interstitial brachytherapy seed. Med. Phys. 28, 2154–2161. Mainegra-Hing, E., Reynaert, N., Kawrakow, I., 2008. Novel approach for the Monte Carlo

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