Corrections to in-air measurements of energy response and calibration of solid state and thick-walled dosimeters for use in phantom

R. P. Hugtenburg1,2 and Z. Yin2
1. Imaging and Medical Physics Group
Queen Elizabeth Hospital
2. School of Physics and Astronomy
University of Birmingham

Corresponding Author:

Dr Richard Hugtenburg
Imaging and Medical Physics Group
Queen Elizabeth Hospital
University Hospital NHS Trust
Birmingham B15 2TH
richard.hugtenburg@university-b.wmids.nhs.uk

Introduction

When a detector is used to determine dose in a phantom material it is useful to know

  1. the energy spectrum of the radiation in the vicinity of the detector, as well as

  2. the differential response of the detector with respect to energy.

Both can be determined using the Monte Carlo method with a knowledge of the source spectrum and the material constituency of the phantom and detector. It is usual to use Monte Carlo to predict the former but the latter is often determined by measuring the response of the detector in air.

Monte Carlo using measured responses is problematic if the detector element is embedded within a sizeable housing, as absorption and scatter occurring in the housing modify the response of the detector to the radiation source in air. Modelling of the changes in the spectrum in the phantom material will include a component which is scatter and absorption that occurs in an equivalent volume to the detector and housing. Subsequently, absorption and scatter are accounted for twice.

In the first section

In the second section

Postscript: Electron slowing-down spectra

Double counting in the determination of quality correction factors

The energy response of a solid-state dosimeter can be determined using a quasi-monoenergetic X-ray source [1,2,3]. The response of a new detector is compared with a detector of a known response or a free-in-air chamber which is energy independent. However, multiple photon scatter occurs in housing of such dosimeters and will lead to a double-counting of the influence of scattered photons if the response data is used to correct dosimeters in a phantom.

The aim of this work is to determine an intrinsic energy response. By intrinsic we mean the response of the dosimeter relative to water considering the elemental composition of the detector, connecters and housing but excluding perturbations to the photon fluence due to scatter and absorption in an equivalent volume of water. These will counted later.



A 48 keV quasi-monoenergetic source (60kVp, 0.6 mm Cu and 3.8 mm Al filtration) and the primary and total fluence components at the position of the sensitive volume in a hypothetical cylindrical detector

Measurements of diamond detectors in air demonstrate that the detector response relative to air and water has a maximum greater than 1 in the kilovoltage range, contrary to the energy absorption of carbon, suggesting that this is due to the influence of contacts (Ag/Cu).

Energy maxima of 80 and 122 keV have been measured, Planskoy (1980) and Seuntjens et al. (1999) respectively, who also report differing changes in magnitude of the response with energy. We suggest the discrepancy between the two measurements might be due, in part, to differences in the sizes of the housings, which are 2.8 mm and 7.3 mm, respectively.

In our own work we have predicted the quality dependence of diamond and silicon diode detectors in phantom on a basis of the mass-absorption ratio of air-to-carbon and air-silicon. Scattering in the detector housing means that the intrinsic energy response of the dosimeter will show greater variation in energy than measured responses.

These results may assist in explaining a shortfall in our estimates of the over-response of diode dosimeters and the under-response of diamond dosimeters to radiation scattered in phantom.

An inverse Monte Carlo method [4,5] is proposed to determine an intrinsic energy response that is independent of the size of the housing and can be used with fluence spectra in phantom to determine quality corrections as a function of depth and field size.

The intrinsic energy response, $r_n(\varepsilon)$, is modified by a double-differential fluence $\psi(\varepsilon,\varepsilon')$, which as a function of the incident fluence and the fluence in the sensitive volume, gives the measured energy response $r_m(\varepsilon)$, according to the integral,

 \begin{displaymath}r_m(\varepsilon)=\int_{\varepsilon'} r_n(\varepsilon') \psi(\varepsilon,\varepsilon') \mbox{d}\varepsilon'\end{displaymath}

(1)



EGS4nrc, with the usercode FLURZnrc, is used to determine the fluence spectra for quasi-monoenergetic photons transported through the volume of water displaced by the dosimeter. A first approximation computes fluence for photons re-emerging from the front entrance face of dosimeter which is typically the location of the sensitive element.

FLURZnrc is used to compute particle fluence differential in energy using the following photon energies, 10, 15, 20, 30, 40, 50, 60, 80, 100, 150, 200 and 300 keV. The differential fluence as a function of incident and outgoing energy, forms a $12\times12$ matrix, $\Psi_{ij}$. Equation 1 in a discrete form is,

 \begin{displaymath}m_i=\sum_j \Psi_{ij}n_j,\end{displaymath}

(2)



where m and n are vectors of the measured and intrinsic quality corrections, respectively.

In the circumstances of negligible statistical error in the Monte Carlo estimate, the equation is easily solved because photon scattering can only decrease the energy of outgoing particles, the matrix is already in row-echelon form and can be inverted using back-substitution. Statistical uncertainty in the Monte Carlo calculation renders the inversion non-trivial and leads to negative entries in $\Psi^{-1}_{ij}$ and blurring out of the intrinsic response. One can test if the measured response is recovered within experimental precision with an uncorrelated calculation of mi from the calculated ni. The simplest method is to treat the intrinsic response, ni, as a spectrum and repeat the simulation.

It is desirable from a point of view of precision to separate the primary radiation which is discrete from the scattered radiation which is distributed, i.e.,

\begin{displaymath}\Psi_{ij}= P_{ij} + S_{ij},\end{displaymath}

(3)



where P is a matrix representing the primary monoenergetic fluence with entries on the diagonal and zero elsewhere, S, the scatter component is still in a row-echelon form. Equation 2 becomes,

 

mi

=

$\displaystyle \sum_j ( S_{ij}n_j+ P_{ij}n_j ),$

 

$\displaystyle \Rightarrow n_k$

=

$\displaystyle \sum _i ( S^{-1}_{ik}m_i - \sum_j S^{-1}_{ik} P_{ij} n_j ),$

(4)



where nk can be determined using an iterative scheme with nj=mj as a first approximation on the right-hand side of the equation.

These methods offer a useful first approximation that assumes the detector housing has near water equivalence but enables one to subsume into the determination of an intrinsic energy response, the influence of the true housing material, metallic connectors and wiring.

The advantage of this technique is that the spectrum in the phantom can be determined independent of the placement of the detector and only one calculation is required. A more detailed model would consider the material construction of the detector and its placement at a specific position in the phantom. Indeed, an in air measurement of energy response can contribute to the precision of a full detail Monte Carlo calculation of detector response.


The intrinsic response of carbon (blue) gives a altered response due to scatter (green) as well as attenuation (red) in water replaced by a PTW diamond detector of typical dimensions (20 mm length, 7.3 mm diameter, including a stem of length 38 mm, diameter 4.3 mm). Calculated using EGSnrc/FLURZ (PCUT = 10 keV, bound Compton, Rayleigh and atomic relaxation switched on, no electron transport)

In the circumstances of an appreciably broad quasi-monoenergetic source, where the spectrum is given,

\begin{displaymath}\Omega_{\overline{\varepsilon},\sigma} = e^{-\frac{(\varepsilon''-\overline{\varepsilon})^2}{2\sigma^2}},\end{displaymath}

(5)



a double integral of over $\varepsilon''$ and $\varepsilon'$ is required. $\Psi_{ij}$ is replaced by $\Psi'_{ij}$ which, without loss of generality, can be computed from within the simulation,

\begin{displaymath}\psi'(\overline{\varepsilon},\varepsilon')=\int \psi(\varepsi......line{\varepsilon},\sigma}(\varepsilon'')\mbox{d}\varepsilon''.\end{displaymath}

(6)



$\Psi'_{ij}$, is, however, not a row-echelon matrix and other inversion methods such as gradient-descent or maximum likelihood are usually required. A positivity constraint can be invoked and a reasonable solution to,

\begin{displaymath}\left\{\sum_i \vert m_i-\sum_j \Psi'_{ij}n_j\vert^2, \Psi'_{ij},n_j\geq 0, \forall i,j\right\} < \sum_i \delta m_i^2\end{displaymath}

(7)



is sought, where $\delta m_i$ is the experimental error in mi.

The intrinsic response is used to compute the detector response, RX, for a detector element whose predominant material is X as a function of the energy fluence, $\phi$, including the influence of attenuation and scatter in the tissue equivalent medium. The calculation of dose response invokes the intrinsic, energy dependent, response to weight the energy absorbed in the detector element accordingly. i.e.,

RX

=

$\displaystyle \int_{\varepsilon} r_{n,X} \left(\frac{\mu_{\mbox{en}}}{\rho}\right)^X\phi\mbox{d}\varepsilon$

 

 

=

$\displaystyle \sum_{i,j} \Psi^{-1}_{ij}m_i\left(\frac{\mu_{\mbox{en}}}{\rho}\right)^X_j \phi_j \Delta\varepsilon_j$

(8)

It would be useful to have the ability in FLURZ to compute a general form of talley such as is available in MCNP such as represented in equation 8. In the meantime we can use DOSRZ to compute pulse-height distributions. This enables the computation of detector response using the more slowly varying ratio of absorption coefficients.

Double counting in the calibration of parallel plate chambers in very low energy x-rays

The 1996 IPEMB code of practice for the dosimetry of x-rays below 300 kV generating potential [6] recommends the use of a parallel plate ionisation chamber for the very-low energy regime (0.035-1.0 mm Al, 80-50 kV). The code defines absorbed dose to water at the surface,

\begin{displaymath}D_{\mbox{w},z=0} = M N_K k_{\mbox{ch}}\left[\left(\frac{\mu_{\mbox{en}}}{\rho}\right)_{\mbox{w/air}}\right]_{z=0,\phi},\end{displaymath}

(9)



where specifically, NK is a calibration factor which converts the reading to air kerma free in air at the reference point of the chamber with the chamber assembly replaced by air and $k_{\mbox{\it ch}}$ is a pertubation factor which accounts for the change in detector response when the chamber is placed in a water phantom. According to the protocol little information is available concerning the value of $k_{\mbox{\it ch}}$ and it is recommended that be assumed to be unity until such information becomes available. It is anticipated in the protocol that information on $k_{\mbox{\it ch}}$ will be forthcoming. It is suggested that the correction will include the effects of stem scatter, displacement correction, energy response of the chamber and back scatter from the chamber housing.

The 1996 protocol represents a departure from earlier codes of practice [] by separating very-low energy X-rays from low-energy X-rays where a backscatter factor is coupled with an in-air kerma measurement. This new categorisation has been motivated by the indeterminate energy response of thimble chambers at very-low energies as well as inconsistencies in backscatter factors.

As the calibration of the parallel plate chamber proceeds with an intercomparison of the chamber, in air, to a standards laboratory free-in-air chamber the perturbative effect of the chamber on the in air fluence has yet to be considered. According to the definition in the code of practice, this effect should be absorbed into NK but contradicts practices with cylindrical chambers of less bulky construction where the displacement correction implicitely includes the in-air correction since the cavity is defined by the outer dimensions of the chamber [7,8]. Nahum comments on work by Lidén [9], the originator of this practice, who describes the displacement correction as consisting of three effects,

  1. the decreased attenuation of primary radiation,

  2. the decreased attenuation of scattered radiation and

  3. the elimination of scatter from the displaced volume,

yet ignoring

  1. the scatter in the chamber construction.

As the parallel plate chamber is more suited to placement in a water equivalent phantom, the chamber is therefore constructed of solid materials which are approximately water equivalent. The scatter contribution from the chamber is consequently much greater and spectral differences in the phantom are likely to challenge the above assumption. It is appropriate to start with the assumption that the chamber is homogeneous and that the displacement effect is secondary. In which case one would calculate

  1. the increased scatter for the in air calibration from a uniform volume defined by the outer dimension of the dosimeter,

  2. the decreased absorption of primary radiation in the phantom,

  3. the decreased absorption of scattered radiation in the and

  4. the elimination of scatter from the displaced volume,

in decreasing order of significance.

As has been mentioned, the protocol stipulates that NK is the air kerma free in air with the chamber assembly replaced by air. It is appropriate then to include the effects of chamber scatter in air and in Nk and not $k_{\mbox{ch}}$.

Some preliminary calculations have been performed with EGS4 with the LSCAT extension firstly for the PTW 23342 parallel plate chamber. It has been treated as an homogeneous water equivalent block of outer dimensions 6.1 cm length 2.2 cm width and 1.44 cm thick irradiated by 45 kVp X-rays with a field larger than than the chamber and 15 cm source focus to chamber distance. The sensitive volume is 0.3 cm in diameter the centre of which is displaced from the end by 0.9 cm. The chamber scatter factor obtain for this arrangement was 1.07. This is our therapy set up but, as previously stated, the correction should be performed for the calibration conditions.

The calibration factor for the field chamber was obtained from an intercomparison with a secondary standard supplied by NRL, a PTW 23344 parallel plate chamber performed in phantom. The outer dimensions of this chamber are 6.1 cm length, 3.0 cm width and 1.44 cm thickness. The sensitive volume is 1.3 cm in diameter the centre of which is displaced from the end by 1.5 cm. An NRL free in air quality standard (2.1.2), 39 kVp with 1.0 mm Be + 0.74 mm Al filtration and 75 cm source focus to chamber distance, was used to calibrate this chamber. The chamber scatter factor obtain for this arrangement was 1.06.

Two energy modalities used in our clinic span the two calibration methods described for low and very-low energy X-rays described in the IPEMB protocol.

BSF values determined for 45 kVp and 100 kVp beams with 5 cm diameter applicator and 15 cm FSD including measurement via the PTW parallel plate chamber in phantom and a 0.6 cc Farmer chamber in air, Monte Carlo computed and BJR-25 (1996) listed values. The uncertainties (in brackets, 2 s.d.) apply to the last significant figure.

Peak voltage

45 kVp

100 kVp

Filtration

0.55 mm Al

1.7 mm Al

First HVL

0.55 mm Al

2.3 mm Al

Measured BSF

1.10(6)

1.18(6)

Monte Carlo

1.111(1)

1.211(2)

BJR 25

1.085

1.189

 

The calculations are based on a set of attenuation measurements combined with the Birch Marshall model are expected to matched more closely to the characteristics of the beam than published data based on the first HVL.

Bound Compton scattering and Doppler broadening, which have not typically been incorporated in BSF calculations, increase the BSF by a small but detectable 0.7% for 45 kVp and 0.4% for 100 kVp.

Electron transport is often neglected in BSF calculations but has a small but detectable influence which would be expected to be larger at the higher of the two tube potentials. Enabling electron transport increases the BSF by 1.0% and 1.1% for the 45 and 100 kVp beams, respectively.

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Electron slowing down spectra calculated for the radiation at the colon of a survivor of the atomic detonation at Hiroshima calculated using EGS4 (ESTEPE=0.1, AE=0.512 keV, AP=0.001 keV). Standard EGS4 (black), EGS4 with the LSCAT low-energy photon scatter distribution (green) and FUDGEMS=0 (pink) are indistinguisable. The Ma/Nahum correction to discrete interaction pathlength sampling (dotted blue) and EGSnrc (dotted red) differ substantially from standard EGS4 below 0.1 MeV. Kawrakow, MP 27(3), 2000, argues that the ammendment suggested by Ma and Nahum is biased