A CRITICAL EXAMINATION OF SPENCER-ATTIX CAVITY THEORY

1. IAEA-CN-182-219 A CRITICAL EXAMINATION OF SPENCER-ATTIX CAVITY THEORY Alan Nahum PhD, Tom Williams, Vanessa Panettieri PhD Physics Department ClatterbridgeCentre for Oncology BebingtonWirral UK This is both a demonstration of the validity of the S-A cavity integral and the existence of d–ray equilibrium below ≈ 50 keV. The marked decrease in the S-A cavity dose with increasing D above ≈ 50 keV is due principally to the non-inclusion of energy deposition from electrons (and positrons) with initial energies below D. It should be noted that both the electron and the positronfluence was used in evaluating (1), and that the latter contributes fully 15% of the total dose. Figure 4 shows the dose to the air cavity computed from (1) using the MC-generated electron (and positron) fluence in the air cavity itself. The perfect agreement with the directly scored energy deposition in the air (solid horizontal line) is demonstrated. Finally, in Fig. 5 the variation of the aluminium-to-air Spencer-Attix stopping-power ratio with D is shown. In this case it is the electron (and positron) fluence in the uniform (aluminium) medium that is employed in both cavity integrals. The value of delta appropriate for a cavity of these dimensions is between 10 and 15 keV [1,4]. The S-A SPR at = 10 keV is 0.849±0.004; this can be compared with the MC-derived dose ratio of 0.837±0.016. Multiplying the SPR by pdis ≈ 0.990 to correct for the upstream position of the effective point of measurement brings these two estimates of the dose ratio within 0.4% of each other. Spencer-Attix theory is highly robust forFarmer-sized air cavities in aluminium irradiated by 10 MeV photons. EABS to 1 keV for all particles. Following Sempau and Andreo [7], two thin “skin” regions were also defined around the air cavity with relatively low parameter values (i.e. for skin 1, WCC and WCR equal to 0.1 keV and C1, C2 equal to 0.01). In the volume of aluminium surrounding the skin regions conventional parameters were assigned. SPENCER-ATTIX CAVITY INTEGRALS The absorbed dose to a small volume or cavity, sometimes known as the Spencer-Attix cavity integral, can be expressed thus [1,3]: (1) where is the total electron fluence at electron kinetic energy E, differential in E, and LD(E)/ris the mass electron collision stopping power restricted to losses less than D. The second term on the r.h.s. is the energy deposition due to track ends at energy D [3]. The Spencer-Attix stopping-power ratio, Al/air, is defined as the ratio of the above cavity integral for aluminium to that for air, with the identical fluence in both the numerator and the denominator, this fluence being computed for the undisturbed medium [1,3]. The purpose of this investigation is firstly to compare the electron fluence computed in the air cavity with that in the aluminium cavity, in order to directly test the Bragg-Gray assumptions, secondly to compare the S-A dose from (1) with the direct Monte-Carlo dose, for the air and aluminium cavities respectively, for a range of different D, and thirdly to compare the S-A stopping-power ratio, also for different D, with the MC dose ratio. RESULTS The electron fluences are compared in Figure 2. There is essentially no difference until close to 2 keV, when the aluminium fluence rises well above that in the air cavity. Figure 3 shows the dose computed from (1) for the case of the homogeneous aluminium ‘cavity’ for varying D. The horizontal line is the direct Monte-Carlo dose; for all D below about 50 keV there is excellent agreement. INTRODUCTION Cavity theory is fundamental to understanding dosimeter response [1], and the Spencer-Attix cavity-size-dependent extension of Bragg-Gray theory, formulated over 50 years ago [2], is still the most advanced form of cavity theory. However, S-A theory involves several approximations e.g. the electron fluence spectrum in the cavity is assumed to be identical to that in the undisturbed medium down to a (cutoff) energy delta, chosen such that electrons of this energy can just cross the cavity. When Spencer and Attix developed their theory they did not possess tools to critically examine their assumptions; today we have Monte-Carlo codes with electron transport schemes specifically designed to yield accurate results from ion chamber simulations [4-8]. This work describes such an examination, and involves computing the electron fluence down to energies far below that of the Spencer-Attixcutoff. METHODS AND MATERIALS Monoenergetic photons of energy 10MeV were incident on an aluminium phantom containing a wall-less air cavity (cylinder, 0.3 cm radius, 2 cm long) at 5 cm depth (see Fig. 1). The electrons were followed down to 1 keV. Aluminium was chosen as its atomic number (Z=13) is significantly different from that of air, and therefore constitutes a more severe test of S-A cavity theory than the wall or phantom materials used in practical radiotherapy dosimetry (i.e. water, graphite, PMMA etc.); furthermore aluminium-walled ion chambers can easily be constructed for experimental purposes. All MC simulations were performed using the 2006 version of PENELOPE [6], a general-purpose MC code widely used in medical radiation physics. PENELOPE is a subroutine package which requires a main program. In this work the general-purpose main program provided with the PENEASY package (v.2008-06-15) (Sempau and Badal 2008) [9] was employed. The transport algorithm involves six user-defined simulation parameters for each material. These values correspond to: cutoff energy for each particle type (EABS), C1 and C2 which determine the cut-off energies for the production of hard inelastic and bremsstrahlung events, respectively, and distance DSMAX, an upper limit on the allowed step length. As explained in [7,8] smaller values of the simulation parameters yield shorter pathlengths, thus increasing the charged particle transport accuracy, at the cost of an increase in calculation time. Low values of the parameters are usually employed in very small regions, such as ion-chamber air cavities. In order to obtain the highest possible accuracy in the air cavity, analogue collision-by-collision simulation was performed by setting the C1, C2, WCC and WCR parameters all to zero, and Absorbed Dose /MeV g-1 particle-1 Spencer-Attix cutoff D / keV Spencer-Attix cutoff D / keV Figure 4. The dose from (1) for varying D.for the air ‘cavity’ compared to the direct MC dose (full horizontal line, 2 s statistical uncertainties shown as dashed-dotted). 10 MeV photons S-A stoppoing-power ratio, Al/air Electron Fluence /cm-2 eV-1 particle-1 Figure 1.A cross-section of the irradiation geometry simulated by Monte-Carlo (not to scale). The cylindrical cavity is 2 cm long and thus has similar dimensions to a Farmer ion chamber. In the homogeneous geometry aluminium replaces air but the cavity dimensions are preserved. Figure 5. Spencer-Attix stopping-power ratio, Al/air. Electron kinetic energy/ eV Figure 2. Monte-Carlo generated electronfluence spectra in the air and aluminium cavities down to 1 keV. CONCLUSIONS ACKNOWLEDGEMENTS We wish to thank Josep Sempau for valuable data on the PENELOPE system and Dave Rogers for running some extremely helpfulparallel EGSnrc simulations. REFERENCES [1] Nahum AE, Cavity Theory, Stopping-Power Ratios, Correction Factors, In Clinical Dosimetry Measurements in Radiotherapy, Editors DWO Rogers and Joanna E Cygler, AAPM 2009 Summer School Proceedings, Medical Physics Publishing, Madison, WI. (ISBN 978-1-888340-84-6). [2] Spencer LV and Attix FH, A theory of cavity ionisation, Radiat. Res., 3, 239–254, 1955. [3] Nahum AE, Water/Air Mass Stopping‑Power Ratios for Megavoltage Photon and Electron Beams. Physics in Medicine and Biology, 23, 24‑38, 1978. [4] La Russa DJ and Rogers DWO, Accuracy of Spencer-Attix cavity theory and calculations of fluence correction factors for the air kerma formalism, Medical Physics36 4173-83 2009. [5] Kawrakow, I., Accurate condensed history Monte Carlo simulation of electron transport: II. Application to ion chamber response simulations, Med. Phys., 27, 499–513, 2000. [6] Salvat F, Fernandez-Varea J M and Sempau J 2006 PENELOPE-2006: A Code System for Monte Carlo Simulation of Electron and Photon Transport Issy-les- Moulineaux, France: OECD Nuclear Energy Agency. Available in pdf format at http://www.nea.fr [7] Sempau J and Andreo P 2006 Configuration of the electron transport algorithm of PENELOPE to simulate ion chambers Phys. Med. Biol. 51 3533–48 [8] Sempau J, Andreo P, Aldana J, Mazurier J and Salvat F 2004 Electron beam quality correction factors for plane-parallel ionization chambers: Monte Carlo calculations using the PENELOPE system Phys. Med. Biol. 49 4427–44 [9] Sempau J and Badal A 2008 PENEASY, a modular main program and voxelised geometry package for PENELOPE [http://www.upc.edu/inte/downloads/penEasy.htm] Track-end depoosition/ % Absorbed Dose /MeV g-1 particle-1 Spencer-Attix cutoff D / keV Spencer-Attix cutoff D / keV Figure 3. The dose from (1) for varying D.for the aluminium ‘cavity’ compared to the direct MC dose (full horizontal line, 2 s statistical uncertainties shown as dashed-dotted); insert – track-end energy deposition as a percentage of the total. INTERNATIONAL SYMPOSIUM ON STANDARDS, APPLICATIONS AND QUALITY ASSURANCE IN MEDICAL RADIATION DOSIMETRY (IDOS), IAEA, Vienna, Austria, 9 – 12 November 2010. Corresponding author: alan.nahum@ccotrust.nhs.uk