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Cavity theory applications for kilovoltage cellular dosimetry
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Download details: IP Address: 130.133.8.114 This content was downloaded on 01/04/2017 at 02:52 Manuscript version: Accepted Manuscript Oliver et al To cite this article before publication: Oliver et al, 2017, Phys. Med. Biol., at press: https://doi.org/10.1088/1361-6560/aa6a42 This Accepted Manuscript is: © 2017 Institute of Physics and Engineering in Medicine During the embargo period (the 12 month period from the publication of the Version of Record of this article), the Accepted Manuscript is fully protected by copyright and cannot be reused or reposted elsewhere. As the Version of Record of this article is going to be / has been published on a subscription basis, this Accepted Manuscript is available for reuse under a CC BY-NC-ND 3.0 licence after a 12 month embargo period. After the embargo period, everyone is permitted to use all or part of the original content in this article for non-commercial purposes, provided that they adhere to all the terms of the licence https://creativecommons.org/licences/by-nc-nd/3.0 Although reasonable endeavours have been taken to obtain all necessary permissions from third parties to include their copyrighted content within this article, their full citation and copyright line may not be present in this Accepted Manuscript version. Before using any content from this article, please refer to the Version of Record on IOPscience once published for full citation and copyright details, as permissions will likely be required. All third party content is fully copyright protected, unless specifically stated otherwise in the figure caption in the Version of Record. When available, you can view the Version of Record for this article at: http://iopscience.iop.org/article/10.1088/1361-6560/aa6a42
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Cavity theory applications for kilovoltage cellular dosimetry
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Cavity theory applications for kilovoltage cellular dosimetry P. A. K. Oliver and Rowan M. Thomson Carleton Laboratory for Radiotherapy Physics, Physics Dept, Carleton University, Ottawa, K1S 5B6
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Revised March 15, 2017
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Abstract
Relationships between macroscopic (bulk tissue) and microscopic (cellular) dose descriptors are investigated using cavity theory and Monte Carlo (MC) simulations. Small, large, and multiple intermediate cavity theory (SCT, LCT, and ICT, respectively) approaches are considered for 20 to 370 keV incident photons; ICT is a sum of SCT and LCT contributions weighted by parameter d. Considering µm-sized cavities of water in bulk tissue phantoms, different cavity theory approaches are evaluated via comparison of Dw,m /Dm,m (where Dw,m is dose-to-water-in-medium and Dm,m is dose-to-medium-in-medium) with MC results. The best overall agreement is achieved with an ICT approach in which d = (1−e−βL )/(βL), where L is the mean chord length of the cavity and β is given by e−βRCSDA = 0.04 (RCSDA is the continuous slowing down approximation range of an electron of energy equal to that of incident photons). Cell nucleus doses, Dnuc , computed with this ICT approach are compared with those from MC simulations involving multicellular soft tissue models considering a representative range of cell/nucleus sizes and elemental compositions. In 91% of cases, ICT and MC predictions agree within 3%; disagreement is at most 8.8%. These results suggest that cavity theory may be useful for linking doses from model-based dose calculation algorithms (MBDCAs) with energy deposition in cellular targets. Finally, based on the suggestion that clusters of water molecules associated with DNA are important radiobiological targets, two approaches for estimating dose-to-water by application of SCT to MC results for Dm,m or Dnuc are compared. Results for these two estimates differ by up to 35%, demonstrating the sensitivity of energy deposition within a small volume of water in nucleus to the geometry and composition of its surroundings. In terms of the debate over the dose specification medium for MBDCAs, these results do not support conversion of Dm,m to Dw,m using SCT.
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1.
Introduction
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Macroscopic dose descriptors (based on computational models involving ∼mm-sized voxels)
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such as dose-to-medium-in-medium (Dm,m ), dose-to-water-in-medium (Dw,m ) or dose-to-
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water-in-water (Dw,w ) are commonly calculated in clinical treatment planning (Beaulieu
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et al 2012; Rivard et al 2004). Monte Carlo (MC) and other model-based dose calculation
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Cavity theory applications for kilovoltage cellular dosimetry
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algorithms (MBDCAs) generally compute Dm,m , which represents dose to a region or voxel in
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which tissue composition is assumed to be locally uniform (homogeneous bulk tissue) within
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a voxelized patient model, ignoring underlying microscopic tissue structure. However, cells
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and their constituents are considered biologically-relevant targets for radiation-induced cell
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death. DNA is contained within the nucleus for most of the cell cycle so that the nucleus
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is often considered a target of interest for understanding radiobiological outcomes (Hall and
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Giaccia 2012; Kuzin 1963). Water molecules located near DNA are clustered into ∼nm-sized
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volumes, and can undergo radiolysis resulting in free radical production and DNA damage
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(Goodhead 2006). Recent work supports the importance of such ∼nm-sized volumes within
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the nucleus (Lindborg et al 2013).
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MC simulations involving multicellular models of human soft tissues demonstrate that
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macro- and microscopic dose descriptors disagree by up to 32% at kilovoltage photon energies,
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suggesting that underlying cellular structure and composition are important considerations
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for accurately determining biologically-relevant dose descriptors (Oliver and Thomson 2017).
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Unfortunately, simulation of radiation transport and energy deposition within detailed, mul-
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ticellular models or on the molecular level is not always practical due to computational
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limitations and time constraints. Therefore, conversion factors relating macro- (e.g., Dm,m )
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and microscopic (e.g., nuclear dose Dnuc ) dose descriptors are potentially useful, and may
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be estimated using cavity theory (see e.g., Nahum 2009). Furthermore, cavity theory may
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provide a theoretical perspective helpful in understanding, predicting and verifying compu-
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tational and experimental results.
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Cavity theory conversion factors relating Dm,m to biologically-relevant dose descriptors
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such as Dnuc have been discussed in the context of the debate over the medium for dose spec-
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ification for MC treatment planning (Beaulieu et al 2012). These factors should be based
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on an appropriate cavity theory method corresponding to the size, density and elemental
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composition of the target of interest. Small, intermediate and large cavity theories (SCT,
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ICT and LCT) have domains of applicability dependent on target size relative to electron
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ranges. Carlsson Tedgren and Alm Carlsson (2013) classified water cavities (sizes from 1 nm
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to 1 mm) in tissue as small, intermediate or large based on an ICT approach, considering
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source energies relevant for brachytherapy. Beyond dose to a water voxel (or voxel composed
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of water-like tissue) in medium, Dw,m may represent dose to DNA-bound water molecules
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(Goodhead 2006), or dose to a subcellular compartment whose elemental composition is
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Cavity theory applications for kilovoltage cellular dosimetry
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closer to that of water than medium (Chetty et al 2007). Motivated by the potential radio-
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biological importance of small volumes of DNA-bound water located within the nucleus and
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the fact that a small water target is independent of cell type, Carlsson Tedgren and Alm
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Carlsson (2013) suggested that conversion of MBDCA-computed dose-to-medium, Dm,m , to
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dose-to-water, Dw,m , via SCT may be “preferred” in the context of brachytherapy. For kilo-
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voltage photon sources, Enger et al (2012) compared SCT and LCT predictions of Dw,m to
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MC-computed doses to nuclear cavities (7 µm radius) in otherwise homogeneous (1 cm)3
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soft-tissue phantoms.
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Thus, despite considerable discussion in the literature regarding applications of cavity
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theory to predict doses to cellular compartments, no such cavity theory predictions have
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been compared with results of MC simulations involving detailed, multicellular microscopic
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tissue models — this is the focus of the current work. We investigate relationships between
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macroscopic (bulk tissue) and microscopic (cellular) dose descriptors using MC simulations
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and cavity theory for kilovoltage photon energies relevant for brachytherapy, as well as ortho-
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voltage treatments and diagnostic radiology. Various cavity theory approaches are evaluated
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by comparing SCT, LCT, and multiple ICT predictions of Dw,m /Dm,m with corresponding
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MC results for a range of source energies, bulk tissue media and several ∼ µm-sized (i.e.,
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nucleus-sized) cavities (part I). The cavity theory approach providing the closest overall
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agreement with MC results for Dw,m /Dm,m is then used to predict nuclear doses which are
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compared with MC simulation results from detailed, multicellular models of healthy and
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cancerous human soft tissues, considering a range of representative subcellular compartment
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sizes and elemental compositions (part II). Finally, SCT conversion factors applied to MC
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simulation results are investigated for prediction of energy deposition within a small volume
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of water in nucleus (part III).
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2.
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Section 2.A provides an overview of the three parts of our study, before details on MC
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simulations and cavity theory methods are given in sections 2.B and 2.C, respectively.
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Methods
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Cavity theory applications for kilovoltage cellular dosimetry
A
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The three parts of our study are schematically depicted in figure 1 and described further Fig 1
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below. In each case, the source is a parallel beam of monoenergetic photons with energies of
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20, 30, 50, 90 or 370 keV, covering the range of photon energies of common brachytherapy
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sources, as well as diagnostic radiology. The ranges of electrons set into motion in soft
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tissues by these kilovoltage photons are comparable to sizes of subcellular compartments
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(Oliver and Thomson 2017). The beam has a circular cross section with 1 cm radius. Five
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bulk tissues are considered: adenoidcystic carcinoma, mammary gland, melanoma, muscle
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and squamous cell lung carcinoma.
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Overview
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In part I, values of Dw,m /Dm,m (figure 1(a)) from MC simulations (section 2.B) are
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compared with corresponding cavity theory predictions, considering five ICT approaches,
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plus SCT and LCT (section 2.C). Three cavity radii, five bulk tissues and five incident
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photon energies are considered, corresponding to a total of N = 75 scenarios. The cavity
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theory method with the lowest root mean square error (RMSE) when compared with MC
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results is used in part II of this study. The RMSE is calculated as follows: v u !2 N u X Dw,m Dw,m u1 , − RMSE = t N i=1 Dm,m i,CT Dm,m i,M C
(1)
where the sum is over the entire parameter space considered (i.e., N = 75), and the quan-
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tities (Dw,m /Dm,m )i,CT and (Dw,m /Dm,m )i,M C are the values (for scenario i) of Dw,m /Dm,m
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according to cavity theory and MC simulation, respectively.
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Nuclear doses (Dnuc ) are considered in part II (figure 1(b)). MC simulation results for
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Dnuc /Dm,m are compared with corresponding cavity theory predictions (using the method
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which minimizes the RMSE from part I). MC simulations for Dnuc involve detailed, multi-
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cellular models of normal and cancerous human soft tissues, described in detail by Oliver
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and Thomson (2017) and illustrated in figure 1(b) (see section 2.B). Three cell radii rcell are
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investigated, along with three nuclear radii rnuc for each rcell ; these radii are based on typical
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values for human cells. Additionally, seven cytoplasm/nucleus elemental compositions, five
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bulk tissues and five source energies are considered, for a total of 1575 scenarios. Cavity
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theory predictions are also compared with Dnuc,m /Dm,m , where Dnuc,m is the dose to a single
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nucleus embedded in an otherwise homogeneous bulk tissue phantom (figure 1(c)). A Overview
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Part II: Cavity theory applied to nuclear doses (Dnuc & Dnuc,m) (b) cell cluster 2.06 µm
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Part I: Evaluation of cavity theory approaches (a) one cavity
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Cavity theory applications for kilovoltage cellular dosimetry
(c) single nucleus
ECM
cytoplasm
Dnuc
nucleus
rcav = 3, 6, 9 µm
nucleus (Dnuc,m)
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rcell / !m rnuc / !m
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Part III: Using Bragg-Gray SCT to estimate dose to small water cavities (e) small water cavity in (d) small water cavity cell cluster
SCT Dw,nuc
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Figure 1: Schematic diagrams summarizing the three parts of the study. Part I: (a) a single cavity (radius = rcav ) consisting of either water or bulk tissue is embedded at the centre of a bulk tissue phantom. Part II: nuclear doses are considered. (b) Cross section of multicellular model consisting of 13 cells in a hexagonal lattice with cells (radius = rcell ) embedded in a sphere of extracellular matrix (ECM) (radius 18 to 33 µm); dose is scored in the central nucleus (radius = rnuc ). (c) A single nucleus embedded in an otherwise homogeneous bulk tissue phantom. Part III: Bragg-Gray SCT is used to estimate dose to a small water cavity embedded in (d) a bulk tissue phantom, or (e) the central nucleus of a cluster of 13 cells. Each geometry is located at the centre of a (2 cm)3 cubic bulk tissue phantom.
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A Overview
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Finally, part III considers the potential use of SCT for conversion of MC-calculated
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Dm,m to Dw,m , motivated by the radiobiological importance and universality (i.e., lack of
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dependence on cell type) of DNA-bound water molecules within the nucleus as a radiation
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target (Carlsson Tedgren and Alm Carlsson 2013; Goodhead 2006). We employ SCT to
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estimate dose to a small volume of water in two different ways, with scenarios illustrated in
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figures 1(d) and 1(e). The small water cavities are not explicitly modelled in MC simulations:
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Dm,m (figure 1(a)) and Dnuc (figure 1(b)) are obtained from MC simulations, and SCT
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conversion factors (stopping power ratios – details provided in section 2.C.b)) are used to
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SCT SCT estimate Dw,m and Dw,nuc , respectively, which we compare via the ratio: " w # . " w # m SCT Dw,nuc Scol Scol Scol Dnuc . = Dnuc × × Dm,m × = SCT Dw,m ρ nuc ρ m Dm,m ρ nuc
(2)
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Elemental compositions and mass densities ρ of mammary gland (ρ = 1.02 g/cm3 ) and
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muscle (ρ = 1.05 g/cm3 ) are from Woodard and White (1986) and ICRU (1989), respec-
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tively. The compositions of the three cancerous tissues (melanoma, adenoidcystic carcinoma
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and squamous cell lung carcinoma) are obtained from Maughan et al (1997), assuming a
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density of 1.04 g/cm3 (Thomson et al 2013). Water is assumed to exist at 22◦ C so that its
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density is taken to be 0.998 g/cm3 (Rivard et al 2004). Seven nucleus and cytoplasm elemen-
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tal compositions representative of animal cells are considered, each of density 1.06 g/cm3
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(Thomson et al 2013). The elemental compositions of these five bulk tissues, seven nuclear
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media and seven cytoplasm media are presented in Table A1 of Oliver and Thomson (2017)
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(page 1434, in the appendix). ECM definitions are determined by requiring that the aver-
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age, mass-weighted composition and density of the cells and ECM are approximately equal
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to those of the surrounding bulk tissue; this calculation is carried out for each bulk tissue,
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cell/nucleus size combination and cytoplasm/nucleus elemental composition.
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B
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MC dose calculations are carried out using the EGSnrc user-code egs chamber, which im-
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plements a variety of variance reduction techniques to efficiently calculate radiation dose to
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Monte Carlo simulations
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small targets (Wulff et al 2008; Kawrakow et al 2011; Kawrakow et al 2009). MC results
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for Dnuc /Dm,m and Dnuc,m /Dm,m are from previous work (Oliver and Thomson 2017), while
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additional MC simulations for Dw,m /Dm,m are carried out for the present work. A sufficient B Monte Carlo simulations
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Cavity theory applications for kilovoltage cellular dosimetry
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number of histories are simulated to obtain statistical uncertainties on doses of < 0.91%.
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Electron and photon fluence spectra are used for calculating the spectrum-averaged mass
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collisional stopping power and mass energy absorption coefficient ratios needed for cavity
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theory estimates (section 2.C), and are obtained from the EGSnrc user-code FLURZnrc
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(Rogers et al 2011). The geometry consists of a cylindrical phantom composed of water or
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bulk tissue of 2 cm height and diameter. The scoring region is also cylindrical (with height
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and diameter equal to 0.05 cm) and is located at the centre of this phantom. The fluence
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spectra are discrete, having 1 keV bin widths, and a minimum energy of 1 keV. Fluence
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spectra were found to be robust under variations in scoring region size. Electron fluence
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spectra are presented in the appendix (figure 8).
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Transport parameters are generally set to EGSnrc defaults, except for Rayleigh scattering
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and electron impact ionization, which are are turned on; the high-resolution random number
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generator is used. Photon and bremsstrahlung cross sections are obtained from the XCOM
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(Berger et al 2010) and NRC (Kawrakow et al 2011) databases, respectively. The transport
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cutoff and production threshold for electrons and photons is 1 keV.
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C
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The three categories of cavity theory (large, small and intermediate) and their implementa-
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tions are described in the following subsections, assuming a monoenergetic photon source.
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a)
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Large cavity theory (LCT) requires that the cavity is large compared to the ranges of primary
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electrons, yet small enough not to disturb the photon fluence in the surrounding medium.
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Charged Particle Equilibrium (CPE) is required. In this case, the dose-to-cavity-in-medium
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Large, small and intermediate cavity theories
Large cavity theory
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to dose-to-medium-in-medium conversion factor is given by (Nahum 2009): h i cav R Emax E µen (E) [ΦE (E)]m dE ρ 0 Dcav,m µen icav h , = = RE µen (E) max Dm,m ρ m [ΦE (E)]m dE E ρ 0
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(3)
m
where Dcav,m is the dose to a cavity embedded in a medium m, Dm,m is the dose to the
medium m at the location of the cavity if the cavity did not exist, Emax is the maximum C Large, small and intermediate cavity theories
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energy in the photon spectrum, and µen (E)/ρ is the mass energy absorption coefficient eval-
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uated at energy E. The photon number fluence in the medium, differential in energy E
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is represented by [ΦE (E)]m , and is assumed to be the same as the fluence in the cavity cav indicates that it is a spectrum-averaged [ΦE (E)]cav . The horizontal line above µρen m
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quantity. As a consequence of using FLURZnrc fluence spectra, lower integration bounds cav are robust to variations in the in equation (3) are 1 keV in practice. Values of µρen m
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maximum fractional energy loss per step (ESTEPE) selected for FLURZnrc calculations.
FLURZnrc provides fluence spectrum values at the centre of each energy bin; µen /ρ val-
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ues are also obtained at these energies, using EGSnrc user-code g (Kawrakow et al 2011)
189
with statistical uncertainties < 0.02%. An in-house script is used to perform the numerical
190
integration needed to evaluate the integrals in equation (3). This script was validated by
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comparison with the same calculation carried out in a spreadsheet as well as with the in-
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dependent results of Carlsson Tedgren and Alm Carlsson (2013). Spectrum averaged µen /ρ
193
ratios for water to medium are presented in the appendix (figure 9(a)).
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b)
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Small cavity theory (SCT) requires that the cavity is small compared to the ranges of pri-
196
mary electrons. It is assumed that the presence of the cavity does not perturb the fluence
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of primary electrons in the surrounding medium, and that δ-ray equilibrium exists. The
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Small cavity theory
conversion factor according to Bragg-Gray SCT is given by (Nahum 2009): h i h i Scol (E) cav R Emax Φe− (E) dE E ρ 0 Scol Dcav,m m h cav i . = RE = Scol (E) − max Dm,m ρ m dE Φe (E)
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0
E
m
ρ
(4)
m
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where Scol (E)/ρ is the unrestricted mass collisional stopping power evaluated at energy E,
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and [ΦeE (E)]m is the primary electron number fluence in the medium, differential in energy
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E (assumed to be the same as the fluence in the cavity [ΦeE (E)]cav ). Unrestricted mass
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collisional stopping powers Scol /ρ used for Bragg-Gray SCT are obtained from the NIST
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−
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ESTAR database (Berger 1992). Again, lower integration bounds in equation (4) are 1 keV cav in practice. Values of Scol were insensitive to variations in ESTEPE used in FLURZnrc ρ
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with the same calculation carried out in a spreadsheet, by comparison with Carlsson Tedgren
208
and Alm Carlsson (2013), and by comparison with the SPRRZBGnrc user-code (Selvam and
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calculations. The in-house script used to evaluate equation (4) was validated by comparison
C Large, small and intermediate cavity theories
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Cavity theory applications for kilovoltage cellular dosimetry
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Rogers 2006). Spectrum averaged Scol /ρ ratios for water to medium are presented in figure
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9(b).
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The Spencer-Attix modification of the Bragg-Gray expression uses the restricted instead
of the unrestricted stopping power, and only requires δ-ray equilibrium for energies below the threshold ∆, but requires that the cavity does not affect the fluence of all electrons with energies greater than ∆ (Spencer and Attix 1955): cav Dcav,m L∆ = Dm,m ρ m i h i h nR o nh i o Emax L∆ (E) tot,e− tot,e− Φ (E) dE + Φ (∆) · [S (∆)/ρ] · ∆ col E E cav ρ ∆ cav m h m i i o nh o i = nR E h L∆ (E) tot,e− tot,e− max dE + ΦE (∆) · [Scol (∆)/ρ]m · ∆ ΦE (E) ρ ∆ m
(5)
m
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m
tot,e−
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where [ΦE
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E, which takes into account all generations of electrons with energy ≥ ∆ (i.e., including
213
δ-rays); L∆ (E)/ρ is the restricted mass collisional stopping power for energy losses less than
214
(E)]m is equal to [Φtot,e (E)]cav . the cutoff ∆ (Nahum 2009). Again, it is assumed that [Φtot,e E E
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The first terms in the numerator and denominator of equation (5) account for energy depo-
216
sition < ∆ due to electrons with E > ∆; the second terms (track-end terms) account for
217
energy deposition by electrons with E ≤ ∆. Following the method of Spencer and Attix
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(1955), ∆ is taken to be the energy of an electron that is capable of just crossing the cavity
219
(according to the mean chord length), using RCSDA values for electrons in water, where
220
RCSDA is the electron range according to the continuous slowing down approximation. Val-
221
ues of RCSDA are calculated according to the procedure outlined in ICRU (1984), using
222
stopping powers from the NIST ESTAR database (Berger 1992). The mean chord length is
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given by L = 4V /S (for a convex cavity), for volume V and surface area S (Kellerer 1971).
(E)]m is the total electron number fluence in the medium, differential in energy
−
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Restricted stopping-power ratios are obtained from the EGSnrc user-code SPRRZnrc
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(Rogers et al 2011), using the same simulation geometry as in the fluence spectrum cal-
226
culations (again, results were found to be independent of changes in scoring cavity size).
227
Restricted stopping-power ratios generated by SPRRZnrc are robust to variations in ES-
228
TEPE. Unrestricted and restricted stopping-power ratios differ by at most 0.4% over the
ce
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range of media, energies and cavity sizes considered herein. This relatively good agreement
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can be interpreted as evidence that δ-ray equilibrium exists, and thus Bragg-Gray SCT is
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used throughout the current work instead of Spencer-Attix SCT (which requires a separate C Large, small and intermediate cavity theories
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MC simulation for each energy, medium and ∆, where ∆ is cavity size-dependent).
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c)
234
Intermediate cavity theory (ICT) describes the case where electron ranges are comparable to
235
the size of the cavity. The dose ratio Dcav,m /Dm,m is the sum of SCT and LCT contributions,
237
Intermediate cavity theory
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weighted by a parameter d (Burlin 1966; Nahum 2009): cav cav Dcav,m Scol µen =d· , + (1 − d) · Dm,m ρ m ρ m
(6)
where d varies between 0 and 1 as a function of cavity size, elemental composition and
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density, as well as with source energy. For certain ICT approaches, cavity surroundings also
240
influence d.
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an
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We consider five approaches that have been presented in the literature for determining d, summarized in table 1 along with convenient labels. For several of the methods considered, Table 1
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d is calculated according to (Burlin 1966):
dM
242
d = (1 − e−βL )/(βL),
244
(7)
where L = 4V /S is the mean chord length of the cavity. Equation (7) is equivalent to
246
the average attenuation factor of electrons generated in the wall as they traverse the cavity
247
(Burlin 1966). The fluence of electrons generated within the cavity is assumed to build up
248
exponentially according to the same coefficient β that defines the exponential attenuation of
249
wall electrons, corresponding to a build-up factor of 1 − d. Therefore, the first (second) term
250
in equation (6) represents the contribution to the cavity dose from electrons originating in
251
the wall (in the cavity).
pte
245
The ‘Burlin-Chan’ and ‘Burlin-Janssens’ approaches (table 1) calculate β according to
253
equations of the form e−βR = x, where x represents the fraction of electrons that reach a
254
depth R. The path of an electron is tortuous so that the depth of penetration is less than
255
the total path length, which is generally assumed to be RCSDA . Burlin and Chan (1969)
256
suggested R = F × RCSDA , where F is a foreshortening factor; F = Rmax /RCSDA ≤ 1, where
ce
252
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257
Rmax is an empirical range. Janssens et al (1974) suggested using the extrapolated range
258
for R. In the current work, F is assumed to be unity so that R = RCSDA . The ‘Burlin-
259
Chan’ method uses x = 0.01, as suggested by Burlin and Chan (1969). Janssens et al (1974) C Large, small and intermediate cavity theories
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Cavity theory applications for kilovoltage cellular dosimetry
Table 1: Summary of intermediate cavity theory approaches implemented in the current work, with convenient labels. Brief description and reference
Burlin-Chan
Equation (7) gives d with β given by e−βRCSDA = 0.01, using RCSDA for electrons of the incident photon energy (Burlin and Chan 1969).
BurlinJanssens
Equation (7) gives d with β given by e−βRCSDA = 0.04, using RCSDA for electrons of the incident photon energy (Janssens et al 1974).
Loevinger
Equation (7) gives d with β (in cm−1 ) given by β = (16ρcav )/(Tmax − 0.036)1.4 where ρcav is the density of the cavity in g/cm3 and Tmax is the maximum kinetic energy of electrons in MeV, taken as the incident photon energy, which must be > 0.036 MeV (Loevinger 1956).
weighted sum
d given by d = (fphoto · dphoto ) + (fincoh · dincoh ), where fphoto = −1 T 1 + σincoh · hν and fincoh = 1 − fphoto ; σincoh is the incoherent cross τ section, τ is the photoelectric cross section, and T is the mean Compton electron energy. Both dphoto and dincoh are given by equation (7) with β given by e−βRCSDA = 0.01; RCSDA for dphoto (dincoh ) corresponds to electrons with energy equal to the incident photon energy hν (the maximum Compton electron energy) (Carlsson Tedgren and Alm Carlsson 2013).
pte
dM
an
Label
NEFW /D NEFW is the cavity dose due d is given by 1 − d = (Dcav cav ), where Dcav to electrons originating from within the cavity (i.e., NEFW = No Electrons NEFW is obtained from a version of the EGSnrc user-code From Wall). Dcav egs chamber that was modified to discard any electrons generated outside of the cavity. Dcav is the cavity dose without any restrictions on electron origin (Mobit et al 1996).
ce
MC estimate
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determined that x = 0.04 is most appropriate, based on their more rigorous model, which
261
accounts for backscatter at the interface between unlike media; hence x = 0.04 is used for
262
the ‘Burlin-Janssens’ method. Values of RCSDA , needed for several of the ICT methods, are
263
obtained as described in section 2.Cb).
us cri
pt
260
The ‘Loevinger’ method was intended to be used for air-filled cavities, and is applica-
265
ble to energies > 36 keV (Loevinger 1956). However, the appropriateness of this method
266
has been questioned for the kilovoltage energy range considered herein (Burlin and Chan
267
1969). Despite this, recent work has considered this approach for brachytherapy-related
268
applications, e.g., classification of different-sized water cavities (Carlsson Tedgren and Alm
269
Carlsson 2013), calculation of doses to thermoluminescent detectors (TLDs) (Patel et al
270
2001) and electron paramagnetic resonance (EPR) detectors (Antonovic et al 2009), and
271
hence it is considered here.
an
264
The ‘weighted sum’ method is based on the work of Carlsson Tedgren and Alm Carlsson
273
(2013), who considered monoenergetic photon sources between 20 and 300 keV, as well as
274
125
275
between 20 and 100 keV; however, we use it over the entire energy range considered herein.
276
The ‘weighted sum’ method involves computing d as the sum of photoelectron and Compton
277
electron contributions, weighted by fphoto and fincoh , respectively (see table 1). The coeffi-
278
cients fphoto and fincoh are calculated using photon cross sections for water; they represent
279
the fraction of energy transferred to electrons that is given to photoelectrons and Compton
280
electrons, respectively.
169
Yb and
192
Ir brachytherapy sources. They used this approach for source energies
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I,
dM
272
The ‘MC estimate’ approach uses MC simulations to obtain a direct estimate of (1 − d)
282
according to its fundamental meaning: (1 − d) is calculated as the fraction of cavity dose due
283
to electrons originating from within the cavity. The simulation geometry used to calculate
284
NEFW Dcav and Dcav (the ratio of these quantities gives (1 − d) — see table 1) consists of a
285
spherical water cavity embedded in a (2 cm)3 cubic water phantom.
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All but two of the ICT approaches produce d values that increase monotonically with
301
energy. ‘Weighted sum’ and ‘MC estimate’ both have a local minimum, a local maximum
302
and an inflection point around 50 keV. Compton interaction probabilities increase with pho-
303
ton energy so that the electron fluence spectrum has an increasing proportion of relatively
304
low-energy electrons (figure 8). In the context of the ‘MC estimate’ method, lower electron
305
energies have shorter ranges so that a larger fraction of the cavity dose is due to electrons
306
NEFW originating from within the cavity; this will increase Dcav /Dcav , resulting in smaller d
307
values with increasing energy (see equation given in table 1). For the ‘weighted sum’ ap-
308
proach, fincoh increases as energy increases, dincoh is generally less than dphoto (because dincoh
309
corresponds to a lower electron energy), and thus the value of d computed will decrease since
310
a higher weight is given to the smaller value of d.
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pt
300
Small, large and intermediate cavity theory estimates of (Dw,m /Dm,m )CT , and MC simulation results for (Dw,m /Dm,m )M C for rcav = 6 µm are presented in figure 3 for all five bulk Fig 3
313
tissues. Similar trends in Dw,m /Dm,m as a function of energy are observed for rcav = 3 or
314
9 µm (results not shown). In general, values of Dw,m /Dm,m are closer to unity for smaller
315
rcav . Trends in (Dw,m /Dm,m )M C can be understood by considering how the radiological
316
parameters of the media considered compare to those of water. For lower energies, photo-
317
electric interactions dominate, and photoelectric interaction cross sections of adenoidcystic
318
carcinoma, melanoma and muscle are higher those of water, resulting in Dm,m values larger
319
than Dw,m . The opposite is true for mammary gland and squamous cell lung carcinoma.
320
Similarly, whether SCT and LCT represent maximum or minimum values may be understood
321
by considering the relative magnitudes of the radiological parameters. For example, mass
322
energy absorption coefficients for mammary gland are lower than those of water, while the
323
opposite is true for the stopping powers (figure 9), hence the LCT and SCT values in figure
324
3(b) represent the maximum and minimum, respectively. The ICT predictions lie between
325
the LCT and SCT values because d is constrained to lie between 0 and 1 (see equation (6)).
326
At higher energies, incoherent interactions dominate, and there are relatively small cross sec-
327
tion variations amongst considered media, hence Dw,m /Dm,m values approach unity. Figure
328
3 demonstrates that differences between MC results and cavity theory predictions vary with
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329
phantom material m. For example, for mammary gland and squamous cell lung carcinoma,
330
most cavity theories overestimate (Dw,m /Dm,m )M C , except for SCT. In contrast, for ade-
331
noidcystic carcinoma and melanoma, most cavity theories underestimate (Dw,m /Dm,m )M C , A Evaluation of cavity theory approaches (part I)
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energy sources, RCSDA values (second column of table 2) are comparable to nuclear dimen-
357
sions, and d is strongly influenced by nucleus size: for cavity radii between 2 and 9 µm,
358
d varies between 0.616 and 0.208 at 20 keV, and between 0.781 and 0.387 at 30 keV. As
359
source energy increases, RCSDA increases toward values that are large compared to cellular
360
dimensions, and d is less strongly influenced by nucleus size; d is always within 2% of unity
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for the 370 keV photon source.
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Energy/keV
362
10−4
8.57 × 1.76 × 10−3 4.32 × 10−3 1.20 × 10−2 1.15 × 10−1
2 0.616 0.781 0.902 0.963 0.996
3 0.500 0.695 0.858 0.945 0.994
dM
20 30 50 90 370
RCSDA /(g·cm−2 )
an
Table 2: Values of d according to the ‘Burlin-Janssens’ method for five energies and eight cavity radii, rcav , corresponding to the eight nuclear radii considered herein. These values are derived using RCSDA for water (taking into account the assumed nuclear media density of 1.06 g/cm3 ), motivated by the relatively small differences in RCSDA (in g/cm2 ) for water and the nuclear media considered herein (
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Download details: IP Address: 130.133.8.114 This content was downloaded on 01/04/2017 at 02:52 Manuscript version: Accepted Manuscript Oliver et al To cite this article before publication: Oliver et al, 2017, Phys. Med. Biol., at press: https://doi.org/10.1088/1361-6560/aa6a42 This Accepted Manuscript is: © 2017 Institute of Physics and Engineering in Medicine During the embargo period (the 12 month period from the publication of the Version of Record of this article), the Accepted Manuscript is fully protected by copyright and cannot be reused or reposted elsewhere. As the Version of Record of this article is going to be / has been published on a subscription basis, this Accepted Manuscript is available for reuse under a CC BY-NC-ND 3.0 licence after a 12 month embargo period. After the embargo period, everyone is permitted to use all or part of the original content in this article for non-commercial purposes, provided that they adhere to all the terms of the licence https://creativecommons.org/licences/by-nc-nd/3.0 Although reasonable endeavours have been taken to obtain all necessary permissions from third parties to include their copyrighted content within this article, their full citation and copyright line may not be present in this Accepted Manuscript version. Before using any content from this article, please refer to the Version of Record on IOPscience once published for full citation and copyright details, as permissions will likely be required. All third party content is fully copyright protected, unless specifically stated otherwise in the figure caption in the Version of Record. When available, you can view the Version of Record for this article at: http://iopscience.iop.org/article/10.1088/1361-6560/aa6a42
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Cavity theory applications for kilovoltage cellular dosimetry
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Cavity theory applications for kilovoltage cellular dosimetry P. A. K. Oliver and Rowan M. Thomson Carleton Laboratory for Radiotherapy Physics, Physics Dept, Carleton University, Ottawa, K1S 5B6
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Revised March 15, 2017
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Abstract
Relationships between macroscopic (bulk tissue) and microscopic (cellular) dose descriptors are investigated using cavity theory and Monte Carlo (MC) simulations. Small, large, and multiple intermediate cavity theory (SCT, LCT, and ICT, respectively) approaches are considered for 20 to 370 keV incident photons; ICT is a sum of SCT and LCT contributions weighted by parameter d. Considering µm-sized cavities of water in bulk tissue phantoms, different cavity theory approaches are evaluated via comparison of Dw,m /Dm,m (where Dw,m is dose-to-water-in-medium and Dm,m is dose-to-medium-in-medium) with MC results. The best overall agreement is achieved with an ICT approach in which d = (1−e−βL )/(βL), where L is the mean chord length of the cavity and β is given by e−βRCSDA = 0.04 (RCSDA is the continuous slowing down approximation range of an electron of energy equal to that of incident photons). Cell nucleus doses, Dnuc , computed with this ICT approach are compared with those from MC simulations involving multicellular soft tissue models considering a representative range of cell/nucleus sizes and elemental compositions. In 91% of cases, ICT and MC predictions agree within 3%; disagreement is at most 8.8%. These results suggest that cavity theory may be useful for linking doses from model-based dose calculation algorithms (MBDCAs) with energy deposition in cellular targets. Finally, based on the suggestion that clusters of water molecules associated with DNA are important radiobiological targets, two approaches for estimating dose-to-water by application of SCT to MC results for Dm,m or Dnuc are compared. Results for these two estimates differ by up to 35%, demonstrating the sensitivity of energy deposition within a small volume of water in nucleus to the geometry and composition of its surroundings. In terms of the debate over the dose specification medium for MBDCAs, these results do not support conversion of Dm,m to Dw,m using SCT.
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1.
Introduction
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Macroscopic dose descriptors (based on computational models involving ∼mm-sized voxels)
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34
such as dose-to-medium-in-medium (Dm,m ), dose-to-water-in-medium (Dw,m ) or dose-to-
35
water-in-water (Dw,w ) are commonly calculated in clinical treatment planning (Beaulieu
36
et al 2012; Rivard et al 2004). Monte Carlo (MC) and other model-based dose calculation
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algorithms (MBDCAs) generally compute Dm,m , which represents dose to a region or voxel in
38
which tissue composition is assumed to be locally uniform (homogeneous bulk tissue) within
39
a voxelized patient model, ignoring underlying microscopic tissue structure. However, cells
40
and their constituents are considered biologically-relevant targets for radiation-induced cell
41
death. DNA is contained within the nucleus for most of the cell cycle so that the nucleus
42
is often considered a target of interest for understanding radiobiological outcomes (Hall and
43
Giaccia 2012; Kuzin 1963). Water molecules located near DNA are clustered into ∼nm-sized
44
volumes, and can undergo radiolysis resulting in free radical production and DNA damage
45
(Goodhead 2006). Recent work supports the importance of such ∼nm-sized volumes within
46
the nucleus (Lindborg et al 2013).
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MC simulations involving multicellular models of human soft tissues demonstrate that
48
macro- and microscopic dose descriptors disagree by up to 32% at kilovoltage photon energies,
49
suggesting that underlying cellular structure and composition are important considerations
50
for accurately determining biologically-relevant dose descriptors (Oliver and Thomson 2017).
51
Unfortunately, simulation of radiation transport and energy deposition within detailed, mul-
52
ticellular models or on the molecular level is not always practical due to computational
53
limitations and time constraints. Therefore, conversion factors relating macro- (e.g., Dm,m )
54
and microscopic (e.g., nuclear dose Dnuc ) dose descriptors are potentially useful, and may
55
be estimated using cavity theory (see e.g., Nahum 2009). Furthermore, cavity theory may
56
provide a theoretical perspective helpful in understanding, predicting and verifying compu-
57
tational and experimental results.
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47
Cavity theory conversion factors relating Dm,m to biologically-relevant dose descriptors
59
such as Dnuc have been discussed in the context of the debate over the medium for dose spec-
60
ification for MC treatment planning (Beaulieu et al 2012). These factors should be based
61
on an appropriate cavity theory method corresponding to the size, density and elemental
62
composition of the target of interest. Small, intermediate and large cavity theories (SCT,
63
ICT and LCT) have domains of applicability dependent on target size relative to electron
64
ranges. Carlsson Tedgren and Alm Carlsson (2013) classified water cavities (sizes from 1 nm
65
to 1 mm) in tissue as small, intermediate or large based on an ICT approach, considering
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source energies relevant for brachytherapy. Beyond dose to a water voxel (or voxel composed
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of water-like tissue) in medium, Dw,m may represent dose to DNA-bound water molecules
68
(Goodhead 2006), or dose to a subcellular compartment whose elemental composition is
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Cavity theory applications for kilovoltage cellular dosimetry
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closer to that of water than medium (Chetty et al 2007). Motivated by the potential radio-
70
biological importance of small volumes of DNA-bound water located within the nucleus and
71
the fact that a small water target is independent of cell type, Carlsson Tedgren and Alm
72
Carlsson (2013) suggested that conversion of MBDCA-computed dose-to-medium, Dm,m , to
73
dose-to-water, Dw,m , via SCT may be “preferred” in the context of brachytherapy. For kilo-
74
voltage photon sources, Enger et al (2012) compared SCT and LCT predictions of Dw,m to
75
MC-computed doses to nuclear cavities (7 µm radius) in otherwise homogeneous (1 cm)3
76
soft-tissue phantoms.
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Thus, despite considerable discussion in the literature regarding applications of cavity
78
theory to predict doses to cellular compartments, no such cavity theory predictions have
79
been compared with results of MC simulations involving detailed, multicellular microscopic
80
tissue models — this is the focus of the current work. We investigate relationships between
81
macroscopic (bulk tissue) and microscopic (cellular) dose descriptors using MC simulations
82
and cavity theory for kilovoltage photon energies relevant for brachytherapy, as well as ortho-
83
voltage treatments and diagnostic radiology. Various cavity theory approaches are evaluated
84
by comparing SCT, LCT, and multiple ICT predictions of Dw,m /Dm,m with corresponding
85
MC results for a range of source energies, bulk tissue media and several ∼ µm-sized (i.e.,
86
nucleus-sized) cavities (part I). The cavity theory approach providing the closest overall
87
agreement with MC results for Dw,m /Dm,m is then used to predict nuclear doses which are
88
compared with MC simulation results from detailed, multicellular models of healthy and
89
cancerous human soft tissues, considering a range of representative subcellular compartment
90
sizes and elemental compositions (part II). Finally, SCT conversion factors applied to MC
91
simulation results are investigated for prediction of energy deposition within a small volume
92
of water in nucleus (part III).
93
2.
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Section 2.A provides an overview of the three parts of our study, before details on MC
95
simulations and cavity theory methods are given in sections 2.B and 2.C, respectively.
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Methods
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A
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The three parts of our study are schematically depicted in figure 1 and described further Fig 1
98
below. In each case, the source is a parallel beam of monoenergetic photons with energies of
99
20, 30, 50, 90 or 370 keV, covering the range of photon energies of common brachytherapy
100
sources, as well as diagnostic radiology. The ranges of electrons set into motion in soft
101
tissues by these kilovoltage photons are comparable to sizes of subcellular compartments
102
(Oliver and Thomson 2017). The beam has a circular cross section with 1 cm radius. Five
103
bulk tissues are considered: adenoidcystic carcinoma, mammary gland, melanoma, muscle
104
and squamous cell lung carcinoma.
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Overview
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In part I, values of Dw,m /Dm,m (figure 1(a)) from MC simulations (section 2.B) are
106
compared with corresponding cavity theory predictions, considering five ICT approaches,
107
plus SCT and LCT (section 2.C). Three cavity radii, five bulk tissues and five incident
108
photon energies are considered, corresponding to a total of N = 75 scenarios. The cavity
109
theory method with the lowest root mean square error (RMSE) when compared with MC
111
dM
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results is used in part II of this study. The RMSE is calculated as follows: v u !2 N u X Dw,m Dw,m u1 , − RMSE = t N i=1 Dm,m i,CT Dm,m i,M C
(1)
where the sum is over the entire parameter space considered (i.e., N = 75), and the quan-
113
tities (Dw,m /Dm,m )i,CT and (Dw,m /Dm,m )i,M C are the values (for scenario i) of Dw,m /Dm,m
114
according to cavity theory and MC simulation, respectively.
pte
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Nuclear doses (Dnuc ) are considered in part II (figure 1(b)). MC simulation results for
116
Dnuc /Dm,m are compared with corresponding cavity theory predictions (using the method
117
which minimizes the RMSE from part I). MC simulations for Dnuc involve detailed, multi-
118
cellular models of normal and cancerous human soft tissues, described in detail by Oliver
119
and Thomson (2017) and illustrated in figure 1(b) (see section 2.B). Three cell radii rcell are
120
investigated, along with three nuclear radii rnuc for each rcell ; these radii are based on typical
121
values for human cells. Additionally, seven cytoplasm/nucleus elemental compositions, five
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bulk tissues and five source energies are considered, for a total of 1575 scenarios. Cavity
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theory predictions are also compared with Dnuc,m /Dm,m , where Dnuc,m is the dose to a single
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nucleus embedded in an otherwise homogeneous bulk tissue phantom (figure 1(c)). A Overview
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Part II: Cavity theory applied to nuclear doses (Dnuc & Dnuc,m) (b) cell cluster 2.06 µm
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Part I: Evaluation of cavity theory approaches (a) one cavity
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Cavity theory applications for kilovoltage cellular dosimetry
(c) single nucleus
ECM
cytoplasm
Dnuc
nucleus
rcav = 3, 6, 9 µm
nucleus (Dnuc,m)
bulk tissue
an
water (Dw,m) or bulk tissue (Dm,m)
rcell / !m rnuc / !m
5 2, 3, 4
7.35 4, 5, 6
10 7, 8, 9
dM
Part III: Using Bragg-Gray SCT to estimate dose to small water cavities (e) small water cavity in (d) small water cavity cell cluster
SCT Dw,nuc
pte
SCT Dw,m
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Figure 1: Schematic diagrams summarizing the three parts of the study. Part I: (a) a single cavity (radius = rcav ) consisting of either water or bulk tissue is embedded at the centre of a bulk tissue phantom. Part II: nuclear doses are considered. (b) Cross section of multicellular model consisting of 13 cells in a hexagonal lattice with cells (radius = rcell ) embedded in a sphere of extracellular matrix (ECM) (radius 18 to 33 µm); dose is scored in the central nucleus (radius = rnuc ). (c) A single nucleus embedded in an otherwise homogeneous bulk tissue phantom. Part III: Bragg-Gray SCT is used to estimate dose to a small water cavity embedded in (d) a bulk tissue phantom, or (e) the central nucleus of a cluster of 13 cells. Each geometry is located at the centre of a (2 cm)3 cubic bulk tissue phantom.
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Finally, part III considers the potential use of SCT for conversion of MC-calculated
126
Dm,m to Dw,m , motivated by the radiobiological importance and universality (i.e., lack of
127
dependence on cell type) of DNA-bound water molecules within the nucleus as a radiation
128
target (Carlsson Tedgren and Alm Carlsson 2013; Goodhead 2006). We employ SCT to
129
estimate dose to a small volume of water in two different ways, with scenarios illustrated in
130
figures 1(d) and 1(e). The small water cavities are not explicitly modelled in MC simulations:
131
Dm,m (figure 1(a)) and Dnuc (figure 1(b)) are obtained from MC simulations, and SCT
132
conversion factors (stopping power ratios – details provided in section 2.C.b)) are used to
us cri
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SCT SCT estimate Dw,m and Dw,nuc , respectively, which we compare via the ratio: " w # . " w # m SCT Dw,nuc Scol Scol Scol Dnuc . = Dnuc × × Dm,m × = SCT Dw,m ρ nuc ρ m Dm,m ρ nuc
(2)
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133
pt
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Elemental compositions and mass densities ρ of mammary gland (ρ = 1.02 g/cm3 ) and
136
muscle (ρ = 1.05 g/cm3 ) are from Woodard and White (1986) and ICRU (1989), respec-
137
tively. The compositions of the three cancerous tissues (melanoma, adenoidcystic carcinoma
138
and squamous cell lung carcinoma) are obtained from Maughan et al (1997), assuming a
139
density of 1.04 g/cm3 (Thomson et al 2013). Water is assumed to exist at 22◦ C so that its
140
density is taken to be 0.998 g/cm3 (Rivard et al 2004). Seven nucleus and cytoplasm elemen-
141
tal compositions representative of animal cells are considered, each of density 1.06 g/cm3
142
(Thomson et al 2013). The elemental compositions of these five bulk tissues, seven nuclear
143
media and seven cytoplasm media are presented in Table A1 of Oliver and Thomson (2017)
144
(page 1434, in the appendix). ECM definitions are determined by requiring that the aver-
145
age, mass-weighted composition and density of the cells and ECM are approximately equal
146
to those of the surrounding bulk tissue; this calculation is carried out for each bulk tissue,
147
cell/nucleus size combination and cytoplasm/nucleus elemental composition.
148
B
149
MC dose calculations are carried out using the EGSnrc user-code egs chamber, which im-
150
plements a variety of variance reduction techniques to efficiently calculate radiation dose to
ce
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dM
135
Monte Carlo simulations
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151
small targets (Wulff et al 2008; Kawrakow et al 2011; Kawrakow et al 2009). MC results
152
for Dnuc /Dm,m and Dnuc,m /Dm,m are from previous work (Oliver and Thomson 2017), while
153
additional MC simulations for Dw,m /Dm,m are carried out for the present work. A sufficient B Monte Carlo simulations
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number of histories are simulated to obtain statistical uncertainties on doses of < 0.91%.
pt
154
Electron and photon fluence spectra are used for calculating the spectrum-averaged mass
156
collisional stopping power and mass energy absorption coefficient ratios needed for cavity
157
theory estimates (section 2.C), and are obtained from the EGSnrc user-code FLURZnrc
158
(Rogers et al 2011). The geometry consists of a cylindrical phantom composed of water or
159
bulk tissue of 2 cm height and diameter. The scoring region is also cylindrical (with height
160
and diameter equal to 0.05 cm) and is located at the centre of this phantom. The fluence
161
spectra are discrete, having 1 keV bin widths, and a minimum energy of 1 keV. Fluence
162
spectra were found to be robust under variations in scoring region size. Electron fluence
163
spectra are presented in the appendix (figure 8).
us cri
155
Transport parameters are generally set to EGSnrc defaults, except for Rayleigh scattering
165
and electron impact ionization, which are are turned on; the high-resolution random number
166
generator is used. Photon and bremsstrahlung cross sections are obtained from the XCOM
167
(Berger et al 2010) and NRC (Kawrakow et al 2011) databases, respectively. The transport
168
cutoff and production threshold for electrons and photons is 1 keV.
169
C
170
The three categories of cavity theory (large, small and intermediate) and their implementa-
171
tions are described in the following subsections, assuming a monoenergetic photon source.
172
a)
173
Large cavity theory (LCT) requires that the cavity is large compared to the ranges of primary
174
electrons, yet small enough not to disturb the photon fluence in the surrounding medium.
175
Charged Particle Equilibrium (CPE) is required. In this case, the dose-to-cavity-in-medium
177
dM
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Large, small and intermediate cavity theories
Large cavity theory
ce
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an
164
to dose-to-medium-in-medium conversion factor is given by (Nahum 2009): h i cav R Emax E µen (E) [ΦE (E)]m dE ρ 0 Dcav,m µen icav h , = = RE µen (E) max Dm,m ρ m [ΦE (E)]m dE E ρ 0
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179
(3)
m
where Dcav,m is the dose to a cavity embedded in a medium m, Dm,m is the dose to the
medium m at the location of the cavity if the cavity did not exist, Emax is the maximum C Large, small and intermediate cavity theories
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energy in the photon spectrum, and µen (E)/ρ is the mass energy absorption coefficient eval-
181
uated at energy E. The photon number fluence in the medium, differential in energy E
183
is represented by [ΦE (E)]m , and is assumed to be the same as the fluence in the cavity cav indicates that it is a spectrum-averaged [ΦE (E)]cav . The horizontal line above µρen m
184
185
us cri
182
pt
180
quantity. As a consequence of using FLURZnrc fluence spectra, lower integration bounds cav are robust to variations in the in equation (3) are 1 keV in practice. Values of µρen m
186
maximum fractional energy loss per step (ESTEPE) selected for FLURZnrc calculations.
FLURZnrc provides fluence spectrum values at the centre of each energy bin; µen /ρ val-
188
ues are also obtained at these energies, using EGSnrc user-code g (Kawrakow et al 2011)
189
with statistical uncertainties < 0.02%. An in-house script is used to perform the numerical
190
integration needed to evaluate the integrals in equation (3). This script was validated by
191
comparison with the same calculation carried out in a spreadsheet as well as with the in-
192
dependent results of Carlsson Tedgren and Alm Carlsson (2013). Spectrum averaged µen /ρ
193
ratios for water to medium are presented in the appendix (figure 9(a)).
194
b)
195
Small cavity theory (SCT) requires that the cavity is small compared to the ranges of pri-
196
mary electrons. It is assumed that the presence of the cavity does not perturb the fluence
197
of primary electrons in the surrounding medium, and that δ-ray equilibrium exists. The
199
dM
Small cavity theory
conversion factor according to Bragg-Gray SCT is given by (Nahum 2009): h i h i Scol (E) cav R Emax Φe− (E) dE E ρ 0 Scol Dcav,m m h cav i . = RE = Scol (E) − max Dm,m ρ m dE Φe (E)
pte
198
an
187
0
E
m
ρ
(4)
m
200
where Scol (E)/ρ is the unrestricted mass collisional stopping power evaluated at energy E,
201
and [ΦeE (E)]m is the primary electron number fluence in the medium, differential in energy
202
E (assumed to be the same as the fluence in the cavity [ΦeE (E)]cav ). Unrestricted mass
203
collisional stopping powers Scol /ρ used for Bragg-Gray SCT are obtained from the NIST
ce
−
−
206
ESTAR database (Berger 1992). Again, lower integration bounds in equation (4) are 1 keV cav in practice. Values of Scol were insensitive to variations in ESTEPE used in FLURZnrc ρ
207
with the same calculation carried out in a spreadsheet, by comparison with Carlsson Tedgren
208
and Alm Carlsson (2013), and by comparison with the SPRRZBGnrc user-code (Selvam and
204
205
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m
calculations. The in-house script used to evaluate equation (4) was validated by comparison
C Large, small and intermediate cavity theories
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Rogers 2006). Spectrum averaged Scol /ρ ratios for water to medium are presented in figure
210
9(b).
pt
209
us cri
The Spencer-Attix modification of the Bragg-Gray expression uses the restricted instead
of the unrestricted stopping power, and only requires δ-ray equilibrium for energies below the threshold ∆, but requires that the cavity does not affect the fluence of all electrons with energies greater than ∆ (Spencer and Attix 1955): cav Dcav,m L∆ = Dm,m ρ m i h i h nR o nh i o Emax L∆ (E) tot,e− tot,e− Φ (E) dE + Φ (∆) · [S (∆)/ρ] · ∆ col E E cav ρ ∆ cav m h m i i o nh o i = nR E h L∆ (E) tot,e− tot,e− max dE + ΦE (∆) · [Scol (∆)/ρ]m · ∆ ΦE (E) ρ ∆ m
(5)
m
an
m
tot,e−
211
where [ΦE
212
E, which takes into account all generations of electrons with energy ≥ ∆ (i.e., including
213
δ-rays); L∆ (E)/ρ is the restricted mass collisional stopping power for energy losses less than
214
(E)]m is equal to [Φtot,e (E)]cav . the cutoff ∆ (Nahum 2009). Again, it is assumed that [Φtot,e E E
215
The first terms in the numerator and denominator of equation (5) account for energy depo-
216
sition < ∆ due to electrons with E > ∆; the second terms (track-end terms) account for
217
energy deposition by electrons with E ≤ ∆. Following the method of Spencer and Attix
218
(1955), ∆ is taken to be the energy of an electron that is capable of just crossing the cavity
219
(according to the mean chord length), using RCSDA values for electrons in water, where
220
RCSDA is the electron range according to the continuous slowing down approximation. Val-
221
ues of RCSDA are calculated according to the procedure outlined in ICRU (1984), using
222
stopping powers from the NIST ESTAR database (Berger 1992). The mean chord length is
223
given by L = 4V /S (for a convex cavity), for volume V and surface area S (Kellerer 1971).
(E)]m is the total electron number fluence in the medium, differential in energy
−
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−
Restricted stopping-power ratios are obtained from the EGSnrc user-code SPRRZnrc
225
(Rogers et al 2011), using the same simulation geometry as in the fluence spectrum cal-
226
culations (again, results were found to be independent of changes in scoring cavity size).
227
Restricted stopping-power ratios generated by SPRRZnrc are robust to variations in ES-
228
TEPE. Unrestricted and restricted stopping-power ratios differ by at most 0.4% over the
ce
224
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range of media, energies and cavity sizes considered herein. This relatively good agreement
230
can be interpreted as evidence that δ-ray equilibrium exists, and thus Bragg-Gray SCT is
231
used throughout the current work instead of Spencer-Attix SCT (which requires a separate C Large, small and intermediate cavity theories
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MC simulation for each energy, medium and ∆, where ∆ is cavity size-dependent).
233
c)
234
Intermediate cavity theory (ICT) describes the case where electron ranges are comparable to
235
the size of the cavity. The dose ratio Dcav,m /Dm,m is the sum of SCT and LCT contributions,
237
Intermediate cavity theory
us cri
236
pt
232
weighted by a parameter d (Burlin 1966; Nahum 2009): cav cav Dcav,m Scol µen =d· , + (1 − d) · Dm,m ρ m ρ m
(6)
where d varies between 0 and 1 as a function of cavity size, elemental composition and
239
density, as well as with source energy. For certain ICT approaches, cavity surroundings also
240
influence d.
241
an
238
We consider five approaches that have been presented in the literature for determining d, summarized in table 1 along with convenient labels. For several of the methods considered, Table 1
243
d is calculated according to (Burlin 1966):
dM
242
d = (1 − e−βL )/(βL),
244
(7)
where L = 4V /S is the mean chord length of the cavity. Equation (7) is equivalent to
246
the average attenuation factor of electrons generated in the wall as they traverse the cavity
247
(Burlin 1966). The fluence of electrons generated within the cavity is assumed to build up
248
exponentially according to the same coefficient β that defines the exponential attenuation of
249
wall electrons, corresponding to a build-up factor of 1 − d. Therefore, the first (second) term
250
in equation (6) represents the contribution to the cavity dose from electrons originating in
251
the wall (in the cavity).
pte
245
The ‘Burlin-Chan’ and ‘Burlin-Janssens’ approaches (table 1) calculate β according to
253
equations of the form e−βR = x, where x represents the fraction of electrons that reach a
254
depth R. The path of an electron is tortuous so that the depth of penetration is less than
255
the total path length, which is generally assumed to be RCSDA . Burlin and Chan (1969)
256
suggested R = F × RCSDA , where F is a foreshortening factor; F = Rmax /RCSDA ≤ 1, where
ce
252
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257
Rmax is an empirical range. Janssens et al (1974) suggested using the extrapolated range
258
for R. In the current work, F is assumed to be unity so that R = RCSDA . The ‘Burlin-
259
Chan’ method uses x = 0.01, as suggested by Burlin and Chan (1969). Janssens et al (1974) C Large, small and intermediate cavity theories
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Cavity theory applications for kilovoltage cellular dosimetry
Table 1: Summary of intermediate cavity theory approaches implemented in the current work, with convenient labels. Brief description and reference
Burlin-Chan
Equation (7) gives d with β given by e−βRCSDA = 0.01, using RCSDA for electrons of the incident photon energy (Burlin and Chan 1969).
BurlinJanssens
Equation (7) gives d with β given by e−βRCSDA = 0.04, using RCSDA for electrons of the incident photon energy (Janssens et al 1974).
Loevinger
Equation (7) gives d with β (in cm−1 ) given by β = (16ρcav )/(Tmax − 0.036)1.4 where ρcav is the density of the cavity in g/cm3 and Tmax is the maximum kinetic energy of electrons in MeV, taken as the incident photon energy, which must be > 0.036 MeV (Loevinger 1956).
weighted sum
d given by d = (fphoto · dphoto ) + (fincoh · dincoh ), where fphoto = −1 T 1 + σincoh · hν and fincoh = 1 − fphoto ; σincoh is the incoherent cross τ section, τ is the photoelectric cross section, and T is the mean Compton electron energy. Both dphoto and dincoh are given by equation (7) with β given by e−βRCSDA = 0.01; RCSDA for dphoto (dincoh ) corresponds to electrons with energy equal to the incident photon energy hν (the maximum Compton electron energy) (Carlsson Tedgren and Alm Carlsson 2013).
pte
dM
an
Label
NEFW /D NEFW is the cavity dose due d is given by 1 − d = (Dcav cav ), where Dcav to electrons originating from within the cavity (i.e., NEFW = No Electrons NEFW is obtained from a version of the EGSnrc user-code From Wall). Dcav egs chamber that was modified to discard any electrons generated outside of the cavity. Dcav is the cavity dose without any restrictions on electron origin (Mobit et al 1996).
ce
MC estimate
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determined that x = 0.04 is most appropriate, based on their more rigorous model, which
261
accounts for backscatter at the interface between unlike media; hence x = 0.04 is used for
262
the ‘Burlin-Janssens’ method. Values of RCSDA , needed for several of the ICT methods, are
263
obtained as described in section 2.Cb).
us cri
pt
260
The ‘Loevinger’ method was intended to be used for air-filled cavities, and is applica-
265
ble to energies > 36 keV (Loevinger 1956). However, the appropriateness of this method
266
has been questioned for the kilovoltage energy range considered herein (Burlin and Chan
267
1969). Despite this, recent work has considered this approach for brachytherapy-related
268
applications, e.g., classification of different-sized water cavities (Carlsson Tedgren and Alm
269
Carlsson 2013), calculation of doses to thermoluminescent detectors (TLDs) (Patel et al
270
2001) and electron paramagnetic resonance (EPR) detectors (Antonovic et al 2009), and
271
hence it is considered here.
an
264
The ‘weighted sum’ method is based on the work of Carlsson Tedgren and Alm Carlsson
273
(2013), who considered monoenergetic photon sources between 20 and 300 keV, as well as
274
125
275
between 20 and 100 keV; however, we use it over the entire energy range considered herein.
276
The ‘weighted sum’ method involves computing d as the sum of photoelectron and Compton
277
electron contributions, weighted by fphoto and fincoh , respectively (see table 1). The coeffi-
278
cients fphoto and fincoh are calculated using photon cross sections for water; they represent
279
the fraction of energy transferred to electrons that is given to photoelectrons and Compton
280
electrons, respectively.
169
Yb and
192
Ir brachytherapy sources. They used this approach for source energies
pte
I,
dM
272
The ‘MC estimate’ approach uses MC simulations to obtain a direct estimate of (1 − d)
282
according to its fundamental meaning: (1 − d) is calculated as the fraction of cavity dose due
283
to electrons originating from within the cavity. The simulation geometry used to calculate
284
NEFW Dcav and Dcav (the ratio of these quantities gives (1 − d) — see table 1) consists of a
285
spherical water cavity embedded in a (2 cm)3 cubic water phantom.
ce
281
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All but two of the ICT approaches produce d values that increase monotonically with
301
energy. ‘Weighted sum’ and ‘MC estimate’ both have a local minimum, a local maximum
302
and an inflection point around 50 keV. Compton interaction probabilities increase with pho-
303
ton energy so that the electron fluence spectrum has an increasing proportion of relatively
304
low-energy electrons (figure 8). In the context of the ‘MC estimate’ method, lower electron
305
energies have shorter ranges so that a larger fraction of the cavity dose is due to electrons
306
NEFW originating from within the cavity; this will increase Dcav /Dcav , resulting in smaller d
307
values with increasing energy (see equation given in table 1). For the ‘weighted sum’ ap-
308
proach, fincoh increases as energy increases, dincoh is generally less than dphoto (because dincoh
309
corresponds to a lower electron energy), and thus the value of d computed will decrease since
310
a higher weight is given to the smaller value of d.
us cri
an
311
pt
300
Small, large and intermediate cavity theory estimates of (Dw,m /Dm,m )CT , and MC simulation results for (Dw,m /Dm,m )M C for rcav = 6 µm are presented in figure 3 for all five bulk Fig 3
313
tissues. Similar trends in Dw,m /Dm,m as a function of energy are observed for rcav = 3 or
314
9 µm (results not shown). In general, values of Dw,m /Dm,m are closer to unity for smaller
315
rcav . Trends in (Dw,m /Dm,m )M C can be understood by considering how the radiological
316
parameters of the media considered compare to those of water. For lower energies, photo-
317
electric interactions dominate, and photoelectric interaction cross sections of adenoidcystic
318
carcinoma, melanoma and muscle are higher those of water, resulting in Dm,m values larger
319
than Dw,m . The opposite is true for mammary gland and squamous cell lung carcinoma.
320
Similarly, whether SCT and LCT represent maximum or minimum values may be understood
321
by considering the relative magnitudes of the radiological parameters. For example, mass
322
energy absorption coefficients for mammary gland are lower than those of water, while the
323
opposite is true for the stopping powers (figure 9), hence the LCT and SCT values in figure
324
3(b) represent the maximum and minimum, respectively. The ICT predictions lie between
325
the LCT and SCT values because d is constrained to lie between 0 and 1 (see equation (6)).
326
At higher energies, incoherent interactions dominate, and there are relatively small cross sec-
327
tion variations amongst considered media, hence Dw,m /Dm,m values approach unity. Figure
328
3 demonstrates that differences between MC results and cavity theory predictions vary with
ce
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dM
312
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329
phantom material m. For example, for mammary gland and squamous cell lung carcinoma,
330
most cavity theories overestimate (Dw,m /Dm,m )M C , except for SCT. In contrast, for ade-
331
noidcystic carcinoma and melanoma, most cavity theories underestimate (Dw,m /Dm,m )M C , A Evaluation of cavity theory approaches (part I)
us cri an dM pte ce
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energy sources, RCSDA values (second column of table 2) are comparable to nuclear dimen-
357
sions, and d is strongly influenced by nucleus size: for cavity radii between 2 and 9 µm,
358
d varies between 0.616 and 0.208 at 20 keV, and between 0.781 and 0.387 at 30 keV. As
359
source energy increases, RCSDA increases toward values that are large compared to cellular
360
dimensions, and d is less strongly influenced by nucleus size; d is always within 2% of unity
361
for the 370 keV photon source.
us cri
pt
356
Energy/keV
362
10−4
8.57 × 1.76 × 10−3 4.32 × 10−3 1.20 × 10−2 1.15 × 10−1
2 0.616 0.781 0.902 0.963 0.996
3 0.500 0.695 0.858 0.945 0.994
dM
20 30 50 90 370
RCSDA /(g·cm−2 )
an
Table 2: Values of d according to the ‘Burlin-Janssens’ method for five energies and eight cavity radii, rcav , corresponding to the eight nuclear radii considered herein. These values are derived using RCSDA for water (taking into account the assumed nuclear media density of 1.06 g/cm3 ), motivated by the relatively small differences in RCSDA (in g/cm2 ) for water and the nuclear media considered herein (
