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An analytical solution for the solid angle subtended by a circular detector for a symmetrically positioned linear source
ARTICLE IN PRESS
Applied Radiation and Isotopes 64 (2006) 603–607 www.elsevier.com/locate/apradiso
An analytical solution for the solid angle subtended by a circular detector for a symmetrically positioned linear source Eduardo Galiano, Christopher Pagnutti Department of Physics, Laurentian University, Ramsey Lake Road, Sudbury, ON, Canada P3E 2C6 Received 14 August 2005; accepted 5 December 2005
Abstract The analytical determination of the solid angle subtended by a circular detector for a line source in a parallel plane, whose midpoint is co-axial with the detector’s center, is of some relevance to the medical and nuclear power fields. No report has been found in the literature of a closed-form solution to this problem. We present a first principles solution to the problem along with a graph, which gives some indication of the behavior of the derived expression. r 2006 Elsevier Ltd. All rights reserved. Keywords: Solid angle; Linear source; Circular detector; Analytical expression for solid angle; Monte Carlo calculation
1. Introduction The problem of analytically determining the solid angle subtended by a circular detector for a line source in a parallel plane—whose midpoint is co-axial with the detector’s center—is of some relevance to the medical and nuclear power fields. In both fields, measuring the radiation intensity from a linear source with a circular detector is frequently required. In order to make absolute measurements, the solid angle subtended by the circular detector must be known from first principles. However, no report has been found in the literature of a closed-form, analytical solution to this problem. In the absence of such a solution, investigators have worked around the problem by ‘‘calibrating’’ the response of their detectors in the radiation field of a linear source of known strength, or by resorting to computational techniques such as the Monte Carlo method. Within the medical field, examples of the problem arise in brachytherapy—the treatment of malignant tumors with surgically implanted radioactive sources—and in therapeutic nuclear medicine. Some examples of investigators within the brachytherapy community who chose the ‘‘calibration’’ method for experimental work include Das and Kenny, Schaeken and Corresponding author. Tel.: +1 705 675 1151; fax: +1 705 675 4868.
E-mail address: [email protected] (E. Galiano). 0969-8043/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.apradiso.2005.12.006
Middelheim, and Rorat et al. Das and Kenny built a circular parallel plate ionization chamber for the purpose of calibrating long Ir-192 brachytherapy linear sources. The diameter of the chamber was 20 cm and the reported operational parameters were: collection efficiency 99.9%, polarity effect 0.25% at 300 V, and a response of 12.4 mC/ MBq–1 (Das and Kenny, 1995). Schaeken and Middelheim demonstrated the clinical feasibility of employing the amino acid L-a-alanine for in vivo dosimetry, by measuring a paramagnetic effect after irradiation using electron spin resonance (ESR). They used 4.8 mm diameter circular L-aalanine crystals for in vivo measurements during brachytherapy treatments of carcinoma of the cervix with linear Ir-192 sources (Schaeken and Middelheim, 1995). Rorat and collaborators have proposed the use of a positron-gamma emitter, 48V (E max;bþ 0:7 MeV, t1=2 ¼ 15:97 d), for intravascular brachytherapy (IVBT). In this technique, a radioactive stent is introduced directly into a blocked artery in an attempt to prevent a restenosis of the vessel. They produced prototype linear sources made of 0.25 mm titanium wire activated with 17 MeV protons in an AIC-144 cyclotron, and measured the radial dose distribution using 2 cm diameter TLD detectors (Rorat et al., 2005). Other brachytherapy investigators have approached the problem from a theoretical perspective by applying Monte Carlo methods, in which case knowledge of the solid angle is irrelevant. In this category we have Coursey
ARTICLE IN PRESS 604
E. Galiano, C. Pagnutti / Applied Radiation and Isotopes 64 (2006) 603–607
and collaborators at NIST (then NBS) who as early as 1992 pointed out that from a standardization point of view, the introduction of new linear sources to replace radium in brachytherapy—along with non-4P detectors—presented fundamental problems. They explicitly pointed out the lack of a closed form solution to the issue of the solid angle, and suggested that perhaps the best approach would be through Monte Carlo methods (Coursey et al., 1992). Dries argued that commercial dose calculation algorithms for brachytherapy treat the sources as points, when in fact their dimensions are large compared to the dose specification distances. This introduces inaccuracies and in order to quantify these inaccuracies, he used the EGS4 Monte Carlo code to generate the dose distribution with 0.5 mm spatial resolution for a single source. The results were compared with those of a commercial brachytherapy planning system. He concluded that secondary electrons have no relevant influence on the dose distribution, and that errors of up to 25% are introduced when using the point source approximation (Dries, 1997). At about the same time, Cheung and collaborators showed that a commercial algorithm-based treatment planning system (GE Target II Sun Sparc) was unable to properly predict the steep dose gradients around 192 Ir brachytherapy linear sources, when compared to experimental measurements with a special type of dosimetry film called GafChromicTM film. However, they found that the EGS4 Monte Carlo code (PRESTA version) correctly predicted the dose gradient with an accuracy of 2%, thus validating the method (Cheung et al., 1997). Seltzer and Soares also of NIST, initially demonstrated the feasibility of using the Monte Carlo method for the dosimetry of betaemitting sources used in intravascular brachytherapy for the prevention of restenosis following balloon angioplasty. They reported using the method to provide a better understanding of the dosimetry of these sources including the relationship of dose distributions in different phantom materials, and the effects of changes in source design and/or measurement geometry. A specific geometry they examined was that of a line source being assayed with a circular detector. Based on comparisons with actual measured data, they concluded that the Monte Carlo method was an effective tool for predicting dose distributions (Seltzer and Soares, 1998). Examples of this problem arise in therapeutic nuclear medicine as well. Culver and Dworkin investigated the development of guidelines—based on direct patient measurements—of when I-131 treated thyroid cancer patients could safely resume close personal contact with healthy individuals after release from the hospital. They measured external exposure rates on 27 patients using a circular 10 cm diameter ionization survey meter, the response of which had previously been calibrated with a line source of known activity. The patients were considered line sources for the interpretation of the measurements, and their exposure rates were measured at the time of release from the hospital and 2–7 days post-hospital discharge. Measurements were taken at 1, 0.6 and 0.3 m from the patient’s upright body axis (Culver and Dworkin, 1992). Influenced by this work, Siegel and
collaborators pointed out that in many nuclear medicine patients, the activity distribution is widely dispersed and does not simulate a point source. Therefore, they proposed a linesource model to more accurately reflect this extended activity distribution. Using circular detectors, they determined the dose rate per unit activity for a point source and for line sources of lengths of 20, 50, 70, 100, and 174 cm. They found that the inverse square law is not valid for a line source until a certain distance is reached, dependent on the length of the line source. In particular, they determined that at short distances, use of the point-source model for a patient with an extended activity distribution will substantially overestimate the absorbed dose to exposed individuals. They concluded that the line-source model is a more realistic and practical approach than the traditional point-source model for determining the dose to individuals exposed to radioactive patients with widespread activity distributions (Siegel et al., 2002). More recently, Matheoud and collaborators examined the radiation dose received by individuals in the immediate vicinity of patients administered I-131 for the treatment of hypothyroidism. They measured external exposure levels from 37 patients. Measurements were made at a distance of 2.5 m from the mid-sternum at 2, 4, 24, 48, 72, and 96 h after I-131 administration using a circular plastic scintillator. The measurements were interpreted using the line-source model proposed Siegel et al. (70 cm line source length) to account for the extended activity distribution (Matheoud et al., 2004). Within the nuclear power industry, the problem has also surfaced from time to time. For example, Holcomb and collaborators at Argonne National Laboratory have measured the activity of high level, non-combustible radioactive solid waste from the reprocessing of reactor fuel at a special facility. The waste was ‘‘packaged’’ in crystallized thin cylindrical rods, and the activity measurements were performed robotically with circular NaI detectors. The source to detector distances were such that the cylindrical sources could effectively be approximated as thin linear sources. The first one hundred packages collectively contained an estimated 6600 kg of waste material with an estimated 1:6 106 curies of activity. By previous calibration of their detector with a linear source of known activity, these investigators concluded that the exposure rate for a given line source waste package could be approximated as 9 10r R=h, where r is distance from the package (Holcomb et al., 1967). Carelli and coinvestigators of the Italian Nuclear Energy Commission developed a new method of studying the coolant mixing and interchannel mixing flow rates in reactor fuel subassemblies. For this purpose, they developed a radioisotope based method. The radioisotope chosen was the b-emitter 32 P, and both plastic scintillator rods and Geiger–Mu¨ller detectors were inserted in the core adjacent to the coolant channels. The channels were linear, and both types of detectors were circular in cross-section. The authors claimed that the advantages of the proposed method were improved sensitivity, absence of perturbing probes, and the possibility of evaluating the mean values of the tracer
ARTICLE IN PRESS E. Galiano, C. Pagnutti / Applied Radiation and Isotopes 64 (2006) 603–607
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concentration in the channels (Carelli et al., 1970). As in the medical field, Monte Carlo techniques have also been used in the nuclear power industry to circumvent the lack of knowledge of the solid angle when faced with this particular geometry. For example, recently Fujimoto and collaborators applied Monte Carlo techniques to investigate the operating parameters of a novel high-temperature engineering test reactor. It is the first block-type hightemperature gas-cooled (HTGR) reactor designed for a 950 1C outlet gas temperature, which uses low-enriched uranium fuel with burnable poison rods. The following operating parameters for fuel rod assemblies were evaluated: uncertainties of effective multiplication factor, neutron flux distribution, burnable poison reactivity worth, control rod worth, and temperature coefficients. All Monte Carlo results were checked against actual measurements on subcritical fuel rod assemblies with circular Ge detectors. The predictive accuracy of the Monte Carlo codes was reported at 5% or better (Fujimoto et al., 2004). 2. Methods The geometry for calculating the solid angle subtended by a circular detector of radius R for a non-coplanar parallel line source of length L, whose midpoint is co-axial with the detector’s center is illustrated in Fig. 1. As a starting point, we take an expression originally derived by Knoll and used by Aguiar and Galiano in previous investigations. This expression describes the solid angle o0 ðr0 Þ subtended by the disk with respect to the midpoint of the source, located at r0 (Knoll, 2000; Aguiar and Galiano, 2004; Aguiar et al., 2005): 2 3 R r0 6 7 o0 ðr0 Þ ¼ 4p 1 cos tan1 ¼ 4p41 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5. r0 2 2 r0 þ R (1) To obtain an expression for the solid angle o subtended by the disk with respect to any other point on the source at position r, two factors must be taken into account: first, the qffiffiffiffiffiffiffiffiffiffiffiffiffiffi point at r on the source is now a distance r ¼ r20 þ y2 from the center of the disk, and second, from this viewpoint the disk appears to be skewed at an angle y with respect to the normal. If we let r0 ! r in Eq. (1) then multiply by cos y to account for the skewing, then the solid angle subtended by the disk as viewed from point r is given as 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 y2 þ r20 7 r0 6 o ¼ o0 ðrÞ cos y ¼ 4p41 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 y2 þ r20 y2 þ r20 þ R 2 3 1 1 6 7 ¼ r0 4p4qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5. 2 2 2 2 2 y þ r0 y þ r0 þ R
ð2Þ
Fig. 1. The geometry used to calculate the solid angle subtended by a circular detector with radius R with respect to a linear radioactive source of length L. The source is oriented parallel to the plane of the detector such that r0 is normal to the plane of the disk. The vector r points to an arbitrary point along the source at a height y above its midpoint.
The solid angle O subtended by the disk with respect to the entire source is obtained by integrating o over the source. That is Z Z L=2 o dy. (3) O ¼ dO ¼ L=2
Substituting Eq. (2) into (3) we get 0 1 Z L=2 1 1 B C O ¼ r0 8p dy@qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA. 2 2 2 2 2 0 y þ r0 y þ r0 þ R
(4)
Note that we have used the fact that the limits of integration are symmetric and the integrand is even. This integral can be solved using the following trigonometric substitution: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Let y ¼ r0 tan a ¼ r20 þ R2 tan b, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi so dy ¼ r0 sec2 a da ¼ r20 þ R2 sec2 b db. Now 8 >
tan1 L=2r0
r0 sec2 a da qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : 0 r20 tan2 a þ r20 9 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > Z tan1 L=ð2pffiffiffiffiffiffiffiffiffi = r20 þR2 Þ r20 þ R2 sec2 b db qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . > 0 ðr20 þ R2 Þtan2 b þ ðr20 þ R2 Þ;
O ¼ r0 8p
ð5Þ
ARTICLE IN PRESS E. Galiano, C. Pagnutti / Applied Radiation and Isotopes 64 (2006) 603–607
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Using the identity tan2 j þ 1 ¼ sec2 j we get the integral (Z
tan1 L=2r0
O ¼ r0 8p
sec a da 0
Z
tan1 L=ð2
pffiffiffiffiffiffiffiffiffi ffi 2 2
)
r0 þR Þ
sec b db ,
ð6Þ
terms of two parameters u ¼ R=r0 and v ¼ L=r0 to get 1 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 L L C Bsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 4þ 2 C B r r R 0 0 C B sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC O ¼ r0 8p lnB 1 þ 2 2 C B r0 L R L A @ þ 4þ4 þ r0 r0 r0
0
(9) or
which gives
pffiffiffiffiffiffiffiffiffiffiffiffiffi O ¼ r0 8p ln 1 þ u2
tan1 L=2r0 O ¼ r0 8p ðln j sec a þ tan ajÞj0 pffiffiffiffiffiffiffiffiffiffi tan1 L=ð2 r20 þR2 Þ ðln j sec b þ tan bjÞj0 2 6 1 L 1 L þ tan tan ¼ r0 8p4ln j sec tan 2r 2r 0
0
In the last step we make use of the right-angled triangles depicted in Fig. 2. Simplifying expression (7) we get
In order to get a quantitative ‘‘feel’’ for the behavior of expression (10), it is instructive to divide it by r08p which gives pffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffiffiffiffiffiffiffiffiffiffiffi v þ 4 þ v2 O pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ¼ ln 1 þ u2 (11) r0 8p v þ 4 þ 4u2 þ v2 The quantity O=r0 8p can then conveniently be plotted as a function of the parameters u and v, as done in Fig. 3. Note that for a given va0, i.e. the case of a constant non-zero L, O increases with an increase in the R/r0 ratio as expected, since the source ‘‘sees’’ the detector disk increase in diameter and therefore cover a greater fraction of solid angle. When v ¼ 0, i.e. the case of a point source, expression (10) correctly predicts that O ¼ 0 when R ¼ 0, and that O increases as R increases. For a given ua0, i.e. the case of constant non-zero R and r0, expression (10) predicts a decrease in O with an increase in v. This result is consistent with the fact that an increase in v implies an increase in L for a constant R. This however is geometrically equivalent to keeping L constant and decreasing R, which results in a
(8)
0
which gives the solid angle in terms of the parameters given in Fig. 1. It is convenient to rearrange this last expression in
Fig. 2. Right triangles used for Eq. (7).
(10)
3. Results and discussion
3 L L 7 1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ tan tan qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 lnsec tan 2 2 2 2 2 r0 þ R 2 r0 þ R 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 L 6 4r0 þ L ¼ r0 8p4 ln þ 2r0 2r0 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4r20 þ 4R2 þ L2 L 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð7Þ ln þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5. 2 r20 þ R2 2 r20 þ R2
0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ R2 2 þ L2 r L þ 4r 0 B 0 C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA, O ¼ r0 8p ln@ r0 L þ 4ðr2 þ R2 Þ þ L2
pffiffiffiffiffiffiffiffiffiffiffiffiffi ! v þ 4 þ v2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . v þ 4 þ 4u2 þ v2
Fig. 3. Plot of O=r0 8p as a function of u ¼ R=r0 and v ¼ L=r0 .
ARTICLE IN PRESS E. Galiano, C. Pagnutti / Applied Radiation and Isotopes 64 (2006) 603–607
decreased solid angle. For u ¼ 0, expression (10) correctly predicts O ¼ 0 regardless of the value of v, since this corresponds to the case R ¼ 0, or zero detector size. 4. Conclusions The analytical determination of the solid angle subtended by a circular detector for a line source in a parallel plane, whose midpoint is co-axial with the detector’s center, is of some relevance to the medical and nuclear power fields. No report has been found in the literature of an analytical, closed-form solution to this problem. We present a first principles solution to the problem along with a graph, which gives some indication of the behavior of the derived expression. The expression correctly predicts that for a given source length, as the detector diameter increases, so does the solid angle. It also correctly predicts that for a given detector size, as the source length increases, the solid angle decreases. For the case of a detector of zero size, the expression predicts a zero solid angle, as expected. References Aguiar, J.C., Galiano, E., 2004. Theoretical estimates of the solid angle subtended by a dual diaphragm-detector assembly for alpha sources. Appl. Radiat. Isot. 61, 1349–1351. Aguiar, J.C., Galiano, E., Arenillas, P., 2005. Determination of the activity concentration of a 238Pu solution by the defined solid angle method utilizing a novel dual diaphragm-detector assembly. Appl. Radiat. Isot. 63, 229–233. Carelli, M., Curzio, G., Renieri, A., 1970. A radioactive tracer technique for measuring coolant mixing in nuclear reactor fuel sub-assemblies. Nucl. Eng. Des. 11 (1), 93–102.
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Cheung, Y.C., Yu, P.K.N., Young, E.C.M., Chan, C.L., Ng, M.F., Tang, F.N.F., Wong, T.P.Y., 1997. The electron-dose distribution surrounding an 192Ir wire brachytherapy source investigated using EGS4 simulations and GafChromicTM film. Appl. Radiat. Isot. 48 (7), 985–990. Coursey, B.M., Goodman, L.J., Hoppes, D.D., Loevinger, R., McLaughlin, W.L., Soares, C.G., Weaver, J.T., 1992. The needs for brachytherapy source calibrations in the United States. Nucl. Method Instrum. A 312 (1–2), 246–250. Culver, C.M., Dworkin, H.J., 1992. Radiation safety considerations for post-iodine-131 thyroid cancer therapy. J. Nucl. Med. 33 (7), 1402–1405. Das, K.R., Kenny, M.B., 1995. A thin parallel plate ionization chamber for calibration of microselectron sources. Radiother. Oncol. 37, S10. Dries, W.J.F., 1997. Monte Carlo calculated dose distribution for endovascular HDR brachytherapy with Ir-192. Radiother. Oncol. 45 (1), 77–82. Fujimoto, N., Nojiri, N., Yamashita, K., 2004. Validation of the nuclear design code system for the HTTR using the criticality assembly VHTRC. Nucl. Eng. Des. 233 (1–3), 255–262. Holcomb, W.F., Paige, D.M., Stevenson, C.E., Graham, D.E., 1967. High level radioactive solid waste burial operations for the EBR-II fuel cycle facility. Nucl. Eng. Des. 5 (2), 221–227. Knoll, G.F., 2000. Radiation Detection and Measurement, third ed. Wiley, New York, pp. 116–118. Matheoud, R., Reschini, E., Canzi, C., Voltini, F., Gerundini, P., 2004. Potential third-party radiation exposure from outpatients treated with I-131 for hyperthyroidism. Med. Phys. 31 (12), 3194–3200. Rorat, E., Petelenz, B., Marczewska, B., Ochab, E., 2005. Thermoluminescence dosimetry of model line sources containing vanadium-48. Radiat. Meas. 39 (5), 495–501. Schaeken, B., Middelheim, A.Z., 1995. EPR-based dose measurements with alanine in clinical dosimetry. Radioth. Oncol. 37, S30. Seltzer, S.M., Soares, C.G., 1998. Monte Carlo calculations for brachytherapy dosimetry. Radiat. Phys. Chem. 53 (3), 345. Siegel, J.A., Marcus, C.S., Sparks, R.B., 2002. Calculating the absorbed dose from radioactive patients: the line-source versus point-source model. J. Nucl. Med. 43 (9), 1241–1244.
Applied Radiation and Isotopes 64 (2006) 603–607 www.elsevier.com/locate/apradiso
An analytical solution for the solid angle subtended by a circular detector for a symmetrically positioned linear source Eduardo Galiano, Christopher Pagnutti Department of Physics, Laurentian University, Ramsey Lake Road, Sudbury, ON, Canada P3E 2C6 Received 14 August 2005; accepted 5 December 2005
Abstract The analytical determination of the solid angle subtended by a circular detector for a line source in a parallel plane, whose midpoint is co-axial with the detector’s center, is of some relevance to the medical and nuclear power fields. No report has been found in the literature of a closed-form solution to this problem. We present a first principles solution to the problem along with a graph, which gives some indication of the behavior of the derived expression. r 2006 Elsevier Ltd. All rights reserved. Keywords: Solid angle; Linear source; Circular detector; Analytical expression for solid angle; Monte Carlo calculation
1. Introduction The problem of analytically determining the solid angle subtended by a circular detector for a line source in a parallel plane—whose midpoint is co-axial with the detector’s center—is of some relevance to the medical and nuclear power fields. In both fields, measuring the radiation intensity from a linear source with a circular detector is frequently required. In order to make absolute measurements, the solid angle subtended by the circular detector must be known from first principles. However, no report has been found in the literature of a closed-form, analytical solution to this problem. In the absence of such a solution, investigators have worked around the problem by ‘‘calibrating’’ the response of their detectors in the radiation field of a linear source of known strength, or by resorting to computational techniques such as the Monte Carlo method. Within the medical field, examples of the problem arise in brachytherapy—the treatment of malignant tumors with surgically implanted radioactive sources—and in therapeutic nuclear medicine. Some examples of investigators within the brachytherapy community who chose the ‘‘calibration’’ method for experimental work include Das and Kenny, Schaeken and Corresponding author. Tel.: +1 705 675 1151; fax: +1 705 675 4868.
E-mail address: [email protected] (E. Galiano). 0969-8043/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.apradiso.2005.12.006
Middelheim, and Rorat et al. Das and Kenny built a circular parallel plate ionization chamber for the purpose of calibrating long Ir-192 brachytherapy linear sources. The diameter of the chamber was 20 cm and the reported operational parameters were: collection efficiency 99.9%, polarity effect 0.25% at 300 V, and a response of 12.4 mC/ MBq–1 (Das and Kenny, 1995). Schaeken and Middelheim demonstrated the clinical feasibility of employing the amino acid L-a-alanine for in vivo dosimetry, by measuring a paramagnetic effect after irradiation using electron spin resonance (ESR). They used 4.8 mm diameter circular L-aalanine crystals for in vivo measurements during brachytherapy treatments of carcinoma of the cervix with linear Ir-192 sources (Schaeken and Middelheim, 1995). Rorat and collaborators have proposed the use of a positron-gamma emitter, 48V (E max;bþ 0:7 MeV, t1=2 ¼ 15:97 d), for intravascular brachytherapy (IVBT). In this technique, a radioactive stent is introduced directly into a blocked artery in an attempt to prevent a restenosis of the vessel. They produced prototype linear sources made of 0.25 mm titanium wire activated with 17 MeV protons in an AIC-144 cyclotron, and measured the radial dose distribution using 2 cm diameter TLD detectors (Rorat et al., 2005). Other brachytherapy investigators have approached the problem from a theoretical perspective by applying Monte Carlo methods, in which case knowledge of the solid angle is irrelevant. In this category we have Coursey
ARTICLE IN PRESS 604
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and collaborators at NIST (then NBS) who as early as 1992 pointed out that from a standardization point of view, the introduction of new linear sources to replace radium in brachytherapy—along with non-4P detectors—presented fundamental problems. They explicitly pointed out the lack of a closed form solution to the issue of the solid angle, and suggested that perhaps the best approach would be through Monte Carlo methods (Coursey et al., 1992). Dries argued that commercial dose calculation algorithms for brachytherapy treat the sources as points, when in fact their dimensions are large compared to the dose specification distances. This introduces inaccuracies and in order to quantify these inaccuracies, he used the EGS4 Monte Carlo code to generate the dose distribution with 0.5 mm spatial resolution for a single source. The results were compared with those of a commercial brachytherapy planning system. He concluded that secondary electrons have no relevant influence on the dose distribution, and that errors of up to 25% are introduced when using the point source approximation (Dries, 1997). At about the same time, Cheung and collaborators showed that a commercial algorithm-based treatment planning system (GE Target II Sun Sparc) was unable to properly predict the steep dose gradients around 192 Ir brachytherapy linear sources, when compared to experimental measurements with a special type of dosimetry film called GafChromicTM film. However, they found that the EGS4 Monte Carlo code (PRESTA version) correctly predicted the dose gradient with an accuracy of 2%, thus validating the method (Cheung et al., 1997). Seltzer and Soares also of NIST, initially demonstrated the feasibility of using the Monte Carlo method for the dosimetry of betaemitting sources used in intravascular brachytherapy for the prevention of restenosis following balloon angioplasty. They reported using the method to provide a better understanding of the dosimetry of these sources including the relationship of dose distributions in different phantom materials, and the effects of changes in source design and/or measurement geometry. A specific geometry they examined was that of a line source being assayed with a circular detector. Based on comparisons with actual measured data, they concluded that the Monte Carlo method was an effective tool for predicting dose distributions (Seltzer and Soares, 1998). Examples of this problem arise in therapeutic nuclear medicine as well. Culver and Dworkin investigated the development of guidelines—based on direct patient measurements—of when I-131 treated thyroid cancer patients could safely resume close personal contact with healthy individuals after release from the hospital. They measured external exposure rates on 27 patients using a circular 10 cm diameter ionization survey meter, the response of which had previously been calibrated with a line source of known activity. The patients were considered line sources for the interpretation of the measurements, and their exposure rates were measured at the time of release from the hospital and 2–7 days post-hospital discharge. Measurements were taken at 1, 0.6 and 0.3 m from the patient’s upright body axis (Culver and Dworkin, 1992). Influenced by this work, Siegel and
collaborators pointed out that in many nuclear medicine patients, the activity distribution is widely dispersed and does not simulate a point source. Therefore, they proposed a linesource model to more accurately reflect this extended activity distribution. Using circular detectors, they determined the dose rate per unit activity for a point source and for line sources of lengths of 20, 50, 70, 100, and 174 cm. They found that the inverse square law is not valid for a line source until a certain distance is reached, dependent on the length of the line source. In particular, they determined that at short distances, use of the point-source model for a patient with an extended activity distribution will substantially overestimate the absorbed dose to exposed individuals. They concluded that the line-source model is a more realistic and practical approach than the traditional point-source model for determining the dose to individuals exposed to radioactive patients with widespread activity distributions (Siegel et al., 2002). More recently, Matheoud and collaborators examined the radiation dose received by individuals in the immediate vicinity of patients administered I-131 for the treatment of hypothyroidism. They measured external exposure levels from 37 patients. Measurements were made at a distance of 2.5 m from the mid-sternum at 2, 4, 24, 48, 72, and 96 h after I-131 administration using a circular plastic scintillator. The measurements were interpreted using the line-source model proposed Siegel et al. (70 cm line source length) to account for the extended activity distribution (Matheoud et al., 2004). Within the nuclear power industry, the problem has also surfaced from time to time. For example, Holcomb and collaborators at Argonne National Laboratory have measured the activity of high level, non-combustible radioactive solid waste from the reprocessing of reactor fuel at a special facility. The waste was ‘‘packaged’’ in crystallized thin cylindrical rods, and the activity measurements were performed robotically with circular NaI detectors. The source to detector distances were such that the cylindrical sources could effectively be approximated as thin linear sources. The first one hundred packages collectively contained an estimated 6600 kg of waste material with an estimated 1:6 106 curies of activity. By previous calibration of their detector with a linear source of known activity, these investigators concluded that the exposure rate for a given line source waste package could be approximated as 9 10r R=h, where r is distance from the package (Holcomb et al., 1967). Carelli and coinvestigators of the Italian Nuclear Energy Commission developed a new method of studying the coolant mixing and interchannel mixing flow rates in reactor fuel subassemblies. For this purpose, they developed a radioisotope based method. The radioisotope chosen was the b-emitter 32 P, and both plastic scintillator rods and Geiger–Mu¨ller detectors were inserted in the core adjacent to the coolant channels. The channels were linear, and both types of detectors were circular in cross-section. The authors claimed that the advantages of the proposed method were improved sensitivity, absence of perturbing probes, and the possibility of evaluating the mean values of the tracer
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concentration in the channels (Carelli et al., 1970). As in the medical field, Monte Carlo techniques have also been used in the nuclear power industry to circumvent the lack of knowledge of the solid angle when faced with this particular geometry. For example, recently Fujimoto and collaborators applied Monte Carlo techniques to investigate the operating parameters of a novel high-temperature engineering test reactor. It is the first block-type hightemperature gas-cooled (HTGR) reactor designed for a 950 1C outlet gas temperature, which uses low-enriched uranium fuel with burnable poison rods. The following operating parameters for fuel rod assemblies were evaluated: uncertainties of effective multiplication factor, neutron flux distribution, burnable poison reactivity worth, control rod worth, and temperature coefficients. All Monte Carlo results were checked against actual measurements on subcritical fuel rod assemblies with circular Ge detectors. The predictive accuracy of the Monte Carlo codes was reported at 5% or better (Fujimoto et al., 2004). 2. Methods The geometry for calculating the solid angle subtended by a circular detector of radius R for a non-coplanar parallel line source of length L, whose midpoint is co-axial with the detector’s center is illustrated in Fig. 1. As a starting point, we take an expression originally derived by Knoll and used by Aguiar and Galiano in previous investigations. This expression describes the solid angle o0 ðr0 Þ subtended by the disk with respect to the midpoint of the source, located at r0 (Knoll, 2000; Aguiar and Galiano, 2004; Aguiar et al., 2005): 2 3 R r0 6 7 o0 ðr0 Þ ¼ 4p 1 cos tan1 ¼ 4p41 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5. r0 2 2 r0 þ R (1) To obtain an expression for the solid angle o subtended by the disk with respect to any other point on the source at position r, two factors must be taken into account: first, the qffiffiffiffiffiffiffiffiffiffiffiffiffiffi point at r on the source is now a distance r ¼ r20 þ y2 from the center of the disk, and second, from this viewpoint the disk appears to be skewed at an angle y with respect to the normal. If we let r0 ! r in Eq. (1) then multiply by cos y to account for the skewing, then the solid angle subtended by the disk as viewed from point r is given as 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 y2 þ r20 7 r0 6 o ¼ o0 ðrÞ cos y ¼ 4p41 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 y2 þ r20 y2 þ r20 þ R 2 3 1 1 6 7 ¼ r0 4p4qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5. 2 2 2 2 2 y þ r0 y þ r0 þ R
ð2Þ
Fig. 1. The geometry used to calculate the solid angle subtended by a circular detector with radius R with respect to a linear radioactive source of length L. The source is oriented parallel to the plane of the detector such that r0 is normal to the plane of the disk. The vector r points to an arbitrary point along the source at a height y above its midpoint.
The solid angle O subtended by the disk with respect to the entire source is obtained by integrating o over the source. That is Z Z L=2 o dy. (3) O ¼ dO ¼ L=2
Substituting Eq. (2) into (3) we get 0 1 Z L=2 1 1 B C O ¼ r0 8p dy@qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA. 2 2 2 2 2 0 y þ r0 y þ r0 þ R
(4)
Note that we have used the fact that the limits of integration are symmetric and the integrand is even. This integral can be solved using the following trigonometric substitution: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Let y ¼ r0 tan a ¼ r20 þ R2 tan b, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi so dy ¼ r0 sec2 a da ¼ r20 þ R2 sec2 b db. Now 8 >
tan1 L=2r0
r0 sec2 a da qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > : 0 r20 tan2 a þ r20 9 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > Z tan1 L=ð2pffiffiffiffiffiffiffiffiffi = r20 þR2 Þ r20 þ R2 sec2 b db qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . > 0 ðr20 þ R2 Þtan2 b þ ðr20 þ R2 Þ;
O ¼ r0 8p
ð5Þ
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Using the identity tan2 j þ 1 ¼ sec2 j we get the integral (Z
tan1 L=2r0
O ¼ r0 8p
sec a da 0
Z
tan1 L=ð2
pffiffiffiffiffiffiffiffiffi ffi 2 2
)
r0 þR Þ
sec b db ,
ð6Þ
terms of two parameters u ¼ R=r0 and v ¼ L=r0 to get 1 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 L L C Bsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 4þ 2 C B r r R 0 0 C B sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC O ¼ r0 8p lnB 1 þ 2 2 C B r0 L R L A @ þ 4þ4 þ r0 r0 r0
0
(9) or
which gives
pffiffiffiffiffiffiffiffiffiffiffiffiffi O ¼ r0 8p ln 1 þ u2
tan1 L=2r0 O ¼ r0 8p ðln j sec a þ tan ajÞj0 pffiffiffiffiffiffiffiffiffiffi tan1 L=ð2 r20 þR2 Þ ðln j sec b þ tan bjÞj0 2 6 1 L 1 L þ tan tan ¼ r0 8p4ln j sec tan 2r 2r 0
0
In the last step we make use of the right-angled triangles depicted in Fig. 2. Simplifying expression (7) we get
In order to get a quantitative ‘‘feel’’ for the behavior of expression (10), it is instructive to divide it by r08p which gives pffiffiffiffiffiffiffiffiffiffiffiffiffi ! pffiffiffiffiffiffiffiffiffiffiffiffiffi v þ 4 þ v2 O pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ¼ ln 1 þ u2 (11) r0 8p v þ 4 þ 4u2 þ v2 The quantity O=r0 8p can then conveniently be plotted as a function of the parameters u and v, as done in Fig. 3. Note that for a given va0, i.e. the case of a constant non-zero L, O increases with an increase in the R/r0 ratio as expected, since the source ‘‘sees’’ the detector disk increase in diameter and therefore cover a greater fraction of solid angle. When v ¼ 0, i.e. the case of a point source, expression (10) correctly predicts that O ¼ 0 when R ¼ 0, and that O increases as R increases. For a given ua0, i.e. the case of constant non-zero R and r0, expression (10) predicts a decrease in O with an increase in v. This result is consistent with the fact that an increase in v implies an increase in L for a constant R. This however is geometrically equivalent to keeping L constant and decreasing R, which results in a
(8)
0
which gives the solid angle in terms of the parameters given in Fig. 1. It is convenient to rearrange this last expression in
Fig. 2. Right triangles used for Eq. (7).
(10)
3. Results and discussion
3 L L 7 1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ tan tan qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 lnsec tan 2 2 2 2 2 r0 þ R 2 r0 þ R 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 L 6 4r0 þ L ¼ r0 8p4 ln þ 2r0 2r0 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4r20 þ 4R2 þ L2 L 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð7Þ ln þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5. 2 r20 þ R2 2 r20 þ R2
0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ R2 2 þ L2 r L þ 4r 0 B 0 C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA, O ¼ r0 8p ln@ r0 L þ 4ðr2 þ R2 Þ þ L2
pffiffiffiffiffiffiffiffiffiffiffiffiffi ! v þ 4 þ v2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . v þ 4 þ 4u2 þ v2
Fig. 3. Plot of O=r0 8p as a function of u ¼ R=r0 and v ¼ L=r0 .
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decreased solid angle. For u ¼ 0, expression (10) correctly predicts O ¼ 0 regardless of the value of v, since this corresponds to the case R ¼ 0, or zero detector size. 4. Conclusions The analytical determination of the solid angle subtended by a circular detector for a line source in a parallel plane, whose midpoint is co-axial with the detector’s center, is of some relevance to the medical and nuclear power fields. No report has been found in the literature of an analytical, closed-form solution to this problem. We present a first principles solution to the problem along with a graph, which gives some indication of the behavior of the derived expression. The expression correctly predicts that for a given source length, as the detector diameter increases, so does the solid angle. It also correctly predicts that for a given detector size, as the source length increases, the solid angle decreases. For the case of a detector of zero size, the expression predicts a zero solid angle, as expected. References Aguiar, J.C., Galiano, E., 2004. Theoretical estimates of the solid angle subtended by a dual diaphragm-detector assembly for alpha sources. Appl. Radiat. Isot. 61, 1349–1351. Aguiar, J.C., Galiano, E., Arenillas, P., 2005. Determination of the activity concentration of a 238Pu solution by the defined solid angle method utilizing a novel dual diaphragm-detector assembly. Appl. Radiat. Isot. 63, 229–233. Carelli, M., Curzio, G., Renieri, A., 1970. A radioactive tracer technique for measuring coolant mixing in nuclear reactor fuel sub-assemblies. Nucl. Eng. Des. 11 (1), 93–102.
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Cheung, Y.C., Yu, P.K.N., Young, E.C.M., Chan, C.L., Ng, M.F., Tang, F.N.F., Wong, T.P.Y., 1997. The electron-dose distribution surrounding an 192Ir wire brachytherapy source investigated using EGS4 simulations and GafChromicTM film. Appl. Radiat. Isot. 48 (7), 985–990. Coursey, B.M., Goodman, L.J., Hoppes, D.D., Loevinger, R., McLaughlin, W.L., Soares, C.G., Weaver, J.T., 1992. The needs for brachytherapy source calibrations in the United States. Nucl. Method Instrum. A 312 (1–2), 246–250. Culver, C.M., Dworkin, H.J., 1992. Radiation safety considerations for post-iodine-131 thyroid cancer therapy. J. Nucl. Med. 33 (7), 1402–1405. Das, K.R., Kenny, M.B., 1995. A thin parallel plate ionization chamber for calibration of microselectron sources. Radiother. Oncol. 37, S10. Dries, W.J.F., 1997. Monte Carlo calculated dose distribution for endovascular HDR brachytherapy with Ir-192. Radiother. Oncol. 45 (1), 77–82. Fujimoto, N., Nojiri, N., Yamashita, K., 2004. Validation of the nuclear design code system for the HTTR using the criticality assembly VHTRC. Nucl. Eng. Des. 233 (1–3), 255–262. Holcomb, W.F., Paige, D.M., Stevenson, C.E., Graham, D.E., 1967. High level radioactive solid waste burial operations for the EBR-II fuel cycle facility. Nucl. Eng. Des. 5 (2), 221–227. Knoll, G.F., 2000. Radiation Detection and Measurement, third ed. Wiley, New York, pp. 116–118. Matheoud, R., Reschini, E., Canzi, C., Voltini, F., Gerundini, P., 2004. Potential third-party radiation exposure from outpatients treated with I-131 for hyperthyroidism. Med. Phys. 31 (12), 3194–3200. Rorat, E., Petelenz, B., Marczewska, B., Ochab, E., 2005. Thermoluminescence dosimetry of model line sources containing vanadium-48. Radiat. Meas. 39 (5), 495–501. Schaeken, B., Middelheim, A.Z., 1995. EPR-based dose measurements with alanine in clinical dosimetry. Radioth. Oncol. 37, S30. Seltzer, S.M., Soares, C.G., 1998. Monte Carlo calculations for brachytherapy dosimetry. Radiat. Phys. Chem. 53 (3), 345. Siegel, J.A., Marcus, C.S., Sparks, R.B., 2002. Calculating the absorbed dose from radioactive patients: the line-source versus point-source model. J. Nucl. Med. 43 (9), 1241–1244.