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System design for in vivo neutron activation analysis measurements of manganese in the human brain: based on Monte Carlo modeling
Applied Radiation and Isotopes 53 (2000) 651±656
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System design for in vivo neutron activation analysis measurements of manganese in the human brain: based on Monte Carlo modeling M.L. Arnold*, F.E. McNeill, W.V. Prestwich, D.R. Chettle Department of Physics and Astronomy, Medical Physics & Radiation Sciences Unit, McMaster University, 1280 Main St. West, Hamilton, ON, Canada L8S 4M1
Abstract Manganese is an essential nutrient required by the human body, but conversely, over exposure to the element may cause central nervous system damage. The technique of in vivo neutron activation analysis, using the McMaster KN-accelerator, is being investigated as a possible method of noninvasively determining manganese concentrations within the human body. Since the brain is the primary target of damage from exposure it would be the ideal site for measurements. Thus, Monte Carlo simulations have been undertaken to de®ne the optimum experimental parameters for such a measurement, examining the use of possible moderator, re¯ector and collimator materials. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Neutron activation analysis; Manganese; In vivo; Brain; Monte Carlo
1. Introduction The toxic eects of manganese (Mn) on the central nervous system have been documented for over 150 years, however as of yet there does not exist a good biomarker which can be used as an indicator of exposure and the possibility of developing adverse eects (Greger, 1999). To date, one of the best biomarkers is T1 weighted MRI, which shows an increase in signal intensity in the basal ganglia region of the brain for exposed nonhuman primates (reviewed by Pal et al., 1999), suggesting an increase in Mn in this region, but unfortunately providing no quantitative information. Recently, there have been studies dealing with the
* Corresponding author. Fax: +1-905-546-1252. E-mail address: [email protected] (M.L. Arnold).
optimization of systems for the technique of in vivo neutron activation analysis (IVNAA) (Ma et al., 1999; Dilmanian et al., 1998; McNeill and Chettle, 1998). IVNAA can be used to quantify the amount of various elements within the body, and the feasibility of this technique for quantitatively measuring Mn within the basal ganglia is currently being investigated. The McMaster KN-accelerator would be used to provide neutrons, by means of the 7Li(p,n)7Be reaction (threshold of 1.88 MeV), and Mn activation would occur via the 55Mn(n,g )56Mn reaction (s=13 barn). Following the irradiation process, the 847 keV gamma rays emitted when 56Mn decays (half-life=2.58 h, branching ratio=100%) would be counted outside of the body using NaI detectors. At present, the irradiation process has been modeled using Monte Carlo simulations to determine the optimum experimental parameters which should be used, and future work will deal with modeling of the entire two step process to
0969-8043/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 9 - 8 0 4 3 ( 0 0 ) 0 0 1 9 9 - 8
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M.L. Arnold et al. / Applied Radiation and Isotopes 53 (2000) 651±656
evaluate system performance and calculate the minimum detectable limit (MDL) for the measurement. 2. Method The Monte Carlo neutron and photon transport code MCNP, version 4B2, was installed on a UNIX workstation at McMaster and used to model the experimental measurements. 2.1. Neutron source Based on published cross section data (Liskien and Paulsen, 1975) and stopping powers (Nuclear Data Tables, 1960) the 7Li(p,n)7Be nuclear reaction has been modeled from analytical calculations of the neutron
yield as a function of the proton beam energy, the neutron angular distribution for several beam energies, and the neutron spectrum at a variety of angles for each beam energy. Similar calculations have previously been carried out for modeling of a boron neutron capture therapy (BNCT) facility at the Massachusetts Institute of Technology (Yanch and Zhou, 1992), however the authors did not explicitly outline their procedures, and thus our calculations are brie¯y summarized here. The calculations assume the accelerator proton beam is unidirectional and incident on the lithium target at an angle of 08 in the laboratory system, the proton energy is less than 2.37 MeV (the threshold energy for the 7Li(p,n)7Be reaction), the lithium target is thick compared to the proton range, and there is no signi®cant moderation of the neutron beam by the
Fig. 1. Modeling of the 7Li(p,n)7Be neutron source. (a) The thick lithium target represented by summing many thin targets with decreasing incident proton energy. (b) Neutron yield as a function of proton energy. (c) Angular distribution of neutrons for the four proton energies. (d) Sample neutron spectra for a proton beam energy of 2.25 MeV, spectra for the other proton energies can be obtained by truncating these spectra at the appropriate neutron energies.
M.L. Arnold et al. / Applied Radiation and Isotopes 53 (2000) 651±656
lithium itself. As illustrated in Fig. 1a, the thick target is represented by several thin targets with varying incident proton energies, ranging from the beam energy down to the reaction threshold, Eth (1.88 MeV). Let dN be the number of neutrons produced from an element of target of thickness dx emitted at angle y into an element of solid angle dO per incident proton. Then, dN
dm dO dx dO
1
where dm/dO is the dierential interaction coecient for angle y in the laboratory system. This calculation can be simpli®ed by performing three transformations. First, the dierential interaction coecient is transformed to the center-of-mass system (dm/dO). The second transformation is the element of thickness dx is transformed into an element of proton energy dEp to allow the calculation to be performed in energy space. The third transformation is from proton energy space to neutron energy space (dEn). This results in the following
dN
dm dO dx dEp dOdEn dO dO dEp dEn
2
The neutron spectrum for a given lab angle is thus
dN dm dO ÿ1 dEp S Ep dEn dO dEn dO dO
3
dm ds n y dO dO
The second term in Eq. (3) is given by
dO 1 g 2 2g cos y 3=2 dO 1 g cos y
4
where n is the lithium target atom density and ds/dO is the center-of-mass dierential cross section at centerof-mass angle y. The center-of-mass dierential cross section at angle y is related to the center-of-mass dierential cross section at 08 through the center-ofmass Legendre coecients. This leads to, " # dm ds ^ X Ai Pi y 5 n 0 dO dO i where the Ai s are the center-of-mass Legendre coecients and the Pi (y)s are the corresponding Legendre polynomials.
6
where s mn mp E p g mBe mBe mn ÿ mp Ep ÿ Eth
7
and cosy is obtained by inverting the following equation
tan y
sin y cos y g
8
Note, mn, mp, and mBe refer to the masses of the respective particles. The stopping power in Eq. (3) was assumed to be proportional to the inverse proton energy, where the constant of proportionality was based on ®tting to the published data. Lastly, dEp/dEn is calculated from the Q-value equation, mp mn En 1 ÿ Ep 1 ÿ mBe mBe ÿ
2 p mn mp En Ep cos y mBe
Q
where S ÿ1(Ep) is the inverse stopping power. Integrating over neutron energy results in the neutron yield for the particular lab angle, and further integrating over solid angle yields the total neutron yield for the incident proton beam energy. The evaluation of each of the four terms in Eq. (3) is outlined below. The center-of-mass dierential interaction coecient can be written as
653
9
which gives p mBe mn ÿ mn mp Ep =En cos y dEp p dEn mBe ÿ mp mn mp En =Ep cos y
10
Based on the above theory, a program was written using Fortran 90 to calculate the neutron yield as a function of proton beam energy (Fig. 1b); neutron angular distribution for incident proton energies of 1.95, 2.05, 2.15, and 2.25 MeV (Fig. 1c); and neutron spectra at several angles for each beam energy (samples for 2.25 MeV are illustrated in Fig. 1d, the spectra for the other beam energies are obtained by truncating these spectra at the appropriate neutron energies). Four independent MCNP sources were created corresponding to the beam energies listed above. The angular neutron distribution was input in MCNP as the probability distribution for determining the source particle direction (u,v,w ), and the particle energy was dependant on the direction and chosen based on the corresponding angular spectral distributions (distributions for every 58 from 08 to 908 were entered). In this way MCNP was able to choose a direction and energy for each source particle and the resulting neu-
654
M.L. Arnold et al. / Applied Radiation and Isotopes 53 (2000) 651±656
tron source was an accurate simulation of the actual neutron source. 2.2. Geometries and tallies The head geometry was based on a model developed by Snyder et al. (1969) and consisted of two nonconcentric ellipsoids. The brain is bounded by the surface (x/6)2+( y/9)2+(z/6.5)2=1, and the skull is the region between this surface and the surface (x/6.8)2+( y/ 9.8)2+[(z + 1)/8.3]2=1, where the positive x, y and z axes are directed to the right, front and top of the head respectively. The elemental composition of the brain and skull were derived from ICRP (23) (1975) and their densities were taken to be 1 and 1.5 g/cm3 respectively. The left and right basal ganglia were represented by ellipsoids with lengths of 4.2 cm, widths of 1.8 cm and heights of 2.2 cm centered at (22.2 cm, 1.5 cm, ÿ1 cm). They were also tilted at 458 in the x±y plane, as can be seen in the schematic diagrams for Figs. 2±4. The composition and density of the basal ganglia were taken to be the same as the rest of the brain. This neutron source was modeled as a point source and positioned at y = 1.5 cm and z=ÿ1 cm, such that the beam axis coincided with the center of both basal ganglia, and the x position was determined by the particular measurement. The number of particles run for all simulations was such that the relative error in the thermal ¯ux density was less than 5% (with the exception of determining the collimator material, in which case it went up to 10% in some instances). The ®rst step was to determine the optimal proton energy. Energies of 1.95±2.25 MeV were considered. With the source placed approximately 0.5 cm to the right of the head the ratio of thermal (<0.5 eV) to fast (>10 keV) ¯ux density, which represents the ratio of Mn activation to dose, was monitored over the
Fig. 2. Ratio of thermal to fast ¯ux density in the right basal ganglia as a function of proton beam energy. A schematic diagram of the simulation is shown: the neutron source was placed approximately 0.5 cm to the right of the head at y = 1.5 cm and z=ÿ1 cm.
Fig. 3. Thermal ¯ux density in the right basal ganglia as a function of moderator thickness (* polyethylene and r heavy water). A schematic diagram of the simulation is shown: the neutron source was ®xed at 11 cm to the right of the head ( y = 1.5 cm, z=ÿ1 cm) and the moderator was placed between it and the head.
volume of the right basal ganglia for beam energies of 1.95, 2.05, 2.15 and 2.25 MeV. Once the beam energy had been determined, the use of a moderator, re¯ector and/or collimator to increase the thermal ¯ux delivered to the basal ganglia was then investigated. Both polyethylene and heavy water were investigate as possible moderators. Cylindrical moderators (radii=25 cm) of varying thicknesses were placed in front of the neutron source, which was ®xed 11 cm to the right of the head, and the thermal ¯ux density over the volume of the right basal ganglia was monitored. It was then examined if placing a lead re¯ector (previously determined to be a suitable neutron re¯ector by Yanch and Zhou (1992)) on the left of the head to backscatter neutrons could be used to increase the thermal ¯ux in the left basal ganglia. The neutron source position was subsequently modi®ed to approxi-
Fig. 4. Thermal ¯ux density in the left basal ganglia as a function of re¯ector (lead) thickness. A schematic diagram of the simulation is shown: the neutron source was placed approximately 0.5 cm to the right of the head ( y = 1.5 cm, z=ÿ1 cm) and the re¯ector was placed on the left of the head.
M.L. Arnold et al. / Applied Radiation and Isotopes 53 (2000) 651±656
mately 0.5 cm and cylindrical re¯ectors (radii=25 cm) of varying thicknesses were placed to the left of the head to monitor the thermal ¯ux density within the left basal ganglia. Finally, it was determined if a collimator could be used to selectively irradiate along the beam axis and decrease the signal from the rest of the head. For these simulations the head was represented by a cylinder (rather than the two ellipsoids) of radius 10 cm and length 15 cm (with the ®rst 0.8 cm corresponding to the skull and the remaining 14.2 cm to the brain), as shown in the schematic diagrams for Figs. 5 and 6). The thermal ¯ux density was measured as a function of radial distance for a depth of 4 cm within the cylinder, which corresponds to the approximate depth of the basal ganglia. The source was initially placed 100 cm to the right of the head and cylindrical collimators (inner radii=1.5, outer radii=50 cm and length=100 cm) of various materials (air, lead, iron, heavy water, polyethylene, borated polyethylene and graphite) were placed between the source and the head to determine the optimal collimator material. Due to ¯attening of the neutron ®eld as the source to head distance increases, once the material had been chosen, the variation of collimation with length was investigated for collimators with lengths of 0, 25, 50, 75 and 100 cm. 3. Results The ratio of thermal to fast ¯ux density in the right basal ganglia as a function of beam energy is shown in
Fig. 5. Variation of thermal ¯ux density with radial distance for dierent collimator materials, at a depth of 4 cm in the cylindrical head phantom (+, air; w, lead; r, iron; t, heavy water; q, polyethylene; r, borated polyethylene; and R, graphite). The data points for no collimator (air) and a borated polyethylene collimator are joined by smooth curves. A schematic diagram of the simulation is shown: the neutron source is placed 100 cm from the cylindrical head phantom (r = 10 cm, l = 15 cm) and 100 cm long collimators (r1=1.5 cm, r2=50 cm) are placed between it and the head phantom.
655
Fig. 2. This ratio is largest for a proton energy of 1.95 MeV, indicating that this energy would result in the maximum Mn activation to dose ratio. Also, since this energy is just above the threshold of the reaction (1.88 MeV) it produces the most forward directed neutron beam (Fig. 1c). Thus it was concluded that 1.95 MeV would be the optimal beam energy for in vivo measurements, which results in a maximum neutron energy of 165 keV. Data for the variation of thermal ¯ux density in the right basal ganglia as a function of moderator (polyethylene and heavy water) thickness, and in the left basal ganglia as a function of re¯ector (lead) thickness are presented in Figs. 3 and 4 respectively. For both moderator materials activation of the desired region decreases with thickness, and for the lead re¯ector the increase in thermal ¯ux density is less than 1% at a thickness of 10 cm. Therefore it was decided that neither a moderator nor re¯ector should be used. The variation of beam collimation at a depth of 4 cm within the cylindrical head phantom is illustrated in Fig. 5 for the various materials investigated (normalized to one at r = 0 cm). It was found that borated polyethylene increased the thermal ¯ux density below a radius of 2 cm (approximate size of the basal ganglia) and then decreased more rapidly than for the other materials, thereby demonstrating its ability to collimate the neutron ®eld. The variation of collimation for dierent lengths of the borated polyethylene collimator is shown in Fig. 6. There is little change in collimation for the dierent lengths, including 0 cm (no collimator present), but as the source to head distance decreased there is a signi®cant increase in absolute thermal ¯ux (2.22 10ÿ2/cm2/n for no collimator compared to 1.09
Fig. 6. Variation of thermal ¯ux density with radial distance for dierent lengths of a borated polyethylene collimator, at a depth of 4 cm in the cylindrical head phantom (+, 2 0 cm; w, 25 cm; r, 50 cm; t, 75 cm; and q, 100 cm). The data points for no collimator (0 cm) and a 100 cm long collimator are joined by smooth curves. A schematic diagram of the simulation is shown: the neutron source is placed at the end of the collimators (r1=1.5 cm, r2=50 cm).
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M.L. Arnold et al. / Applied Radiation and Isotopes 53 (2000) 651±656
10ÿ5/cm2/n for a 100 cm collimator). For these reasons, within the parameters studied it appears that no collimator should be used for in vivo measurements of the basal ganglia region of the head. 4. Discussion From Monte Carlo simulations it has been determined that the optimal proton energy is 1.95 MeV for IVNAA measurements of the basal ganglia, and that no moderator, re¯ector or collimator should be used to maintain the maximum thermal ¯ux within the basal ganglia. It is not surprising that a moderator is not required as the maximum neutron energy is relatively low (165 keV) and the basal ganglia lies beneath several centimeters of tissue which acts as a natural moderator. It was hoped that a collimator would increase the thermal ¯ux along the beam axis at the depth of the basal ganglia, but since the neutron ®eld for a proton energy of 1.95 MeV is already highly forward directed the ¯attening of the neutron ®eld, which is caused as the source is moved further from the head, counterbalances the collimation of the ®eld by the collimator. The issue of dose was only considered in the initial step of choosing the beam energy. The use of ®lters, to reduce the thermal dose to the skin and/or high energy neutrons which do not signi®cantly contribute to the thermal ¯ux in the basal ganglia, was not investigated at this time. This is because a ®lter, even though preferentially eliminating undesired neutrons, would also cause some decrease in the thermal ¯ux in the basal ganglia region, and at this time the main goal is to maintain the maximum thermal ¯ux to achieve the optimal MDL. It is also important to note that the ®nal system design would also consist of shielding to protect the patient's body from the neutron ®eld, but before this is considered it is necessary to determine the feasibility of this system for in vivo measurements. Future research will consist of modeling the actual irradiation and counting processes for various geometries of NaI detectors, and determining the corresponding MDLs of the system. Dose calculations will also be carried out to estimate the resulting dose to the patient from this procedure, and if appropriate the design of neutron and photon shielding will also be investigated. 5. Conclusions MCNP neutron sources have been developed which model the production of neutrons at the McMaster KN-accelerator via the 7Li(p,n)7Be reaction, for a number of dierent proton energies. The experimental
operating conditions for IVNAA measurements of Mn in the basal ganglia have been investigated using Monte Carlo simulations and it has been ascertained that measurements should be conducted at a proton beam energy of 1.95 MeV, and that no moderator, re¯ector or collimator be used if the maximum thermal ¯ux to this region is to be obtained. Future research will evaluate this system in terms of dose delivered to the patient and the minimum detectable limit which can be achieved.
Acknowledgements The authors wish to acknowledge Merek Kiela for his assistance with computer software related concerns. The Natural Sciences and Engineering Research Council provided funding for this research in the form of a Postgraduate Scholarship B for M.L. Arnold and research grants for F.E. McNeill and D.R. Chettle.
References Dilmanian, F.A., Lidofsky, L.J., Stamatelatos, I., Kamen, Y., Yasumura, S., Vartsky, D., Pierson Jr, R.N., Weber, D.A., Moore, R.I., Ma, R., 1998. Improvement of the prompt-gamma neutron activation facility at Brookhaven National Laboratory. Phys. Med. Biol. 43 (2), 339±349. Greger, J.L., 1999. Nutrition versus toxicology in humans: evaluation of potential biomarkers. Neurotoxicology 20 (23), 205±212. ICRP, 1975. Report of the Task Group on Reference Man, ICRP No.23. Pergamon Press, Oxford. Liskien, H., Paulsen, A., 1975. Neutron production cross sections and energies for the reactions 7Li(p,n)7Be and 7 Li(p,n)7Be. Atomic Data Nucl. Tables 15 (1), 57±84. Ma, R., Ellis, K.J., Yasumura, S., Shypailo, R.J., Pierson Jr, R.N., 1999. Total body-calcium measurements: comparison of two delayed-gamma neutron activation facilities. Phys. Med. Biol. 44 (6), N113±118. McNeill, F.E., Chettle, D.R., 1998. Improvement to the in vivo measurement of cadmium in the kidney by neutron activation analysis. Radiat. Isot. 49 (56), 699±700. Nuclear Data Tables, 1960. Nuclear Data Tables, Part 3. National Academy of Science, Washington, DC. Pal, P.K., Sam III, A., Calne, D.B., 1999. Manganese neurotoxicity: a review clinical features, imaging and pathology. Neurotoxicology 20 (23), 227±238. Snyder, W.S., Fisher Jr, H.L., Ford, M.R., Warner, G.G., 1969. Estimates of absorbed fractions for monoenergetic photon sources uniformly distributed in various organs of a heterogeneous phantom. J. Nucl. Med. Suppl. 3, 7±52. Yanch, J.C., Zhou, X.-L., 1992. Accelerator-based epithermal neutron beam design for neutron capture therapy. Med. Phys. 19 (3), 709±721.
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System design for in vivo neutron activation analysis measurements of manganese in the human brain: based on Monte Carlo modeling M.L. Arnold*, F.E. McNeill, W.V. Prestwich, D.R. Chettle Department of Physics and Astronomy, Medical Physics & Radiation Sciences Unit, McMaster University, 1280 Main St. West, Hamilton, ON, Canada L8S 4M1
Abstract Manganese is an essential nutrient required by the human body, but conversely, over exposure to the element may cause central nervous system damage. The technique of in vivo neutron activation analysis, using the McMaster KN-accelerator, is being investigated as a possible method of noninvasively determining manganese concentrations within the human body. Since the brain is the primary target of damage from exposure it would be the ideal site for measurements. Thus, Monte Carlo simulations have been undertaken to de®ne the optimum experimental parameters for such a measurement, examining the use of possible moderator, re¯ector and collimator materials. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Neutron activation analysis; Manganese; In vivo; Brain; Monte Carlo
1. Introduction The toxic eects of manganese (Mn) on the central nervous system have been documented for over 150 years, however as of yet there does not exist a good biomarker which can be used as an indicator of exposure and the possibility of developing adverse eects (Greger, 1999). To date, one of the best biomarkers is T1 weighted MRI, which shows an increase in signal intensity in the basal ganglia region of the brain for exposed nonhuman primates (reviewed by Pal et al., 1999), suggesting an increase in Mn in this region, but unfortunately providing no quantitative information. Recently, there have been studies dealing with the
* Corresponding author. Fax: +1-905-546-1252. E-mail address: [email protected] (M.L. Arnold).
optimization of systems for the technique of in vivo neutron activation analysis (IVNAA) (Ma et al., 1999; Dilmanian et al., 1998; McNeill and Chettle, 1998). IVNAA can be used to quantify the amount of various elements within the body, and the feasibility of this technique for quantitatively measuring Mn within the basal ganglia is currently being investigated. The McMaster KN-accelerator would be used to provide neutrons, by means of the 7Li(p,n)7Be reaction (threshold of 1.88 MeV), and Mn activation would occur via the 55Mn(n,g )56Mn reaction (s=13 barn). Following the irradiation process, the 847 keV gamma rays emitted when 56Mn decays (half-life=2.58 h, branching ratio=100%) would be counted outside of the body using NaI detectors. At present, the irradiation process has been modeled using Monte Carlo simulations to determine the optimum experimental parameters which should be used, and future work will deal with modeling of the entire two step process to
0969-8043/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 9 - 8 0 4 3 ( 0 0 ) 0 0 1 9 9 - 8
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M.L. Arnold et al. / Applied Radiation and Isotopes 53 (2000) 651±656
evaluate system performance and calculate the minimum detectable limit (MDL) for the measurement. 2. Method The Monte Carlo neutron and photon transport code MCNP, version 4B2, was installed on a UNIX workstation at McMaster and used to model the experimental measurements. 2.1. Neutron source Based on published cross section data (Liskien and Paulsen, 1975) and stopping powers (Nuclear Data Tables, 1960) the 7Li(p,n)7Be nuclear reaction has been modeled from analytical calculations of the neutron
yield as a function of the proton beam energy, the neutron angular distribution for several beam energies, and the neutron spectrum at a variety of angles for each beam energy. Similar calculations have previously been carried out for modeling of a boron neutron capture therapy (BNCT) facility at the Massachusetts Institute of Technology (Yanch and Zhou, 1992), however the authors did not explicitly outline their procedures, and thus our calculations are brie¯y summarized here. The calculations assume the accelerator proton beam is unidirectional and incident on the lithium target at an angle of 08 in the laboratory system, the proton energy is less than 2.37 MeV (the threshold energy for the 7Li(p,n)7Be reaction), the lithium target is thick compared to the proton range, and there is no signi®cant moderation of the neutron beam by the
Fig. 1. Modeling of the 7Li(p,n)7Be neutron source. (a) The thick lithium target represented by summing many thin targets with decreasing incident proton energy. (b) Neutron yield as a function of proton energy. (c) Angular distribution of neutrons for the four proton energies. (d) Sample neutron spectra for a proton beam energy of 2.25 MeV, spectra for the other proton energies can be obtained by truncating these spectra at the appropriate neutron energies.
M.L. Arnold et al. / Applied Radiation and Isotopes 53 (2000) 651±656
lithium itself. As illustrated in Fig. 1a, the thick target is represented by several thin targets with varying incident proton energies, ranging from the beam energy down to the reaction threshold, Eth (1.88 MeV). Let dN be the number of neutrons produced from an element of target of thickness dx emitted at angle y into an element of solid angle dO per incident proton. Then, dN
dm dO dx dO
1
where dm/dO is the dierential interaction coecient for angle y in the laboratory system. This calculation can be simpli®ed by performing three transformations. First, the dierential interaction coecient is transformed to the center-of-mass system (dm/dO). The second transformation is the element of thickness dx is transformed into an element of proton energy dEp to allow the calculation to be performed in energy space. The third transformation is from proton energy space to neutron energy space (dEn). This results in the following
dN
dm dO dx dEp dOdEn dO dO dEp dEn
2
The neutron spectrum for a given lab angle is thus
dN dm dO ÿ1 dEp S Ep dEn dO dEn dO dO
3
dm ds n y dO dO
The second term in Eq. (3) is given by
dO 1 g 2 2g cos y 3=2 dO 1 g cos y
4
where n is the lithium target atom density and ds/dO is the center-of-mass dierential cross section at centerof-mass angle y. The center-of-mass dierential cross section at angle y is related to the center-of-mass dierential cross section at 08 through the center-ofmass Legendre coecients. This leads to, " # dm ds ^ X Ai Pi y 5 n 0 dO dO i where the Ai s are the center-of-mass Legendre coecients and the Pi (y)s are the corresponding Legendre polynomials.
6
where s mn mp E p g mBe mBe mn ÿ mp Ep ÿ Eth
7
and cosy is obtained by inverting the following equation
tan y
sin y cos y g
8
Note, mn, mp, and mBe refer to the masses of the respective particles. The stopping power in Eq. (3) was assumed to be proportional to the inverse proton energy, where the constant of proportionality was based on ®tting to the published data. Lastly, dEp/dEn is calculated from the Q-value equation, mp mn En 1 ÿ Ep 1 ÿ mBe mBe ÿ
2 p mn mp En Ep cos y mBe
Q
where S ÿ1(Ep) is the inverse stopping power. Integrating over neutron energy results in the neutron yield for the particular lab angle, and further integrating over solid angle yields the total neutron yield for the incident proton beam energy. The evaluation of each of the four terms in Eq. (3) is outlined below. The center-of-mass dierential interaction coecient can be written as
653
9
which gives p mBe mn ÿ mn mp Ep =En cos y dEp p dEn mBe ÿ mp mn mp En =Ep cos y
10
Based on the above theory, a program was written using Fortran 90 to calculate the neutron yield as a function of proton beam energy (Fig. 1b); neutron angular distribution for incident proton energies of 1.95, 2.05, 2.15, and 2.25 MeV (Fig. 1c); and neutron spectra at several angles for each beam energy (samples for 2.25 MeV are illustrated in Fig. 1d, the spectra for the other beam energies are obtained by truncating these spectra at the appropriate neutron energies). Four independent MCNP sources were created corresponding to the beam energies listed above. The angular neutron distribution was input in MCNP as the probability distribution for determining the source particle direction (u,v,w ), and the particle energy was dependant on the direction and chosen based on the corresponding angular spectral distributions (distributions for every 58 from 08 to 908 were entered). In this way MCNP was able to choose a direction and energy for each source particle and the resulting neu-
654
M.L. Arnold et al. / Applied Radiation and Isotopes 53 (2000) 651±656
tron source was an accurate simulation of the actual neutron source. 2.2. Geometries and tallies The head geometry was based on a model developed by Snyder et al. (1969) and consisted of two nonconcentric ellipsoids. The brain is bounded by the surface (x/6)2+( y/9)2+(z/6.5)2=1, and the skull is the region between this surface and the surface (x/6.8)2+( y/ 9.8)2+[(z + 1)/8.3]2=1, where the positive x, y and z axes are directed to the right, front and top of the head respectively. The elemental composition of the brain and skull were derived from ICRP (23) (1975) and their densities were taken to be 1 and 1.5 g/cm3 respectively. The left and right basal ganglia were represented by ellipsoids with lengths of 4.2 cm, widths of 1.8 cm and heights of 2.2 cm centered at (22.2 cm, 1.5 cm, ÿ1 cm). They were also tilted at 458 in the x±y plane, as can be seen in the schematic diagrams for Figs. 2±4. The composition and density of the basal ganglia were taken to be the same as the rest of the brain. This neutron source was modeled as a point source and positioned at y = 1.5 cm and z=ÿ1 cm, such that the beam axis coincided with the center of both basal ganglia, and the x position was determined by the particular measurement. The number of particles run for all simulations was such that the relative error in the thermal ¯ux density was less than 5% (with the exception of determining the collimator material, in which case it went up to 10% in some instances). The ®rst step was to determine the optimal proton energy. Energies of 1.95±2.25 MeV were considered. With the source placed approximately 0.5 cm to the right of the head the ratio of thermal (<0.5 eV) to fast (>10 keV) ¯ux density, which represents the ratio of Mn activation to dose, was monitored over the
Fig. 2. Ratio of thermal to fast ¯ux density in the right basal ganglia as a function of proton beam energy. A schematic diagram of the simulation is shown: the neutron source was placed approximately 0.5 cm to the right of the head at y = 1.5 cm and z=ÿ1 cm.
Fig. 3. Thermal ¯ux density in the right basal ganglia as a function of moderator thickness (* polyethylene and r heavy water). A schematic diagram of the simulation is shown: the neutron source was ®xed at 11 cm to the right of the head ( y = 1.5 cm, z=ÿ1 cm) and the moderator was placed between it and the head.
volume of the right basal ganglia for beam energies of 1.95, 2.05, 2.15 and 2.25 MeV. Once the beam energy had been determined, the use of a moderator, re¯ector and/or collimator to increase the thermal ¯ux delivered to the basal ganglia was then investigated. Both polyethylene and heavy water were investigate as possible moderators. Cylindrical moderators (radii=25 cm) of varying thicknesses were placed in front of the neutron source, which was ®xed 11 cm to the right of the head, and the thermal ¯ux density over the volume of the right basal ganglia was monitored. It was then examined if placing a lead re¯ector (previously determined to be a suitable neutron re¯ector by Yanch and Zhou (1992)) on the left of the head to backscatter neutrons could be used to increase the thermal ¯ux in the left basal ganglia. The neutron source position was subsequently modi®ed to approxi-
Fig. 4. Thermal ¯ux density in the left basal ganglia as a function of re¯ector (lead) thickness. A schematic diagram of the simulation is shown: the neutron source was placed approximately 0.5 cm to the right of the head ( y = 1.5 cm, z=ÿ1 cm) and the re¯ector was placed on the left of the head.
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mately 0.5 cm and cylindrical re¯ectors (radii=25 cm) of varying thicknesses were placed to the left of the head to monitor the thermal ¯ux density within the left basal ganglia. Finally, it was determined if a collimator could be used to selectively irradiate along the beam axis and decrease the signal from the rest of the head. For these simulations the head was represented by a cylinder (rather than the two ellipsoids) of radius 10 cm and length 15 cm (with the ®rst 0.8 cm corresponding to the skull and the remaining 14.2 cm to the brain), as shown in the schematic diagrams for Figs. 5 and 6). The thermal ¯ux density was measured as a function of radial distance for a depth of 4 cm within the cylinder, which corresponds to the approximate depth of the basal ganglia. The source was initially placed 100 cm to the right of the head and cylindrical collimators (inner radii=1.5, outer radii=50 cm and length=100 cm) of various materials (air, lead, iron, heavy water, polyethylene, borated polyethylene and graphite) were placed between the source and the head to determine the optimal collimator material. Due to ¯attening of the neutron ®eld as the source to head distance increases, once the material had been chosen, the variation of collimation with length was investigated for collimators with lengths of 0, 25, 50, 75 and 100 cm. 3. Results The ratio of thermal to fast ¯ux density in the right basal ganglia as a function of beam energy is shown in
Fig. 5. Variation of thermal ¯ux density with radial distance for dierent collimator materials, at a depth of 4 cm in the cylindrical head phantom (+, air; w, lead; r, iron; t, heavy water; q, polyethylene; r, borated polyethylene; and R, graphite). The data points for no collimator (air) and a borated polyethylene collimator are joined by smooth curves. A schematic diagram of the simulation is shown: the neutron source is placed 100 cm from the cylindrical head phantom (r = 10 cm, l = 15 cm) and 100 cm long collimators (r1=1.5 cm, r2=50 cm) are placed between it and the head phantom.
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Fig. 2. This ratio is largest for a proton energy of 1.95 MeV, indicating that this energy would result in the maximum Mn activation to dose ratio. Also, since this energy is just above the threshold of the reaction (1.88 MeV) it produces the most forward directed neutron beam (Fig. 1c). Thus it was concluded that 1.95 MeV would be the optimal beam energy for in vivo measurements, which results in a maximum neutron energy of 165 keV. Data for the variation of thermal ¯ux density in the right basal ganglia as a function of moderator (polyethylene and heavy water) thickness, and in the left basal ganglia as a function of re¯ector (lead) thickness are presented in Figs. 3 and 4 respectively. For both moderator materials activation of the desired region decreases with thickness, and for the lead re¯ector the increase in thermal ¯ux density is less than 1% at a thickness of 10 cm. Therefore it was decided that neither a moderator nor re¯ector should be used. The variation of beam collimation at a depth of 4 cm within the cylindrical head phantom is illustrated in Fig. 5 for the various materials investigated (normalized to one at r = 0 cm). It was found that borated polyethylene increased the thermal ¯ux density below a radius of 2 cm (approximate size of the basal ganglia) and then decreased more rapidly than for the other materials, thereby demonstrating its ability to collimate the neutron ®eld. The variation of collimation for dierent lengths of the borated polyethylene collimator is shown in Fig. 6. There is little change in collimation for the dierent lengths, including 0 cm (no collimator present), but as the source to head distance decreased there is a signi®cant increase in absolute thermal ¯ux (2.22 10ÿ2/cm2/n for no collimator compared to 1.09
Fig. 6. Variation of thermal ¯ux density with radial distance for dierent lengths of a borated polyethylene collimator, at a depth of 4 cm in the cylindrical head phantom (+, 2 0 cm; w, 25 cm; r, 50 cm; t, 75 cm; and q, 100 cm). The data points for no collimator (0 cm) and a 100 cm long collimator are joined by smooth curves. A schematic diagram of the simulation is shown: the neutron source is placed at the end of the collimators (r1=1.5 cm, r2=50 cm).
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10ÿ5/cm2/n for a 100 cm collimator). For these reasons, within the parameters studied it appears that no collimator should be used for in vivo measurements of the basal ganglia region of the head. 4. Discussion From Monte Carlo simulations it has been determined that the optimal proton energy is 1.95 MeV for IVNAA measurements of the basal ganglia, and that no moderator, re¯ector or collimator should be used to maintain the maximum thermal ¯ux within the basal ganglia. It is not surprising that a moderator is not required as the maximum neutron energy is relatively low (165 keV) and the basal ganglia lies beneath several centimeters of tissue which acts as a natural moderator. It was hoped that a collimator would increase the thermal ¯ux along the beam axis at the depth of the basal ganglia, but since the neutron ®eld for a proton energy of 1.95 MeV is already highly forward directed the ¯attening of the neutron ®eld, which is caused as the source is moved further from the head, counterbalances the collimation of the ®eld by the collimator. The issue of dose was only considered in the initial step of choosing the beam energy. The use of ®lters, to reduce the thermal dose to the skin and/or high energy neutrons which do not signi®cantly contribute to the thermal ¯ux in the basal ganglia, was not investigated at this time. This is because a ®lter, even though preferentially eliminating undesired neutrons, would also cause some decrease in the thermal ¯ux in the basal ganglia region, and at this time the main goal is to maintain the maximum thermal ¯ux to achieve the optimal MDL. It is also important to note that the ®nal system design would also consist of shielding to protect the patient's body from the neutron ®eld, but before this is considered it is necessary to determine the feasibility of this system for in vivo measurements. Future research will consist of modeling the actual irradiation and counting processes for various geometries of NaI detectors, and determining the corresponding MDLs of the system. Dose calculations will also be carried out to estimate the resulting dose to the patient from this procedure, and if appropriate the design of neutron and photon shielding will also be investigated. 5. Conclusions MCNP neutron sources have been developed which model the production of neutrons at the McMaster KN-accelerator via the 7Li(p,n)7Be reaction, for a number of dierent proton energies. The experimental
operating conditions for IVNAA measurements of Mn in the basal ganglia have been investigated using Monte Carlo simulations and it has been ascertained that measurements should be conducted at a proton beam energy of 1.95 MeV, and that no moderator, re¯ector or collimator be used if the maximum thermal ¯ux to this region is to be obtained. Future research will evaluate this system in terms of dose delivered to the patient and the minimum detectable limit which can be achieved.
Acknowledgements The authors wish to acknowledge Merek Kiela for his assistance with computer software related concerns. The Natural Sciences and Engineering Research Council provided funding for this research in the form of a Postgraduate Scholarship B for M.L. Arnold and research grants for F.E. McNeill and D.R. Chettle.
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