936 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 38, NO. 3, JUNE 1991 Microcomputer Monte Carlo Electron Transport Codes for Beta Skin Dose Calculations Manho Chung, Anthony H. Foderaro, William A. Jester, and Samuel H. Levine Abstract-The one-dimensional Monte Carlo electron trans- port program called ZEBRA was successfully adapted for use this adaptation was to reduce the cost of the Monte Carlo Eltran2 has in turn been modified into a two-di- mensional program called Eltrans for computing the dose from a point or a disk source to a cylindrical target. For Monte Carlo kernels that describe the distribution of absorbed dose around a point source, one can determine the spatial pattern of absorbed dose in media in which a beta-emitting radionuclide is distributed in some specified form. Application of the Point-kemel method is limited to an U h u n d e d homogeneous medium with uniform density. The well-known VARSKIN on microcomputersin a Program called Eltran2. The Purpose of calculations, theoretical beta energy spectra are calculated based on the Fermi beta decay theory. The calculated average energies of spectra agree with the values in the related publications within six percent. An extended study has been done using the Monte Carlo programs Eltran2 and Eltran3 to facilitate the design of beta/gamma skin dose monitor, which we are develop- ing using silicon detectors. The programs calculate the effects of angular distribution of source electrons and the radial distribu- tion of the hot particle dose. It is found that the hot particle dose averaged over a live skin area of 1 cm2 significantly underestimates the real dose value at the very small area just under the hot particle by a factor of about 1OOO. INTRODUCTION TO THE MONTE CARLO METHODS IN BETA DOSIMETRY ANY different methods have been used to solve elec- M tron transport problems in beta dosimetry [l]. The momentum method was the first technique used to solve the Boltzman equation for radiation shielding applications. An extensive table of the dose distribution function has been reported by Spencer [2]. The table includes the dose distribu- tion function for the slowing down of electrons in carbon, aluminum, copper, tin, lead, air, and polystyrene for electron energies ranged from 25 keV to 10 MeV. If the source has a continuous energy spectrum, the dose distribution can be calculated by dividing the source spectrum into a group of monoenergetic electrons to calculate individual dose distribu- tions. A serious drawback of this method is its limitation to plain geometries, i.e., a very simple configuration of a source and an absorbing medium. It is usually applied to infinite homogeneous plane or point source. Another method is the so-called point-kernel method. By the superposition of Manuscript received October 30, 1990; revised January 9, 1991. This work was supported in part by Project F ed (a group of nuclear utilities and other industries that provide funds to support selected research at the Nuclear Engineering Department of Perm State), Duquesne Light Company, GPU Nuclear Corporation, and The Pennsylvania Power and Light Company. The authors are with the Nuclear Engineering Department. The Pennsyl- vania State University, 231 Sackett Building, University Park, PA 16802. A. H. Foderaro is currently retired. IEEE Log Number 9143160. code [3] is a microcomputer code to calculate the skin dose due to a point or disk source using the point-kernel method. In most practical applications, the Monte Carlo method is superior than the other methods. However, in contrast to the transport of neutrons or photons, electrons suffer an enor- mous number of scatterings along their path through matter because of the long-range Coulomb interactions. Thus, there is a problem of requiring a tremendous computation time when individual history of electrons is followed. To solve this problem, the electrons are transported in a schematized random walk in which each step or path increment accounts for the effects of a multitude of collisions, instead of follow- ing every individual event. M. J. Berger [4], [5] used the so-called ‘‘condensed history scheme’ ’ instead of completing the beta particle histories. In the condensed history scheme, he preset an energy grid and performed the Monte Carlo calculations at the discretized energies only. The random walk for an electron is continued until the electron is re- moved from the target by reflection or transmission, or until it loses almost all of its kinetic energy; the electron cutoff energy used in our codes is 7.5 keV. When an electron history is terminated, another history is begun until the required number of histories have been followed. Using this procedure, there is no difficulty in modeling any spatial system. All that is needed is simply the computer time required to handle enough histories to get an acceptable result. All existing Monte Carlo codes mix theoretical analysis and a purely numerical technique of construction of random walk. Thus, they consist of two parts: 1) a data program that computes the probability distributions including stopping powers for range and multiple scattering and their correc- tions; and 2) a Monte Carlo program, which also handles the boundary conditions. ONE-DIMENSIONAL MONTE CARLO ELECTRON TRANSPORT PROGRAMS Two microcomputer Monte Carlo codes of electron trans- port are being used in our research to aid in the development 0018-9499/91/0600-0936$01.00 0 1991 IEEE
Transcript
936 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 38, NO. 3, JUNE 1991
Microcomputer Monte Carlo Electron Transport Codes for Beta Skin
Dose Calculations Manho Chung, Anthony H. Foderaro, William A. Jester, and
Samuel H. Levine
Abstract-The one-dimensional Monte Carlo electron trans- port program called ZEBRA was successfully adapted for use
this adaptation was to reduce the cost of the Monte Carlo Eltran2 has in turn been modified into a two-di-
mensional program called Eltrans for computing the dose from a point or a disk source to a cylindrical target. For Monte Carlo
kernels that describe the distribution of absorbed dose around a point source, one can determine the spatial pattern of absorbed dose in media in which a beta-emitting radionuclide is distributed in some specified form. Application of the Point-kemel method is limited to an Uhunded homogeneous medium with uniform density. The well-known VARSKIN
on microcomputers in a Program called Eltran2. The Purpose of
calculations, theoretical beta energy spectra are calculated based on the Fermi beta decay theory. The calculated average energies of spectra agree with the values in the related publications within six percent. An extended study has been done using the Monte Carlo programs Eltran2 and Eltran3 to facilitate the design of beta/gamma skin dose monitor, which we are develop- ing using silicon detectors. The programs calculate the effects of angular distribution of source electrons and the radial distribu- tion of the hot particle dose. It is found that the hot particle dose averaged over a live skin area of 1 cm2 significantly underestimates the real dose value at the very small area just under the hot particle by a factor of about 1OOO.
INTRODUCTION TO THE MONTE CARLO METHODS IN BETA DOSIMETRY
ANY different methods have been used to solve elec- M tron transport problems in beta dosimetry [l]. The momentum method was the first technique used to solve the Boltzman equation for radiation shielding applications. An extensive table of the dose distribution function has been reported by Spencer [2]. The table includes the dose distribu- tion function for the slowing down of electrons in carbon, aluminum, copper, tin, lead, air, and polystyrene for electron energies ranged from 25 keV to 10 MeV. If the source has a continuous energy spectrum, the dose distribution can be calculated by dividing the source spectrum into a group of monoenergetic electrons to calculate individual dose distribu- tions. A serious drawback of this method is its limitation to plain geometries, i.e., a very simple configuration of a source and an absorbing medium. It is usually applied to infinite homogeneous plane or point source. Another method is the so-called point-kernel method. By the superposition of
Manuscript received October 30, 1990; revised January 9, 1991. This work was supported in part by Project F e d (a group of nuclear utilities and other industries that provide funds to support selected research at the Nuclear Engineering Department of Perm State), Duquesne Light Company, GPU Nuclear Corporation, and The Pennsylvania Power and Light Company.
The authors are with the Nuclear Engineering Department. The Pennsyl- vania State University, 231 Sackett Building, University Park, PA 16802. A. H. Foderaro is currently retired.
IEEE Log Number 9143160.
code [3] is a microcomputer code to calculate the skin dose due to a point or disk source using the point-kernel method.
In most practical applications, the Monte Carlo method is superior than the other methods. However, in contrast to the transport of neutrons or photons, electrons suffer an enor- mous number of scatterings along their path through matter because of the long-range Coulomb interactions. Thus, there is a problem of requiring a tremendous computation time when individual history of electrons is followed. To solve this problem, the electrons are transported in a schematized random walk in which each step or path increment accounts for the effects of a multitude of collisions, instead of follow- ing every individual event. M. J. Berger [4], [5] used the so-called ‘‘condensed history scheme’ ’ instead of completing the beta particle histories. In the condensed history scheme, he preset an energy grid and performed the Monte Carlo calculations at the discretized energies only. The random walk for an electron is continued until the electron is re- moved from the target by reflection or transmission, or until it loses almost all of its kinetic energy; the electron cutoff energy used in our codes is 7.5 keV. When an electron history is terminated, another history is begun until the required number of histories have been followed. Using this procedure, there is no difficulty in modeling any spatial system. All that is needed is simply the computer time required to handle enough histories to get an acceptable result.
All existing Monte Carlo codes mix theoretical analysis and a purely numerical technique of construction of random walk. Thus, they consist of two parts: 1) a data program that computes the probability distributions including stopping powers for range and multiple scattering and their correc- tions; and 2) a Monte Carlo program, which also handles the boundary conditions.
ONE-DIMENSIONAL MONTE CARLO ELECTRON TRANSPORT PROGRAMS
Two microcomputer Monte Carlo codes of electron trans- port are being used in our research to aid in the development
0018-9499/91/0600-0936$01.00 0 1991 IEEE
CHUNG et U/.: MONTE CARLO ELECTRON TRANSPORT CODES
of a skin dose monitoring system. The original Monte Carlo electron transport code ZEBRA was developed by M. J. Berger and L. D. Buxton of the National Bureau of Standards ( N B S ) in 1971 [4]. The original code, which requires the use of a large mainframe computer, was adapted to run on the Macintosh-SE computer to reduce the cost of the Monte Carlo calculations. The new code consists of three compo- nent programs: Datapac6, Eltran2, and MCA. Datapac6 was modified from the Datapad of the ZEBRA code to generate electron transport data for the Monte Carlo program Eltran2. Thus, Datapac6 generates the necessary data for Eltran2. Eltran2 is a Macintosh Basic version of the Fortran Monte Carlo program ZEBRA-1 [4]. Eltran2 is basically the same as ZEBRA-1, but some simplifications and enhancements were made during the adaptation. Program MCA reads the output of Eltran2 and displays the results in a graphical multichannel form that is similar to the screen display of a multichannel analyzer.
The program Datapac6 generates the electron transport data for the Monte Carlo program Eltran2. The data include path length differentials, sets of angular deflection cumulative distributions, and average deflection cosines. These are tabu- lated as a function of electron energy for each material of interest. Eltran2 computes the deposited energy and its spec- trum in each layer of a multilayer slab and the transmitted and reflected energies. The problem is limited to the geome- tries in which the target can be assumed to have infinite lateral dimensions in comparison with its thickness. The incident electron beam can either be monoenergetic or have a continuously distributed energy spectrum.
The initial direction of the electrons has the following four options: normal incident, monodirectional with a specified angle, isotropic, and cosine distribution.
1) normal
2) monodirectional
3) isotropic
4) cosine
All electrons are emitted in the same direction, which is normal to the tar- get front face. All electrons are emitted in the same direction with a specified angle with the normal axis to the target front face. If the angle is zero, this distri- bution becomes the normal distribu- tion. The initial electron directions are dis- tributed isotropically out from the source. The initial electron directions are dis- tributed as a form of cosine function of the direction angles measured from the axis normal to the target front face.
The four source distributions will be referred to as normal, monodirectional, isotropic, and cosine source distributions in this paper.
The output of Eltran2 contains the following information:
1) listing of all input data with identification headings, date qf run, and running time;
931
reflection and transmission of electrons by the target; percentages of the total number of electrons transmitted and reflected, percentages of the incident energy trans- mitted and reflected, and average energy of the incident beam; deposition of energy by electrons in each layer of the target; percentage of the incident energy deposited is given for each layer; number of counts in each channel of total 2048 chan- nels in the energy range 0 to 4 MeV.
Accuracy of Monte Carlo calculation depends on the ran- domness of random numbers generated and the total number of random numbers used in each run. The period of a random number generator should be greater than the number of random numbers used. For all the computers used in this study, periods of the random number generators are so sufficiently large that they do not need to be considered as a source of statistical error. Even in the Macintosh-SE micro- computer, the built-in random number generator has a period of 1.7 x lo’, which is greater than the number of random numbers normally used in each run. Eltran2 was run for a specific case with different numbers of histories. From this calculation, it was found that the standard deviation of de- posited dose was inversely proportional to square root of the number of histories.
Eltran2 has been verified by performing calculations and comparing the results with those obtained by ZEBRA and, in some cases, with measurements. To ensure that Eltran2 is functioning correctly, several identical cases were run by Eltran2 on the Macintosh-SE computer and ZEBRA-1 on the IBM 3090-600s mainframe computer for 10 OOO histories. As a result, the fraction of incident electron energy deposited in each layer of the slab was recorded. Eltran2 and ZEBRA-1 agreed within 0.5 % , which can be explained by the statistical fluctuations caused by the different random number se- quences. Another Monte Carlo electron/photon transport code system, the Integrated TIGER Series (ITS) [6], was tried to compare with Eltran2. Similar to the ZEBRA code, ITS is based primarily on the ETRAN model 171, which computes microscopic photon transport with a macroscopic random walk for electron transport. Presently, ITS has not run on the IBM 3090-600s mainframe computer, but it has been possible to compare the example output of TIGER, which was in the original tape, with the Eltran2 calculation of a composite of two layers of material. Even though the material of the first layer is different, i.e., tantalum for TIGER and tungsten for Eltran2, the two programs agreed within about two percent of the total dose.
The Fortran program ZEBRA was also tried on a VAX 8850 computer to compare the running time with that on the IBM 3090-600s mainframe computer. The Macintosh Mi- crosoft Basic program Eltran2 was also transferred to an IBM compatible microcomputer, the Tandy 4000SX. Since El- tran2 was originally written by Microsoft Basic on the Mac- intosh-SE computer, it needed some minor modifications in order to be able to run on the IBM compatible microcom- puter. This version of Eltran2 utilizes either Turbo Basic or
938 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 38, NO. 3, JUNE 1991
TABLE I COMPARISON OF RUNNING SPEED OF MONTE CARLO PROGRAMS
Computer CPU Software Program Relative Speed
Macintosh-SE or MC 68000 Microsoft Basic Eltran2 1 .o Macintosh-Plus
Macintosh-II MC 68020 Microsoft Basic Eltran2 3.6
Tandy 4000SX Intel 80386SX Turbo Basic 1 .O Eltran2 10.9 Macintosh-SE/30 MC 68030 Microsoft Basic Eltran2 4.7
Microsoft Quick Basic for the Tandy 4000SX computer, which is equipped with a math coprocessor. Table I shows the running speed of the Monte Carlo programs of different softwares on the different computers, normalized to one for Eltran2 on the Macintosh-SE or Macintosh-Plus computer. The relative speed value is inversely proportional to the program running time.
TWO-DIMENSIONAL MONTE CARLO PROGRAM Since Eltran2 is a one-dimensional program, it calculates
the energy deposited in a semi-infinite multilayer slab for an incident electron beam. Thus it would not be accurate, for example, for the case of a point source, a small detector, and a short source-to-detector distance. To calculate the dose distribution more accurately for that case, Eltran2 has been modified to a two-dimensional program called Eltran3. El- tran3 calculates radial dose distribution for each layer under a point source or a disk source. Since Eltran3 needs more time for two-dimensional calculations, the running time of Eltran3 is about twice as long as that of Eltran2 for a similar case with the same number of histories.
Eltran3 has been used to study different beta source distri- butions and the hot particle dose and its radial dose distribu- tion, as explained later in this paper. It is not difficult to expand Eltran3 into a three-dimensional code for analyzing a specified geometry. For example, we are in the progress of modifying Eltran3 for the RO-2 ion chamber monitor of Eberline to compare with measurements.
CALCULATION OF EETA ENERGY SPECTRUM Individuals working with or near sources of ionizing radia-
tion are exposed to high energy electrons, which can damage the sensitive cell forming layers of the skin. From the radionuclides, electrons are emitted as internal conversion electrons, Auger electrons, or beta particles from beta decay. Internal conversion electrons and Auger electrons have dis- crete energies, while beta particles have a continuous energy spectrum.
The conversion electron or Auger electron source can be treated as monoenergetic electron source in the beta dose calculations. But a knowledge of the energy distribution is required to calculate the dose for radionuclides emitting
0 1 2 3 4 5
End point energy of beta spectrum (MeV)
Fig. 1. Distribution of end point energies of spectra of 121 common beta sources.
continuous spectra of beta particles. There are two types of beta decay: 0- decay and 0' decay. In 0- decay a neutron in the nucleus is transformed into a proton, and in 0' decay a proton is transformed into a neutron. The energy distribu- tion for a given beta transition ranges from zero to a maxi- mum endpoint energy. In 1934, the shape of /3 spectra and the lifetime of &ray emitters were first explained by Fermi using Pauli's neutrino hypothesis [8]. The energy distribution function for beta decay can be calculated from the Fermi theory [9] using an approximation [lo].
There are many radionuclides that emit beta radiations. Fig. 1 shows the distribution of end point energies of spectra of 121 beta sources, which are commonly found and used in the nuclear industries and laboratories [11]-[14]. If the aver- age energy is assumed to be approximately one-third the end point energy, then most beta sources have an average energy ranging between 0 and 2 MeV.
The spectrum equation derived from the Fermi beta decay theory can be calculated easily on the computer. Table II shows results of calculation of spectra for various sources. Most beta sources have more than one basic transition. In that case, each basic spectrum was calculated and added together after multiplying by the fraction of each transition. The calculated average energies are compared with literature values [11]-[14]. However, the spectrum of 204T1 was cor- rected further following the method of W. G. Cross et al. [141. The calculated average energies agreed with the litera- ture values with a relative error of less than six percent. The calculated spectra are used in the Monte Carlo calculations.
CHUNG er al. : MONTE CARLO ELECTRON TRANSPORT CODES 939
TABLE II COMPARISON OF THE CALCULATED AVERAGE ELECTRON
ENERGIES (MeV) WITH LITERATURE VALUES
Number Calculated Literature ofBasic Value Value Relative
CONVERTING SILICON DOSES TO TISSUE-EQUIVALENT DOSES
This portion of the study investigated the difference be- tween the electron absorbed dose in human tissue and in silicon. Using Eltran2, pairs of physical configurations, iden- tical except for the dosed region, which was either the silicon detector material or human tissue, were compared. The first configuration has three layers of tissue with different thick- nesses, and the second configuration has the same three layers except the second layer, i.e., the dosed region, is replaced by silicon with the same mass thickness.
In the two material configurations, the mass thicknesses of the first and the second layer were varied to find the relation- ships between the energy deposited in the second region and the mass thicknesses of the two layers. Throughout the study, a cosine-distributed monoenergetic source was employed with the energy of the incident electrons varied to investigate the source-energy dependence of the absorbed dose. For 30 cases with different mass thickness of shield and/or detector, the average ratio of the deposited energies in tissue and in silicon was found to be 1.16 with standard deviation of 0.04. The minimum and maximum ratios were 1 .08 and 1.25, respec- tively. The most important thing to note about the results is the relative insensitivity of the tissue/silicon dose ratio to the incident electron energy. This allows one to assume that the dose deposited in a silicon detector, regardless of the (un- known) incident electron spectrum, may be converted to dose in tissue with acceptable accuracy through multiplication by a single conversion factor. For example, if the detector is a 7 mg/cm2 (30 pm) silicon totally depleted detector, then multi- plication of its measured dose or dose rate by the conversion factor C = 1.20 would produce the dose or dose rate in tissue within 5 parts in 120, an accuracy which is far better than that obtained in most dosimetry.
COMPARISON OF SOURCE DISTRIBUTIONS Eltran3 was used to predict the hot particle dose measured
by a small cylindrical detector with finite volume. Fig. 2 shows the source-target geometries that can be used for
(b) . , Fig. 2. Source-target geometries for (a) Eltran2 and (b) Eltran3.
N
I. al n U
c :: m
3
----- normal ..........................
2
1
0 0 1 0 2 0 3 0 4 0 5 0 6 0
- Iaotropfc
cosine ----- normal
........
..........................
2
1
0 0 1 0 2 0 3 0 4 0 5 0 6 0
Eltran2
Eltran3
z (mg/cm2)
Fig. 3. Attenuation of 0.25 MeV electrons in tissue with different source- target geometries (s = source to target front face distance in cm).
Monte Carlo calculations with Eltran2 and Eltran3. Fig. 2(a) shows the three options of source distributions in program Eltran2, i.e., isotropic, cosine, and normal source distribu- tions. Fig. 2(b) shows the source-target geometry for Eltran3, which includes an isotropic point source and a cylindrical detector or target with finite volume. In the calculation with Eltran3, the detector was assumed to have a radius of 2.28 g/cm2 (0.977 cm) in the units of mass thickness, which is equal to the radius of the passivated ion-implanted planar silicon (PIPS) detectors now being used in our research. For both Eltran2 and Eltran3, the target was divided into 17 layers, and % energies absorbed in each layer were calcu- lated for monoenergetic electron sources with energies 0.125, 0.25, 0.5, 1, and 2 MeV. For the calculations with Eltran3, rmX and e,,, are the radius of the target and the maximum angle of deviation from the normal axis connecting the isotropic point source and the center of the target, respec- tively.
Figs. 3 and 4 show the dose attenuations in tissue for monoenergetic electrons with energies 0.25 and 1 MeV, respectively, for several different source-target geometries. Cases with isotropic, cosine, or normal source distribution were simulated by Eltran2, and the others with a certain source-to-target distance (s) were simulated by Eltran3, both on the Tandy 4000SX computer.
It can be concluded that if an isotropic point source such as a hot particle is located just on or very near to the small cylindrical detector (s C 0.1 cm), the isotropic source distri- bution of Eltran2 predicts the dose attenuation better than the other source distributions. If the source is located quite far from the target (s > 10 cm), the normal source distribution
940 IEEE TFUNSACTlONS ON NUCLEAR SCIENCE, VOL. 38, NO. 3, JUNE 1991
- Iaotroplc -.----.. CWIM Enran2 ----- normal 2 0.4 - a d . 1 c m - a=00.3cm r a
Fig. 4. Attenuation of 1 MeV electrons in tissue with different source-target geometries (s = source to target front face distance in cm).
predicts the dose attenuation better. The cosine distribution is the best approximation when s is between 0.3 and 0.5 cm. Although the conclusions are based on the calculations per- formed for a tissue target, they should be true for any target of low atomic number. This work demonstrates the impor- tance of using Eltran3 rather than Eltran2 for particle dosime- try.
MONTE CARLO CALCULATION OF THE HOT PARTICLE DOSE
The hot particle dose has been defined as the dose from a hot particle on the skin surface to a live skin area of 1 cm2 at a depth of 7 mg/cm2 [ 151. A hot particle having a relatively low activity may lead to rather large doses in excess of the skin dose limits if the particle remains attached to the skin for a significant time.
Radial dose distributions at 7 mg/cm2 in tissue under a hot particle were calculated using the two-dimensional electron transport program Eltran3. For the geometry shown in Fig. 5, the fraction of dose absorbed in the skin was calculated in each concentric annulus about the position of the hot particle. The width of the annular region absorbing the dose is 0.001 cm or 1 mg/cm2. Dose distributions at 7 mg/cm2 under the skin surface were calculated due to five different hot parti- cles: 14’F”, %o, 204T1, 19’Au, and 90Sr/90Y.
Values of % energy absorbed, obtained from the Monte Carlo calculations, can be converted to values of absorbed dose rate, for instance, in the unit of rad/h (0.01 Gy/h), by assuming a certain value for the source strength of radiation. For a 1 pCi (37 OOO Bq) hot particle,
Skin surface
/lr 3 mg/cm2
at 7
Fig. 5. Geometry used for calculation of radial dose distribution from a hot particle.
1 0 0 0 0 0 ~ ’ ’ ’ ’ ’ ’ ’ ’ ’
\ , 1
0 0 . 0 2 0.04 0 . 0 6 0 . 0 6 0.1
r (cm) Fig. 6. Skin dose rate in rad/h due to a 1 pc1 (37000 Bq) hot particle on
the skin surface.
TABLE III COMPARISON OF DOSE RATE AT CENTER WITH DOSE RATE
AVERAGED OVER 1 C M ~ FOR A 1 pCi (37 000 Bq) HOT PARTICLE
over 1 cm2 (rad/h) 1.9 3.6 5.3 5.6 5.5 Dc, Dose rate at center,
0.0003cm2 (rad/h) 4100 6OOO 4900 6200 3600 DC / D l 2200 1700 920 1100 650
a For a 1 pCi WSr/WY source, i.e., 0.5 pCi %r plus 0.5 pCi
19’Au is not shown because it is nearly the same as that of T1. The dose to the skin falls off dramatically as the radius
increases beyond about 0.03 cm. It can be seen that most of the energy is deposited within a circular area of 1 cm2 with the center just under the hot particle. This confirms, in some sense, the hot particle dose described above; the dose from a hot particle on the skin surface to a live skin area of 1 cm2 at a depth of 7 mg/cm2. However, the dose value averaged over a live skin area of 1 cm2 greatly underestimates the real
204
dose rate in rad/h (0.01 Gy/h) = [average energy of the source electrons in keV]
x [ % energy absorbed in the detecting volume, ( A cm2) x ( d g/cm2)]
[ 1.332 x 10’ (electrons/h)/pCi] x (1,602 x lou9 erg/keV) X 2 x (100%) x ( A cm2) x (d g/cm2) x [ 100 (erg/g)/rad] ’ (1)
CHUNG et al. : MONTE CARU) ELECTRON TRANSPORT CODES 941
value at the very small area just under the hot particle by a factor of about 1OOO.
CONCLUSION In this work, Monte Carlo electron transport codes were
developed for use on the several types of available computers [ 161. First, the one-dimensional Monte Carlo program called ZEBRA was adapted to run on the microcomputers. The microcomputer version of Monte Carlo program is called Eltran2, which computes the deposited dose and its energy spectrum in each layer of a multilayer slab and the transmit- ted and reflected energies. Turbo Basic and Microsoft Quick Basic were used on the IBM compatible Tandy 4000SX computer, and Microsoft Basic was used on Macintosh com- puters to run Eltran2. Eltran2 has been verified by perform- ing calculations and comparing the results with those ob- tained by the ZEBRA code. Next, Eltran2 was modified to become a two-dimensional program called Eltran3. Eltran3 has been used to predict the hot particle dose measured by a small cylindrical detector with finite volume. It is found that for an isotropic point source and a cylindrical target with a radius of 2.28 g/cm2 in mass thickness, the source distribu- tion is close to an isotropic distribution when the source-to- detector distance is less than 0.1 cm. It is close to a cosine distribution when the distance is between 0.3 and 0.5 cm and to a normal distribution when the distance is greater than 10 cm. Using Eltran3, radial dose distributions at 7 mg/cm2 in tissue were calculated due to different hot particles. As a result, it was found that most of the energy was deposited within a circular area of 1 cm2 and the dose falls off dramatically as the radius increases beyond about 0.03 cm. The dose value averaged over a live skin area of 1 cm2 significantly underestimates the real dose value at the very small area just under the hot particle by a factor of about 1OOO.
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