HeaEth Physics Pergamon Press 1976. Vol. 31 (August), pp. 115-118. Printed in Northern Ireland

PROTON-TISSUE DOSE BUILDUP FACTORS JOHN W. WILSON

NASA-Langley Research Center, Hampton, VA 23665 and

GOVIND S. KHANDELWAL

Old Dominion University, Norfolk, VA 23508 (Received 17 November 1975; accepted 22 January 1976) Abstract-Calculational methods for estimation of dose from external proton exposure of arbitrary convex bodies is briefly reviewed and all of the necessary information for the estimation of nuclear reaction effects in soft tissue is presented. nal proton radiation is inferred from calculations in slab geometry (Ne72). In the present note, the dose conversion factors for protons in tissue are represented using buildup factors. A parametric form for the buildup factors is presented. The values of the parameters are derived from Monte Carlo calculations of various authors. A11 of the necessary information to estimate nuclear reaction effects in proton irradiation of convex objects of arbitrary shape is contained herein.

INTRODUCTION

WHENan object is exposed to external radiation, the dose field within the object is a complicated function of the character of the external radiation, the shape of the object (including orientation), and the object’s material composition. Calculation of dose within the object involves solution of the appropriate Boltzmann transport equation where the external radiation source imposes boundary conditions on the solution. Although general purpose computer programs exist for making such estimates, they are seldom used in practice when the object is bounded by a complicated surface as, for example, is the human body. Instead, calculations are usually made for simple geometric shapes from which inferences are then made for more general geometries and the resultant errors are uncertain. In the case of external proton radiation such as that encountered near high-energy accelerators, in space, and in high-altitude aircraft, it was found that the problem of dose estimation could be greatly simplified (Wi74) and still include the effects of nuclear reactions, which impose the major hurdles in any accurate calculations, with a high degree of accuracy. Furthermore, it was shown that the method, when in error, was always conservative. Required for such calculations is a knowledge of the transition of protons in semiinfinite slab geometry which is the simplest geometry for existing transport computer programs. Indeed, almost everything that is known about the dose in humans due to exter-

BUILDUP FACTORS

In passing through tissue, energetic protons interact mostly through ionization of atomic constituents by the transfer of small amounts of momentum to orbital electrons. Although the nuclear reactions are far less numerous, their effects are magnified because of the large momentum transferred to the nuclear particles and the struck nucleus itself. Unlike the secondary electrons formed through atomic ionization by interaction with the primary protons, the radiations resulting from nuclear reactions are mostly heavily ionizing and generally have large biological effectiveness. Many of the secondary particles of nuclear reactions are sufficiently energetic to promote similar nuclear reactions and thus cause a buildup of secondary radiations. The description of such processes requires solution of the transport equation. The approximate solutions for the transition of protons in 30cm thick slabs of soft tissue for fixed incident energies have been made (A170; Ar71; Ar70; Sn69;

115

PROTON-TISSUE DOSE RUIL~DUPFACTORS

116

Tu64; Wr71; WI-69; Ze65). ‘The results of such calculations are dose conversion factors for relating the primary monoenergetic proton fluence to dose or dose equivalent as a function of position in a tissue slab. Whenever the radiation is spatially uniform, the dose at any point x in a convex object may be calculated (Wi74) by

D(x)

;i’JRn[z,(0),E ] + ( 0 ,E ) d 0 d E I1

R

ment and extension of that work will be discussed. The conversion factor R , ( z , E ) is composed of two terms representing dose due to the primary beam protons and the dose due to secondary particles produced in nuclear reaction. Thus,

R”(Z,E ) = RPb, E ) + Rs(z,E ) (2) where the primary dose equivalent conversion (1) factor is

where R,(z, E ) is the dose at depth z for normal incident protons of energy E on a tissue slab, 4(0,E) is the differential proton fluence along direction 0, and z,(fi) is the distance from the boundary along fi to the point x. It has been shown that equation (1) always overestimates the dose, but is an accurate estimate when the ratio of the proton beam divergence due to nuclear reaction to the bodies radius of curvature is small. Equation (1) is a practical prescription for introducing nuclear reaction effects into calculations of dose in geometrically complex objects such as the human body. The main requirement is that the dose conversion factors for a tissue slab be adequately known for a broad range of energies and depths. Available information on conversion factors are for discrete energies from 100MeV to 1 TeV in rather broad energy steps and for depths from 0 to 3 0 c m in semi-infinite slabs of tissue (A170; Ar70; Tu64; Ze65). The nuclear reaction data used for high-energy nucleons is usually based on Monte Carlo (Be63; Be69; Be7 1) estimates with low-energy neutron reaction data taken from experimental observation. The quality factor as defined by the ICRP (ICRP64) is used for protons. The quality factor for heavier fragments and the recoiling nuclei is arbitrarily set to 20 which is considered conservative but the average quality factor obtained by calculation is comparable to estimates obtained through observations made in nuclear emulsion (Sc70). T o fully utilize equation (l),a parametrization of the conversion factors was introduced by Wilson and Khandelwal (Wi74) which allowed reliable interpolation and extrapolation from known values. In the following, a refine-

RP(GE ) = P(E)QF[S(E,)IS(E,)/P(E,) (3) with the reduced energy given by E, = E [ R ( E ) -Z ]

(4) with the usual quality factor QF defined as a function of LET, with LET denoted here by the symbol S, and total nuclear survival probability for a proton of energy E given by P ( E )=exp

[

-

JOE

(T(E’) dE’/S(E’)]

(5)

where the macroscopic cross-section a ( E ) for tissue as calculated by Bertini is given by Alsmiller et al. (A172). The R ( E ) is the usual range-energy relation for protons in tissue and F ( X ) is the inverse of R ( E ) . The proton total optical thickness given by

T(E)=

JOE

c+(E’)dE’/S(E’)

(6)

is tabulated in Table 1 for purposes of numerical interpolation. In the case of conversion factors for absorbed dose, the R p ( z , E ) is taken as R P ( zE , ) = P(E)S(E,)/P(E).

(7) The representation of the conversion factors is simplified (Wi74) by rewriting equation (2) as

R ( z , E ) = [ l+ R s ( z ,E ) / R P (E)IRp(z, ~, E) = F ( z , E ) R p ( zE , ) (8) where F(z, E ) is recognized as the dose buildup factor. The main advantage for introducing the buildup factor into equation (8) is that unlike R , ( z , E ) , the buildup factor is a smoothly varying function of energy at all

117

J. W. WILSON and G . S. KHANDELWAL Table 1 . Total tissue optical thickness for protons

(El

E,GeV

?(E)

1.3

6.57

.01

.0033

1.5

8.03

,025

.0171

1.7

9.52

.05

.Oslo

2.0

11.76

.1

,135

2.2

13.27

.15

.2 39

2.4

14.78

.2

,362

2.6

16.29

.25

.so1

2.8

17.79

E,GeV

0.

I

7

0.

.3

.655

3.0

19.29

.35

.a22

4.0

26.62

1.004

5.0

33.81

.5

1.429

6.0

40.84

.7

2.471

7.0

47.75

.4

.9

3.743

8.5

57.91

1.1

5.143

10.0

67.85

I

L

7

Rad Buildup Factor (Tissue)

1

Proton Energy, GeV

FIG. 1. Rad buildup factor for several depths in tissue as a function of incident proton energy.

13

Hem Buildup Factor (Tissue)

i

I1

I M o n t e Carlo -Parametric

depths in the slab and can be approximated the simple function

E ) (A, + A ~ +ZA , . z ~exp ) (-A~Z)

~ ( 2 , r=

where the parameters A, are understood to be energy dependent. The A, coefficients are found by fitting equation (9) to the values of the buildup factors as estimated from the Monte Carlo calculations of proton conversion factors. The resulting coefficients are shown in Table 2. The coefficients for 100, 200 and

; I

1 1

1 Proton Energy, GeV

FIG. 2. l&m buildup factor for several depths in tissue as a function Of incident proton energy.

Table 2. Buildup factor parameters

I

II

Rem

.03* 1.00

.o

.06*

1.20

.O

.10

1.40

,020

.is*

1.50

.070

.20

1.60

,090

.30

1.70

,110

.40

1.90

.73

.o .o .o

Rad

.o

1.00

.o

.0130

1.07

.010

,0300

1.10

,040

.o .n .n .o

.ooo

,032

.0150

,010

,026

.o .o

,0385

1.12

.060

,0400

1.1s

,062

.a330

1.20

,068

,130

.o .o

.0228

1.24

,071

.o .o .o

3.40

,156

.00035

.OlSO

1.40

,090

,0001

1.2'

4.32

,167

,00145

,0130

1.67

,094

,0008

,0122

1.5

4.60

,170

,00250

. n iz o

1.80

,095

.0015

3.0

5.35

,190

.on300

.0100

2.00

,100

.0020

.01a .01m

10.0

6.20

.280

,00350

,0100

2.30

.111

.oozns

.01m

Values obtained by interpolation.

,031

.026 ,0228

10

118

PROTON-TISSUE DOSE BUILDUP FACTORS

300MeV protons were obtained using the Ar70 Armstrong T. W. and Chandler K. C., 1970, Calculation of the Absorbed Dose and Monte Carlo data of Turner et al. (Tu64). The Dose Equivalent from Neutrons and Protons in values at 400, 730, 1500 and 3000 MeV were the Energy Range from 3.5 GeV to 1 TeV. Oak obtained from the results of Alsmiller et al. Ridge National Lab, ORNL-TM-3758. (A170). The 10 GeV entry was obtained from the calculations of Armstrong and Chandler Be63 Bertini H. W., 1963, Phvs. Rev. 131, 1801. Bertini H. W., 1969, Phys. Retr. 188, 1711. (Ar70). Values noted in Table 2 by an asterisk Be69 Be71 Bertini H. W. and Guthrie hl. P.. 1971, on the corresponding energy were obtained by Nucl. Phys. A169. 670. interpolating between data points or smoothly ICRP64 International Commission on Radioextrapolating to unit buildup factor at proton biological Protection (ICRP) Publication 4, 1964 energies near the Coulomb barrier for tissue (New York: Macmillan). nuclei (4 2 MeV). The coefficients are found Ne72 Neufeld J. and Wright H. A., 1972, Health Phys. 23, 183. for all energies to 1 0 G eV by using second order Lagrange interpolation between the val- Sc70 Schaefer N. J. and Sullivan J. J., 1970, Nuclear Emulsion Recordings of the Astronauts' ues shown in Table 2. The resulting buildup Radiation Exposure on the First Lunar Landing factors are shown in Fig, 1 and 2 in compariMission Apollo XI. National Aeronautics and son to the Monte Carlo results where the Space Administration, NASA CR-115804. error bars were determined by drawing Sn69 Snyder W. S., Wright H. A., Turner J. E. smooth limiting curves so as to bracket the and Neufeld J., 1969, Nucl. Applic. 6, 336. Monte Carlo values and to follow the general Tu64 Turner J. E., Zerby C. functional dependence. These uncertainty I-., Wright H. A,, Kinney W. limits should, therefore, be interpreted as apand Neufeld J., 1964, Health Phys. 10, 783. prox 20- limits, rather than la ranges usually Wi74 Wilson J . W. and Khandelwal G. S., 1974, Nucl. Techn. 23, 298. used in expressing uncertainty limits. REFERENCES

A170 Alsmiller R. G. and Armstrong 'I'. W.. 1970, Nucl. Sci. Eizgng 42, 367. A172 Alsmillcr R. G., Santoro R. T.. Barish J . and Claiborne H. C., 1972, Shielding of Manned Space Vehicles Against Protons and Alpha Particles. Oak Ridge National Lab. ORNL-RSIC-35. Ar71 Armstrong T. W. and Bishop R . L.. 1971. Radial. Res. 47, 581.

Wr71 Wright H. A., Hamm R. N. and Turner J. A,, 197 1, Proc. int. Congr. for Protection Against Accelerator and Space Radiation, CERN 71-16, pp. 207-219. Wr69 Wright H. A,, Anderson V. E., Turner J. E., Neufeld J . and Snyder W. S., 1969, Health Phys. 16, 13. Ze65 Zerby C. D. and Kinney W. E., 1965, Nucl. Instrum. Meth. 36, 125.