Calculated dose response of Gafchromic MD55 film for 103Pd and 125I relative to 60Co

Radiation Measurements 32 (2000) 173±179

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Calculated dose response of Gafchromic MD55 ®lm for 103 Pd and 125I relative to 60Co W.V. Prestwich*, R.J. Murphy Department of Physics and Astronomy, McMaster University, Hamilton, ON, Canada, L8S 4K1 Received 23 June 1999; accepted 5 October 1999

Abstract Generalized cavity theory is used to calculate the dose±response of the radiochromic dye ®lm MD55 to the Xand g-radiations from 103Pd and 125I. The analysis indicates that for ®lm calibrated using 60Co, the correction factor giving the dose to water is 1.81 2 0.05 for 103Pd and 1.71 2 0.05 for 125I. 7 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction

2. Theory

Radiochromic dye ®lm is a dosimetric medium with high spatial resolution which may be used for imaging dose distributions. An extensive review has recently been published (Niroomand-Rad et al., 1998). In medical applications two radionuclides of particular interest for prostate brachytherapy are 125I and 103Pd. These isotopes are both low energy photon emitters. In order to interpret correctly the measurements made with any dosimeter it is necessary to know the response to the radiation spectrum to which the dosimeter is exposed. Moreover, it is necessary that the results presented correspond to the dose produced by the ®eld in a standard medium rather than in the dosimetric material itself, and in medical applications the standard is water. In this work an analysis of the dose-response of MD-55 Gafchromic ®lm based upon generalized cavity theory is presented. The response is calculated relative to the response to 60Co, a common isotopic radiation source used in medical dosimetric calibration.

Assume that the creation of active chromophores is characterized by an average energy deposition independent of radiation quality. Then if the total number density of target molecules in the ®lm is N and the number of active chromophores created by an element of dose dDF to the sensitive layer, dn, is proportional to both the dose and the number of inactivated targets

* Corresponding author.

dn ˆ a…N ÿ n† dDF

…1†

giving n ˆ N…1 ÿ eÿaDF †:

…2†

This would lead to an optical density of O…l† ˆ cN…1 ÿ eÿaDF †E…l†

…3†

where E(l ) is the extinction coecient and c is a conversion factor. The relation between ®lm dose and optical density is then

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W.V. Prestwich, R.J. Murphy / Radiation Measurements 32 (2000) 173±179

 DF ˆ aÿ1
ln

 cNE…l† : cNE…l† ÿ O…l†

…4†

This development ignores interactions between chromophores which could make the extinction coecient a function of n. Nevertheless it illustrates the point that in this model the optical density is a determinant of the dose to the ®lm, ie the energy per unit mass left in the sensitive ®lm material. Under speci®ed conditions the radiation ®eld producing DF will produce a linearly related dose to water DW. This relationship will depend upon the spectrum of the ®eld. In particular designating the parameters of the relationship for the 60Co ®eld by k(Co) and for the general ®eld by k(G), then DW …Co† ˆ k…Co†DF

1 si ˆ Ei

…5†

  N X m pi en Ei : r i iˆ1

…6†

The ®lm consists of three media, a polyester base having a mylar-like composition (Klassen, 1997), the sensitive material designated S and an adhesive layer. There are two 15 mm sensitive layers each with a 67 mm polyester base on the exterior side and a 44.5 mm adhesive layer on the interior side. The center layer between the two adhesive layers is of the polyester base and is 25 mm thick. The radiological properties of the base and adhesive layers are similar and the system is approximated as a standard cavity arrangement with the cavity consisting of the sensitive material and a surround consisting of a material with mass energy absorption coecient and stopping power equal to the average of the adhesive and base values. This e€ective homogeneous layer is designated B. The dose to the ®lm may then be written (Burlin, 1959) DF ˆ

N X si fi KB …i † ‡ …1 ÿ fi †KS …i †

SS …E, D† dE SB …E †

…8†

#

for each component i, K(i ) is the contribution to the kerma from component i in each medium, and f represents the fractional contribution from electrons created by interactions external to the cavity. In Eq. (8), R(E,Ei) is the ratio of the electron degradation spectrum to the slowing down spectrum, SS(E,D) is the restricted stopping power for the sensitive region and D is the cut-o€ energy determined as the energy of an electron with range equal to the thickness of the sensitive region. From Eqs. (5) and (7)

1 ˆ k

N X pi Ei …si fi hmen …i †=riB ‡ …1 ÿ fi †hmen …i †=riS iˆ1

:

N X pi Ei hmen …i †=riW

…9†

iˆ1

For a radiation ®eld characterized by a ¯uence F and spectrum { pi,Ei; i = 1,N } where pi is the probability of observing energy Ei the kerma produced in a given medium is KˆF

D

R…E, Ei †

SS …D† ‡ R…D, Ei † D SB …D†

DW …G† ˆ k…G†DF k…G† DW …Co†: DW …G† ˆ k…Co†

"… Ei

…7†

iˆ1

where si is the average ratio of the mass stopping powers of the sensitive material to the surround (Spencer, 1955; Nahum, 1978)

The mass energy coecients and stopping power ratios may be determined using compiled elemental values combined according to the Bragg additivity rule. The remaining quantity to be determined is f. First it should be noted that scattering of photons in the 27±35 keV range results in negligible energy transfer so the dominant contribution to the dose for 125I and 103 Pd is from the photoelectric e€ect. This is made quantitative by noting as a typical case that at 30 keV in mylar the photoelectric cross section accounts for 94% of the mass energy absorption cross section. In the calculation, 1ÿf is identi®ed as the fraction of the dose arising from electrons created in the cavity. The cavity may be considered an isotropic uniform slab source of electrons having thickness t and source density tKF, where tK is the macroscopic cross section for the K-shell photoelectric e€ect. The problem now becomes one-dimensional and can be analyzed using the plane dose kernel F(x ), de®ned as the dose produced at a distance x from a plane source with unit strength per unit area. This function must satisfy …1 ÿ1

F…x†rdx ˆ E:

…10†

For the slab source of thickness t, using a coordinate system centered in the slab, the dose at any point x can be written D…x† ˆ tK F

… t=2 ÿt=2

F…x ÿ x 0 †dx 0 ˆ tK F

… x‡t=2 xÿt=2

F…u†du

…11†

W.V. Prestwich, R.J. Murphy / Radiation Measurements 32 (2000) 173±179

175

men …E † ˆ exp…a ‡ bE ‡ cE 2 †: r

…15†

where u=xÿx'. If the scaled kernel of unit area is de®ned by g(x )=EF(x )/r then the above can be written as … x‡t=2  tK D…x† ˆ
E
F g…u†du: r xÿt=2 

…12†

The average dose to the slab is then 1 D ˆ t

… t=2 ÿt=2

D…x†dx:

…13†

Combining Eqs. (12) and (13) and identifying the factor multiplying the integral in Eq. (12) as the Kerma for the photoelectric e€ect, gives the approximation fˆ1ÿ

1 t

… t=2 … x‡t=2 ÿt=2 xÿt=2

g…u†dudx:

…14†

3. Results The composition of the media involved as used in these calculations is given in Table 1. The fraction by weight of each element is listed for the three components, the polyester base, the adhesive layer and the sensitive layer. The 125I spectrum was represented by three groups, the Ka X-rays with average energy of 27.4 keV and emission probability of 1.14, the Kb X-rays at 31.1 keV with P = 0.26 and the g-ray at 35.5 keV with P = 0.07. For the 103Pd spectrum consideration was restricted to the Ka X-rays at 20.2 keV with P = 0.57 and the Kb X-rays at 22.8 keV with P = 0.12. The weak g-rays (P < 10ÿ3) were not included. The mass energy absorption coecient for each ®lm component and water was calculated as the weighted average of the elemental values, the weighting factor being the fraction by weight. Values were calculated from the compiled elemental values at 20, 30 and 40 keV (Seltzer, 1993). In order to interpolate to the energies comprising the iodine spectra the coecients were represented by the functional form

Table 1 Composition of ®lm components Element

Polyester base

Adhesive

Sensitive layer

H C N O

0.04 0.61 0.00 0.35

0.070 0.551 0.00 0.379

0.090 0.606 0.112 0.192

The energy variations so obtained are illustrated in Fig. 1. The curves represent Eq. (15) for the sensitive layer, the polyester base and water. The values for the adhesive layer di€ered from the polyester base by less than 2% and are not shown. The points represent the compiled values from which the parameters a, b, and c of Eq. (15) were obtained. As can be seen from the ®gure there is a rapid energy variation in the region below 30 keV. The values for the strongest group in 125 I at 27.4 keV and the 103Pd Kb X-ray group are thus the most dicult to determine. The in¯uence of interpolation error on the calculation is somewhat complicated. The result depends essentially on the ratio of the values for water to that of the ®lm components. Since the interpolation procedure for each medium is performed using the same approach there will be a strong correlation between the errors which should ameliorate the situation. The calculated mass energy coecients for the 125I spectrum are listed in Table 2. The column labeled B represents the average of the polyester base and adhesive layer. At the 60Co energy the variation of the mass energy absorption coecient is only approx. 1% per 100 keV. In this case a linear interpolation was used. The ratio of the mass energy coecient for water to that for the ®lm, averaged over the base and adhesive materials is 1.053 for both the 1.17 and 1.33 MeV radiations. Stopping powers were calculated according to the ICRU Report 37 (1984) methodology using the NIST1 ESTAR program (ICRU37), and the ratio calculated according to Eq. (8). The cut-o€ energy D corresponds to 30 keV. Thus for the low energy radiations there is no distinction between the restricted and total stopping powers and the simpler Bragg±Gray expression si ˆ

1 Ei

… Ei 0

SS …E † dE SB …E †

…16†

was used. For the 125I components the ratio was calculated to be 1.057, and for 103Pd, 1.062. For 60Co the stopping power ratio was calculated from Eq. (8) for the average Compton electron energy of approx. E0=0.6 MeV. To perform the calculation the equation was re-arranged as follows. First the ratio r(E,D)=S(E,D)/S(E ) is introduced. This quantity is relatively insensitive to material and values for C taken from ICRU Report 35 (1985) were used. The values at D=30 keV were obtained by logarithmic interpretation 1 Available at Website: physics.nist.gov/PhysRefData/Star/ Text/Estar-t.html

176

W.V. Prestwich, R.J. Murphy / Radiation Measurements 32 (2000) 173±179

Fig. 1. The variation of the mass energy absorption coecients in the low energy region, for water,W, the mylar base, M, and the sensitive layer, S. The points are calculated at the tabulated energies. The curves indicate the ®tted variation used for interpolation to the required spectral energies.

of the tabulated values for D=10 and 100 keV. Substitution gives 1 s0 ˆ E0

"…

E0

D

R…E, E0 †r…E, D†

SS …E † dE SB …E †

# SS …D† ‡ R…D, E0 † D : SB …D†

E0 ˆ

D

R…E, E0 †r…E, D†dE ‡ R…D, E0 †D:

The stopping power ratio can then be written

s0 ˆ

D

SS …E † SS …D† dE ‡ R…D, E0 † D R…E, E0 †r…E, D† SB …E † SB …D† : … E0 R…E, E0 †r…E, D†dE ‡ R…D, E0 †D D

…19† …17† Table 2 Mass energy absorption coecients

Applying this equation to the trivial case where B and S are identical and leads to the requirement … E0

… E0

…18†

E (keV) hmen/riB (cm2/g) hmen/riS (cm2/g) hmen/riW (cm2/g) 20.2 22.8 27.4 31.1 35.5

0.3470 0.2390 0.1351 0.09205 0.06364

0.2914 0.2011 0.1144 0.07862 0.05512

0.5371 0.3693 0.2021 0.1394 0.09447

W.V. Prestwich, R.J. Murphy / Radiation Measurements 32 (2000) 173±179

177

Fig. 2. The plane dose kernel for 27 keV electrons. When plotted against distance as a fraction of the CSDA range the shape is relatively insensitive to energy and medium composition for the cases considered in this work.

This form is numerically more stable than Eq. (17) since R(E,E0) must also be estimated. This function was calculated by Spencer and Attix (1955) for C at E0=0.64 MeV and scaled to 0.6 Mev by taking the variation to be identical in the ratio of E/E0. A numerical evaluation gives s0=1.049, compared to 1.049 for Eq. (16). The plane kernel was calculated for 30 keV electrons and is shown in Fig. 2. The calculation was performed using the Tiger series Monte Carlo simulation. The shape when plotted against the ratio of distance x to CSDA range R was insensitive to material and energy for the 125I and 103Pd radiations. The integration in Eq. (14) was performed numerically. The values of f were found to be 0.20, 0.27 and 0.33 for the Ka, Kb and g-ray photoelectrons of 125I and 0.05 and 0.10 for the Ka and Kb photoelectrons of 103Pd. For the 60Co radiation ®eld the electron range is very much greater than the thickness of the cavity and the limit of the Bragg±Gray approximation with f = 1 is reached. From Eq. (9) the calculated values are k(I)=1.718,

k(Pd)=1.818 and k(Co)=1.003. Hence the calculated relationship between the dose to water calibration for 125 I and for 60Co is DW …I† ˆ …1:71†DW …Co†: For Pd, DW …Pd † ˆ …1:81†DW …Co†:

4. Discussion It is dicult to completely assess the uncertainty in the correction factor. The quantity depends in large measure on relative values of the radiological parameters for the di€ering media. These would be expected to be much more accurate than the absolute values, re¯ecting the relatively well-understood variations with atomic number and mass. It is possible to

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W.V. Prestwich, R.J. Murphy / Radiation Measurements 32 (2000) 173±179

examine the sensitivity with respect to f, the only absolute quantity in the expression. The response for 125I is most sensitive to this quantity so this case is examined. In order to achieve this, Eq. (9) is approximated by the equation for a single component radiation ®eld. Average mass energy coecients are introduced as weighted by the product of the energy and emission probability for each of the three components. The three fractions are also replaced by an average calculated with the same relative weighting. This leads to the numerical result 1  ˆ …0:6939†f ‡ …0:5553†…1 ÿ f† k…I† in which the average fraction is left as a variable. The weighted average is 0.214 leading to k(I)=1.716 as compared with 1.718 calculated from the correct expression. Analysis of this equation indicates that a 50% change in the average fraction induces a 2.4% change in k(I). The conventional estimate of f is based upon an assumed exponential kernel with the decay constant being determined by requiring the exponential to reach a predetermined value at the range. Values of 0.01 (Burlin, 1959) and 0.04 (Attix, 1986) have been suggested. These lead to estimates of 0.17 and 0.24 for f respectively in this instance. These values would alter the value calculated here by less than 1%. The value is similarly not very sensitive to the stopping ratios at low energies, since the external contribution in which they are involved is only about 20%. Varying the ratio by 10% induces about a 1% change in k(I). A second consideration is the implication that only the ®lm materials are involved in the 60Co calibration. In practice a layer of low Z-material is always introduced to ensure that charged particle equilibrium is introduced. In the calculation presented here it is assumed that the build up layer is that of the ®lm surround, essentially mylar. More common materials are (Lucite) polymethyl methylacrylate or polystyrene. Were either of these to entirely replace the base material k(Co) would become 1.018 and the response constants would decrease to k(I)=1.69 and k(Pd)=1.79. In reality the sensitive layers would receive a mixture of contributions from the build-up and ®lm materials. Since the average range of the Compton electrons is approx. 1.5 mm the contribution from the build-up material would dominate. It is estimated that the calculated response should be accurate to 3%. The calculated correction factor relating the dose to water calibration for 125I and for 60Co is supported by the results of McLaughlin et al. (1996). McLaughlin et al. have demonstrated that the optical density of MD55 ®lm irradiated to a given absorbed dose in water is 40% lower if the irradiation energy is on the order of

20±30 keV. The optical density increases with increasing photon energy up to approx. 100 keV where it then reaches a plateau. The calculated ratio of mass energy absorption coecients for the photon energies were found to follow the measured trend in optical density very closely. Investigations by Chiu-Tsao et al. (1994) and Sayeg et al. (1990) have also found the ®lm response to be less sensitive at lower energies. Their results also support the calculations herein with measured ®lm responses of 44 and 46% lower sensitivity for 125I radiation and 30 keV photons respectively.

5. Conclusions Generalized cavity theory may be used to calculate the relative response of a dosimeter. The response of Gafchromic ®lm to 125I based upon a 60Co calibration is calculated to be DW …I† ˆ …1:7120:05†DW …Co†: For

103

Pd the calculation gives

DW …Pd † ˆ …1:8120:05†DW …Co†: In practice, if a build-up layer not matching the base material is used for the 60Co irradiation then the above response values would need correcting. For the commonly used materials, polystyrene or lucite, the values both decrease by approx. 1% to 1.6920.05 and 1.792 0.05 respectively.

Acknowledgements The authors wish to thank the Natural Sciences and Engineering Research Council of Canada for their ®nancial support.

References Attix, F.H. 1986. Cavity theory. In: Introduction to Radiological Physics and Radiation Dosimetry. Wiley, New York, pp. 231±263. Burlin, T.E., 1959. The measurement of exposure dose for high energy radiation with cavity ionization chambers. Phys. Med. Biol. 3, 197±206. Chiu-Tsao, S., et al., 1994. High-sensitivity GafChromic ®lm dosimetry for 125I seed. Med. Phys. 21, 651±657. ICRU Report 35, 1985. Radiation dosimetry: electron beams with energies between 1 and 50 MeV. ICRU, Washington, DC. ICRU Report 37, 1984. Stopping powers for electrons and positrons. ICRU, Washington, DC.

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