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Measurements of the Fe3+ diffusion coefficient in Fricke Xylenol gel using optical density measurements
Author's Accepted Manuscript
Measurements of the Fe3 þ diffusion coefficient in Fricke Xylenol gel using optical density measurements Lucas Nonato de Oliveira, Francisco Glaildo Almeida Sampaio, Marcos Vasques Moreira, Adelaide de Almeida
www.elsevier.com/locate/apradiso
PII: DOI: Reference:
S0969-8043(14)00137-7 http://dx.doi.org/10.1016/j.apradiso.2014.04.004 ARI6659
To appear in:
Applied Radiation and Isotopes
Received date: 27 October 2013 Revised date: 2 April 2014 Accepted date: 6 April 2014 Cite this article as: Lucas Nonato de Oliveira, Francisco Glaildo Almeida Sampaio, Marcos Vasques Moreira, Adelaide de Almeida, Measurements of the Fe3 þ diffusion coefficient in Fricke Xylenol gel using optical density measurements, Applied Radiation and Isotopes, http://dx.doi.org/10.1016/j.apradiso.2014.04.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Measurements of the Fe3+ diffusion coefficient in Fricke Xylenol gel using optical density measurements
Lucas Nonato de Oliveira1*, Francisco Glaildo Almeida Sampaio2, Marcos Vasques Moreira3, Adelaide de Almeida2 1
Instituto Federal de Educação, Ciência e Tecnologia de Goiás-IFG, 75400-000, Inhumas, GO, Brazil
2
Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto-FFCLRP – Universidade de São Paulo-USP, 14040-
901, Ribeirão Preto, SP, Brazil 3
Instituto de Radioterapia e Megavoltagem de Ribeirão Preto-IRMEV, 14010-180, Ribeirão Preto, SP, Brazil
1*
Corresponding author: Instituto Federal de Educação, Ciência e Tecnologia de Goiás-IFG,
Departamento de Áreas Acadêmicas, Av. Universitária, s/n, CEP 75400-000, Vale das Goiabeiras, Inhumas, GO, Brazil.
Phone: +55-62-3514-9500 Fax: +55-62-3514-9595 E-mail address: [email protected] (Lucas Nonato de Oliveira, Ph.D)
Abstract In Fricke dosimetry, optical density measurements are made some time after dosimeter irradiation. Values of the diffusion coefficient of Fe3+ in Fricke Xylenol gel (FXG) are necessary for determining the spatial distribution of the absorbed dose from measurements of the optical density. Five sets of FXG dosimeters, kept at different constant temperatures, were exposed to collimated 6 MV photons. The optical density profile, proportional to the Fe3+ concentration, at the boundary between the irradiated and non irradiated parts of each dosimeter was measured periodically over a period of 60 h. By comparing the experimental data with a function that
1
accounts for the unobserved initial concentration profile of Fe3+ in the FXG, we obtained diffusion coefficients 0.30 ± 0.05, 0.40 ± 0.05, 0.50 ± 0.05, 0.60 ± 0.05 and 0.80 ± 0.05 mm2/h for the temperatures 283.0 ± 0.5, 286.0 ± 0.5, 289.0 ± 0.5, 292.0 ± 0.5, and 296.0 ± 0.5 K, respectively. The activation energy of Fe3+ diffusion in the gel, 0.54 ± 0.06 eV, was determined from the temperature dependence of the diffusion coefficients.
1. Introduction We have studied the Fricke Xylenol gel (FXG) dosimeter with Xylenol Orange (XO) dye and swine skin gelatin. Previous papers describe its preparation technique (Bero et al., 2000), and the temperature dependence of the response, and some of the features for measuring its optical density (Caldeira et al., 2007). The optical density of the FXG dosimeter at 585 nm depends linearly on the concentration of the Fe3+ ions produced by the oxidation of Fe2+, which is proportional to the absorbed dose of ionizing radiation (Gambarini et al., 2004; Kelly et al., 1998; Saur et al., 2005). We determine the optical density at 585 nm with a scanner previously described (Felipe, 2003) and used satisfactorily also for applications of FXG dosimetry to 2D and 3D Quality Assurance (QA) measurements in radiation therapy (Calcina et al., 2007; Moreira et al., 2004). When a FXG dosimeter is used to quantify a nonuniform radiation exposure, diffusion of the Fe3+ causes gradients of concentration to decrease with time, degrading the spatial resolution. Knowledge of the diffusion coefficient is necessary to infer the radiation dose profile from an optical density profile observed some time after irradiation. Examination of the temporal dependence of the spatial distribution of a one dimensional concentration of the Fe3+-XO complex C(x,t) reveals the temperature dependent Fe+3 diffusion coefficient D in the material in accordance with Fick’s second law (Olsson et al., 1992): 2
∂2 ∂ C ( x , t ) = D 2 C ( x, t ) ∂x ∂t
(1)
To diffuse in the gel, Fe3+ ions need to overcome an energetic barrier (activation energy) (Oliveira et al., 2009) whose value may be determined by measuring the temperature dependence of the D. The diffusion coefficient of Fe3+ in gels has been extensively investigated, with magnetic resonance imaging ((Kron et al., 1997; Rae et al., 1996, Pasquale et al., 2006) and with other methods (Schulz et al., 1990; Oliveira et al., 2009; Tseng et al., 2005). The goal of this work is to describe an alternative method, not yet reported in the literature, to determine the Fe3+ diffusion coefficient in a gel and its temperature dependence. The solution of Equation 1 for an initial abrupt discontinuity in concentration is an error function that reveals the diffusion coefficient.
The next section describes our method of
observing the Fe3+ diffusion across a less than ideally abrupt boundary between irradiated and non irradiated portions of individual FXG dosimeters.
2. Materials and methods Thirty acrylic cuvettes with internal dimensions 1.0 × 1.0 cm square and 3.5 cm long were filled with FXG. Fifteen of them served as non irradiated reference dosimeters. The other fifteen were exposed, three at a time, half the length protected by an absorber as shown in Figure 1, in a uniform field of 6 MV photons (Siemens/Mevatron/6MD; SSD = 100 cm; field size 10 × 10 cm2 with a 1.5 cm acrylic buildup plate) to 9.5 gray.
3
Five groups of three irradiated together with three non irradiated dosimeters were isolated and placed in five separate temperature controlled environments (283.0 ± 0.5, 286.0 ± 0.5, 289.0 ± 0.5, 292.0 ± 0.5 and 296.0 ± 0.5 K). For each group the 585 nm light intensity transmitted I ( x , t ) in the radiation direction was measured at 1 mm intervals at values of x along the 35 mm
dosimeter length. The intensity I o ( x , t ) transmitted through the three non irradiated dosimeters was also measured. Extracting the logarithm of the ratio of the average of each three intensity measurements effectively yields the increment of the optical density OD owing to Fe3+ produced by irradiation.
⎛ I ( x, t ) ⎞ ⎟⎟ OD( x, t ) = log10 ⎜⎜ ⎝ I o ( x, t ) ⎠
(2)
Measurements of the dosimeters in each group were repeated at appropriate later times t. This procedure also eliminates the effect of the time-dependent natural oxidation of Fe2+ to Fe3+. Manipulation of the dosimeters during the measurements did not change their temperatures significantly. The transmitted light intensity measurements were made with a light-emitting diode and a photodiode sensor, each chosen for optimum performance near 585 nm. Computer control of the dosimeter movement and the data collection reduced the acquisition time for a complete profile of the light transmitted through each dosimeter. Figure 2 shows optical density profiles of dosimeters held at 283 K and 296 K.
3. Analysis The non uniformly irradiated samples yield optical density profiles, representative of the Fe+3-XO concentration profiles. The concentration profiles are well represented at position x and
4
time t after irradiation by the error function solution (Feynman et al.1994; Jefrey and Dai 1995) to Equation 1: ⎛ x − xo ⎞ ⎟ OD ( x, t ) or C ( x, t ) = Co erfc⎜ ⎜ 2 (t + t ' ) D ⎟ ⎝ ⎠
(3)
where C0 is a virtual initial step function concentration at xo. C (x, t ) is the concentration at time t.
The inflection point xo and the initial concentration Co are chosen to conform to the
experimental optical density profiles as shown in Figures 2b and 2c. We wish to determine the diffusion coefficient D.
The additional time t’ accounts for the delay in beginning timed
measurements after irradiation. However, it does more than that. The edge of the irradiated half of the dosimeter is almost an ideal step function, but necessarily has instead a penumbra whose dimension is determined by the photon beam and absorber geometry. Thus t’ is an empirical time elapsed since a “virtual initial condition” step function. A step function would diffuse in a time t’ to the first observed profile that defines t = 0. This artifice eliminates the effect of the penumbra and the effect of diffusion during irradiation and before equilibrating at the desired temperature. Thus, both D and t' are parameters to be determined from the experimental OD(x.t) profiles which also obey Equation 3. The time t’ is chosen for a set of profiles, such that the diffusion constant D is independent of the time t when a single profile is measured. Our analysis of the time evolution of the optical density profiles reveals accurate values of the diffusion constants for Fe3+ in FXG at five temperatures.
Using the experimentally
determined diffusion constants D1, D2, D3, D4 and D5 for the dosimeters maintained at temperatures T1, T2, T3, T4 and T5, respectively, we may calculate the activation energy E for diffusion of Fe3+ ions in the FXG matrix, using the Arrhenius equation (Jones and Atkins, 1999): E D(T ) = Do exp kT −
(4)
5
where k is the Boltzmann constant (8.617 × 10-5 eV K-1). Figure 4 is a graph of the logarithm of the diffusion coefficients D as a function of the inverse of the temperatures 1/T of the dosimeters. The slope of this line gives –E/k, from which we extract the value of the activation energy E.
4. Results and discussion Figure 2a shows a representative image of an irradiated FXG dosimeter. Typical optical density measurements in Figures 2b and 2c show how the Fe+3-XO concentration varies at each coordinate along the length of the dosimeter owing to diffusion. Figure 2b shows a comparison of the optical density for the dosimeters kept at 296 K at times t = 1h and t = 6 h compared with Equation 3 using values of D = 0.80 mm2/h and t’ =4.6 h. Similarly, Figure 2c is the optical density for dosimeters kept at 283 K at times t = 10 h and t = 60 h compared with Equation 3 with values D = 0.30 mm2/h and t’ = 50 h. The high value of t’ for the dosimeters measured at 283 K is because the initial virtual step function concentration would have to diffuse a long time at 283 K to agree with our first measurement at t = 0. If it were possible to have kept the dosimeters at 283 K during and after irradiation, t’ would be much smaller. Figure 3 shows diffusion coefficients D chosen to fit Equation 3 to the incremental optical density measurements OD for the dosimeters shown in Figure 2 as a function of the time t of measurement. The open symbols show diffusion coefficient values that approach the correct value only if t >> t’, demonstrating an inadequate choice of t’ (for illustration, we chose t’ = 0). The closed symbols show constant, and accurate values of the diffusion coefficients when t’ is chosen empirically to account properly for the profiles and diffusion prior to initiating the timed sequence of measurements. 6
Table 1 shows our diffusion coefficients determined from experimental optical density profiles of partially irradiated FXG dosimeters held at five temperatures between 283 K and 296 K. Figure 4 is an Arrhenius plot of the values in Table 1. In accordance with Equation 4, the slope of the plot yields an activation energy E = 0.54 ± 0.06 eV. This energy is related to the barrier for diffusion of the Fe3+ ions along a concentration gradient in the gel matrix (Oliveira et al., 2009). 5. Conclusions Some authors determine the coefficient of diffusion by making measurements after a long waiting period in order to minimize effects of the initial conditions (Balcom et al., 1995; Olsson et al., 1992; Pasquale et al., 2006). We have determined accurate diffusion coefficients with optical density measurements in FXG dosimeters with an analytic technique that uses an ideal concentration step function as a virtual initial condition (Oliveira et al., 2009). The photon penumbra produced by the electron beam spot size and the collimator and the effects of diffusion prior to beginning measurements are accounted for in our analysis. Careful temperature control and accurate diffusion constants allow better estimates of non uniform radiation fields in 2D and 3D FXG dosimeters when measurements cannot be made immediately after irradiation. Our report of an accurate value of the activation energy E = 0.54 ± 0.06 eV for Fe3+ ion diffusion in a gel matrix, provides renewed incentive to develop a matrix that binds more tightly the Fe3+ and a more thermally stable Fricke dosimeter.
7
Acknowledgments The present work was supported by grants from the Brazilian funding agencies CAPES (the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) and FAPEG (the Fundação de Amparo à Pesquisa do Estado de Goiás).
References Balcom, B.J., Lees, T.J., Sharp, A.R., Kulkarni, N.S. and Wagner, G.S., 1995. Diffusion in Fe(II/III) radiation dosimetry gels measured by magnetic resonance imaging Phys. Med. Biol. 40 1665-76. Bero, M.A., Gilboy, W.B., Glover, P.M., El-masri, H.M., 2000. Tissue-equivalent gel for noninvasive spatial radiation dose measurements. Nucl. Instr. Meth. Phys. Res. B 166-167, 820-825. Calcina, C.S., Oliveira, L.N., Almeida, C.E., Almeida, A., 2007. Dosimetric parameters for small field sizes using Fricke xylenol gel, thermoluminescent and film dosimeters, and an ionization chamber. Phys. Med. Biol. 52, 1431-1439. Caldeira, A.M., Neto, A.M., Bento, A.C., Baesso, M.L., Silva, M.A., Almeida, A., 2007. Behavior of oxidation in the radiochromic gel dosimeter through photoacoustic technique measurements. Appl. Radiat. Isot. 65, 605-609. Felipe, D.S., 2003. Development, test and application of a prototype reader for FXG dosimetry. Master Thesis - Universidade de São Paulo, FFCLRP/DFM (in Portuguese). Feynman, R.P., Leighton, R., Sands, M., 1994. Feynman Lectures on Physics, Vol 3. AddisonWesley, Boston. Gambarini, G., Birattaria, C., Marianic, M., Marchesini, R., Pirola, L., Prestini, P., Sella, M., Tomatis, S., 2004. Study of light transmittance from layers of Fricke-xylenol-orange-gel dosimeters. Nucl. Instr. Meth. Phys. Res. B 213, 321–324. Jefrey, A., Dai, H., 1995. Handbook of Mathematical Formulas and Integrals. Academic Press, San Diego. Jones, J., Atkins, P., 1999. Chemistry: Molecules, Matter and Change. W H Freeman, New York. Kelly, R.G., Jordan, K.J., Battista, J. J., 1998. Optical CT reconstruction of 3D dose distributions using the ferrous-benzoic-xylenol (FBX) gel dosimeter. Med. Phys. 25, 1741-1750. Kron, T., Jonas, D., Pope, J.M., 1997. Fast T1 imaging of dual gel samples for diffusion measurements in NMR dosimetry gels. Magn. Reson. Imag. 15, 211-221. Moreira, M.V., Almeida, A., Costa, R.T., Perles, L.A., 2004. FXG mass attenuation coefficient evaluation for radiotherapy routine. J. Phys. D: Appl. Phys. 3 146-149. Oliveira, L.N., Zimmerman, R. L., Moreira, M.V., Ila, D., Almeida, A., 2009. Determination of diffusion coefficient in Fricke Xylenol gel dosimeter after electron beam bombardment. Surf. Coat. Technol. 203, 2367-2369. Olsson, L.E., Westrin, B.A., Fransson, A., Nordell, B., 1992. Diffusion of ferric ions in agarose dosimeter gels. Phys. Med. Biol. 37, 2243-2252. Pasquale, F., Barone, P., Sebastiani, G., d'Errico, F., Egger, E., Luciani, A. M., Pacilio, M, 8
Guidoni, L. Viti. V., 2006. Ion diffusion modelling of Fricke-agarose dosimeter gels. Rad. Prot.Dosim. 120, 151-154. Rae, W.I., Willemse, C.A., Lotter, M.G., Engelbrecht, J.S., Swarts, J.C., 1996. Chelator effect on ion diffusion in ferrous-sulfate-doped gelatin gel dosimeters as analyzed by MRI. Med. Phys. 23, 15-23 Saur, S., Strickert, T., Wasboe, E., Frengen, J., 2005. Fricke gel as a tool for dose distribution verification: optimization and characterization. Phys. Med. Biol. 50, 5251-5261. Schulz, R.J., de Guzman, A.F., Nguyen, D.B., Gore, J.C., 1990. Dose-response curves for Frickeinfused agarose gels as obtained by nuclear magnetic resonance. Phys. Med. Biol. 35, 1611-1622. Tseng, Y.J., Huang, S.C., Chu, W.C., 2005. A least-squares error minimization approach in the determination of ferric ion diffusion coefficient of Fricke-infused dosimeter gels. Med. Phys., 32, 1017-1023.
9
Figure Captions Figure 1. Schematic of the arrangement used for the simultaneous irradiation of three FXG dosimeters with 6 MV photons. An absorber blocked the beam from half of each dosimeter. Figure 2. a) Optical image of an irradiated dosimeter. Experimental optical density profiles of irradiated FXG dosimeters held at 296 K b) and 283 K c) compared with error function solutions of the diffusion equation. Figure 3. Diffusion coefficients obtained from the experimental data shown in Figs. 2 b and c. The upper points (open symbols) were obtained without properly accounting for the profile of the Fe3+ concentration when measurements started. The diffusion coefficient determined with the lower points (solid symbols), were obtained using time measured from a step function virtual initial condition. Figure 4. Dependence of the logarithm of the diffusion coefficient versus inverse of the FXG gel temperature. The slope yields the activation energy of the Fe3+ ions in the gel matrix.
10
Figures:
Figure 1.
11
a)
b)
c)
Figure 2.
12
Figure 3.
13
Figure 4.
14
Table 1 Diffusion constants determined at each temperature. Temperature (K)
Diffusion constant (mm2/h)
283.0 ± 0.5
0.30 ± 0.05
286.0 ± 0.5
0.40 ± 0.05
289.0 ± 0.5
0.50 ± 0.05
292.0 ± 0.5
0.60 ± 0.05
296.0 ± 0.5
0.80 ± 0.05
Highlights - A new analytical method to determine diffusion coefficients of ions in gels is proposed. - The method is applied to measurements of the diffusion coefficients of Fe3+ ions in a Fricke gel dosimeter. - Activation energy of the Fe3+ ions in the gel was found to be 0.54 ± 0.06 eV.
15
Keywords Fricke xylenol gel; Diffusion coeficiente; Optical density
Measurements of the Fe3 þ diffusion coefficient in Fricke Xylenol gel using optical density measurements Lucas Nonato de Oliveira, Francisco Glaildo Almeida Sampaio, Marcos Vasques Moreira, Adelaide de Almeida
www.elsevier.com/locate/apradiso
PII: DOI: Reference:
S0969-8043(14)00137-7 http://dx.doi.org/10.1016/j.apradiso.2014.04.004 ARI6659
To appear in:
Applied Radiation and Isotopes
Received date: 27 October 2013 Revised date: 2 April 2014 Accepted date: 6 April 2014 Cite this article as: Lucas Nonato de Oliveira, Francisco Glaildo Almeida Sampaio, Marcos Vasques Moreira, Adelaide de Almeida, Measurements of the Fe3 þ diffusion coefficient in Fricke Xylenol gel using optical density measurements, Applied Radiation and Isotopes, http://dx.doi.org/10.1016/j.apradiso.2014.04.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Measurements of the Fe3+ diffusion coefficient in Fricke Xylenol gel using optical density measurements
Lucas Nonato de Oliveira1*, Francisco Glaildo Almeida Sampaio2, Marcos Vasques Moreira3, Adelaide de Almeida2 1
Instituto Federal de Educação, Ciência e Tecnologia de Goiás-IFG, 75400-000, Inhumas, GO, Brazil
2
Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto-FFCLRP – Universidade de São Paulo-USP, 14040-
901, Ribeirão Preto, SP, Brazil 3
Instituto de Radioterapia e Megavoltagem de Ribeirão Preto-IRMEV, 14010-180, Ribeirão Preto, SP, Brazil
1*
Corresponding author: Instituto Federal de Educação, Ciência e Tecnologia de Goiás-IFG,
Departamento de Áreas Acadêmicas, Av. Universitária, s/n, CEP 75400-000, Vale das Goiabeiras, Inhumas, GO, Brazil.
Phone: +55-62-3514-9500 Fax: +55-62-3514-9595 E-mail address: [email protected] (Lucas Nonato de Oliveira, Ph.D)
Abstract In Fricke dosimetry, optical density measurements are made some time after dosimeter irradiation. Values of the diffusion coefficient of Fe3+ in Fricke Xylenol gel (FXG) are necessary for determining the spatial distribution of the absorbed dose from measurements of the optical density. Five sets of FXG dosimeters, kept at different constant temperatures, were exposed to collimated 6 MV photons. The optical density profile, proportional to the Fe3+ concentration, at the boundary between the irradiated and non irradiated parts of each dosimeter was measured periodically over a period of 60 h. By comparing the experimental data with a function that
1
accounts for the unobserved initial concentration profile of Fe3+ in the FXG, we obtained diffusion coefficients 0.30 ± 0.05, 0.40 ± 0.05, 0.50 ± 0.05, 0.60 ± 0.05 and 0.80 ± 0.05 mm2/h for the temperatures 283.0 ± 0.5, 286.0 ± 0.5, 289.0 ± 0.5, 292.0 ± 0.5, and 296.0 ± 0.5 K, respectively. The activation energy of Fe3+ diffusion in the gel, 0.54 ± 0.06 eV, was determined from the temperature dependence of the diffusion coefficients.
1. Introduction We have studied the Fricke Xylenol gel (FXG) dosimeter with Xylenol Orange (XO) dye and swine skin gelatin. Previous papers describe its preparation technique (Bero et al., 2000), and the temperature dependence of the response, and some of the features for measuring its optical density (Caldeira et al., 2007). The optical density of the FXG dosimeter at 585 nm depends linearly on the concentration of the Fe3+ ions produced by the oxidation of Fe2+, which is proportional to the absorbed dose of ionizing radiation (Gambarini et al., 2004; Kelly et al., 1998; Saur et al., 2005). We determine the optical density at 585 nm with a scanner previously described (Felipe, 2003) and used satisfactorily also for applications of FXG dosimetry to 2D and 3D Quality Assurance (QA) measurements in radiation therapy (Calcina et al., 2007; Moreira et al., 2004). When a FXG dosimeter is used to quantify a nonuniform radiation exposure, diffusion of the Fe3+ causes gradients of concentration to decrease with time, degrading the spatial resolution. Knowledge of the diffusion coefficient is necessary to infer the radiation dose profile from an optical density profile observed some time after irradiation. Examination of the temporal dependence of the spatial distribution of a one dimensional concentration of the Fe3+-XO complex C(x,t) reveals the temperature dependent Fe+3 diffusion coefficient D in the material in accordance with Fick’s second law (Olsson et al., 1992): 2
∂2 ∂ C ( x , t ) = D 2 C ( x, t ) ∂x ∂t
(1)
To diffuse in the gel, Fe3+ ions need to overcome an energetic barrier (activation energy) (Oliveira et al., 2009) whose value may be determined by measuring the temperature dependence of the D. The diffusion coefficient of Fe3+ in gels has been extensively investigated, with magnetic resonance imaging ((Kron et al., 1997; Rae et al., 1996, Pasquale et al., 2006) and with other methods (Schulz et al., 1990; Oliveira et al., 2009; Tseng et al., 2005). The goal of this work is to describe an alternative method, not yet reported in the literature, to determine the Fe3+ diffusion coefficient in a gel and its temperature dependence. The solution of Equation 1 for an initial abrupt discontinuity in concentration is an error function that reveals the diffusion coefficient.
The next section describes our method of
observing the Fe3+ diffusion across a less than ideally abrupt boundary between irradiated and non irradiated portions of individual FXG dosimeters.
2. Materials and methods Thirty acrylic cuvettes with internal dimensions 1.0 × 1.0 cm square and 3.5 cm long were filled with FXG. Fifteen of them served as non irradiated reference dosimeters. The other fifteen were exposed, three at a time, half the length protected by an absorber as shown in Figure 1, in a uniform field of 6 MV photons (Siemens/Mevatron/6MD; SSD = 100 cm; field size 10 × 10 cm2 with a 1.5 cm acrylic buildup plate) to 9.5 gray.
3
Five groups of three irradiated together with three non irradiated dosimeters were isolated and placed in five separate temperature controlled environments (283.0 ± 0.5, 286.0 ± 0.5, 289.0 ± 0.5, 292.0 ± 0.5 and 296.0 ± 0.5 K). For each group the 585 nm light intensity transmitted I ( x , t ) in the radiation direction was measured at 1 mm intervals at values of x along the 35 mm
dosimeter length. The intensity I o ( x , t ) transmitted through the three non irradiated dosimeters was also measured. Extracting the logarithm of the ratio of the average of each three intensity measurements effectively yields the increment of the optical density OD owing to Fe3+ produced by irradiation.
⎛ I ( x, t ) ⎞ ⎟⎟ OD( x, t ) = log10 ⎜⎜ ⎝ I o ( x, t ) ⎠
(2)
Measurements of the dosimeters in each group were repeated at appropriate later times t. This procedure also eliminates the effect of the time-dependent natural oxidation of Fe2+ to Fe3+. Manipulation of the dosimeters during the measurements did not change their temperatures significantly. The transmitted light intensity measurements were made with a light-emitting diode and a photodiode sensor, each chosen for optimum performance near 585 nm. Computer control of the dosimeter movement and the data collection reduced the acquisition time for a complete profile of the light transmitted through each dosimeter. Figure 2 shows optical density profiles of dosimeters held at 283 K and 296 K.
3. Analysis The non uniformly irradiated samples yield optical density profiles, representative of the Fe+3-XO concentration profiles. The concentration profiles are well represented at position x and
4
time t after irradiation by the error function solution (Feynman et al.1994; Jefrey and Dai 1995) to Equation 1: ⎛ x − xo ⎞ ⎟ OD ( x, t ) or C ( x, t ) = Co erfc⎜ ⎜ 2 (t + t ' ) D ⎟ ⎝ ⎠
(3)
where C0 is a virtual initial step function concentration at xo. C (x, t ) is the concentration at time t.
The inflection point xo and the initial concentration Co are chosen to conform to the
experimental optical density profiles as shown in Figures 2b and 2c. We wish to determine the diffusion coefficient D.
The additional time t’ accounts for the delay in beginning timed
measurements after irradiation. However, it does more than that. The edge of the irradiated half of the dosimeter is almost an ideal step function, but necessarily has instead a penumbra whose dimension is determined by the photon beam and absorber geometry. Thus t’ is an empirical time elapsed since a “virtual initial condition” step function. A step function would diffuse in a time t’ to the first observed profile that defines t = 0. This artifice eliminates the effect of the penumbra and the effect of diffusion during irradiation and before equilibrating at the desired temperature. Thus, both D and t' are parameters to be determined from the experimental OD(x.t) profiles which also obey Equation 3. The time t’ is chosen for a set of profiles, such that the diffusion constant D is independent of the time t when a single profile is measured. Our analysis of the time evolution of the optical density profiles reveals accurate values of the diffusion constants for Fe3+ in FXG at five temperatures.
Using the experimentally
determined diffusion constants D1, D2, D3, D4 and D5 for the dosimeters maintained at temperatures T1, T2, T3, T4 and T5, respectively, we may calculate the activation energy E for diffusion of Fe3+ ions in the FXG matrix, using the Arrhenius equation (Jones and Atkins, 1999): E D(T ) = Do exp kT −
(4)
5
where k is the Boltzmann constant (8.617 × 10-5 eV K-1). Figure 4 is a graph of the logarithm of the diffusion coefficients D as a function of the inverse of the temperatures 1/T of the dosimeters. The slope of this line gives –E/k, from which we extract the value of the activation energy E.
4. Results and discussion Figure 2a shows a representative image of an irradiated FXG dosimeter. Typical optical density measurements in Figures 2b and 2c show how the Fe+3-XO concentration varies at each coordinate along the length of the dosimeter owing to diffusion. Figure 2b shows a comparison of the optical density for the dosimeters kept at 296 K at times t = 1h and t = 6 h compared with Equation 3 using values of D = 0.80 mm2/h and t’ =4.6 h. Similarly, Figure 2c is the optical density for dosimeters kept at 283 K at times t = 10 h and t = 60 h compared with Equation 3 with values D = 0.30 mm2/h and t’ = 50 h. The high value of t’ for the dosimeters measured at 283 K is because the initial virtual step function concentration would have to diffuse a long time at 283 K to agree with our first measurement at t = 0. If it were possible to have kept the dosimeters at 283 K during and after irradiation, t’ would be much smaller. Figure 3 shows diffusion coefficients D chosen to fit Equation 3 to the incremental optical density measurements OD for the dosimeters shown in Figure 2 as a function of the time t of measurement. The open symbols show diffusion coefficient values that approach the correct value only if t >> t’, demonstrating an inadequate choice of t’ (for illustration, we chose t’ = 0). The closed symbols show constant, and accurate values of the diffusion coefficients when t’ is chosen empirically to account properly for the profiles and diffusion prior to initiating the timed sequence of measurements. 6
Table 1 shows our diffusion coefficients determined from experimental optical density profiles of partially irradiated FXG dosimeters held at five temperatures between 283 K and 296 K. Figure 4 is an Arrhenius plot of the values in Table 1. In accordance with Equation 4, the slope of the plot yields an activation energy E = 0.54 ± 0.06 eV. This energy is related to the barrier for diffusion of the Fe3+ ions along a concentration gradient in the gel matrix (Oliveira et al., 2009). 5. Conclusions Some authors determine the coefficient of diffusion by making measurements after a long waiting period in order to minimize effects of the initial conditions (Balcom et al., 1995; Olsson et al., 1992; Pasquale et al., 2006). We have determined accurate diffusion coefficients with optical density measurements in FXG dosimeters with an analytic technique that uses an ideal concentration step function as a virtual initial condition (Oliveira et al., 2009). The photon penumbra produced by the electron beam spot size and the collimator and the effects of diffusion prior to beginning measurements are accounted for in our analysis. Careful temperature control and accurate diffusion constants allow better estimates of non uniform radiation fields in 2D and 3D FXG dosimeters when measurements cannot be made immediately after irradiation. Our report of an accurate value of the activation energy E = 0.54 ± 0.06 eV for Fe3+ ion diffusion in a gel matrix, provides renewed incentive to develop a matrix that binds more tightly the Fe3+ and a more thermally stable Fricke dosimeter.
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Acknowledgments The present work was supported by grants from the Brazilian funding agencies CAPES (the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) and FAPEG (the Fundação de Amparo à Pesquisa do Estado de Goiás).
References Balcom, B.J., Lees, T.J., Sharp, A.R., Kulkarni, N.S. and Wagner, G.S., 1995. Diffusion in Fe(II/III) radiation dosimetry gels measured by magnetic resonance imaging Phys. Med. Biol. 40 1665-76. Bero, M.A., Gilboy, W.B., Glover, P.M., El-masri, H.M., 2000. Tissue-equivalent gel for noninvasive spatial radiation dose measurements. Nucl. Instr. Meth. Phys. Res. B 166-167, 820-825. Calcina, C.S., Oliveira, L.N., Almeida, C.E., Almeida, A., 2007. Dosimetric parameters for small field sizes using Fricke xylenol gel, thermoluminescent and film dosimeters, and an ionization chamber. Phys. Med. Biol. 52, 1431-1439. Caldeira, A.M., Neto, A.M., Bento, A.C., Baesso, M.L., Silva, M.A., Almeida, A., 2007. Behavior of oxidation in the radiochromic gel dosimeter through photoacoustic technique measurements. Appl. Radiat. Isot. 65, 605-609. Felipe, D.S., 2003. Development, test and application of a prototype reader for FXG dosimetry. Master Thesis - Universidade de São Paulo, FFCLRP/DFM (in Portuguese). Feynman, R.P., Leighton, R., Sands, M., 1994. Feynman Lectures on Physics, Vol 3. AddisonWesley, Boston. Gambarini, G., Birattaria, C., Marianic, M., Marchesini, R., Pirola, L., Prestini, P., Sella, M., Tomatis, S., 2004. Study of light transmittance from layers of Fricke-xylenol-orange-gel dosimeters. Nucl. Instr. Meth. Phys. Res. B 213, 321–324. Jefrey, A., Dai, H., 1995. Handbook of Mathematical Formulas and Integrals. Academic Press, San Diego. Jones, J., Atkins, P., 1999. Chemistry: Molecules, Matter and Change. W H Freeman, New York. Kelly, R.G., Jordan, K.J., Battista, J. J., 1998. Optical CT reconstruction of 3D dose distributions using the ferrous-benzoic-xylenol (FBX) gel dosimeter. Med. Phys. 25, 1741-1750. Kron, T., Jonas, D., Pope, J.M., 1997. Fast T1 imaging of dual gel samples for diffusion measurements in NMR dosimetry gels. Magn. Reson. Imag. 15, 211-221. Moreira, M.V., Almeida, A., Costa, R.T., Perles, L.A., 2004. FXG mass attenuation coefficient evaluation for radiotherapy routine. J. Phys. D: Appl. Phys. 3 146-149. Oliveira, L.N., Zimmerman, R. L., Moreira, M.V., Ila, D., Almeida, A., 2009. Determination of diffusion coefficient in Fricke Xylenol gel dosimeter after electron beam bombardment. Surf. Coat. Technol. 203, 2367-2369. Olsson, L.E., Westrin, B.A., Fransson, A., Nordell, B., 1992. Diffusion of ferric ions in agarose dosimeter gels. Phys. Med. Biol. 37, 2243-2252. Pasquale, F., Barone, P., Sebastiani, G., d'Errico, F., Egger, E., Luciani, A. M., Pacilio, M, 8
Guidoni, L. Viti. V., 2006. Ion diffusion modelling of Fricke-agarose dosimeter gels. Rad. Prot.Dosim. 120, 151-154. Rae, W.I., Willemse, C.A., Lotter, M.G., Engelbrecht, J.S., Swarts, J.C., 1996. Chelator effect on ion diffusion in ferrous-sulfate-doped gelatin gel dosimeters as analyzed by MRI. Med. Phys. 23, 15-23 Saur, S., Strickert, T., Wasboe, E., Frengen, J., 2005. Fricke gel as a tool for dose distribution verification: optimization and characterization. Phys. Med. Biol. 50, 5251-5261. Schulz, R.J., de Guzman, A.F., Nguyen, D.B., Gore, J.C., 1990. Dose-response curves for Frickeinfused agarose gels as obtained by nuclear magnetic resonance. Phys. Med. Biol. 35, 1611-1622. Tseng, Y.J., Huang, S.C., Chu, W.C., 2005. A least-squares error minimization approach in the determination of ferric ion diffusion coefficient of Fricke-infused dosimeter gels. Med. Phys., 32, 1017-1023.
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Figure Captions Figure 1. Schematic of the arrangement used for the simultaneous irradiation of three FXG dosimeters with 6 MV photons. An absorber blocked the beam from half of each dosimeter. Figure 2. a) Optical image of an irradiated dosimeter. Experimental optical density profiles of irradiated FXG dosimeters held at 296 K b) and 283 K c) compared with error function solutions of the diffusion equation. Figure 3. Diffusion coefficients obtained from the experimental data shown in Figs. 2 b and c. The upper points (open symbols) were obtained without properly accounting for the profile of the Fe3+ concentration when measurements started. The diffusion coefficient determined with the lower points (solid symbols), were obtained using time measured from a step function virtual initial condition. Figure 4. Dependence of the logarithm of the diffusion coefficient versus inverse of the FXG gel temperature. The slope yields the activation energy of the Fe3+ ions in the gel matrix.
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Figures:
Figure 1.
11
a)
b)
c)
Figure 2.
12
Figure 3.
13
Figure 4.
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Table 1 Diffusion constants determined at each temperature. Temperature (K)
Diffusion constant (mm2/h)
283.0 ± 0.5
0.30 ± 0.05
286.0 ± 0.5
0.40 ± 0.05
289.0 ± 0.5
0.50 ± 0.05
292.0 ± 0.5
0.60 ± 0.05
296.0 ± 0.5
0.80 ± 0.05
Highlights - A new analytical method to determine diffusion coefficients of ions in gels is proposed. - The method is applied to measurements of the diffusion coefficients of Fe3+ ions in a Fricke gel dosimeter. - Activation energy of the Fe3+ ions in the gel was found to be 0.54 ± 0.06 eV.
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Keywords Fricke xylenol gel; Diffusion coeficiente; Optical density