Modelling of post-irradiation events in polymer gel dosimeters

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Phys. Med. Biol. 46 (2001) 28272839 PII: S0031-9155(01)22521-4

Modelling of post-irradiation events in polymergel dosimeters

M Lepage1,5, A K Whittaker2, L Rintoul3, S A J Back1,4and C Baldock1,6

1 Centre for Medical, Health and Environmental Physics, Queensland University of Technology,GPO Box 2434, Brisbane, Qld 4001, Australia2 Centre for Magnetic Resonance, University of Queensland, Brisbane Qld 4072, Australia3 Centre for Instrumental and Developmental Chemistry, Queensland University of Technology,GPO Box 2434, Brisbane Qld 4001, Australia4 Department of Radiation Physics, Lund University, Malmo University Hospital, SE-205 02Malmo, Sweden

E-mail: c.baldock@qut.edu.au

Received 5 March 2001, in final form 6 August 2001Published 5 October 2001Online at stacks.iop.org/PMB/46/2827

AbstractThe nuclear magnetic resonance (NMR) spinspin relaxation time (T2) isrelated to the radiation-dependent concentration of polymer formed in polymergel dosimeters manufactured from monomers in an aqueous gelatin matrix.Changes in T2 with time post-irradiation have been reported in the literaturebut their nature is not fully understood. We investigated those changes with timeafter irradiation using FT-Raman spectroscopy and the precise determination ofT2 at high magnetic field in a polymer gel dosimeter. A model of fast exchangeof magnetization taking into account ongoing gelation and strengthening ofthe gelatin matrix as well as the polymerization of the monomers with time ispresented. Published data on the changes of T2 in gelatin gels as a function ofpost-manufacture time are used and fitted closely by the model presented. Thesame set of parameters characterizing the variations of T2 in gelatin gels and theincreasing concentration of polymer determined from FT-Raman spectroscopyare used successfully in the modelling of irradiated polymer gel dosimeters.Minimal variations in T2 in an irradiated PAG dosimeter are observed after 13 h.

1. Introduction

Polymer gel dosimeters are typically composed of acrylic monomers dissolved in a hydrogelmatrix. In the most widely used polymer gel dosimeter (i.e. the polyacrylamide gel5 Present address: Department of Diagnostic Radiology, School of Medicine, Yale University, 330 Cedar Street,Fitkin B, New Haven, CT 06510, USA.6 Author to whom correspondence should be addressed.

0031-9155/01/112827+13$30.00 2001 IOP Publishing Ltd Printed in the UK 2827

2828 M Lepage et al

(PAG) dosimeter), the gelling agent is gelatin. Upon irradiation, the monomers undergoa copolymerization reaction and the 1H-NMR properties of the dosimeter are changed(Lepage et al 2001b). The production of solid polymer in the dosimeter causes a decrease inthe NMR spinspin relaxation time (T2) that can then be related to the absorbed radiation dose.Initially, the observation of continuing changes in T2 post-irradiation was believed to arisesolely from a continuing polymerization reaction (Maryanski et al 1994, McJury et al 1999).Changes with time in the properties of gelatin gels have been extensively studied (Djabourovand Leblond 1987, Djabourov et al 1985, 1988, Maquet et al 1986, Normand et al 2000). Onlyrecently has it been recognized that these phenomena are also present in gelatin-based polymergel dosimeters (De Deene et al 2000). In a study of the gelation of gelatin gels, a continuousdecrease in T2 was observed until a quasi-steady state was obtained after approximately 30 h(De Deene et al 2000). It was postulated that post-irradiation evolution of the PAG dosimeterswas due to both continuing polymerization as well as gelation. In that paper a qualitativedescription of the processes was provided.

In PAG dosimeters, the relationship between T2 determined long after the absorbtion ofa radiation dose has been quantitatively described using a three-proton pool model for fastexchange of magnetization (Lepage et al 2001b). FT-Raman spectroscopy was used in thatstudy to quantify the fraction of protons belonging to the polymer network as a function of theabsorbed dose, long after irradiation. In the present paper, we apply the same relaxation timemodel to the evolution of T2 as a function of time and for different absorbed doses. Previouslypublished data for the changes in T2 in gelatin gels (De Deene et al 2000) are reanalysed.In addition, an FT-Raman spectroscopy study of the continuing formation of copolymerpost-irradiation is presented. Taking both the variations originating from the gelatin matrixand from the formation of the polymer network into account, the model is shown to fit closelythe evolution of T2 with time in PAG dosimeters. Continuing polymerization, ongoing gelationand strengthening of the gelatin matrix are used to explain the experimental results.

2. Materials and methods

2.1. Gel preparation and irradiation

PAG dosimeters were manufactured from acrylamide (AA) and N,N -methylene-bis-acrylamide (BIS) (99+%, electrophoresis grade, Aldrich), gelatin (300 bloom, Aldrich) andwater (deionized). The monomers (AA and BIS) were dissolved in an aqueous gelatin matrix.The proportions were AA 3%, BIS 3%, gelatin 5% and H2O 89%, all by weight. In themanufacture process, gelatin was added to water at room temperature and left to soak for10 min. The solution was then heated and maintained at a temperature of 45 C. AA andBIS were subsequently added and magnetically stirred for typically 15 min until completedissolution was achieved. The solution was poured into glass vials having a teflon-linedscrew-top cap or into 5 mm NMR tubes sealed with epoxy resin and enclosed in heat-sealedBarexTM pouches. After gelation of the solution, the monomers were assumed to be uniformlydispersed throughout the gel. The PAGs were manufactured and sealed under a controlled N2atmosphere inside a glove box. The concentration of oxygen in the glove box was monitoredwith an oxygen meter (Quest Technologies, USA) and was maintained below 0.2%. Noinhibition of polymerization of the monomers could be detected at this level.

The PAGs were irradiated at 22 C in a Gammacell 200 (Atomic Energy Canada Limited)delivering photons from 60Co. The dose rate had been calibrated as 0.27 Gy s1 (Baldocket al 1999).

Post-irradiation events in polymer gel dosimeters 2829

2.2. Post-irradiation NMR spectroscopy

Sealed NMR tubes containing PAG dosimeter from the same batch were irradiated to 5 and7 Gy, respectively. The 5 Gy sample was irradiated 150 min after manufacture and the 7 Gysample was irradiated 3 weeks after manufacture. The tubes were transferred to a BrukerMSL-300 NMR spectrometer operating at 300 MHz within 10 min of -irradiation and thevariation in T2 monitored over a period of time extending up to 2000 min. The spectrometerwas equipped with a static 7 mm double resonance probe. The 90-pulse time for 1H NMR was6 s, the spectrum width was 5 kHz and 8192 data points were acquired. T2 was determinedusing the Carr-Purcell-Meiboom-Gill sequence. A single point was collected at the echomaximum for 2048 echoes in a pulse train. The time between successive 180 pulses was setto 600 s. The recycle delay was 5 s. T2-relaxation decays were collected every 20 min andthe standard uncertainty was 0.01 s.

2.3. Post-irradiation FT-Raman spectroscopy

Five vials filled with PAG dosimeter were irradiated from 2 to 10 Gy in steps of 2 Gy,respectively. The samples were transported from the irradiation source to the FT-Ramanspectrometer within 30 min and scanned thereafter for 1500 min. The concentration of doublebonds from the monomers still remaining after the gel had absorbed a given radiation dosewas measured from the FT-Raman spectra (Baldock et al 1998). The FT-Raman spectrometer(System 2000, Perkin-Elmer, Beaconsfield, UK) comprised a continuous wave Nd:YAGlaser emitting at 1064 nm. The scattered radiation was detected with an InGaAs solid-state detector. Four hundred scans at a resolution of 8 cm1 and laser power of 400 mWwere acquired for each sample. The time interval between successive acquisitions was30 min.

3. Model for NMR relaxation, polymerization and gelation

3.1. Fast exchange of NMR magnetizationFor systems for which the magnetization residing in all possible pools is in fast exchange, theexperimentally determined T2 (T2,exp) can be written as (Zimmerman and Brittin 1957)

1T2,exp

=i

f Hi

T2,i(1)

where the fraction of protons in the ith proton pool is fiH having an apparent T2 of T2,i. Theapparent T2 values result from the intrinsic rate of spinspin relaxation in each proton pool (i.e.in absence of exchange with other pools) and from a contribution of magnetization exchangewith other proton pools. This model has been used successfully to describe the changesof T2 as a function of the absorbed dose long after irradiation in PAG dosimeters (Lepageet al 2001a, 2001b). Three proton pools corresponding to free and quasi-free protons (denotedas mobile, mob), a growing polyacrylamide network (poly) and a gelatin matrix (gela) wererequired:

1T2,exp

= fHmob

T2,mob+

fHpolyT2,poly

+f HgelaT2,gela

. (2)

The fraction of mobile protons initially contains the protons from water and the monomers.The latter are gradually transferred to the polymer pool on irradiation. The fraction of protons

2830 M Lepage et al

in the gelatin pool is kept constant. The concentration of monomers remaining long afterabsorption of a given dose, and hence the concentration of polymer, was determined usingFT-Raman spectroscopy. Note that this model is an alternative to the more conventional modelincluding bound or hydration water protons (Koenig and Schillinger 1969). The existenceof bound water has been disputed from the thermal analysis of p(HEMA) hydrogels with anadiabatic calorimeter (Roorda et al 1988). An equivalence between the two models could beachieved if it were considered that some water protons were instead included in the gelatin orthe polymer pool. However, this remains an open question that is beyond the scope of thiswork. As it will be shown, the simple model of equation (2) provides a good description ofthe events taking place in a polymer gel dosimeter post-irradiation.

3.2. Kinetics of gelationThe evolution with time of the NMR spectra of diluted solutions of -gelatin chains hasbeen investigated (Finer et al 1975). The only change in the spectra during gelation was amono-exponential decrease in the intensity of the high-resolution (or mobile gelatin)component. The time constant for this decrease extracted from their data is approximately93 min at 15 C. The authors proposed that this time constant described the transition fromsingle -gelatin molecules in random-coil conformation to nucleated random coils. Thiswas supposed to be followed by a rapid transition to rigid collagen-like triple helices, whichcould then slowly reorganize to more ordered configurations. This is in agreement with thesuggestion that a continuous breaking and remaking of junction zones could lead to a slowreorganization of gelatin gels (Rees 1969).

The subsequent observation of a progressive decrease in T2 in gelatin gels aftermanufacture was taken as evidence that protons are progressively incorporated into a rigidstructure (Maquet et al 1986). In addition, an increase in shear modulus was reported toresult from the formation and reorganization of triple helices, proceeding towards a stableequilibrium (Djabourov and Leblond 1987). In the latter study, results for the changes inT2 as a function of time were explained using the ZimmermanBrittin formalism, wherethree different water pools were considered. However, the evolution of T2 with time wasnot quantitatively modelled. A direct relationship was established between 1/T2 and thehelix content, derived from optical rotation measurements, for a gelatin concentration of21%.

The gelation kinetics of gelatin was recently studied for samples with controlled molecularweight distributions and at different temperatures during gelation (Normand et al 2000). Inall cases, the elastic modulus was observed to increase linearly with the logarithm of timefor the time interval between 1 and 100 h. The time constant for this process extractedfrom their data, for a 6.66% gelatin gel at 20 C, is 445 min. The extension of existingcrosslinks (defined as the formation of segments of intermolecular triple helix (Normandet al 2000)) within the gelatin matrix, rather than the formation of new crosslinks, has beeninvoked to explain the observation. This means that a large proportion of gelatin moleculeshave already reached the nucleated coil conformation and proceed towards the formationand/or the reorganization of triple helices, thereby increasing the elastic modulus. A smallamount of new crosslinks cannot, however, be ruled out. Since an increase in gel strength canbe expected to lead to more immobilized gelatin molecules, a lower T2,gela can therefore beexpected.

In the model of fast exchange of magnetization used here, we consider that all waterprotons are included in one mobile proton pool. As mentioned earlier, considering that somewater protons instead belong to the gelatin or the polymer pool would be consistent with

Post-irradiation events in polymer gel dosimeters 2831

models using bound water populations. The apparent T2,gela becomes time-dependent to takeinto account the changes in the gelatin matrix. Ongoing gelation is also considered. Thismeans that a small fraction of the final f Hgela is initially mobile and eventually nucleates andforms a triple helix. Newly formed crosslinks are then able to reorganize (and thus strengthen)with time. The same overall idea has been previously applied by Djabourov et al (1985).

Normand et al (2000) have recently studied the well-known temperature sensitivity ofgelatin gels. They reported that the gel strength decreased with an increase in the gelationtemperature. However, the kinetics of gelation was found to be only slightly dependent onthe gelation temperature within the range 520 C . In the study of De Deene et al (2000), thegelation temperature was constant but the manufacture temperature was varied. In consequencefor their samples, it is reasonable to expect that the kinetics of gelation will be constant butthat the parameter T2,gela, indicative of the gel strength as argued below, may be affected bythe manufacture temperature.

3.3. Evolution of the copolymerization reactionA study of the kinetics of the chemically initiated polymerization of AA and BIS in waterhas been reported (Gelfi and Righetti 1981). The absorbance of light at 600 nm for an initialsolution of AA 4.25% and BIS 0.75% was characterized by an initial plateau followed bya decrease consistent with the rapid formation of precipitated polymer particles. The initialabsence of change reflects the fact that the gelation point had not been reached. After this initialplateau, we found that the decrease in absorbance they published could be empirically fittedwith a two-exponential function having time constants of 1 and 6 min. When increasing theproportion of BIS, these authors found that the rate of polymerization decreased dramatically,so the time constants for the decrease in absorbance are getting longer. Consequently,significantly higher time constants are expected for the present samples containing 3% ofAA and 3% of BIS.

4. Results and discussion

4.1. Evolution of the gelatin matrixAn experimental study of the stability of PAG dosimeters has been published recently(De Deene et al 2000). The reproducibility in the manufacture of gelatin gels and the effectof temperature during manufacture was investigated. The values of T2,exp from this study arereproduced in figure 1. The upper panel demonstrates the reproducibility in the manufactureof a 6% gelatin gel at 50 C. The lower panel shows the effect of varying the temperatureduring manufacture of a 6% gelatin gel.

Our proposed model for gelation was applied to these data and the results are shown infigure 1 as solid curves. Using an MRI scanner with the same magnetic field (1.5 T) as thatused by De Deene et al (2000), it was previously determined that T2,mob was 3.0 s (Lepage et al2001a, 2001b). The relevant parameters relating to the instantaneous concentration of gelatinand the changes in gelatin strength were adjusted to provide best fits to the data. However,since the gelation temperature was the same for all samples, it is required that the parametersrelated to the kinetics of gelatin are kept constant.

The values of the fitting parameters found to provide the best description of the data ofDe Deene et als results are now detailed. Ninety-four percent of the total gelatin protonfraction was considered as gelled when the gel began to set, the time which defines the origin

2832 M Lepage et al

0 500 1000 1500 2000 2500 3000

0.82

0.84

0.86

0.88

0.90

0.92

0.94

0.96

0.98 Gelatin 6%, 64 MHz

Maximum manufacture temperature

40C 50C 60C 70C 80C 90C

T 2 (s)

TIME (mins)

0 500 1000 1500 2000 2500 3000

0.82

0.84

0.86

0.88

0.90

0.92

0.94

Heated at 50Cduring manufacture

Tube 1 Tube 2 Tube 3 Tube 4

Gelatin 6%, 64 MHz

T 2 (s)

Figure 1. Experimentally determined T2 (T2,exp) from 6% gelatin gels (reproduced from De Deeneet al (2000)). The upper panel shows the reproducibility for four different preparations of the gelat a temperature of 50 C (tubes 14). The lower panel shows T2,exp for different manufacturetemperatures. The solid curves are derived from the model and only the final T2,gela value ischanged.

of the post-manufacture time scale. The remaining 6% of gelatin molecules, which are stillcontributing to the mobile proton pool, underwent continuing gelation (i.e. the formation ofnew crosslinks) with a time constant of 700 min. This process is slower than the reorganizationof the triple helices and involves only a small proportion of the gelatin molecules. It is possibletherefore that this effect is superimposed on the overall increase of optical rotation and istherefore unnoticed in those optical experiments (Djabourov and Leblond 1987). However, in

Post-irradiation events in polymer gel dosimeters 2833

the present paper, the consideration of this small effect proved to be important to the qualityof the fits obtained.

We now examine the parameters representing the increase in gelatin strength. The value ofT2,gela for newly gelled gelatin molecules was initially 17% higher than its final value (table 1).The precise explanation for this change in T2,gela must be related to the geometricalconformation change of the triple helices and/or the extension or the lengthening of crosslinks,but further work would be necessary to elucidate this. The value of T2,gela then decreasesexponentially, in accordance with the previously observed linear increase in gel strength withthe logarithm of time (Normand et al 2000). The time constant found in the present paper(340 min) is 25% lower than the time constant of 445 min for the change in elastic modulusof a 6.66% gelatin gel at 20 C between 1 and 30 h post-gelation, as extracted from the workof Normand et al (2000). Differences of 10% were obtained in the same work for theirdifferent gelatin samples studied at 10 C (upper panel in figure 1). The small discrepancycould therefore be due to different gelatin batches and/or a different temperature during themanufacture and/or the measurement procedure. Since it can be assumed that the dynamicsof plasticized swollen gelatin will be comparable to those of a rubbery polymer, we canmake comparisons between our gelatin results and those of rubbers for which a theory exists(Gotlib et al 1976). It has been found that in the high-temperature limit (well above the glass-transition temperature) T2 of rubbery polymers decreases in a regular manner with increasedcrosslink density. This supports the apparent inverse proportionality between T2,gela and theelastic modulus of gelatin, a theory (Gotlib et al 1976) based on the calculation of the scaleddipolar interaction for anisotropic motion of chain segments in a Kuhn chain (Kuhn and Grun1942). A Kuhn chain replaces the detailed polymer by an equivalent chain of segments,each consisting of a number of monomer units. Their work shows that T2 is proportional tothe rigid-lattice T2 multiplied by a constant z, the number of statistical segments betweenjunction points which constrain the motion of the chain. Although many assumptions are usedin developing the full theory, it has been tested with success a number of times in the literature(Fry and Lind 1988, Litvinov et al 1998).

Table 1. T2,gela used in the model for different temperatures of gel manufacture. All values in ms 0.2 ms.

Tube 1 Tube 2 Tube 3 Tube 4 Tmax Tmax Tmax Tmax Tmax Tmax50 C 50 C 50 C 50 C 40 C 50 C 60 C 70 C 80 C 90 C

T2,gela 57.0 58.1 57.2 57.8 56.5 56.8 58.0 57.4 59.4 61.7

The parameters described were hereafter kept constant, except for the final value of T2,gela(table 1) which was expected to change for different temperatures, and were used to model thecurves shown in figure 1. It can be seen that the model closely fits all experimental data. Themodel includes only two proton pools, i.e., a gelatin pool and a mobile pool. This is a simpleralternative to the model proposed by Maquet et al (1986) which includes three different waterpools.

The values of the fraction of gelatin protons as a function of post-manufacture time andfor T2,gela as a function of the post-gelation time for tube 2 (from De Deene et al (2000)) areshown in figure 2. The post-gelation time is defined as the time after which a new chain isformed. Therefore, for 94% of the total gelatin proton fraction, the post-manufacture time andthe post-gelation time are the same. The remaining 6% of gelatin protons undergo gelation ata given post-manufacture time and this sets the origin of their post-gelation time.

2834 M Lepage et al

0 500 1000 1500 2000 2500 3000

58

60

62

64

66

68

T 2,g

ela

(ms)

POST-MANUFACTURE TIME (mins)

0 500 1000 1500 2000 2500 3000

4.80

4.85

4.90

4.95

5.00

5.05

5.10MODEL DATA FOR6% GELATIN GEL

f gelaH

(%

)

POST-GELATION TIME (mins)

Figure 2. Fraction of gelatin protons(fHgela

)and apparent T2 for the gelatin pool (T2,gela) as

computed from the theoretical model.

The expected overall increase of T2,gela with increasing temperature can be noted intable 1. However, no further quantitative conclusions on the values of T2,gela with manufacturetemperature can be drawn from this experiment since the samples have not been heated to asingle temperature for a constant time (De Deene et al 2000).

4.2. Post-irradiation changes in PAG dosimeters

The parameters describing the kinetics of gelation of gelatin determined in section 4.1 areused here in an analysis of the time evolution of relaxation times in PAG dosimeters. Thefraction of gelatin protons was calculated for a PAG dosimeter containing 3% AA, 3% BIS,5% gelatin and 89% H2O. The value of T2,gela for this system was found to be 48 ms at 64MHz and 35 ms at 300 MHz in a previous publication (Lepage et al 2001a). The slightlylower T2,gela of 48 ms compared to those extracted from the results of De Deene et al (2000)may simply be due to the amount of time for which the gelatin was heated during manufactureor due to a difference in gelatin strength, not unexpected for different batches of gelatin. Thekinetic parameters obtained from the modelling of T2,exp for the 6% gelatin gels are used forPAG dosimeters containing 5% gelatin. This follows the finding of Normand et al (2000) thatthe ratio of the kinetic constant of crosslink formation and crosslink melting does not changesignificantly with a small increase in gelatin concentration.

Post-irradiation events in polymer gel dosimeters 2835

The decrease in intensity with time of the peaks in the FT-Raman spectra associated withthe vinyl groups of the two monomers (AA and BIS) was found to be adequately described bya two-exponential function. Similarly the data of Gelfi and Righetti (1981) can be fitted to atwo-exponential function, although in their case a plateau could be observed before the decayin monomer concentration. The initial slow rate of polymerization occurs prior to gelationof the system. The two-exponential decay in monomer concentration could be due to twopossible reasons. First, there may be regions of different densities within the precipitatingpolymer network, as was observed for materials having lower BIS content by small angle x-rayscattering (Cohen et al 1992). It would be expected that the rate of diffusion of the monomerswithin these two regions will be different, but that in each region the polymerization reactionwould follow first-order kinetics. The overall rate of disappearance of monomers wouldtherefore be described by the sum of two exponential decays. Second, a first-order processassumes that the radical concentration is constant (steady-state condition) and sufficient toinitiate polymerization until all the monomers have reacted. Given that all the free radicalsfrom water react quickly, one must consider the concentration of growing macroradicals. Itwas established that these macroradicals precipitate during the polymerization reaction andhence become less accessible to the unreacted monomers (Lepage et al 2001a). The effectiveor accessible radical concentration thus decreases with time, and the result is a second-orderdecrease in monomer concentration. Although the analytical solution to this problem cannot bederived simply, the overall form of the decay would be similar to a two-exponential function.It is likely that either or both of these mechanisms contribute to the observed kinetics ofmonomer reaction, and thus there is no need to invoke the participation of supposed radicalstrapped on gelatin sites.

In all cases examined in the present study, the decrease in concentrations of AA and BISwas described by a two-exponential function. However, large variations in the time constantsand the relative amplitudes of the short and long decay components were observed. Theshort time constant varied from 20 to 46 min while the long one varied from 170 to 500 min.These time constants are longer than those of Gelfi and Righetti (1981), as expected fromthe much larger BIS concentration in the present samples. We have suggested elsewhere thatan increase in BIS concentration leads to more rapid precipitation of growing macroradicals,leading to a lower polymerization rate (Lepage et al 2001a). This has the effect of makingit more difficult for monomers remaining in solution to diffuse to the macroradical andparticipate in the propagation reaction, leading to polymerization extending to unusuallylong periods (Chapiro 1962). Alternatively, the lower polymerization rate may partly arisefrom the presence of gelatin. An increasing gelatin concentration can lead to either increasedscavenging of initiator fragments or increased chain transfer reactions to gelatin molecules orboth (Lepage et al 2001a).

The fraction of polymer(fHpoly

)formed in the 7 Gy sample as a function of time,

determined from the decrease in monomer concentration using FT-Raman spectroscopy, isshown in figure 3. Scatter in the experimental data can be observed, and is believed to be dueto temporal variations in the laser power and increasing opacity of the samples with time. Thedecay time constants obtained for the decay of the monomers were 21 10 and 273 21 minfor this sample. In the model, decay constants of 30 and 200 min were found to give the bestfit to the results presented below. The fraction of polymer calculated using the latter decayconstants is shown as a solid curve in figure 3.

The values of T2,exp at 300 MHz for the two PAG dosimeters irradiated at 5 and 7 Gy,respectively, are shown in figure 4. It is apparent that T2,exp varies rapidly in the first 500 min.This observation is in agreement with the results of De Deene et al (2000) but is in sharp

2836 M Lepage et al

0 500 1000 1500 2000 2500 3000

1.2

1.6

2.0

2.4

2.8

3.2

Model FT-Raman

3% AA, 3% BIS,5% gelatin, 89% H2O

f poly

H (%

)

POST-IRRADIATION TIME (mins)Figure 3. Variation in the fraction of polymer protons

(fHpoly

)for PAG dosimeters () with

time. fHpoly is taken as the difference between 1 and the fraction of monomers left in the PAG, asdetermined from FT-Raman spectroscopy. The curve (solid) used in the theoretical model is alsoshown.

contrast with those of McJury et al (1999) who reported a saturation of R2 (1/T2) only afterapproximately 18 000 min.

The fraction of monomers remaining in the PAG dosimeter depends upon the absorbeddose and can be quantified using FT-Raman spectroscopy (Lepage et al 2001a). Thecorresponding final fHpoly along with the variations in T2,exp arising from the ongoing gelationand change in conformation in gelatin were included in the model. The value of T2,mob was setto 3.0 s as determined previously (Lepage et al 2001b). The apparent T2 of the polymer pool(T2,poly) was fixed at 14.5 ms, a value that has been used previously for the same formulationof PAG dosimeters (Lepage et al 2001a).

The output of the model is shown as solid curves passing through the data points in figure 4.Of relevance to the stability of PAG dosimeters, it can be concluded that T2,poly is constant withtime, suggesting that the polymer morphology and topology does not change over the courseof these experiments. This means that the additional polymer material formed with time hasthe same general structure as that formed at short times, but that the overall conversion topolymer increases with time.

The importance of considering the changes in the gelatin matrix in the analysis of thetime evolution of the relaxation times can be judged from a comparison of the results forthe samples irradiated at different times after manufacture. The 5 Gy sample was irradiated150 min after manufacture while the 7 Gy sample was irradiated 20 days later, where thevariations in the gelatin matrix can be neglected. For comparison, the output of the model for

Post-irradiation events in polymer gel dosimeters 2837

0 500 1000 1500 2000 2500 3000

0.28

0.30

0.32

0.34

0.36

0.38

Gelation included not included

5 Gy 7 Gy

POST-IRRADIATION TIME (mins)

PAG dosimeter 300 MHz

T 2 (s)

10 100 1000

0.280.300.320.340.360.38

Figure 4. Post-irradiation variation in T2,exp with time in PAG dosimeters, measured at 300 MHz.The two samples were irradiated to 5 and 7 Gy, respectively. Both the effects of ongoing gelationand strengthening of the gelatin matrix and the polymerization of the monomers are included in themodel. The calculations for the 5 Gy sample when the changes in the gelatin matrix are neglectedare also shown (dashed curves). The difference is more clearly seen in the inset where the time isplotted on a logarithmic scale.

the 5 Gy sample, ignoring the variations in the gelatin matrix, has been added to figure 4. Itcan be immediately seen that neglecting the changes in the gelatin matrix leads to a poorer fitto the experimental data. The fit to the 7 Gy data appears superior than the fit to the 5 Gy data.This is due to the changes in gelatin which are present in the latter but absent in the former.Our parameters were derived from the study of De Deene et al (2000) who used a differentbatch of gelatin. Although the magnitude of the changes in T2,exp arising from the variations

2838 M Lepage et al

in the gelatin matrix is smaller than those arising from the polymerization reaction, these mustbe taken into account to describe the system fully.

5. Conclusions

A theoretical model of the time dependence of the NMR relaxation in PAG dosimeters hasbeen presented and validated using experimental results from different techniques. The modelassumes fast exchange of magnetization between three different proton pools. The mobilepool initially contains the protons from water, the monomers and ungelled gelatin molecules.The polymer proton pool is initially empty and is gradually filled as a polyacrylamide networkis formed. The formation of polymer is characterized by two time constants derived fromresults of FT-Raman spectroscopy measurements. The gelatin pool also evolves with time,initially containing 94% of all gelatin protons, the remaining 6% being added with time asongoing gelation proceeds. The parameter T2,gela depends on the inverse of the gelatin matrixstrength, which increases with time. The kinetic parameters for the gelation of gelatin gelswere extracted from the results published by De Deene et al (2000) on the changes in T2 indifferent gelatin gels obtained from a clinical MRI scanner.

Finally, the changes in T2 with time were monitored using an NMR spectrometer operatingat 300 MHz, and were modelled using the information described above. While the details ofthe exchange of magnetization between the pools cannot be ascertained from this work, anexcellent agreement was found between the simple model used and the experimental results.Since the study of De Deene et al (2000) was performed on a different batch of gelatin, themodel may be generally applicable to polymer gel dosimeters.

From a practical point of view, the conclusions from De Deene et al (2000) arecorroborated. Minimal variations in T2 in an irradiated PAG dosimeter are observed after13 h. This is in sharp contrast with the conclusions of McJury et al (1999). Further, itbecomes apparent that imaging calibration vials and a phantom at different times a short timeafter irradiation could lead not only to an absolute error in the dose calibration but also todistortions in dose distributions.

Acknowledgments

We thank Y De Deene for kindly providing us with the experimental data of figure 1. Thesupport of Southern X-Ray Clinics and the Wesley Research Institute are acknowledged. SAJBacknowledges the support from The Swedish Foundation for International Cooperation inResearch and Higher Education (STINT).

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