In Fricke-agarose gels, an accurate determination of the spatial dose distribution is hindered by the diffusion of ferric ions. In this work, a model was developed to describe the diffusion process within gel samples of finite length and, thus, permit the reconstruction of the initial spatial distribution of the ferric ions. The temporal evolution of the ion concentration as a function of the initial concentration is derived by solving Fick's second law of diffusion in two dimensions with boundary reflections. The model was applied to magnetic resonance imaging data acquired at high spatial resolution (0.3 mm) and was found to describe accurately the observed diffusion effect

Barone, P.

d'Errico, F.

de Pasquale, F.

Egger, E.

Guidoni, L.

Luciani, A. M.

Pacilio, M.

Sebastiani, G.

Viti, V.

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ION DIFFUSION MODELLING OF FRICKE-AGAROSEDOSEMETER GELSF. de Pasquale1, P. Barone1, G. Sebastiani1, F. d’Errico2, E. Egger3, A. M. Luciani4, M. Pacilio4,L. Guidoni4 and V. Viti4,1Istituto per le Applicazioni del Calcolo, CNR, viale del Policlinico 137, 00161 Rome, Italy2Yale University School of Medicine, New Haven, CT, USA3Paul Scherrer Institut, Villigen, Switzerland4Dipartimento di Tecnologie e Salute, Istituto Superiore di Sanita` and INFN-Sez.Roma 1,viale Regina Elena 299, 00161 Rome, ItalyIn Fricke-agarose gels, an accurate determination of the spatial dose distribution is hindered by the diffusion of ferric ions. Inthis work, a model was developed to describe the diffusion process within gel samples of finite length and, thus, permit thereconstruction of the initial spatial distribution of the ferric ions. The temporal evolution of the ion concentration as a functionof the initial concentration is derived by solving Fick’s second law of diffusion in two dimensions with boundary reflections.The model was applied to magnetic resonance imaging data acquired at high spatial resolution (0.3 mm) and was found todescribe accurately the observed diffusion effects.INTRODUCTIONThe diffusion of ferric ions is one of the main prob-lems that limits the applicability of Fricke-agarosegels in radiation dosimetry. A significant degrada-tion of the recorded dose distributions can occursince the timescale of the diffusion process is com-parable with the duration of the experimentalprocedures. The latter include irradiation, transferof irradiated samples to the imaging system, MRsystem set-up and image acquisition. In some cases,an effective solution to the diffusion problem canbe provided by the use of ferric ion chelators,which contrast ion mobility. In particular, the useof Xylenol Orange, proposed by Appleby andLeghrouz(1), drastically reduces the diffusion effectswhile causing a limited reduction in sensitivity of theFricke-gel system. This compound specifically bindsto ferric ions, thus, slowing down their diffusion.However, when strong concentration gradients arepresent, diffusion effects cannot be neglected and areconstruction of the initial ion distribution at theend of irradiation is necessary. This can be achievedby using an appropriate model for diffusion(2), whichdepends on the unknown diffusion coefficient. Inthis work, we propose a simple method to estimatethis coefficient.MATERIALS AND METHODSFricke-agarose gels were prepared in cylindricalPerspex holders according to previously publishedmethods(3). The final concentrations were: 1.5 mMferrous ammonium sulphate, 1 mM sodium chloride,50 mM sulphuric acid and 1% agarose by weight.Hereafter, this sample will be referred to as ‘Noxyl’.Samples containing Xylenol Orange were preparedwith the same procedure except for adding Frickesolution containing the chelator agent. The final fer-rous ammonium sulphate concentration was 1 mMand the final Xylenol Orange concentration was0.25 mM. These gels will be referred to as ‘Xyl’.The gel samples were always kept at room temper-ature and irradiated between 20 and 28 h afterpreparation.A first set of samples had a diameter of 45 mm anda height of 40 mm (Figure 1). These were coaxiallyirradiated at the Paul Scherrer Institut in Villigen,Switzerland, with a 62 MeV proton beam, which wasmodulated to yield a spread-out Bragg peak. Refer-ence dosimetry was performed using a parallel plateFigure 1. Geometry of the irradiated Fricke-gel sample,the shadowed area represents the acquired slice, and theinternal cylinder is the region that was irradiated.Corresponding author: viti@iss.itRadiation Protection Dosimetry (2006), Vol. 120, No. 1–4, pp. 151–154doi:10.1093/rpd/nci683 Advance Access published on April 27, 2006 The Author 2006. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.orgMarkus chamber. The proton irradiations generateda sharp concentration gradient along the axis of thesamples. Images of these samples were acquired athigh resolution (0.3 mm) with a 4.7 T Sisco-VarianMR research unit. An Inversion Recovery (IR)sequence with an echo time TE of 30 ms and adelay time tD of 3 s was used. A second set ofsamples with a diameter of 7 cm and a height of19 cm was irradiated inside a water phantom usinga radiotherapy beam of 6 MV X rays. The irradi-ation geometry was again coaxial. These X-ray irra-diations generated smooth concentration gradientsalong the axis of the samples. Imaging in this casewas done with a whole-body GE-Echospeed systemoperating at 1.5 T. Axial slices were acquired with athickness of 0.5 cm using a head coil.A series of MR scans was performed at varioustime intervals producing R1 maps of the two sets ofsamples. The scans started 20 min after the end ofirradiations, and each acquisition lasted 45 min.METHODSOwing to the proportionality between ‘relaxationrate’ R1 ¼ 1/T1 and local concentrations 2C, 3C offerrous and ferric ions we have, under suitable hypo-theses,DR1 tð Þ ¼ R1 tð ÞR1b ¼ a bð Þ3C tð Þ, ð1Þwhere a and b are, respectively, the effective and thetrue relaxivities of ferric and ferrous ion solutions,and R1b is the relaxation rate before irradiation.It has been previously shown(2) that, assumingthat there is no diffusion along the z axis, in theplane z ¼ 0, the concentration C(x,y,t) ¼ C(x,y,0,t) is a solution of the Fick’s second law:@C@t¼ D @2C@x2þ @2C@y2 ðÞwith Cauchy’s initial condition C(x,y,t0) ¼ C0(x,y)and boundary reflection conditions:@C@x x¼0¼ @C@x x¼h¼ @C@y y¼0¼ @C@y y¼2r1¼ 0, 8twhere h and r1 correspond to the physical boundar-ies of the sample and are shown in Figure 1 and D isthe unknown diffusion coefficient.Assuming that we can neglect the contributionat point (x, y) of the points ~x; ~yð Þwhose distanceis such that jx ~xj  3s and jy ~yj  3s, where s ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2DDtpand Dt ¼ t t0ð Þ, in which t0 is the time ofirradiation of the gels at which the diffusion processbegins, an approximate solution of () 8t  t0, isgiven by:C x,y,tð Þ¼ 14pDDtZ MxmxZ MymyC ~x,~y,t0ð Þexp  x~xð Þ2þ y~yð Þ24DDt" #·1þ exp  x~xDDt þ exp  hxð Þ h~xð ÞDDt  ·1þ exp  y~yDDt þ exp  2r1yð Þ 2r1~yð ÞDDt  d~xd~yð1Þwhere: mx ¼ max (x 3s, 0), Mx ¼ min (xþ 3s, h),my ¼ max (y 3s, 0) and My ¼ min (yþ 3s, 2r1).Using this expression for C (x,y,t), an estimate ofthe diffusion coefficient D can be obtained by solvingthe minimisation problemD^ ¼ arg minDL Dð ÞwithL Dð Þ ¼Xi;jð Þ2AjCtheor i,j,tð ÞCmeas i,j,tð Þj2,where A is the region of a 2-D image over whichlarge variations of ferric ion concentration areobserved in points close to each other. Cmeas(i, j, t)is the measured ferric ion concentration derivedfrom DR1(t) (t time after irradiation). Ctheor(i, j, t)is the ferric ion concentration at a time t, calculatedby discretising the approximate solution of Fick’sEquation 1 with initial values Cmeas(i, j, t0), t> t0.The estimation of D was performed with a numer-ical approach. First, we chose a finite number ofpoints presenting a wide range of D values. Afterchecking that L (D) presents only one minimum inthis range, we restricted the range around this valueand used a higher resolution. The process wasrepeated 200 times starting from the same data andrandomly altering the location of A.RESULTSThe mean values of the diffusion coefficientscalculated for the Noxyl and Xyl gels are reportedin Table 1, where the associated error is the stan-dard deviation. These values were used to modelTable 1. Estimated values for the diffusion coefficient for the‘Noxyl’ and ‘Xyl’ samples by using MR data at 4.7 T.Sample D (mm2 h1)Noxyl 1.96 0.01Xyl 1.14 0.02F. DE PASQUALE ET AL.152numerically the evolution of the DR1 images at dif-ferent times after the irradiations.In turn, experimental DR1 images were generatedas the difference of the R1(t) and R1b maps fromthe MR scans, using a previously reported Bayesianapproach(3). The DR1 images were obtained at vari-ous times following irradiation, where the time tassociated with each image is the average valuefrom the end of irradiation.Figure 2, relative to the proton irradiations, showsa comparison between Markus chamber measure-ments and the absorbed dose profile derived fromthe first DR1 map acquired at time t1 40 min afterthe irradiation of a Xyl sample. In the smooth prox-imal region, preceding the Bragg peak, the agree-ment between ion chamber measurements andrelaxation rate difference is quite good. Conversely,the diffusion effects cause the discrepancy in thesharp gradient of the distal region.Based on our previously described model of thediffusion process and on the estimated diffusioncoefficient D (Table 1), the DR1(t5) profile shownin Figure 3 was computed for a time t5 ¼ 249 min.Results from the computational modelling areclearly in agreement with the correspondingmeasured DR1(t5) map. It should be noted that itis important to include boundary reflections in thediffusion model. In their absence, a clear discrepancyarises between measured and computed data (alsoshown in Figure 3). Varying the spatial resolutionin our data analysis had no impact on the results,this was shown by generating R1 maps at lowerresolution (1 mm) by averaging high-resolutionmaps within a 3 3 local window and then perform-ing three-pixel image-subsampling.The Xyl and Noxyl samples irradiated with X-raybeams were characterised by smaller concentrationgradients both in longitudinal and transverse direc-tions. Dose maps based on images acquired at aspatial resolution of 1 mm showed negligible differ-ences with the dose profiles from reference dosi-metry, indicating that diffusion effects are muchsmaller. This was also confirmed by the simulations,indicating minimal changes of the dose profiles as afunction of time from the irradiation.CONCLUSIONSOur investigations indicate that ion diffusion is not acritical issue for dose reconstruction from MRI withspatial resolution of 1 mm when images are takenwithin a few hours of the irradiation and when theFigure 2. Comparison between dose profiles in water from MR relaxation rate variations (DR1) of the Fricke gels(solid line) and ion chamber measurements (dashed line).DIFFUSION MODELLING IN FRICKE GELS153irradiation does not generate strong concentrationgradients (0.1 s1 mm1 in R1 maps), e.g. withconventional X-ray irradiations. However,when large concentration gradients are generated(0.6 s1 mm1 in R1 maps), e.g. following protonbeam irradiations, diffusion effects are not negligibleand must be taken into account. In this case, ourproposed model is extremely useful, since the initialion concentration can be accurately reconstructedbased on measurements performed even severalhours after irradiation.REFERENCES1. Appleby, A. and Leghrouz, A. Imaging of radiation doseby visible color development in ferrous-agarose-xylenolorange gels. Med. Phys. 18, 309–312 (1991).2. Barone, P. and Sebastiani, G. Solving an inverse dif-fusion problem for Magnetic Resonance dosimetry by afast regularization method. Real Time Imag. 7, 21–29(2001).3. de Pasquale, F. et al. Bayesian estimation of relaxationtimes T1 in MR images of irradiated Fricke-agarose gels.Magn. Reson. Imag. 18, 721–731 (2000).Figure 3. Comparison between dose profiles at a time t5 based on measured DR1(t5) values (solid lines) and on thesimulated temporal evolution of the initial DR1(t1) image either with or without boundary reflections (bold and dotted-dashed lines, respectively).F. DE PASQUALE ET AL.154

Ion diffusion modelling of Fricke-agarose dosemeter gels

Ion diffusion modelling of Fricke-agarose dosemeter gels

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