Magnetic
Resonance Imaging, Vol. 1.5, No. 2, pp. 211-221, 1997 Copyright 8 1997 Elsevier Science Inc. Printed in the USA. AI1 rights reserved ~73~-~2~~97 $17.00 + .m
PI1 SO730-725X( 96)00352-9
ELSEVIER
* Original Contribution FAST T1 IMAGING OF DUAL GEL SAMPLES FOR D~FUSION MEASUREMENTS IN NMR DOSIMETRY GELS TOMAS I&ON,* DAVID JONAS,~ AND JAMES M. POPE+ *Dept. of Radiation Oncology, Newcastle Mater Misericordiae Hospital, Locked Bag 7, Hunter Region Mail Centre, NSW 2310, Australia, @chool of Physics, University of New South Wales, Sydney, NSW 2052, Australia, and $School of Physics, Queensland University of Technology, GPO Bos 2434, Brisbane, QLD 4001, Australia Diffusion of iron is one of the major problems limiting the usefulness of NMR gel dosimetry. Thii was studied in dual gel samples using a 4.7T micro-~a~ng MR scanner and a fast Tz imaging sequence which allowed the acquisition of a 64 x 128 x 8 data sets (phase encoding x frequency encoding x number of inversion times) in less than 15 min. The procedure enabled us to obtain relative relaxation times for any region of interest within the sample. After the two differently doped gels were brought into contact in the dual gel samples (diameter 12mm), the diffusion could be observed on subsequent images as a function of time. An inverse square root function was used to fit the change of liTI across the junction between the two gel phases. A diiusion constant of 0.014 i 0.003 cm2/h was dete~ined for Fe” in a typical dosimetry gel (1.5% agarose, 50mM H2S04). This could be lowered by adding a chelating agent such as xylenol orange to the gel. It was also found that diiusion was slower in gelatine gels, however these gels tended not to set properly when H2S04 was added as required for NMR dosimetry. From the present results we propose that a gel consisting of 1.5% agarose (for stability), 3% gelatine and O.lmM xylenol orange (to combat diff~ion and allow a visual ev~uation) is a suitable base for NMR dosimetry gels. The use of a fast T1 imaging sequence reduces acquisition times and therefore the potential impact of diiusion. 0 1997 Elsevier Science Inc. Keywords:
Agarose gels; Diffusion;
Fast T, measurements;
NMR doshnetry.
The oxidation of ferrous iron (Fe2’) to ferric iron ( Fe3’) due to ionising radiation has been used for chemical dosimetry since 1927. 1 The two different ionic states of iron have a different magnetic moment, different ionic radii and correlation times which influence the proton relaxation times in a nuclear magnetic resonance (NMR) experiment.2-3 If the spatial position of the iron is fixed by embed~ng it in a gel matrix, this effect can be used to obtain a three dimensional image of the dose distribution after irradiation of the sample by mapping T I or T 2. Even though T2 measurements have been exploited for NMR dosimetry4-5 variations in T, can be assessed more accurately, and they are predominantly employed for NMR dosimetry.“-’ More recently, also the direct use of MR signal intensi-
ties for dosimetry via “magnetization to dose” conversion curves has been proposed.g-‘o Many applications have been suggested in the last five years for NMR dosimetry ranging from electron beam,” superficial beam’ and bmch~~apy dos~e~g,12-‘3 to the verification of stereotactic radiosurgery7”4 and dynamic wedges.15The NMR dosimetric system is particularly of interest for the three-dimensional mapping of dose distributions,7~‘4 and for the dosimetry of low energy x-rays since detector and ph~tom are identic~ and virtually tissue equivalent. **16In the steep dose gradients produced in many of these applications, diffusion of the ions in the gel matrix is a major problem limiting the usefulness of NMR dosimetry. 17-‘* This problem is often aggravated by the long time typically required to gain access to an MR scanner in a clinical environment and then to map relaxation times throughout the sample.
3~20~96~ACCEPTED 81’30196. Address correspondence to Tomas Kron, PhD, Dept. of Radiation Oncology, Newcastle Mater Misericordiae Hospi-
tal, Locked Bag 7, Hunter Region Mail Centre, NSW 2310, Australia, Phone: 61-49-211171, Fax: 61-49-602566, Email:
[email protected]
RECEIVED
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Magnetic Resonance Imaging l Volume 15, Number 2, 1997
However, reports on clinical applications of NMR dosimetry are appearing slowly.7~9~‘2 In the present study the impact of a varying gel composition on iron diffusion in dosimetry gels was studied using dual gel samples. Two gels with different iron concentrations brought in contact constitute the closest approximation to an infinitesimally steep dose gradient in real dosimetry gels. Of particular interest to combat diffusion are chelating agents such as xylenol orange. *l-z3 The in~uence of xylenol orange was studied as well as the use of combination agarose/gelatine gels. In addition to this, the use of a fast T1 image acquisition was evaluated for its capability to speedup the measurement procedure thereby reducing both diffusion and costs. The results of the study should help to optimise the composition of gels used for NMR dosimetry as well as making NMR dosimetry more suitable for a busy clinical environment. MATERIAL
AND METHODS
I. Preparation of Gels A variety of different gel compositions was studied as can be seen in Table 1. For agarose gels agarose powder (SIGMA chemicals: Type I) was weighed into
a 250 ml glass vessel with narrow neck (diameter 3 cm), and distilled water was added to make up half the desired volume of the gel. The colloid solution was well mixed, placed in a water bath, and brought close to boiling (T < 98°C) in a microwave oven over approximately six min stirring the solution twice. A loose fitting lid was used to limit water losses. In a separate glass vessel ammonium ferrous sulfate, Fe( NH,),( SO,),, was dissolved in dilute sulfuric acid to make the other half of the final gel volume. The acid concentration was double that of the final concentration which is given for the different gels in Table 1. In the dose response experiments, the ferrous sulfate solution was aerated well. In the diffusion experiments also ferric sulfate, Fe( NH)( SO,),, was added to some samples to mimic an irradiated gel. To help in dissolving the ferric sulfate powder, the acid solution was placed in an ultrasonic bath for approximately five min and heated modera~ly (T < 50°C) if required. In some experiments xylenol orange (Sigma Chemicals) was added to the acidic iron solution to yield a final concentration between 0.1 and 0.25mM. Immediately after taking the agarose solution out of the microwave oven it was mixed with the acidic iron solution. The resulting gel solution was allowed to cool
Table 1. Resultsof the dual gel diffusion experiments.It shouldbe noted that the data from literature quotedin the second
half of the table was obtainedwith different experimetalset-ups. Diffusing agent
n
Fe3’ Fe3+ Fe3+ Fe3+ Fe3+ Fe3+ Fe3” Fe3+ Fe3+ Fe3+
3 3 1 1 2 1 2 1 1 2
Fe” Fe’” Fe3’ Fe3+ Fe3+
1
Fe3’ Fe3+ Fe” Fe3” Fe3+
2
Gel type and concentration(%) a 1.5 a 1.5
Other constituents(r&I)
al agar 1
S 50, Fe” 0.5 S 100 Fe”05 S 200: Fe’+ 015 x0 0.1 s 50, x0 0.25 S 50 and 100 Fe2’ 0.5 S 50, Fe” 0.5 S 200, Fe’+ 0.5, X0 0.2 S 50 and 100, Fe*+0.5, X0 0.1 and 0.25 S 12.5 S 25 S 50, NaCl 1
a 1.5 al al 84 84
S 50 Fe’+ 1 S = ‘30*, Fez+ 1 S - 30* Fe’+ 1 S 26, Fe’+ S 26, Fe’+, X0 0.02
a a a a g a
1.5 1.5 1.5 1.5 10
1.5, g 3 a 1, g 2 a 1.5, g 3 a 1.5 g5 andg 10 al
D (10e3cm* hr-‘) (mean + SD) 14 k3 20 +5 22 9 6.5 rt 2 11 5 kl 9 9 3 +1
5 4000ms, TI 320ms, TE 7.5ms, flip angle 15”). The agarose gel on the left side of the image is doped with Fe*+ and Fe3+ (“irradiated part”) and the right side only with Fe*+ ( “non-irradiated” background). II. MR Imaging Images were obtained on a 4.7T Bruker MSL 200 NMR scanner using a 1.5 cm diameter MR micro-
KRON
ETAL.
213
Fig. 2. Images 1, 3, 5 and 7 of a dual gel sample as acquired using the fast T, imaging sequence depicted in Figure 3. The respective TI values are 320ms, 960ms, 1600ms and 2240ms. The images were acquired 15 min after the gels have been poured (TR > 4000ms, TE 7Sms, slice thickness lmtn, flip angle 15’). The agarose gel on the left side of the image is doped with Fe’+ and Fe3+ (“irradiated part”) and on the right side only with Fe’+ ( “non-irradiated” background).
imaging probe at room temperature (T = 22” t 2”). As the temperature exchange in the small aqueous samples was fast, the samples were at this temperature when imaging took place. An inversion recovery sequence without gradients was employed in the dose response experiments to determine bulk T1 of the samples ( 16 points, 4 NEX). In the diffusion experiments, a Look-Locker type fast T1 imaging sequenceU was employed. The pulse sequence used is depicted in Figure 3. This sequence acquires a gradient echo at different times after the inversion of the spin system. A small flip angle is used for each interrogation to minimise the perturbation of the recovery process. The recovery in this case depends on the flip angle LY,the time difference between two read-out pulses td and the relaxation time T, . Therefore, the recovery is governed by a modified relaxation time T,* which is related to T1 as follows:24 td
T; =
(1) (tdfll)-ln(cosck!)
In the present study a! was set to about 15” and td
Magnetic Resonance Imaging 0 Volume 15, Number 2, 1997
214
with each other. An example Figure 4.
xgradient
of this is shown in
ygradiint
When paramagnetic ions are present in the vicinity of water protons, the relaxation rate l/T, is directly proportional to the concentration of the dopants according to: 25
I
zgradient
Imp 1: ph8se c
emzodng
steps,
n = 128 l
kmp 2: TI irtcmneti,
n =8
Fig. 3. Schematic diagram of the fast ‘Ii imaging pulse sequence used in the study. The timing is shown in the bottom
line with arrow heads indicating loops.
to appro~mately half T1. With these settings TT is approximately 7.5% shorter than Tr . Eight images were acquired after a single inversion pulse using the Look-Locker sequence at different times after the inversion of the magnetisation (TI). The spatial resolution was 128 X 64 (frequency X phase encoding) pixels. Figure 2 shows the images 1, 3, 5 and 7 out of such a sequence at different times after the inversion. The two gel phases can be clearly distinguished by their different recovery after the inversion. AU images displayed are magnitude images and the grey scale in the four images depicted in Figure 2 is identical. The signal intensities of all images were transferred to a personal computer (IBM compatible 486) and spatial signal averaging performed using in-house software. The program developed allows specification of any rectangular region for the spatial averaging, however in most cases a 4 x 16 matrix from a 128 X 64 pixel image was found to be a good compro~se between optimal spatial resolution and processing time for the T; c~c~ation. The latter was pe~o~ed using a three parameter fit for the signal intensity in all eight images using SigmaPlot (Jandel Scientific) software on an IBM compatible 486 PC. If T1 values do not vary too much, T1 and T; can be regarded as proportional to each other. As such, no correction was made for the difference between T1 and TT. For simplicity of notation the asterix will be dropped from T, throughout the rest of the manuscript. The T I imaging sequence was repeated at various times t after the gels were brought into contact
with c as the concentration of the dopant ions. In the case of dosimetry gels this could be Fe3+ or Fe2+ or both in which case the respective cont~butions are The additive (R, c = RiFe2+ c&2& + RIFe3+ c&3+)relaxivity Ri is approximately 20 times larger for Fe3’ than for Fe2+ ‘,*. When investigating the diffusion of Fe3+ all parameters except for the Fe3+ doping are identical on both sides of the dual gel system. Therefore, l/T1 is directly proportional to the Fe3’ concentration. The variation of l/T1 across the junction between
Fig. 4. Four images of the same dual gel system at different times after pouring the gels. Shown is the second image (TI 64Oms) of the fast T, sequence for each time. Gels and all other parameters were as in Figure 2.
Fast T, imaging* T. IRON
215
ETAL.
between the gels x0. The latter was typically well below lmm and allows for the fact that it was usually impossible to visualise precisely the position of the junction between the gels in the MR image. As the zero location depends on the exact position of the container in the NMR coil its initial guess is affected by a degree of uncertainty. Therefore (x-x0) replaced x in Equation 3 for the fitting procedure. The value of x0 should stay constant for consecutive images of the same gel. This was checked in the evaluation, An illustration for the fit is shown in Figure 5 which shows data from the dual gel sample shown in Figure 2. The error bars indicate 2 a single standard error of the fit of Tit and the magnitude of the spread of the T, values is typical for most experiments. If one dimensional diffusion is assumed, the curvature parameter n is proportional to the time after pouring as shown in the Appendix. The one dimensional diffusion of paramagnetic ions across the junction between the two gels can be determined from the slope of the change in n with time by:
-
distance from the jun~ion x (mm) Fig. 5. Fit of the spatial variation of 1/T, in the dual gel sample shown in Figure 2 using the inverse square root function given in equation 3. The steepness (slope) of the step depends on the doping of the two gels and will decrease due to diffusion with increasing time after the two gels have been brought into contact. The error bars represent the standard mean error for T, as derived from the fit of T1.
D = (3% - I>n - = O.l27~[cm’/ho~] 4ln2 t
This is demonstrated in Figure 6 which shows the
q
. 0 -
the two gels was fitted using an inverse square root (ISQR) function:
where x is the distance from the boundary between the gels (x = 0), Tf and T? are the relaxation times of the two gels at time t = 0 and n(t) a fitting parameter which describes the curvature of the step. The smaller n is, the steeper is the step in the curve. A similar function has been used to describe the conformational transition of very long chains of biopolymersz6 and the dose ~s~bution at the pen~bra of megavoltage photon beams. 27The ISQR function was used instead of the theoretically more correct error function as it is analytical and can easily be fitted to the data as discussed in the Appendix. It was fitted to the variation of l/T1 with position for different times after pouring of the gel. Four parameters were fitted: l/T?, (l/T? - 1 /T?), n and a potential mismatch of the junction
(41
0 L
0
~--
1st experiment and regression 2nd experiment and regression 3rd experiment and regression
50
100
150
200
250
time after pouring the gels t (min) Fig. 6. Variation of the curvature parameter n in equation 4 with time after pouring a dual gel sample.This is given for three experimentswith identical gel compositions.
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Magnetic Resonance Imaging l Volume 15, Number 2, 1997
variation of the curvature parameter n in equation 4 with time after pouring a dual gel sample. When the gels are initially brought into contact one would expect an infinitively steep step. As such one would expect that all regression lines shown in the figure would go through the origin. However, this was not required in the fit as the gel temperature was higher directly after pouring the gels. This and other potential initial boundary effects, such as an iron depletion at the very surface of the first gel, will affect the initial diffusion. Figure 6 shows the results for three experiments with identical gel compositions. The error bars indicate 2 a single standard error as derived from the fit of n using Equation 3. The spread of n values increases as time evolves. This was reproduced in other experiments and may be due to the fact that in this case the assumption of one-dimensional diffusion in a “large” container breaks down. In general, the data spread and the reproducibility depicted in Figure 6 is typical for the experiments performed and is reflected in the uncertainty of the diffusion coefficients given in Table 1. IV, Dose Response Assessment In the dose response measurements the whole sample was irradiated and bulk Tr determined using an inversion recovery sequence as detailed above. The gels were irradiated in plastic test tubes (inner diameter 9mm, wall thickness lmm) using a Philips RT 100 superficial irradiation unit. Superficial x-ray radiation was chosen because of its high dose rate (approximately lOGy/min) and its easy accessibility. A radiation quality 2.4 mm aluminium (Al) half value thickness ( 100 kVp, 1.7 mm Al Filter, 8 mA) was chosen. All experiments were made using an open end steel cone (7 X 7 cm2) at a focus to surface distance (FSD) of 10 cm. The dose distribution throughout the sample was homogenous within &lo% of the stated dose which refers to the mean dose in the sample. The dose distribution in different samples and the accuracy of the mean dose delivered within each sample was within 55% of the stated dose. In the evaluation it was assumed that a bulk T, measurement was integrating over all iron ions which would be a measure for the mean dose. RESULTS
AND DISCUSSION
I. ~re~uruti~n of Gels Preparation of the gels using a microwave oven proved to be easy and fast. While microwaves should not affect agarose gels, it is possible that they may denature proteins in the gelatine gel. This was not specifically investigated in the present study. The dual gel system is easily prepared without air bubbles using the
syringe filling technique. It mimics the an infinitesimaly steep dose gradient in a dosimeter gel. Due to lateral secondary electron disequilibrium2’ and photon scatter this cannot be achieved using a blocked photon beam. If these Iimitations are taken into account, also a blocked photon beam can be employed for diffusion measurements in larger samples as recently demonstrated by Baldock et a1.29 The boundary between the two gels in dual gel samples is difficult (if at all) to ascertain on an MR image as can be seen in Figures 2 and 4. However, even several days after the experiment the gels could be separated at their boundary. It appears that a thin layer of water separates the gels and there is no cross linking of the polymer chains between the two. Agarose forms gels upon cooling by forming a double helix structure. Several of these aggregate to form fibres some 3 to 6nm in diameter.30 The pores between these are up to 3OOnm in size. As such, small molecules and ions with a thin hydrate shell would perceive agarose as an open porous medium.“’ In this context, it was assumed that the effect of a thin boundary between the two gels could be neglected. In the present experiments, straight gelatine gels were slow in setting in the presence of sulfuric acid and the resulting gels had limited mechanical strength. This is in contrast to other reports in the literature5’9 and may be contributed to the relatively high temperature of 70°C used to melt the gelatine. Also, the use of a microwave oven for the production of gels which has been verified for agarose gels”j could have denatured proteins in the gelatine gels. These effects were not further investigated in the present study. All gels including sulphuric acid proved difficult to set once reheated. As can be seen in Table 1 only small concentrations of xylenol orange were added to the gels. However, the resulting colour change indicated the oxidisation state of the iron in the solution clearly (only Fezi : orange to yellow; Fe’+ and Fe3+ : blue to purple). The difference in colour also confirmed that even in the ten fold presence of Fe”, Fe3+ was efficiently bound by xylenol orange. ZZ.Fast T, Imaging The gels used were doped to yield a T1 of the order of one second for the unit-radiated sample containing only Fe’+. This results in T1 values between 300ms and 1OOOms throughout a typical irradiated sample, a range of relaxation times which is well suited for T, measurements. With a repetition time of about 8 s, a spacing between images of 0.3 s and two averages (2 NEX) the acquisition of the eight data sets with a spatial resolution of O.lmm in a 12mm sample took
Fast T, imagingl T. %oN =AL.
appro~mately 12 min. This has to be compared to normal T, measurements using either multiple inversion recovery images with varying T14 or spin-echo images with varying TRs. 8*31As at least four images have to be acquired to allow a three-parameter Tr fit the time required would be of the order of 30 min. Another way of reducing the acquisition time in NMR dosimetry has recently been proposed by L.J. Schreiner, et al.’ Using a magnetization to dose (M& D) conversion curve this group has demonstrated that it is possible to convert the image intensity of a single image to dose values by establishing a relevant calibration curve. In this technique only one image with a short to medium repetition time is required which reduces the acquisition time even further than in fast T, measurements. The advantage of the fast Tl sequence proposed here is that more raw data is utilised for the dose dete~ination which bears the potential for better reproducibility. Similarly, the present approach does not rely on a calibration curve as T1 changes are directly related to dose according to Equation 2. It has previously been debated whether changes of T, or Tz would be better used to evaluate chemical change due to ionising radiation.4*3” Even if T2 determination from images is generally less reliable33 it has been suggested that this disadvantage would be outweighed by a higher dose response and the possibility of assessment of T2 from multi-echo images in a single acquisition4*5Y” I It appears that at least the latter argument against the use of T, maps would be overcome by the use of a fast T1 imaging sequence such as the one depicted in Figure 3. It has also been suggested in the past that quantitative evaluation of relaxation times may have an important role in the clinical diagnosis of a variety of diseases. 35-36As such, clinical scanners of the future may offer a fast T 1 imaging sequence which could also be employed for NMR dosimetry. III. DifSuusion Measurements Figure 7 shows a typical variation in the shape of the 1 IT r fit with ongoing diffusion observed at different times after bringing two gels in contact. Both gels investigated are straight agarose gels with the gel on the left side containing Fe3+. The 1/T, curves are from the gel depicted in Figure 4 and the values for the gel system 15 min after pouring were calculated from eight images with differenl: TIs, four of which are shown in Figure 2. Due to the small sample size and the relatively fast diffusion no meas~ments were taken later than four h after bringing the gels into contact. As can be seen in Figure 7, already 117 min after pouring the second gel some of the Fe3+ had diffused throughout the whole
217
0 -5
0
5
distance from the junction x (mm) Fig. 7. Variation in the shape of the l/T, fit with ongoing diffusion observed at different times after bringing two gels in contact. T, values were obtained from the images of the gel pictured in Figure 4. The gel with the short T1 (left side of the Figure) was in contact with air. sample. The assumption of one-dimensional diffusion between semi-in~nite slabs of gel is then invalid and an additional error introduced in the inverse square root fit. This may also be reflected in the large scatter of the points in Figure 6 more than 150 min after the gels were poured. In other gels with lower diffusion coefficients (see Table 1)) the experiments continued for a longer period of time. Figure 7 also shows a general increase of the baseline of l/T1 with time. This is most likely due to spontaneous oxidisation of Fe2+ which appears to be equal throughout the sample. The gel on the left hand of the figure with the shorter T, was exposed to air. In most experiments the spontaneous oxidisation was less noticeable even if no particular steps (e.g. working in dim lighting’) were taken to reduce this. The diffusion coefficient for one-dimensional diffusion was calculated from the change with time of the shape parameter n describing the 1/T r distribution detailed in the materials and methods section. Table 1 lists the diffusion coefficients for all dual gel experiments performed. The influence of the choice of imaging and evaluation p~meters was tested by varying the parameter and evaluating its influence on the calcu-
218
Magnetic Resonance Imaging l Volume 15, Number 2, 1997
lated diffusion coefficient. Neither the choice of the interval between different images on the inversion curve nor the size and shape of the regions of interest (ROI) in the T i evaluation had a significant impact on the results as long as they were kept within reasonable limits. For most experiments the TI increment was chosen to be of the order of 300ms. With a range of T, between 300ms and 1OOOms no discernible difference in the diffusion coefficient could be seen by altering TI to 150ms. The ROI size and shape is a compromise between obtaining maximum spatial resolution and minimising noise and evaluation time. It was typically chosen to average four points along the sample axis resulting in a spatial resolution of 0.4mm. For the diffusion coefficients observed in most samples this proved to be satisfactory. Averaging less points improved the spatial resolution but it did not impact on the results obtained in the ISQR function fit. The results given in Table 1 for the diffusion coefficient are the mean + single standard deviation of multiple experiments using the same gel compositions. Systematic errors arise from the fact that diffusion in the dual gel samples is treated as one dimensional between semi-infinite slabs of gel. In reality, the gel samples are relatively small to fit the MR scanner and achieve the desired spatial resolution. Therefore, the limited reservoir of Fe3+ and edge effects will impact on the measurement. However, the mean dislocation of an ion with a diffusion coefficient of 0.02cm2 h-’ is less then 3mm ([ 2Dt] -I”) which is still small compared to the container size of 12mm. As such, the calculations presented in the Appendix can be considered to be valid at least up to experimental times of two h. In addition to this, the aim of the present study was primarily to assess the image degradation due to diffusion in NMR dosimetry samples and compare different gel compositions rather than obtaining accurate diffusion coefficients. This justifies also the use of the ISQR function for the fitting process instead of the more correct error function. Interestingly, the ISQR function provided generally a better fit than various analytical solutions for the error function. This could possibly be due to the softer edges in the ISQR function which would suit the boundary conditions better. Table 1 lists also data compiled from the literature for comparison. For similar gels these results are of the same order of magnitude. It should be noted however, that the data from the literature was obtained with a variety of different experimental designs, none of them involving dual gel samples. As demonstrated by the work of Baldock et al. also temperature has an impact on diffusion in dosimetry gels.*’ The results shown in Table 1 reveal a tendency of decreasing diffusion with reduction of the sulfuric acid
concentration. Also the addition of a chelating agent such as xylenol orange can slow down diffusion, in agreement with results obtained by W. Rae, et a1.23As stated by J. Hazle, et a1.6diffusion is slower in gelatine gels but these gels tended not to set properly when the acid required for dosimetry was added. Therefore, a mixed agarose and gelatine gel was produced which has the favourable diffusion properties of gelatine while retaining mechanical strength. IV. Dose ResponseAssessment
Besides diffusion, dose response is an important consideration in gels used for NMR dosimetry. This is reflected in Table 2 which lists the dose response of various NMR dosimetry gels. It is expressed as change of l/T1 per 1000 Gy delivered dose in s-l kGy -‘. The dose response in gels proved to be difficult to reproduce as it depends on a variety of different factors such as gel composition, acidity, oxygenation/aeration, magnetic field strength and time between irradiation and measurement. 3,17Due to the multitude of parameters involved in NMR dosimetry it is most useful as a relative dosimetric technique for the mapping of dose distributions or the comparison of doses. This requires the knowledge of background and dose response for the gel which is best established for each individual batch of gel samples. This approach is similar to the magnetization to dose conversion from signal intensities.9 However, the evaluation of relaxation times in the present technique only requires one calibration point as opposed to a whole calibration curve. Table 2 also contains data from the literature. It has to be noted that they were acquired at a different field strength which would influence the dose response.“17 In general, the dose response of gelatine gels is lower than that of agarose gels with the same doping. This can be attributed to a lower G ( Fe3+ ) value in gelatine gels 37 and to a decrease of acidity due to gelatine38 which would also affect the dose response. ” The dose response given in Table 2 can be compared to a dose response of 200 s-l kGy -’ achieved by T2 measurements recently Gambarini 1994 in the same Fricke based gel systems and 250 s -’ kGy -’ observed in a new type of NMR gel dosimetry system, Bis Acrylamide Nitrogen Gelatine (BANG) .39 CONCLUSIONS Dual gel samples were found to be well suited to study the one dimensional diffusion of ions in gels similar to the diffusion encountered in NMR dosimetry at steep dose gradients (e.g. beam penumbra). The use of a fast Tl imaging sequence reduces acquisition time and proved to be useful in these experiments. It was
Fast T, imaging * T. KROWETAL
219
Table 2. Doseresponsefor various gel systems Gel type and
concentratian (%)
Other constituents (mM)
n
a 1.5
S 100 Fe2’ 0.5
g 10
S iO0: Fe’+ 1
3 3
g 10
S, Fe2+1, X0 S, Fe”” 1
2
S 1003 Fe*+ 1 X0 0.2 various combinations
1
g5 g 5, a 1
a
al
al
2
S 30, Fe”+ 0.7
Field strength (T) 4.7 4.7 4.7 4.7 4.7
84
S 50, Fe” 1.5, NaCi 1 S SO,Fe2+1, NaCl 1 S 50, Fe’+ 1.5, NaCl I S 125 Fe”05 S 50, be’+ l.;, NaCl 1 S 50, Fe’* 1.5, NaCl 1
g8
S 50, Fe2+ 1.5, NaCl 1
0.25 0.25
g5
Fe’+ 2 0.5, NaCl I S 50, Fe’+ 1, NaCl 1 S 26, Fe*” S 26, Fe’+, X0 0.2
1.5 1.5 1 1
al
al a 1.5
a 1.5
83 84 g4
0.5 0.4
1.5 0.5
(s-l kGy-I)
Commentsor reference
standard agarose dosimetry gel
80 + 10 423
11 24 13 f 2 14
0.5 and 2 0.2 0.25
Dose response
17
up to 120 48 74 47 zt 8 66
20 37 42 43 19
84
31
108 41 31 42 22
37
12.9
23
37
6 4
9.3
23
Expressed as change of l/T, per 1000 Gy delivered dose in [s.“ kGy-‘I. All gels listed in the first part of the table were aerated prior to setting. a = agarose, g = gelatine, S = H,SO,, X0 = xylenol orange, n = number of experiments.
found that diffusion of the irradiation product Fe3+ could be reduced by adding xylenol orange and gelatine to agarose gels which provide mechanical strength. As such, we propose that a gel consisting of 1.5% agarose, 3% gelatine and 0.1 mM xylenol orange forms a suitable base for NMR dosimetry. In addition to reducing diffusion, xylenol orange provides a visible evaluation of the radiation effects prior to MR imaging. Acknow~ed~e~enfs-We would like to thank the Australian National Health and Medical Research Council for financial support (project grant 930734) and Patrick Burke and Vas Rajanayagam for help with the meas~ments and many discussions. We are also indebted to the Department of Radiation Oncology in the Prince of Wales Hospital, Sydney for the use of their superficial radiation treatment unit and Nathalie Perret for helpful hints regarding the error function.
The diffusion coefficient D can then be calculated as40:
s x
e xl2V’Dt
The solution of the integral is the error function which is defined as:
Using the error function complement z Equation 5 can be written as:
erfc z = 1 - et-f
APPENDIX: CALCULATION OF THE DIFFUSION COEFFICIENT It is assumed that the data gathered from regions of interest (ROIs) located along the central axis of the cylindrical sample represents the diffusion data of ferric iron on a homogenous and time inde~ndent background of Fe2+. In this case 1 /T? - 1/T? = C is proportional to the Fe3+ concentration. If one assumes further that the dual gel experiments represent the diffusion between semi-infinite sources brought together at time t = 0, the border conditions at t = 0 can be given as C = Co for x < 0 and C = 0 for x > 0.
C(x,t)
= 2 erfc -&2di5i
(7)
The error unction and its complement are not analytical. Several analytical approximations can be given41. However, while polynomial approximations typically fit the step well they do not approximate the horizontal parts of the data shown in Figure 5 appropriately. Therefore, the inverse square root (ISQR) function given in Equation 3 in the text was used for the fitting
Magnetic ResonanceImaging0 Volume 15, Number 2, 1997
220 of the step. The derivative function can be given as:
n
d(lflt)
s =-
of the inverse square root
dx
=
‘4x2
+
---
(8)
@3/2
This describes a peak function which are similar to the Gaussian peak function under the integral of Equation 5. The ISQR peak function can be integrated and yields the ISQR function which was used for the fitting (Equation 3). If one assumes that the FWHM of both functions is related equally to the diffusion coefficient one can find:
= 2&&4-l
(9)
or
4ln2
t
0.127:
[
cm2/hour
1 .
(4)
This is the equation used to derive the one-dimensional diffusion coefficient D from a plot of the curvature parameter n versus time t.
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