Bayesian estimation of relaxation times T 1 in MR images of irradiated Fricke-agarose gels

Bayesian estimation of relaxation times T1 in MR images of irradiatedFricke-agarose gels

Francesco de Pasqualea,b,* Giovanni Sebastiania, Emmanuel Eggerc, Laura Guidonib,Anna Maria Lucianib, Pasquina Marzolad, Riccardo Manfredie, Massimiliano Paciliob,

Angelo Piermatteif, Vincenza Vitib, Piero Baronea

aIstituto per le Applicazioni del Calcolo, Consiglio Nazionale delle Ricerche, viale del Policlinico 137, Rome, ItalybLaboratorio di Fisica, Istituto Superiore di Sanita` and INFN-Sez. Roma 1, Gr. Coll. Sanita`, Rome, Italy

cDepartment or Radiation Medicine, Paul Scherrer Institut, Villigen, SwitzerlanddIstituto di Anatomia e Istologia, Universita` degli Studi di Verona, Verona, Italy

eIstituto di Radiologia, Universita` Cattolica del Sacro Cuore, Rome, ItalyfIstituto di Fisica, Universita` Cattolica del Sacro Cuore, Rome, Italy

Received 18 January 2000; accepted 29 March 2000

Abstract

The authors present a novel method for processing T1-weighted images acquired with Inversion-Recovery (IR) sequence. The method,developed within the Bayesian framework, takes into account a priori knowledge about the spatial regularity of the parameters to beestimated. Inference is drawn by means of Markov Chains Monte Carlo algorithms. The method has been applied to the processing of IRimages from irradiated Fricke-agarose gels, proposed in the past as relative dosimeter to verify radiotherapeutic treatment planning systems.Comparison with results obtained from a standard approach shows that signal-to noise ratio (SNR) is strongly enhanced when the estimationof the longitudinal relaxation rate (R1) is performed with the newly proposed statistical approach. Furthermore, the method allows the useof more complex models of the signal. Finally, an appreciable reduction of total acquisition time can be obtained due to the possibility ofusing a reduced number of images. The method can also be applied toT1 mapping of other systems. © 2000 Elsevier Science Inc. All rightsreserved.

Keywords:Relaxation times; Relative dosimetry; Fricke-agarose gels; Bayesian statistics

1. Introduction

Quantitative knowledge of relaxation times is a relevantinformation in different applications of magnetic resonanceimaging (MRI), e.g., for discriminating among pathologicand normal tissues in clinical applications [1], for optimiz-ing contrast [2], for monitoring oxygenation state of bloodand tissues [3], for perfusion studies with either exogenousor endogenous contrast agents [4], and for studying thestructure of materials [5]. Another application of preciseMR relaxation times measurements has been proposed inthe field of relative dosimetry [6,9]. In fact, gel systems,either Fricke-agarose gels [6,7] or polymeric gels [8,9],

change their MR relaxation rates upon irradiation in a lineardose-dependent way. As a consequence, these gel systemshave been proposed as three-dimensional dosimeters to ver-ify radiotherapeutic treatment planning systems. Our atten-tion has been devoted mainly to the study of Fricke-agarosegels, as the possibility of modifying gel composition andpreparing phantoms with cavities of different densities tosimulate different tissues, seemed very attractive. The dif-fusion of ferric ions is still the main limiting factor inFricke-gel systems, and the use of fast sequences and ofmathematical corrections would be of great help in solvingthis problem. It has been also proposed to add XylenolOrange to the gel, as this molecule partially hinders diffu-sion by chelating ferric ions [10]. In this case, the limitedsensitivity of the Fricke-gel systems, shared with otherpolymeric gel systems, is further reduced.

To cope with the problem of the low signal-to-noise ratio

* Corresponding author. Tel.:11-39-06-88470236; fax:11-39-06-4404306.

E-mail address:[email protected] (F. de Pasquale).

Magnetic Resonance Imaging 18 (2000) 721–731

0730-725X/00/$ – see front matter © 2000 Elsevier Science Inc. All rights reserved.PII: S0730-725X(00)00149-1

(SNR) of these gel systems, mainly due to both low Fricke-gelsystem sensitivity and random noise in the images, we havedeveloped a novel method based on the Bayesian approach toobtain spatial distribution of relaxation times. Bayesianmethods have been widely applied in image processing (fora review, see [11]) and start appearing also in the field ofMRI [12,14]. Some examples can be found also in the fieldof image restoration [15,16], but few have been developedto the purpose of retrieving relaxation time maps [17].

In the first part of this article, we present the experimen-tal procedure and the mathematical basis of the new ap-proach. In the second one, we show and discuss someresults on the application of the method to estimate relax-ation rateR1 images from Inversion-Recovery (IR) imagesof irradiated Fricke-agarose gel samples. A comparison ofthe results with those from a standard approach is alsoperformed. To show the performances of the method indifferent situations, two series of data have been processed.In the first series, gel samples were irradiated with a mod-ulated proton beam. Therefore, irradiated and not irradiatedgel portions were present in the sample at the same time,providing a wide spread ofR1 values. T1-weighted IRimages were acquired with a research instrument operatingat 4.7 T. Samples containing Xylenol Orange (“Xyl”) andwithout it (“Noxyl”) were both measured. In the secondseries, the samples were irradiated with a therapeutic x-raybeam. T1-weighted IR images were acquired with a wholebody clinical 1.5 T system, providing much lower SNR thanin the former case. The reduction of random distortions inthe estimatedR1 maps, without appreciable systematic dif-ferences from the standard maps, and the possibility ofusing more complex models are the main advantages of theproposed method.

2. Experimental procedures

All reagents were purchased from Sigma Chemicals (St.Louis, MO, USA). Agarose powder (type VII) was dis-solved in triple distilled water and brought to boiling until aclear solution was obtained. It was subsequently oxygenatedby bubbling the gas through it until it cooled down to 36°C,then Fricke solution was added. All the steps after boilingwere conducted under constant stirring. The solution wasthen poured into cylindrical Perspex holders and allowed toset at room temperature. The final concentrations were: 1.5mM ferrous ammonium sulfate, 1 mM sodium chloride, 50mM sulfuric acid, and 1% agarose by weight. We will callthis sample “Noxyl” in the followings. Samples containingXylenol Orange were prepared with the same procedureadding Fricke solution containing the chelator. In this gel,the final ferrous sulfate concentration was 1 mM and thefinal Xylenol Orange concentration was 0.25 mM. This gelwill be referred to as “Xyl” in the followings.

Some of the gel samples had a diameter of 45 mm and aheight of 40 mm. They were kept at room temperature and

irradiated from 20 to 28 h after preparation with a 62 MeVmodulated proton beam (penetration in water 3.2 cm) at adose rate of 1 Gy/s for doses ranging from 0 to 30 Gy. Thebeam had a circular section with a radius of 1.7 cm and itsaxis was allineated to the one of the sample. The beam wasmodulated to give a 30% “spread out Bragg peak” (see [18]for the relative depth-dose distribution). A second series ofsamples had a diameter of 7 cm and a height of 19 cm. Theywere irradiated with a therapeutic 6 MV x-ray beam at doserate of 2 Gy/min with a maximum dose of 30 Gy (see [19]for qualitative depth-dose distribution). The beam had asquare field of 43 4 cm2 and its axis was allineated withrespect to the sample axis.

The images of the first series of samples were acquiredwith an MR system Sisco-Varian (Varian associated, Inc.,Palo Alto, CA, USA) operating at 4.7 T. An IR sequencewas used with an echo time TE of 30 ms and a delay timetD of 3 s. Six IR times TI of 40, 80, 200, 400, 800, and 4000ms were used. The field of view was 80 mm by 80 mm onan acquisition matrix of 1283 256. Coronal slices wereacquired with a thickness of 0.2 cm by using a saddleshaped coil with a receiver bandwidth of 35 KHz. Thenumber of excitations (NEX) was equal to 2. The total timefor the acquisition was approximately 51 min. We note that,when processing only the three images with TI of 80, 800,and 4000 ms, the acquisition time is reduced to 30 min. Theimages of the second series of samples were acquired witha whole body system GE-Echospeed (General Electrics,Milwaukee, WI, USA) operating at 1.5 T. The IR times usedwere 100, 200, 300, 400, 500, and 600 ms, and the acqui-sition matrix 2563 256. Axial slices were acquired with athickness of 0.5 cm by using a head coil. The NEX wasequal to 1. The total acquisition time was the same as beforewhile the repetition time was 3 s and the echo time was 20ms. The pixel dimensions in this case were 0.973 0.97mm2. All images correspond to a longitudinal slice along aplane containing the axis of the sample.

3. Bayesian image data analysis

The image data consist here ofm 2563 256 matrices ofpositive integer numbers in the range (0,255) correspondingto different inversion timesTIi (i 5 1, . . . ,m). The modeladopted to describe the “measured” MR imageyi in terms ofthe “true” MR imagexi is:

y i 5 x i 1 j i , (1)

wherexis is the value ofxi at thesth pixel, s 5 1, . . . , S,S 5 2562 andj is zero mean white noise. The “true” imagexi can be represented by means of a suitable physical modelfi:

xis 5 fi~R1s, h1s, . . . , hls! , (2)

whereR1s is the sth component of the longitudinal relax-ation rate imageR1. The quantityhks is the sth component of

722 F. de Pasquale et al. / Magnetic Resonance Imaging 18 (2000) 721–731

hk (k 5 1, . . . , l), which represents one of the otherlparameters on which the physical model depends. Amongthese parameters, there is an amplitude factor depending onboth the nuclear spin density and the transverse relaxationrate. Some other correcting factors take into account imper-fect experimental conditions. The explicit form offi and themeaning ofhk will be discussed later in this section.

The image pixel valuesyis are the modulus of the ac-quired complex MR images whose real and imaginary partsare affected by additive white Gaussian noise with meanzero and variances2. The noise in the modulus imagefollows a Rice distribution [20,21]. Hence, the conditionalprobability ofy 5 y1, . . . , ym) givenR1 andh 5 (h1, . . . ,hl), is given by:

P~yzR1, h! 5 Pi51

m Ps51

uSu yis

s2

z expH2@ yis 2 fi ~R1s, h1s, . . . , hls!#

2

2s2 Jz I0S yis fi~R1s, h1s, . . . , hls!

s2 D , (3)

whereI0 is the modified Bessel function of first kind [22].The image data model in Eq. (3) is adopted here for eachpixel where estimation is performed.

In the “background” regions, i.e., where the expectedvalues of the real and imaginary parts are zero, the noise isRayleigh distributed [21]. In the classic approach the imagemodel in Eq. (3) for a specific functionf is maximized withrespect toh. Due to the form of Eq. (3), the maximization isperformed separately for each pixel. This approach is usedhere to compare the results from the proposed method.

In the Bayesian approach, some knowledge about thequantities to be estimated, available before data acquisition(the a priori knowledge), plays a fundamental role in theestimation process. We can take into account this informa-tion by consideringR1 andh to be random, following theprior distributionP (R1, h):

P~R1, h! 5 PR1~R1) Pk51

l

Ph~hk!. (4)

for the prior distribution in Eq. (4), we used

PR1~R1! } expF2bR1Oî, j&

~R1i 2 R1j!2G

Ph~hk! } exp2 FbhOî, j&

~hki 2 hkj!2G , (5)

where the sum is extended over all the pairs of pixels thatare horizontal and vertical first order neighbors andbR1, bh

are two positive parameters [23]. The distribution in Eq. (5)is a Gibbs distributionP } exp [2b¥î,j&Uij ] with potential:

Uij 5 UR1~R1i 2 R1j! 5 bR1 z ~R1i 2 R1j!2. (6)

This model penalizes configurations with large differencesbetween the values ofR1 andhk in neighboring pixels andreaches its maximum value for constant configurations. Inthis way, we take into account a priori knowledge about thespatial regularity of the parameters to be estimated.

We can now obtain, using the Bayes theorem, the pos-terior probability distributionP(R1, hzy) that is the condi-tional probability ofR1 andh given the measured datay1,. . . , ym:

P(R1, huy) 5P(yuR1, h) P(R1, h)

P(y). (7)

A convenient way to estimate the quantitiesR1 andh consistsof calculating their mean values underP. Such an estimator iscalled Minimum Mean Square Estimator (MMSE):

R1MMSE5 ^R1& 5 ER1,h

R1 P~R1, huy!dR1 dh

hMMSE5 ^h& 5 ER1,h

h P~R1, huy! dR1 dh. (8)

The exact calculation of Eq. (8) is impossible because of thedependency among variables and of their very high number.However, we can obtain a good approximation of Eq. (8)using Markov chain Monte Carlo simulation [24]. In par-ticular, we used the Metropolis Algorithm [25].

A spin-echo IR sequence has been used to acquire theMR images. If the repetition timeTR between two succes-sive p pulses is such that:

TR 2 TIM 2TE

2.. T1M (9)

whereT1M is the largest longitudinal relaxation time of thesample,TIM is the maximumTI used andTE is the Echotime, then an appropriate physical model for the acquiredimage will be:

fk~R1s, rs, bs! 5 rsu1 2 2bs exp $2TIkR1s%u, (10)

wherers is the “weighted” spin density at pixel “s,” whichalso incorporates a factor depending on the transverse re-laxation time, andbs takes into account spatial inhomoge-neity of the Radio-Frequency (RF) field. An extension ofmodel in Eq. (10) is:

fk~R1s, rs, bks! 5 rsu1 2 2bks exp $2TIkR1s%u, (11)

where, the correcting factorsbks (k 5 1, . . . ,m, s 5 1, . . . ,uSu) take also into account possible optimization proceduresof the MR pulses parameters. In this case, we haveh 5 (r,b) 5 (r, b1, . . . , bm).

It is important to note that the model in Eq. (11) couldnot be used to estimate the MR parameters with a classicapproach since the number of parameters to be estimated forevery pixel (m12), is greater than the corresponding num-ber of data (m).

723F. de Pasquale et al. / Magnetic Resonance Imaging 18 (2000) 721–731

The value ofs2 appearing in Eq. (3) has been estimatedby the formula:

s2 5 Oi51

m Ob51

uBu yib2

muBu(12)

whereB is a background region of the image. This is anunbiased estimator fors2 with minimal variance [21].

The “hyperparameter”b in the prior model has to beassigned in some way. Large values ofb determine over-smooth (r, R1, b) images. Conversely, small values ofb

determine only small amount of noise reduction. Here, wefirst obtained an initial guess for theb’s by minimizing thedifference between the energy terms of Eq. (3) and Eq. (4)after algorithm convergence had been reached. Then, withthese values we estimated the MR parameters (r̂, R̂1, b̂)images (“true” images). By using the physical model in Eq.(10) and the noise model in Eq. (3), we simulated theTIkimages and performed from these the estimation forbvalues on a regular grid. Finally, we choose theb’s valuesthat minimize the distance between the “true” and the esti-

Fig. 1. The six T1-weighted IR images (2563 256) for the irradiated Xyl sample. The sample was a cylinder with a diameter of 45 mm and a height of 40mm. The images correspond to a longitudal slice along a plane containing the cylinder axis. The sample was irradiated along its axis with a 62 MeV modulatedproton beam with a radius of 1.7 cm (from top to bottom in the figure). The field of view was 80 mm. TheTI times used were 40 ms (a), 80 ms (b), 200ms (c), 400 ms (d), 800 ms (e), and 4000 ms (f ).


mated R1 images. These values have then been used toperform the estimation based on the real image data.

4. Results

In Fig. 1, the six measured IR images for the irradiatedXyl sample are shown for the differentTIk used. As alreadysaid in the previous section, this sample was irradiated by a62 MeV modulated proton beam (see [18] for the relativedepth-dose distribution). In Fig. 2a, we can observe theestimatedDR1 5 (R1after2 R1before) image, obtained fromthe images in Fig. 1 and the corresponding images acquiredbefore irradiation by the classic approach described in theprevious section with model in Eq. (10). When assuming thephysical model in Eq. (10),h 5 (r , b), l 5 2 and for thesth

pixel the parameters to be estimated areR1s, rs andbs (s 51, . . . ,S). The Bayesian approach can be applied, alwaysassuming model in Eq. (10), in order to increase imageregularity. Data processing with the new method is effectivein reducing noise in theDR1 images, as it is shown in Fig.2b. It is evident the regularization influence of the prior termin Eq. (4) in the estimation process. Since now we do notestimate theR1s values pixel by pixel independently, theestimates are less noisy than with the classic approach. Thisis one of the main advantages of using an a priori model andthis point is really important for dosimetric applications. InFig. 2c, the “difference” image between the classic and theBayesian approach is shown. We note that differences ap-pear with a very small degree of aggregation. This indicatesthat systematic differences are present to a very low extent,as it can be better appreciated in Fig. 3. Moreover, from theFig. 3, we argue that the choice of theb’s values is correctsince the slopes of the two curves are essentially the same.As said before, the choice of a large value ofb would leadto over-smoothed estimates.

In Fig. 4a, we compare a column of one acquired IRimage (TI 5 TI1) and the corresponding simulation calcu-lated from Eq. (3) with model in Eq. (10) with the (r, R1,b) values estimated with the classic approach. The disagree-ment in this figure can also be observed for allTIk but k 54,6. The same amount of discrepancies shown in Fig. 4aappears also when the Bayesian approach with model in Eq.(10) is used (not shown). Moreover, this disagreement isstatistically significant (e.g.,x2/# points5 10.36 for the firstsignal). We modified the physical model in Eq. (10) in orderto eliminate these discrepancies. To this aim, we adoptedmodel in Eq. (11) that is more flexibile since it uses moreparameters. In this case, for each pixels, we hadm 1 2parameters to be estimated andm data since we had toestimaters, R1s andbs 5 (b1s, . . . , bms). At this stage wehadm5 6. As said before, the classic approach could not beapplied in this case. To compensate the lack of data, weused some “a priori” information onbi, modeled as in Eq.(5). The computational cost was in this case very high andthe number of iterations needed for algorithm convergence

Fig. 2. TheDR1 images for the Xyl sample of Fig. 1 from the classical (a)and the Bayesian (b) approach. The difference in image between (a) and (b)is shown in (c). The sample was irradiated with a 62 MeV modulatedproton beam. The direction of the beam is from top to bottom in the figure.


was ten times larger than before. In order to assess thegoodness of model in Eq. (11), we simulated the six mea-sured IR images by means of the model in Eq. (11) on theestimated (r, R1,b) and obtained a very good agreement forevery k. In Fig. 4b the comparison between acquired andsimulated data are shown. The good agreement was alsoconfirmed from a statistical point of view (e.g.,x2/#points 5 0.84 for the first signal). At the moment, thephysical model in Eq. (11) has not strong physical justifi-cation, though we may think that the spatial inhomogeneityof the RF pulses could vary among images acquired atdifferent times with differentTI. Further investigations areneeded to validate or not model in Eq. (11). This is out ofthe aims of this work. We want to point out the possibilityof using more complex models within the Bayesian ap-proach. This could be useful in many situations since thereare many potential sources of systematic errors in the MRexperiment: temperature, diffusion, inhomogeneity of themagnetic field inside the finite sample, etc. [26].

The IR sequence is one of the most suitable for estima-tion of R1 but it has the disadvantage of long acquisitiontime (51 min. to acquire the six images used here). Hence,we selected only three TI values (TI1 5 80, TI2 5 800, TI35 4000). In this way the acquisition time (30 min) isstrongly reduced (41%). The observed ranges of T1 valueswere 0.320.7 s and 0.421.0 s for the Noxyl and Xyl

samples respectively. In Fig. 5, it is presented the compar-ison between a column ofR1after and R1before estimatedimages using six and threeTI images. Only small differ-ences appear.

The method has been also applied successfully to themeasured IR images from the sample without Xylenol Or-ange (not shown).

The whole procedure without any change has also beenused to process IR image data from a sample measured witha whole body clinical 1.5 T MR system. This sample wasirradiated by a therapeutic 6 MV x-ray beam (see [19] forqualitative depth-dose distribution). The results from theclassic and the Bayesian approach using model in Eq. (10)are shown in Figs. 6a and 6b, respectively. In Fig. 6c, thedifference image between the classic and the Bayesian ap-proaches is shown. Due to the smaller value of the staticmagnetic field used in this case (1.5 T instead of 4.7 T), thenoise in the image estimated by the classic approach ismuch larger than before. Furthermore, noise reduction isobtained in this case by the Bayesian method to a largerextent than in the former case. This can be seen by com-paring Figs. 7 and 3.

We also considered the errors associated to the estimatedvalues ofR1. Theoretically, one must replicate the experi-ments many times under the same conditions. Unfortu-nately, this cannot be done, even on the same sample,

Fig. 3. Comparison between a column ofR1after image obtained using Eq. (10) with the proposed (bold line) and the classical method (solid line). Data arereferred to the Xyl sample of Fig. 1 after irradiation with a 62 MeV modulated proton beam. The beam direction is from left to right.


Fig. 4. Comparison between the same column of the first measured IR image of Fig. 1 (solid lines) and the simulated image (bold lines) based on the (r,R1, b) images estimated with the classical method using the model in Eq. (10) (a) and the Bayesian approach using the model in Eq. (11) (b). TheTI timeused was 40 ms. Dara are referred to the Xyl sample after irradiation with a 62 MeV modulated proton beam. The beam direction is from left to right.


because of the diffusion of ferric ions. Repetition of theexperiment on different samples could instead introducesome variability due to different gel preparations and/orirradiation conditions. Hence, we again resorted to simula-tion. The parameter values estimated from the data by theBayesian approach with model in Eq. (10) were consideredas “true” values. From these values we generated “true” IRimages using model in Eq. (10). Noisy images were finallyobtained by adding Rice distributed white noise with thesame standard deviation as calculated on the acquired im-ages. In the same way, we generated IR images using modelin Eq. (10) from the values estimated for the parameters bythe classic method. For each of the two methods, we gen-erated five repetitions of the measured images correspond-ing to independent errors. We then performed the estimationof R1 for each of the five repetitions. The Bayesian (classic)approach was used for the estimation of the Bayesian (clas-sic) based simulated data. Finally, we calculated the MeanSquare Error (MSE):

MSE5(s51

uSu (k515 ~R1s

k 2 R1strue!2

uSu z 5, (13)

whereR1sk is the estimated value for thesth pixel and thekth

repetition andR1strue is the corresponding “true” value. The

values estimated using Bayesian and classic methods of theRoot Mean Square Error (RMSE5 =MSE) for the R1aft

and R1bef are reported in Table 1) for the Noxyl and Xylsamples measured with the 4.7 T system. In this table wereport also the values ofRMSEfor DR1 calculated as:

RMSEDR1Bayes,det. 5 ÎMSEbef

Bayes,det. 1 MSEaftBayes,det. (14)

We note that theRMSEDR1 obtained for the Bayesianmethod is halved with respect to the one relative to theclassic approach. For the data from the 1.5 T whole bodysystem theRMSEDR1 with the proposed method is reducedby a factor of four.

5. Discussion

Uncertainties in relaxation rate measurements of dosi-metric gel systems must be kept as low as possible becausethe practical use of these systems in conformal radiotherapydose verification is prevented by inaccuracy and unprecisiongreater than 3%–5%. Inaccuracies in dose reconstruction byMR may be caused by field inhomogeneities, non-linearityof gradients, eddy currents etc., inducing spatial errors aswell as dose errors. In this paper, we addressed the problemof the stochastic noise present in MR images. The presenceof noise prevents high-resolution dose maps with a goodSNR to be obtained. Long scanning times may help insolving this problem but the unavoidable heating of the

Fig. 5. Comparison between a column fromR1afterand fromR1beforefor the Xyl sample of Fig. 1 estimated with sixTI images (solid line) and threeTI images(bold line). The sample was irradiated with a 62 MeV modulated proton beam. The beam direction is from left to right.


sample possibly causing sample degradation makes theseexperiments not feasible.

In this work, we propose a new method to provideR1maps with higher precision than from classic approach with-out appreciable systematic differences between the mapsfrom the two approaches. This would allow satisfactorydose reconstruction also in the presence of low SNR signalsfrom the Fricke-gel systems. The method is based on theBayesian approach and uses a priori“ information about thespatial regularity of the physical quantities of interest. In theclassic approach, relaxation ratesR1 are estimated pixel-by-pixel and because of data noise, results are affected bypercentage errors greater than those of measured images. Inthe proposed approach, it is possible to retrieve the param-eters of interest, namely the longitudinal relaxation rates,taking into account the a priori knowledge about the spatialregularity of the parameter to be estimated, which allows thenoise to be sensibly reduced. Furthermore, with the Bayes-

ian approach it is possible to use more complex models ofthe signal. Finally, the method allows total acquisition timeto be reduced by estimatingR1 from a reduced number ofimages acquired with different IR times.

The improvement in precision ofR1 maps obtained bythis statistical approach has been evaluated by applying themethod to images of Fricke-agarose gels irradiated withparticles of different quality (namely protons and x-rays)leading to qualitatively differentR1 distribution. Further-more, the IR images were acquired with two MR systemsoperating at different frequencies. By comparing Figs. 2aand 2b as well as Figs. 6a and 6b, one can see that imageprecision is significantly improved in the images processedwith the Bayesian method with respect to the classic one.The regularization effect obtained by the method can bebetter appreciated from the examination of a single imageprofile as shown in Figs. 3 and 7. In the images relative to4.7 T, SNR was increased by a factor of 2 while the effectis even greater for the images at 1.5 T (a factor of 4). Verynotably, the total time of acquisition can be reduced ascomparable results are obtained by using only 3 out of the6 acquired images (see Fig. 5). A further advantage of themethod is the possibility of incorporating a signal modelwith a number of parameters exceeding the number of data.This turned to be particularly important for the data of theexperiments at 4.7 T. In this case, the estimation of thesignal with the simple model in Eq. (10) could not repro-duce the signal at a satisfactory level from a statistical pointof view (Fig. 4). It was beyond the purpose of this paper todraw final considerations on the accuracy in obtaining therelative dose values. Preliminary considerations also takinginto account the diffusion of ferric ions are currently pre-sented elsewhere. We want here emphasize that the im-provement in SNR ofDR1 images will allow to betterexploit the good spatial resolution performances offered byMR systems.

The elaboration technique proposed here can be impor-tant in the dosimetric use of the Fricke-agarose gel system.We want to point out that this method, without any modi-fication, has been applied to the treatment of images ac-quired with different MR systems from samples irradiatedwith totally different techniques, generating different dosedistributions and therefore differentR1 maps. Moreover,this statistical approach can be applied in other fields when-ever exact T1 values are required. On the other hand, theproposed approach requires a much higher computationalcost than the classic one. This could be a problem when realtime image processing is needed. However, the very fastimprovements in computer power will make this problemless and less serious in the near future. In this work, we havetaken advantage of using a priori information about spatialregularity of theR1 maps to be estimated. We have seenwith the examples shown in this paper that this allowed toprovide R1 maps with very small systematic differenceswith the classic approach. The Bayesian approach can beapplied with success also in situations when very steep

Fig. 6. TheR1after image for the Xyl sample. The sample was a cylinderwith a diameter of 7 cm and a height of 19 cm and was irradiated along itsaxis with a 6 MV x-ray beam with a square section of 43 4 cm2. Theimage corresponds to a longitudal slice along a plane containing the sampleaxis. The direction of the beam is from left to right in the figure. (a) TheR1after image estimated by the standard method with the model in Eq. (10).(b) TheR1after image estimated by the Bayesian method with the model inEq. (10). (c) The difference in images between (a) and (b).


gradients are present in some regions of the sample. In thosesituations, a more complex a priori model can be adoptedwhich takes into account both regularity in most of the imageand presence of discontinuities in some regions of it [27].

Acknowledgments

The help of Guido Kuehne in taking images at 4.7 T isgreatly acknowledged. The authors also wish to thank thereferees for their valuable comments and suggestions.

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Fig. 7. Comparison between a row from theR1after of the Xyl sample images in Fig 6, (a) and (b) obtained from implementing the model in Eq. (10) withthe classical method (solid line) and the Bayesian method (bold line). The sample was irradiated with a 6 MV x-ray beam. The direction of the beam is fromleft to right.

Table 1Comparison between the RMSE values obtained for the Noxyl and Xylsamples using the deterministic and the Bayesian methods with model inEq. (10)

RMSE values

Xyl sample

RMSE (s21)

Bayes. met. Det. met.

R1after 0.01 0.03R1before 0.02 0.02DR1 0.02 0.04

Noxyl sampleR1after 0.02 0.05R1before 0.01 0.03DR1 0.03 0.06


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Bayesian estimation of relaxation times T 1 in MR images of irradiated Fricke-agarose gels

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