1989 Calculation of T1, T2, And Proton Spin Density Images in Nuclear Magnetic Resonance Imaging

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    JOURNAL OF MAGNETIC RESONAN CE t&95- 110 ( 1989)

    Calculation of T1, T2, and Proton Spin Density Imagesin Nuclear Magnetic Resonance magingJUWHANLIU, ANTTIO.K.NIEMINEN,ANDJACKL.KOENIG*

    Department ofMacromolecular Science, Case Western Reserve University, C jevekm d, Ohio 441&jReceived September 13, 1988; revised February 27, 1989

    For nuclear magnetic resonance imaging, the spin-latt ice relaxation time T, , the spin-spin relaxation time Tz and the spin density images are computed from a series of T, -and Tz-weighted images on a pixel-by-pixel basis. In the most general ca se, T, and T2values are related to the image intensity nonlinearly. This fact together with the largenumber of pixels involved renders the computation a time-consuming process. The effectof pulse profi le on the intensity should also be considered. A rapid and stable algorithmis presented to optimize the computation process based on a detailed analysis of theintensity expressions. The pulse sequence used is a spin-echo type, and the proposedmethod is applicable to any type of pulse profi le. A series of images taken at differentrepetition times b ut the same echo time are used to calculate the r, image. Similarly,images taken at the same repetition time bu t different echo times are used to computethe T2 image. Finally these T, and T, images are incorporated in the calculation of thespin density image. The slice profile function and its derivatives are tabulated by numeri-cal integration and interpolations are applied when the evalua tions are required. Num eri-cal analyses are given based on the simulations, and experimental results are alsopresented. 0 1989 Academic Press, I nc .

    Flourishing in the field of medicine, nuclear magnetic resonance imaging (NMRI)is also recognized as an important tool of nonmedical applications, especially in ma-terials research (I, 2). In the latter case, the problems related to sample movementare not generally involved, and the imaging time is not a crucial factor shaping theoverall experimental procedure. Thus, one improves the S/N ratio via signal averag-ing. Moreover, other experimental conditions, typical ly temperature control, canreadily be applied to the material samples. As a result of a series of efforts to applythe NMRI technique to the investigation of materials, especially polymers, we reporta systematic approach to extract the relaxation times and the proton spin densityinformation from NMR images. Although the method is generally applicable to med-ical as well as nonmedical images, the simulations and experimental results are givenfor the latter.The images produced by NMRI experiments are determined by various factors.Extrinsic parameters include such operator-controlled parameters as field strengths,radiofrequency pulses, and pulse-sequence timing. On the other hand, intr insic pa-rameters are the proton spin density p, spin-lattice relaxation time Tr, spin-spin

    * To whom correspondence should be addressed.95 0022-2364189 $3.00

    Copyright 0 1989 by Academ ic Press, Inc.Al l rights of reproduction in any form reserved.

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    96 LIU, NIEMINEN, AND KOENIGrelaxation time T2, chemical shift, and flow velocity. The signal intensity is thus aresult of a complex interplay of these various parameters. However, this versatilitycan give a lot of information if well controlled and optimized.

    In NMRI, the positions of resonating nuclei in a sample are encoded by magneticfield gradients. As a consequence, the intr insic parameters are rendered spatially de-pendent making possible the physical and chemical analysis of the sample in termsof the local environments that determine these parameters.In the study of a system in which no flow is involved and the chemical-shift effectis ignored or eliminated, the material-dependent parameters that need considerationare the relaxation times T, and T2 and the spin density p. The distribution of signalsof the images can be directly used to investigate the nature of the detected species.However, information on the intr insic parameters is important when the absolutediscrimination of the resonating species is pursued and/or when one tries to assignoptimal delay times to a given pulse sequence with a view to obtaining maximumcontrast between the species under consideration. Once the values of T, , T,, and pare obtained from one experiment, expected images from other experiments with adifferent set of operator-controlled parameters and/or with different pulse sequencescan by synthesized (3-7). In this way the experimental design of the imaging experi-ment is facilitated.For a given spin-echo or inversion-recovery experiment, a mathematical expres-sion for the signal intensity as a function of the experimental and intr insic parameterscan be derived. In the case of multiple-spin-echo and several other modified pulsesequences (5,8-11) where more than one image is obtained simultaneously, the indi-vidual image intensities can be defined using suitable physical expressions.Here, a numerical procedure is applied to fit the experimental intensities to theappropriate equation on a pixel-by-pixel basis using the experimental parameters asindependent variables. The intr insic parameters can be obtained by nonlinear least-squares fittings. An iteration method based on the linearization of the intensity ex-pressions, that is, the Gauss-Newton method, has been reported ( I I ). It is, however-,the usual practice in view of speed and efficiency to calculate the T, and the T2 imagesseparately and then calculate the spin density image from these images and the signalintensity expression as well as the intensity images ( 12, 13).

    T, images can be computed iteratively from images taken at different repetitiontimes. There have also been efforts to obtain the T, image as well as the T2 and thespin density images in real time, either by the combined use of saturation-recoveryspin-echo (SE) and inversion-recovery (IR) sequences ( Z2), or by an SE sequenceonly ( 14). In these fast methods of calculation, TI images are generally obtainedusing look-up tables that define the ratio of two signal intensities as a function of T,Recently a method combining ratios and linear least squares was presented and ap-plied to the SE -t- IR sequence (15). One of the factors that needs to be consideredwhen computing the T, images is the effect of slice profile, which can cause systematicerrors of over 50% ( 16).For the calculation of T2 images, usually the SE type of experiment is adopted, inwhich the signal intensity can be approximated or assumed to be an exponentiallydecaying function with respect to the spin-echo delay time (TE). Thus a plot of loga-rithms of the spin echoes vs TE yields the negative reciprocal of T2 as its slope. In th:

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    CALCULATION OF NMR IMAGES 9790'(a) 180' 90(a)

    TE

    signal

    :!1--.------m:. ',',

    ij' FID spin echo

    FIG. 1. A pulse sequence for the spin-echo experiment. The pulse angle (Y s introduced to study the sliceprofile effect.

    case of multislice imaging experiments, the slice profile also affects the estimation ofT2 (17, 18). From the T,, T2, and intensity images, the spin density image is easilyobtained.In this paper, a procedure for calculating T, , T2, and spin density images from theimages obtained by a spin-echo sequence is presented. Rather than simply using aminimum data set to save time, an effort i s made to increase the accuracy of compu-tation.

    MATHEMATICAL MODELSOne of the spin-echo pulse sequences used in our laboratory is shown in Fig. 1. The90 and the 180 pulses are selective and nonselective, respectively. The intensities ofthe spin-echo signals are given by

    s = pe-WT2 { 1 - e-WT, } El1s = pe-WT2 1_ &-(TR-TWWT, e-WT, }, [21where TR and TE represent the repetition and the spin-echo times, respectively.Equation [ 1 is the result of neglecting the effect of the 180 pulse on the longitudinalrelaxation process, and Eq. [ 21 is obtained assuming the slice profile is ideal.A mathematical model for the signal from the selective 90 and the nonselective180 pulse sequence has been presented (16) under the assumption that the trans-verse components of magnetization decay completely before the next 90 pulse. Here

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    98 LIU, NIEMINEN, AND KOENIGmore detailed equations will be given with a view to describe the inhomogeneityfactor as well as the slice profile effect.The behavior of the magnetization is described by solving the Bloch equationstaking the effect of the select ive 90 pulse into consideration. This approach is similarto the treatments applied to the steady-state free-precession methods ( 19-22). Actu-ally in these derivations, the transverse relaxation time T2 is implicitly assumed to beessentially the same as the effective relaxation time T: (23). Introducing the pulseangle CY or the 90 pulse to investigate the slice profile effect and solving the Blochequations under steady-state conditions, one obtains the following equation for thetransverse component of the magnetization, M,, at the spin echo (see Fig. 1).

    where p is the equilibrium magnetization, and F, is the factor due to the imperfect90 pulse:

    Fe =cos a sin aemTRi Tz- 1 + cos ae-WT2 + sin N

    1+ sin 2(y e-TRIT21 + cos ae-TRIT2 e-TRIT, + cos ,,--TR!T, .Note that the above equations are not dependent on any specific slice profile. If thepulse envelope has a distribution, the signal is obtained by integrating M:E over theslice profile ( 16, 24), i.e.,

    where z is the distance from the center of the slice along the direction perpendicularto the plane of the excited slice. When (Y s 90, Fa becomes

    and the signal for the ideal square pulse is given by

    The factor Fgv can be found in an equation derived for two spin echoes (25), aX-though the sign in front of the exponential terms is different.To consider the inhomogeneity factor, one may express 1 T: as (26)1/T:= l/T,+yAHj2,where y is the gyromagnetic ratio and AH the field inhomogeneity. In general, L / 7:can be divided into two parts, 1 T2 + 1 T2i, where T2i is the time constant governingthe relaxation process due to various kinds of inhomogeneities. While the naturallinewidth 1 T2 is related to the incoherent processes inherent to the material, 1 7Z,,is related to the coherent processes which can be eliminated by refocusing (2 7). Nowusing a phenomenological description for the spin echo (28), one may showthat the only modification needed of Eqs. [ 31 and [ 61 when the inhomogeneity fac-tor is to be included is to replace the terms of e-TRTz in the factor F;, by

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    CALCULATION OF NMR IMAGE S 99e-TRT2e-1cTR-TE)r211fl.Here the exponent IZ s 1 when the decay due to the inhomoge-neity is assumed to produce a Lorentzian line broadening in the frequency domain.On the other hand, it takes the value of 2 for a Gaussian line broadening.

    The equations thus far described may be used without approximation in moregeneral situations, for example, when T: is comparable with TR in magnitude. Un-der the usual spin-echo experiment conditions, however, TT is on the order of severalmill iseconds and TR is set to a value of an order of several hundred mill iseconds toseveral seconds. This is illustrated schematically in Fig. 1. Thus the condition T:4 TR was adopted in this study. In that case, the desired expression for the signal inEq. [ 51 is simplified to

    s = pe-WT2 { 1 - 2e-(TR-rVT, + e-WT, } SF,where the slice-dependent function SF is given by

    [71

    +a0F = ssin ff

    --co 1 + cos (ye-TRT1dz-Equation [ 71 shows the basic model of this study. Note that the same equation canalso be derived simply by ignoring the transverse components of the magnetiza-tion(l6).

    ALGORITHMS TO CALCUL ATE T, IMAG ESA. Ideal Slice

    T, images can be calculated from the images taken at the same TE but differentTRs. In this case, Eq. [ 21 can be rewritten asS(TRi) = Cf(TRi, k), [81where C = peeTEIT2 , f( TR, k) = { 1 - 2e-(TR-TE2)k + epTRk}, k = 1 / T, , and theindex i means the ith image.Although any nonlinear least-squares method can be used in general, the largenumber of pixels in an image (typically 256 X 256 in our laboratory) requires anefficient algorithm and a method to choose reliable initial estimates for the Tl values.Initial runs using the widely accepted Levenverg-Marquardt algorithm which is amodified Gauss-Newton method were not satisfactory, because it took about onehour in CPU time on a MicroVAX II computer. Rather than using an algorithmbased only on the first derivative information as in the Levenverg-Marquardt

    method, algorithms using second derivatives are more efficient. Calculation of thesecond derivative is simple for these equations since they are mainly exponentialterms whose derivatives contain the original functions,

    The algorithm adopted currently is based on an overrelaxation technique of theNewton-Raphson method (29). The algorithm treats Eq. [ 81 as if it had only oneparameter k or T, instead of the two parameters C and k. For a given k, C is givenby the least-squares equation:c = xi St T;, k)S(TR;) .i [.f(Tk, k)12 191

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    100 LIU, NIEMINEN, AND KOENIGThe parameter k at the y1h iteration is improved to k* by the formula

    wherek* = k - ( 1 + p)g(k)lg(k).

    g(k) = d{ C; [S(TRi) - CftTR, k)121dk

    Il(rl

    and P is a control parameter to regulate the iteration.Since the algorithm given above sometimes results in negative k or T, which is notacceptable, it is modified to start the iteration again with the control parameter resetwhen such a situation occurs. Also for initial estimates of T, which are too small,g(k) and g(k) tend to become zero. In that case the init ial estimates are increasedby a factor of 10.0. With these changes the algorithm converges for a wide range ofinitial estimates. For example, a simulation with T, = 1.0 s showed that the samevalue is returned for init ial estimates ranging from I .O X 1O-5 to 100 s. To furtherfacili tate the computation, the init ial estimates are obtained as follows:

    1. Calculate S = S,,,,,( 1 - e-l), where S,,,, is the intensity of the image withlongest TR.2. A time that corresponds to s is estimated from the data of signal vs TR by linearinterpolation. This time is the desired initial estimate for T, .

    This is simply approximating C by the signal of the most relaxed image since thesignal is equal to C for infinite TR.B. Slice with Pro&le

    The basic algorithm is the same as that presented in the previous subsection, andthe following description is applicable to any type of pulse envelope. A scheme toapproximate the slice-dependent function SF in E@. [ 71 using piecewise polynumialshas been presented (24). In the context of the current algorithm in which evaluationof the second derivative is required, however, a different approach that involvesdirectevaluation of the relevant integrations to form integral tables was taken.Like Eq. [ 81, Eq. [ 7 ] can be recast into the form

    S(TR;) = Cf(TRi, k)SF(TR,, k),where the independent variables of the function SF are explicitly written out forclarity.Again one needs to evaluate g(k) and g(k) to apply the iteration formuia (Eq.[ lO])and they aregiven by Eq. [ 1 ] but withf( TRi, k) replaced byf(TRi, k)SF(TRj,k). Actual differentiation shows that g(k) and g(k) have terms including SF and itsfirst (SF) and second (SF ) derivatives with respect to k as well as various exponen-tials. Here SF and SF are given by

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    102 LIU, NIEMINEN, AND KOENIGTABLE 1

    Resu lts of Fitt ing Using Eq. [ 11; he Signals AreGenerated by F.q. [2] for the Case of Ideal S lice

    T, (ms) Calculated Tl (ms) Error(W)100 117 17200 230 14300 341 14400 452 13500 563 Ii600 674 12700 786 I2800 900 13900 1015 13

    it is straightforward to calculate the spin density p with known or computed T, andT2 using following equation similar to Eq. [ 91:p = C; f(TRi, TEi)S(TRi, TEi)

    i [f(TR, TE i)lRESULTS AND DISCUSSIONS

    A. T, Image CalculationFirst, signal intensities were generated using Eq. [2] for the ideal slice case. Six

    repetition times of 192,292,492,892,1292, and 2058 ms and nine T, values farmi?;ifrom 100 to 900 ms were used. TE and T2 were set to 32 and 50 ms, respectively, forall the simulations. Then the fitting was done using Eq. [ 1] to see the error involvedin this mathematical model. The result is given in Table 1, which shows that system-atic errors above 10% are introduced in this example, even though no measurementerrors are assumed. As a consequence, the real time and the other techniques that arebased on Eq. [ l] need reevaluation especially when the TE is not negligible comparedwith TR.Next, to simulate the case of a slice with a profile, the signal intensities were gener-ated using Eq. [ 7 ] for the same set of values of TR and T, as above. Altbougb thealgorithm treats the slice profile as one of the inputs via cr and thus independent ofthe choice of the excitation pulse profile, the Gaussian pulse was used to illustrate theslice profile effect, mainly because it yields a slice profile distinctive from an idealsquare profile. Because the response of the magnetization to a selective pulse is givenby the Fourier transform of the pulse envelope in the time domain as a first approxi-mation (31), and the FT of a Gaussian function is also a Gaussian, the flip angle Ncan be approximated by a Gaussian function with respect to the distance z definedbefore: a! = 90 6.Fitting with Eq. [ 71 to check the algorithm returned the desired T1 vaIues within aprecision of 1 O X 10Y4. Then both Eqs. [I] and [ 21 were used in the fitting procedure

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    CALCULATION OF NMR IMAGES 103TABLE 2

    Resu lts of Fitt ing Using Eqs. [I] and [2]; the Signals Are Generated by Eq. [7]for the Case of Slice with Profile

    T (ms)Eq. 111 Eq. 121

    Calc. T, (ms) Error(%) Calc. T, (ms) Error (%)100 133 33 11 4 14200 276 38 242 21300 422 41 373 24400 571 43 508 27500 727 45 648 30600 892 42 793 32700 1066 52 944 35800 1250 56 1100 38900 1442 60 1261 40

    to see the effect of ignoring a term involving TE and/or the influence of slice profile.The results are summarized in Table 2. In both cases, the T, values are overestimatedwell up to 60%. These results demonstrate that the slice profile correction is crucialin calculating T, images when a shaped 90 pulse is used in the experiment.Calculationspeed. he CPU times for experimental images with 256 X 256 pixelsare typically 2 min in the case of the ideal s lice and 10 min in the case of the slicewith profile. They are substantial improvements in terms of the calculation speedcompared with the methods based on the Levenverg-Marquardt algorithm.B. T, Image Calculations

    It is well known that, in the exponential decay processes, the weak signals becomenoisier upon taking logarithms than the strong signals (30). In statistical terms, thestandard deviation in the logarithmic scale, u( log S), is related to that of the originalsignal, a(S) , bya(log S) = o(S)/S

    through the relation d log S = dS/S (30). Here Sis the estimated or noise-free signal.Thus log S does not exhibit the same noise level throughout the data even for S withuniform random noise. In fact the weaker the signal, the greater the noise will be-come. Moreover, unlike the case of, for example, photon decay in which the numberof data points is an order of 100, the number of spin-echo images is only a few,typical ly four. Thus the extraction of the involved parameter will be more influencedby the presence of noise in the spin-echo experiment than in the photon decay.The weight of the ith data of log S is given by ( 7,30)

    w,(logS) = l/a~(logS) = $/a?(S).Since it is assumed that ai( S) is same throughout all the images and a constant multi-plication of the weight does not affect the calculation of T2, one can write

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    104 LIU, NIEMINEN, AND KOENIGTABLE 3

    The Average and Standard Deviation of T2 Values in Mill iseconds Obtained from the Four-Point Data:Effect of Logarithmic Linearization and Weighting

    T2 (input) T2 1. For data with noise of kO.05 CSD TZ6 SD h Tz SD-.

    40 41.77 4.60 39.95 7.12 40.35 4.5 160 61.23 5.90 60.1 I 7.47 60.45 5.8280 81.46 7.8 1 80.8 I 8.79 80.84 7.17100 101.66 10.27 100.96 11.20 101.05 10.23120 122.18 13.00 121.59 14.01 121.61 12.98

    T2 (input) Tz2. For data with noise of 20.0 I CSD T*h SD T2 SD-I_

    40 40.04 0.87 39.96 I .39 39.99 0.8760 60.05 1.10 60.03 1.35 60.02 1 0980 80.08 1.50 80.06 1.72 80.06 I,50

    100 100.03 2.00 99.96 2.19 100.00 2.00120 120.07 2.63 120.05 2.81 120.05 2.63il Obtained using linearized equation with weighting by square of signal. Obtained using linearized equation with no weigh ting.Obtained using original expone ntial e quation.

    w;(log S) = s?.However, $ cannot be obtained until they are actually estimated. One alternative isto replace ,!$ by the measured intensity data Si. Although it is possible to use Si itera-tively, i.e., first estimate $ either with no weighting or with weighting by S: and thenapply it in the calculation, a test run shows no significant gain as a result. It is partlybecause, while perfect or noise-free data need no weighting scheme, T2 and C values(see Eq. [ 121) estimated from a noisy set of data are not true values but can stil l bederived from them.To simulate the effect of weighting on the calculation of T, from noisy data, a seriesof signals were generated at TE = 25,50,75, and 100 ms for five T2 anging from 40to 120 ms. Then random noise was added to these noise-free signals to create two setsof intensity data, one set having a noise ranging from -0.0X to i-0.05C and theother set from -0.0 1C to +O.O 1C. As defined above, the pseudo-density C corre-sponds to the intensity at TE = 0. The fitting schemes were as follows: (a) use linear-ized equation by taking logarithms where each datum is weighted by the square ofintensity, (b) use linearized equation with no weighting, or (c) directly use the expo-nential function, Eq. [ 121. The results of T2 calculation are given in Table 3. Eachreported T, and standard deviation (SD) value was obtained from 1000 T2 valuescalculated from 1000 sets of intensity data generated by adding random noise of thecorresponding level. These 1000 values were divided into 10 groups, each group con-sisting of 100 values of calculated T2. Then an average and standard deviation werecalculated from each group. When 10 averages and standard deviations were ob-

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    CALCULATION OF NMR IMAGES 105TABLE 4

    The Average and Standard Deviation of T2 Values Obtainedfrom the Two-Point Data; Effect of Limited Data

    T, (input)

    I. For data with noise of +0.05 CTE = 2550 ms TE = 25,100 ms

    T2 SD i-2 SD40 41.38 7.84 40.16 7.7460 62.37 12.79 60.14 8.0180 84.15 19.78 80.9 1 9.26

    100 106.47 28.86 101.11 12.06120 129.23 39.68 121.62 14.67

    T, (input)406080100

    120

    2. For data with noise of +O.OlCTE = 25,50 ms TE=25,100ms

    T2 SD T2 SD39.98 1.50 39.95 1.5360.14 2.33 60.05 I .4680.22 3.47 80.10 1.84100.25 4.81 99.94 2.32

    120.22 6.45 120.04 2.91

    tained as a result, these values were again averaged and given as T, and SD in Table3, respectively. Table 3 shows that the direct implementation of the exponential func-tion produces the most reliable results as one expects, but the performance is compa-rable with that of the weighted linear least-squares method especially when the noiseis not severe. The advantage of util izing weighting over no weighting becomes clearwhen the noise level decreases.Also other tests were made with two sets of selected data corresponding to TE= 25, 50 ms and TE = 25, 100 ms, respectively, to simulate the calculation of T2using only two images. Because the three schemes gave the same results up to fivesignificant figures, only the results obtained by the weighted linear least-squaresmethod are given in Table 4, where the T, and SD were obtained in the same way asabove. Comparing Table 4 with Table 3 indicates that using only two images givesworse results than using all the images.On the basis of this simulation, it can be concluded that the weighting improves theaccuracy of calculation of T2 mages except in very noisy data, where the exponentialfunction is a more reliable mathematical model to implement. Typical ly it takesabout one minute in CPU time to calculate T, by the weighted linear least-squaresmethod and spin density images by Eq. [ 131, respectively, from experimental imageswith 256 X 256 pixels.C. ExperimentalResults

    The experimental and computational results are given in the next four figures.Plasticized poly( vinyl acetate) (PVAc) emulsion with a solids content of 40% and a

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    106 LIU, NIEMINEN, AND KOENIG

    FIG . 2. NM R image of PVAc emulsion (upper half) and PVAc emulsion-wood powder ( lower half)surrounded by tap water. The image was taken at TRIT E = 4.0 s/ 12 ms .

    viscosi ty of 15 Pas was used to construct a sample. A plast ic vial with a diameter of20 mm was spli t to make two semicirc le containers. One section was filled with thePVAc emulsion, while the other one was filled with a mixture of 9 parts af PVAcemulsion and 1 part of wood powder. Then they were put together, wrapped withparafilm, and placed in a bigger vial with a diameter of 30 mm containing tap water. Aseries of 256 X 256 images ofthe sample were taken on a Bruker MSL300 instrumentoperating at 300 MHz equipped with mini imaging accessories by using a spin-echosequence with CYCLOPS phase cycling. A typical image taken at TR/TE = 4 s/ 12ms is shown in Fig. 2. Tap water was used as a reference, which appears as an annulusin the image. The pure emulsion and the mixture appear in the upper and lowerhalves of the image. The images taken at four TR values of 0.5, 1.0, 2.0, and 4.0 sand the same TE value of 12 ms were used to calculate the r, image shown i,n Fig. 3.On the other hand, the images taken at four TE values of 12.24,48, and 96 ms and

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    CALCULATION OF NMR IMAGES 107

    FIG. 3. Calculated T, image. W hite corresponds to a T, value of 3.0 s.

    the same TR value of 4 s were used to calculate the T, image shown in Fig. 4. Theimage taken at TR / TE = 4 s/ 12 ms ( Fig. 2 ) was used in both calculations; thus sevenimages were used in the overall procedure. The proton spin density image was alsocomputed and is given in Fig. 5 using the same gray scale range as that in Fig. 2 forcomparison. The spots, especially in the upper half, indicate the presence of air bub-bles which were apparently moving slowly upward during the imaging. The heteroge-neity in the mixture of emulsion and wood powder is well reflected in the three calcu-lated images which subsequently suggests incomplete mixing.

    CONCLUSIONA procedure to compute T, , T2, and spin density images in NMRI is described, inwhich T, and T, images are calculated separately and then the spin density image iscalculated util izing these two images. A general mathematical model of the signal

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    108 LIU, NIEMINEN, AND KOENIG

    FIG. 4. Calculated T, image. T z values of up to 200 ms are shown.intensity for the spin-echo pulse sequence that includes terms arising from the heldinhomogeneity and the slice profile effect is presented. Ti images are calculated fromthe images taken at the same TE but different TRs by an overrelaxation technique ofthe Newton-Raphson method. For the case of a slice with a profile as a result of theimperfect selective 90 pulse, an efficient and fast way to implement the iterationscheme to compute T, images is proposed based on the linear interpolation of thediscrete integral tables of the slice-dependent function and its derivatives. Also it ispointed out that the equation of type (1 - e-TRTl >eeTElT2 obtained ignoring theeffect of 180 pulse on the longitudinal relaxation process results in noticeable system-atic errors. T2and the spin density images are calculated by linear least-squares meth-ods. In the case of the T2calculation, the importance of weighting when an exponen-tial decay process is linearized by taking logarithms is demonstrated especially forspin-echo experiments which normally involve only several data points or images. Inthis approach of computing T,, T2, and spin density images, an overall saving ofcomputation time while maintaining the accuracy is significant.

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    CALCULATION OF NMR IMAGES 109

    FIG. 5. Calculated proton spin density shown in the same gray level range as that in Fig. 2.

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