Vol. 11, No. 1/2 69

Medical Applications of NMR

Wladyslaw T. Sobol

Radiology DepartmentThe Bowman Gray School of Medicine

300 S. Hawthorne RoadWinston-Salem, NC 27103

ContentsI. Introduction 69

II. In-vivo studies: Magnetic Resonance Imaging 70A. Spin warp (2DFT) method in MRI 70B. Determination of NMR parameters in MRI 73

1. Saturation Recovery protocol 732. Inversion Recovery protocol 733. Determination of tissue parameters from MR images 75

III. In-vitro studies 78

A. NMR relaxation studies of tissues . 78

IV. Theoretical modelling of the NMR properties of tissues 80

V. Conclusions 81

VI. References 83

I. Introduction

The use of Nuclear Magnetic Resonance (NMR) tostudy living tissues began in 1955 with preliminaryhydrogen NMR measurements of human and rabbitblood fractions, yeast cells, rabbit and rat livers,muscle and fat, calf cartillage and human Achillestendons (1).

In 1972, the first concepts of the use of NMRmethods to produce images of living organismsemerged (2,3). These basic ideas lead to the de-velopment of Magnetic Resonance Imaging (MRI).Since then, tremendous progress has been made inboth technological improvements of the method it-self and its applications in everyday clinical diag-nostic practice.

The most important diagnostic factor in an MRIimage is the CNR- contrast to noise ratio (i.e. thedifference in signal intensity normalized to the im-

age rms noise level) for two special regions of inter-est (ROI): the abnormal ROI (lesion, tumor, cyst,hemorrage etc.) and the normal, healthy adjacenttissue within the same organ (brain, spinal cord,liver, muscle etc.). The CNR for any image is deter-mined by many factors, which can be divided intotwo groups: instrumental factors (related to MRIscanner hardware configuration and software controlparameters), and intrinsic tissue parameters (spin-lattice relaxation time Ti, spin-spin relaxation timeT2, proton spin density, and diffusion constants).These intrinsic tissue parameters depend on the mo-tion of the nuclei, the temperature, the viscosity ofthe tissue and the magnetic effects from nearby nu-clei (4). Therefore, since they depend on the localtissue conditions, they are believed to carry infor-mation about the disease state of tissues (5). Itis important then to correlate MRI clinical studies

70 Bulletin of Magnetic Resonance

with basic studies of tissue relaxation in order to im-prove the efficacy of clinical MRI protocols and in-crease our knowledge of the relaxation mechanismsin biological materials.

II. In-vivo studies: MagneticResonance Imaging

Several methods may be used to produce an MRIimage; however, the spin-warp technique (6), an im-provement of the original two-dimensional FourierTransform (2DFT) imaging idea (7), is most com-monly used in everyday clinical practice.

A. Spin warp (2DFT) method in MRIThe basic protocol for this imaging method is

shown in Figure 1. The top line shows the timingof rf pulses that are applied to the system duringa single phase encoding cycle. The first pulse hasa special profile and is applied in the presence ofmagnetic field gradient along the axis, perpendicu-lar to the desired image plane (slice select gradientGz). This assures that only a thin slice of samplematerial will be saturated (the nominal tip angle ofthis rf pulse is close to n/2), as illustrated in Figure2 (8). If the Bloch equations are solved for the casewhen the rf field is turned on, the first order solutionpredicts that only spins with a resonance frequencywithin the range covered by nonvanishing FourierTransform (FT) of the rf pulse envelope will be af-fected. If the envelope of the rf pulse is tailored sothat its FT is rectangular, only the spins with reso-nance frequency within the interval Aw centered at<j0 will be irradiated. When this irradiation occursin the presence of a uniform magnetic field gradi-ent (MFG) oriented along the z direction, a slice ofthickness d= Au>/(7Gz), where 7 is the gyromag-netic ratio, will be irradiated (Figure 2). The posi-tion of the excited slice along the z direction can beadjusted by changing the carrier frequency u)o.

In practical applications of this method the sliceselectivity and its profile definition is far from be-ing perfect. Several pragmatic methods to improveslice selectivity have been proposed (9-11). One ofthe common "tricks" is shown in Figure 1: an addi-tional, negative gradient lobe has been added to theslice-select gradient G2, which improves the magne-tization's phase distribution across the slice.

After the magnetization in the selected plane hasbeen excited, two additional gradient pulses are ap-plied to the system: one along the so-called "phaseencoding direction", the other along the "frequencyencoding direction" (Gy and Gx in Figure 1, respec-tively). The purpose of the preliminary Gx lobe isto generate suitable initial conditions for the phaseevolution along the x direction within the imagedplane. Since the signal is read in the presence ofthe Gx gradient, the phase distribution along x di-rection must vanish in the middle of the readoutperiod so that a full echo signal could be accumu-lated. The "phase encoding" gradient codes the spindistribution information along the y direction; theamplitude of this gradient is constant for a singlecycle (shown in Figure 1), but the whole cycle is re-peated many times (64, 128, 256 or 512, dependingon the resolution desired) with the amplitude of thisgradient lobe changing in constant steps.

To assure proper echo formation, a second rfpulse (shown in Figure 1 as a short, rectangular,non-selective pulse) of nominal tip angle ir is ap-plied before the readout portion of the Gx gradient.

In order to better understand the describedmethod, consider an object with a square cross-section in the imaged x-y plane:

—a < x < a (la)

-a<y<a (Ib)The signal phase modulation in the x direction (thereadout gradient) has two components: the firstcomes from the initial gradient lobe put before thejr pulse:

= +yGoxT, (2)the other accumulates during the proper readout pe-riod; since the TT rf pulse inverted the phase accu-mulated earlier, then during readout

= - $ x + 7Goi(t + T) = (3)

The amplitude of the readout gradient, Go, may beconveniently used to calibrate the phase encodinggradient as well, to ensure proper geometrical rela-tions along the X and Y directions within the gener-ated image. The whole sequence is repeated N timeswith the amplitude of the phase encoding gradient

Vol. 11, No. 1/2H

71

rf

Gx

SLICE

PHASE EHC0D1HG

FREQUENCY ENCODIMG

SIGNAL

-T

Figure 1. Basic MR imaging protocol. This is a so called "spin warp" modification of Ernst's 2DFT imagingmethod. Ernst proposed phase encoding gradients of varied length, but the spin warp method varies theiramplitude; also, echo signals are commonly measured instead of FIDs.

rf amplitude

excitedslice

Figure 2. Slice selective rf excitation. A long, soft pulse (typical duration a few ms) of modulated amplitudeis applied in the presence of a slice-select magnetic field gradient. This results in NMR excitation of spinswithin a thin slice of imaged body only.

72 Bulletin of Magnetic Resonance

changing in equal steps; therefore the phase accumu-lation due to the presence of the Gv gradient lobe isequal to

4>v = fTyGoym; -AT/2 < m < (AT/2) - 1. (4)

It is straightforward to write the solution of Bloch'sequations for such a case. The resulting signalstrength Sm(t) = Mx(t) + iMs(t) is (12):

(<)= / /J—aJ—

+Goxt + Bz(x, y)t - Bot) -(t + Td)/T2]dxdy (5)where, for simplicity, it was assumed that the se-lective excitation was perfect and the timing wasadjusted so that the spin echo from magnetic fieldinhomogeneity coincides with the spin echo from theencoding gradients. A(x,y) is the spin density dis-tribution within the slice, Br(x,y) is the receivercoil sensitivity, B2(x,y) describes the actual distri-bution of the z-component of the main magneticfield within the imaged plane, and T* is the timedelay from the middle of the first rf pulse to the be-ginning of the signal readout (Figure 1.). To keepthis formula simple, the local phase distribution ofthe receiver pickup coil was neglected.

This continous signal is sampled in discrete in-tervals:

t = nAt; -N/2 <n< {N/2) - 1 (6)

and the proper Nyquist cryteria must be fulfilled toassure the desired FT properties:

= 2w/N

= 2w/N.

(7a)

(76)where Ad is the pixel size (square pixels were as-sumed).

As a result of such a procedure, after N phaseencoding steps are completed, the raw data array isobtained with a signal intensity S(n,m) assigned toeach element:

S{n,m) =N/2-1 JV/2-1

Br{u, v)A{u, v)exp\2tri/N[un + vm + n(?{u, v)]xexp[-2nn/Ncr{u,v)] (8)

where:

x = uAd; -N/2<u<{N/2)-l (9a)

y = vAd; -N/2< v < {N/2) - 1 (96)and

er(u, v) = 1(10)

(11)

describe the effects of spin-spin relaxation and mag-netic field inhomogeneities on the signal intensity,respectively.

This raw data array is subjected to an image re-construction algorithm, which is nothing more thana 2DFT, performed with the assumption that theimage space is sampled at equal cartesian incre-ments (proper scalings of Gj, and Gx gradients haveto be performed to assure this). Therefore, the re-constructed image can be repesented as

A'(u,v) = ^n tn

(12)Full details of further calculations are presented

elsewhere (12). It has been found that spin-spinrelaxation effects present during the data acquisi-tion period cause picture element (pixel) blurring,while field inhomogeneity effects cause geometricaldistortions of the image; if one can assume that botheffects are negligible, from Eqns. 8 and 12 follows

A'(u,v) = p £ E exp[-Td/r2(U', v')]Br(«', v')n,m u'lV'

xA(u', v')exv(-2wi/N[n(u' -u) + m(u' - v)]) (13)Since

-27ri7/v'n(u' - u)] = N6U,U

then

u=-2V/2 v=-N/2 '(u, v) = Br(u,v)A(u, V (u, v)}. (14)

r-q

Vol. 11, No. 1/2 73

This is the final formula for the image array re-sulting from the spin warp protocol; elements of thisarray contain information about the pixel intensitywithin the imaged plane and, when displayed on agraphics monitor with a gray intensity scale, forman MRI image. As it is seen from Eqn. 14, the MRIpixel pseudodensity data represent true spin densityin the corresponding voxel A(u,v) weighted with thereceiver coil sensitivity Br(u,v) and spin-spin relax-ation effects.

Eqn. 14 was derived without taking into accountany rf phase distribution effects. The phase effectsare caused by phase shifts due to nonequivalent elec-trical paths of the sample signal and the rf referencesignal used by the quadrature detectors (constantterm), rf excitation nonuniformity across the imageplane and receiver pickup coil sensitivity distribu-tion (pixel position dependent). The standard wayto eliminate these effects is to reconstruct a mag-nitude image from the complex data array A'(u,v).This step, by definition, cancels all the phase infor-mation within the image. In some cases, however,(e.g. some inversion recovery protocols, flow studies,etc.) the real image is needed. Special phase correc-tion algorithms were designed to solve this problem.

B. Determination of NMR parametersin MRI

Clinical applications of MRI are based on theassumption that the image contrast between thepathological condition and the surrounding normaltissue is determined by their different NMR charac-teristics. Therefore, studies of NMR parameters ofbiological materials in vivo are important for bet-ter tissue characterizations as well as for other im-provements of clinical efficacy of this new imagingmodality.

Imaging time restrictions force the typical imag-ing protocols to be very different from standardNMR experiments. Although basic imaging proto-cols (Saturation Recovery - SR and Inversion Re-covery - IR) originated from their respective NMRmethods, the results are much more difficult to anal-yse. Because each basic sequence has to be repeatedmany times (usually 128 or 256) in order to cover afull range of phase encoding gradient increments, inMRI the repetition time (TR - calculated from thebeginning of a sequence segment to the beginning ofa new segment with a different phase encoding gra-

dient value) is almost never long enough to allow thesystem to return to its thermal equilibrium state. Asa result, steady-state solutions of the Bloch Eqns.13-15 are commonly used in MRI, in contrast toeveryday NMR practice.

1. Saturation Recovery protocol

The saturation recovery (SR) protocol is verysimilar to the basic spin warp sequence (Figure 3).The only significant modification commonly used isa selective ir pulse-to allow for multislice data col-lection in a short time. Essentially, since the wholesequence is now restricted to a single slice, it is pos-sible to repeat it with an rf offset (to collect thedata from another slice) during the "wait" intervalfor the previous slice. This allows for data acquisi-tion of multiple slices in greatly reduced time and isnow a common practice in MRI.

The theoretical pixel signal intensity for an SRsequence is given by the formula (16):

5 = Mosinaexp(-T£/r2) x

[1 + exv(TE/2T2) - (exp(-TE/2Tl) + cos/? x

[exp(-~TE/2Tl) - l])exp(-TR/Tl)]x

{1 - cos(/? - a)exp[-TE/2:Tl - TE/2T2 - TiJ/Tl]}"(15)

where Tl is the local spin-lattice relaxation time, T2is the local spin-spin relaxation time, Mo is the localproton spin density, TE is the echo time (countedfrom the beginning of sequence to the maximum ofthe echo signal), TR is the sequence repetition time,a is the nominal tip angle of the first rf excitation(usually close to ir/2) and /? is the nominal tip angleof the second rf pulse (close to ir).

There are some general properties of SR gener-ated MR images. Since TR is commonly consid-erably shorter than 5*T1, the image intensity dataare both Tl and T2 dependent, or "weighted". Se-quences with shorter TRs provide more Tl-relatedcontrast, or are more Tl weighted. Sequences withlonger TE yield smaller signal intensities (or smallersignal-to-noise values), but are more T2 weighted.

2. Inversion Recovery protocol

The basic sequence for Inversion Recovery (IR)protocol is shown in Figure 4. Again, all the rf

74 Bulletin of Magnetic Reson an ce

Saturation Recovery sequence

rf

2 r

G

s igria 1

A :£_V7 \J

TE

TR

Figure 3. Saturation Recovery sequence protocol. This imaging protocol is based on its (TT/2 - TR)n NMRanalog and sometimes is called a partial saturation protocol, since it is very difficult to achieve a completespin saturation with single, slice-selective pulses.

Inversion Recovery sequence

rf

aignal

TI

Ar\

TR

TE

Figure 4. Inversion Recovery sequence protocol. This is an analog to a (ff-r-7r/2TR) NMR inversion recoveryexperiment.

f

Vol. 11, No. 1/2 75

pulses used are applied in slice selective mode to al-low for multislice data collection, as it was describedfor SR protocol. The pixel signal intensity in thiscase is equal to (16):

5 = Mosinaexp(-7\E/r2) x

1 + (cos/? - l ) e x p ( - r i / r i ) - cos/?exp(-rfl/ri)1 + cosf3cosaexp(-TR/Tl)

(16)where all the variables are the same as in the SRcase, except for the angle /?, which now denotes thenominal tip angle of the first, inverting pulse. Thethird pulse, needed for echo generation, is assumedto be an exact ir pulse. The inversion waiting time,TI, is counted from the beginning of the sequenceto the second (monitoring) rf pulse. Generally, theIR protocol is more difficult to apply in standardclinical practice, mainly because of problems withphase correction and increased image noise, whichoccur when TI is chosen to produce small signals.

3. Determination of tissue parameters fromMR images

The standard method of computing relaxationtime data from MR images involves two-point fits(17-20). For T2 determination, a double-echo SRprotocol is used, so two different images (with dif-ferent TEs) are acquired. The intensity ratio for cor-responding pixels is then computed and T2 is deter-mined from the known analytic expression for thisratio. To determine TI, two different images are ac-quired, either both SR with different TR or one SRand the other IR with the same TR. It is importantto acquire both images with the same TE, so thatT2 weighting effects will influence the correspond-ing pixel in exactly the same way and cancel outwhen intensity ratios are calculated. Again, fromthe intensity ratio the pixel values of TI are com-puted. Since analytical formulas are quite complexand their inverse functions can be difficult to calcu-late, it is common practice to speed up the comput-ing process by generating TI look-up tables priorto the computation; the computing algorithm sim-ply interpolates the look-up table data to fit theexperimental intensity ratios.

This standard technique has been criticized forits lack of accuracy and stability; numerical valuesfor relaxation times obtained by this method tend to

fluctuate wildly and they are commonly not trusted.Apart from the fact that the algorithm, based on thetwo-point fit, has inherent instability when noise ispresent in the input data, there are several addi-tional factors which influence the precision of signalintensity measurements in MR images. The truepixel signal intensity is affected by several factors:

• nonuniform rf excitation within the imagedplane,

• spatial sensitivity distribution of the receivercoil,

• slice profile distortion caused by selective ex-citation pulses,

• partial volume effects,

• magnetic field inhomogeneity,

• eddy current effects,

• signal coherence losses in the echo formation,

• flow artifacts (blood, CSF),

• motion artifacts (breathing, bowel motion,body movements).

Attempts to eliminate these problems face onecrucial obstacle: they usually require far more scan-ning time than standard protocols and for this rea-son are unacceptable in clinical practice. Fortu-nately, recent progress in the development of special,fast imaging protocols which use small tip angle ex-citation has opened new possibilities (15,21-26).

One of them is based on the method, first pro-posed by Look and Locker (27). It uses a seriesof rf excitations, closely spaced together, applied tothe sample which was initially in thermal equilib-rium; this MR imaging protocol is shown in Figure5 (28,29). If the experimental conditions are chosento assure that no transverse coherence is left at thetime of subsequent rf pulses, it can then be shown(solving Bloch equations for this protocol) that thepixel signal intensity in consecutive images is de-scibed by simple formula:

my{t,T,Tl,a) = a + b * exp(- t / r e / / ) (17)

where t=(n-l)T , n is the pulse number, T is timeinterval between rf pulses, TI is the local spin-lattice

f i

76 Bulletin of Magnetic Resonance

Previous Cycle

< »

Hp« n» p

Ul U | Ull u l u lp

Dlt

1 1 1 I fl i I I 1 I ITl

it

RF

J_G,

to,

*- Next Cycle

Figure 5. Schematic diagram of the fast imaging protocol, designed for Tl determination from the magnetiza-tion evolution towards its steady-state value. Gz indicates the sliced-select gradient, Gy is a readout gradient,and Gx is the phase encoding gradient. The straight arrows indicate positions of the random amplitude spoilergradient lobes placed along slice-select direction; the feathered arrows indicate positions of similar spoiler gra-dients along the readout gradient direction. Spoiler gradients are used to destroy transverse magnetizationcoherence effects.

i

relaxation time, or is the nominal tip angle of the rfpulses (0 < ct < 7r/2) and Te// is the time constantwhich describes the evolution of the magnetizationtoward its steady state value. The parameter "a"represents the well known steady-state componentof the solution (13,14) while the parameter "b" isthe amplitude of the transient component. The de-pendence of these parameters on T, Tl and a isshown in Figure 6; for compactness of the plot, areduced variable T l /T was used. For a standardNMR experiment with a long waiting time (so thatT l /T is close to zero), the parameter "b" is zero andthe parameter "a", which represents the signal am-plitude after each pulse, is simply equal to Mosina.When a — ir/2, the full value of Mo is registeredafter each pulse. As the pulse interval gets shorter,the steady-state component decreases; for short T(large Tl/T) there is an optimum angle a for whichthe steady-state response is maximum - it is thewell known Ernst angle (13). As the steady-statevalue decreases, the transient amplitude grows, so

that the magnetization after the first pulse appliedto the system is always equal to Mosina. This fea-ture is clearly seen in Figure 6a,b. The effective de-cay time Te// varies with the tip angle a, decreasingfrom Tl to 0 as a varies from 0 to TT/2.

One can monitor the transient evolution of voxelmagnetizations towards steady-state values by col-lecting data after each consecutive pulse. A threeparameter nonlinear fit of Eqn.17 to these data willproduce experimental values of a,b and Te// . Fromthose values a local tip angle a, a local pseudoden-sity Mo, and a local Tl can be determined, usingfollowing relations:

a = arccosa + b

T

a + b * exp{T/Teff)

1f-\n[cosa]

Mo = ( a+ 6)/sin a

(18a)

(186)

(18c)

Vol. U, No. 1/2 77

a/Mo

Tl/T

Rlpha b/Mo

Rlpha

Rlpha

Figure 6. The dependence of the fitted coefficients on the rf pulse-nominal tip angle a (a changes between 0and v/2), relaxation time Tl, interpulse spacing T (the ratio T l /T varies between 0 and 5), and pseudodensityMo:a) the amplitude of the steady-state signal a;b) the amplitude of the transient decay b;c) the effective decay time constant Te/y.

78 Bulletin of Magnetic Resonance

This approach has substantial advantages. The mul-tiple point fitting algorithm appreciably increasesthe precision of parameter determinations. In addi-tion, this process determines three local parameters:the local pseudodensity (the spin density weightedwith both the local receiver coil sensitivity and thespin-spin relaxation effects), the local tip angle a(which is determined by the Bi field distribution ofthe transmitter coil) and the local spin-lattice re-laxation time Tl. Thus the Tl map is substantiallyunaffected by the two factors listed earlier as majorsources of decreased accuracy of Ti determined withtwo-point algorithms.

At the end of the described procedure, a set ofcomponent images is available. Each one of them,when considered individually, is usually of inferiorquality for clinical diagnosis. However, they can befurther processed (using, for example, a weighted-sum algorithm (30) to optimize signal-to-noise ra-tios) to obtain a final image of diagnostic quality.An estimation of the image contrast for a simplesum of six component images is shown in Figure 7(the weighted sum method is likely to improve thisestimated results). In the shadowed regions, the pa-rameter values are going to produce a final image ofthe quality inferior to that achievable by a "single-shot", routine image. In addition, it can be seenthat small tip angle protocols are likely to produceimages with very little Tl-related contrast (isointen-sity lines are almost parallel to the relaxation timeaxis for small values of a). However, medium angleprotocols look promising from the point of view ofboth signal-to-noise and predicted contrast. Largetip angle protocols are impractical, because Tejj be-comes so short that it would be difficult to measureit accurately.

III. In-vitro studiesNMR studies of biological materials have a

long history; an impressive volume of both experi-mental data and theoretical considerations has beenpublished, including several reviews (31-40). Withthe commercial introduction of powerful, high fre-quency, computer driven FT NMR spectrometersin the 1970s, biophysical and biochemical NMR re-search focused on studies of model systems. In thosemodel systems some molecular groups move fastenough so that most of their dipolar interactions are

suppressed. This allowed the use of High Resolution(HR) FT NMR techniques such as the solvent sup-pression method or the very powerful 2DFT tech-nique (38,41,42). With the introduction of MRI inthe early 1980s, a new field of practical applicationswas created, which led to a significant research ac-tivity commited to improve MRI technologies, ad-vance the applications and technical knowledge ofMRI diagnostic equipment and to develop the in-formation, data and results required to advance theclinical efficacy of MRI. This also generated a re-newed interest in the relaxation properties of tissuematerials and the correlation between their in vivoand in vitro properties. In a recent review the au-thors stressed the dearth of available informationon the temperature and frequency dependence ofpathological tissue relaxation (43,44).

A. NMR relaxation studies of tissuesThe study of relaxation times of intact excised

tissues represents a challenging problem. Most ofthe dipole-dipole interactions between the protonsin macromolecules do not average out since the ther-mal motion of these huge aggregates is relativelyslow. On the other hand, the exchange processesconsiderably affect the observed "apparent" relax-ation times. In fact, the observed relaxation timesof protons in tissue water are substantially shorterthan in pure water. The lines, resulting in the *HFT NMR spectra, are therefore quite broad, so thathigh frequency NMR spectrometers are needed toobtain the best possible resolution of the spectrallines (in order to be able to calculate accurately therelative spin weights and relaxation times Tl andT2). So far, this spectroscopy has been successfulin the resolution of the signals from water moleculesand from other highly mobile small molecules orhighly mobile small groups within macromolecules(45,46). A very good discussion of these problemsfor intact muscle tissue has been published (47).

The applications of these findings focus mainlyon water and lipid resolution (48-54). However,studies of NMR relaxation in tissues at lowerfrequencies (1-100 MHz) are also important fortwo reasons: first, all MRI diagnostic equipmentpresently operates within this frequency range;second-the frequency dispersion of relaxation timesfor tissues is practically confined to this range andits characteristics are important for theoretical mod-

Vol. 11, No. 1/2 79

MINIMUM:. 1

MflXIMUM:2.4

INTERVflL:.1

«• T l / T •»Figure 7. The calculated noise-normalized signal amplitudes within-the composite image generated by addingtogether first six component images. The shadowed areas indicate ranges where the signal amplitude in thecomposite image is smaller than the signal obtained from a single standard acquisition image. Note thatsmall tip angles are likely to produce images with very little Tl contrast (isodensity lines are almost parallelto T l /T axis for small tip angles a).

elling.A FID signal from a tissue sample has a mul-

ticomponent structure consisting of a small, gaus-sian component characterized by T2 of several ps(it can be attributed to protons in large macro-molecules within tissue structure) and several muchlarger Lorenzian components arising from the pro-tons with much longer spin-spin relaxation times-usually between 1 and 80 ms (mostly water) (55-57).The analysis of a pulsed NMR relaxation experi-ment with such a FID signal can be substantiallyenhanced by using the so called "spin-grouping"method [58,59], which identifies magnetization com-ponents with different relaxation times Tl and/orT2. A modification of this technique, adapted toFT NMR data, has recently been proposed (60).An alternative and powerful method of analysis ofmultiexponential decay curves has been proposed byProvencher (61-63) and successfully applied to somestudies of tissues (64).

In addition, Edzes and Samulski (65) have pro-posed a modification of relaxation experiments to

measure the dynamics of interactions between differ-ent proton spin reservoirs in tissues. This method,combined with spin grouping analysis, has recentlybeen applied to the NMR studies of mouse lung andmuscle tissue (66,67). These experiments confirmedthe important fact that the macromolecular reser-voir can not be neglected in an overall study of re-laxation mechanisms in tissues and that a carefulplanning of the experimental conditions (especiallywhen selective excitation pulses are used) is impor-tant for proper data analysis.

The experimental information obtained for agiven tissue (spin-spin and spin-lattice relaxationtimes, spin densities of different molecular groups,translational diffusion coefficients, relaxation timesin the rotating frame, etc.) could be used in theo-retical modelling of the MR properties of biologicalmaterials. Better theoretical understanding of themechanisms responsible for inherent tissue proper-ties (tissue characterization) could improve the tech-nology and diagnostic efficacy of MRI.

80 Bulletin of Magnetic Resonance

IV. Theoretical modelling of theNMR properties of tissues

Most of the theoretical modelling of the NMRproperties of biological materials is based on theconcepts, initially proposed by Zimmerman andBrittin (68). Their approach considers the relax-ation pattern observed in a system of several spinreservoirs coupled together. Similar problems havebeen studied later by others (69-73). This math-ematical model has simple analytic solutions onlywhen two coupled reservoirs are considered. For agreater number of spin reservoirs, the detailed pro-cedure requires a use of numerical methods. Twolimiting cases can still be solved analytically, how-ever; a rather trivial case of very slow exchange be-tween reservoirs and a case of very fast exchange.This later limit has been commonly used in themodelling of relaxation phenomena in tissues. Sincethere is a large number of papers reporting mod-els of NMR properties of many different biologicalmaterials, only some most representative are quotedhere (74-89).

Although the two-site or fast exchange modelshave produced interesting results, it has been appar-ent for some time that they are too crude to reason-ably reproduce all the experimental data available,especially the observed frequency dispersion of re-laxation times.

A preliminary attempt to build a more sophisti-cated, three site exchange model has been recentlyreported (90). This model was built using a linearthree site exchange scheme first proposed by Noack(91) and is shown in Figure 8.

The macromolecular reservoir has many protonsites with very different molecular dynamics, it istherefore characterized by the distribution of cor-relation times. The intrinsic spin-lattice relaxationrate Ria of this group is essentially determined bytwo processes: a molecular reorientation of macro-molecules and their segments and a cross-couplingwith the bound water protons. The former can bemodelled using a log-normal distribution of correla-tion times with a median correlation time rr and adistribution width /?r (92). The latter is dependenton the correlation time T\.

Ri{, is the intrinsic relaxation rate of a boundwater reservoir. It is assumed that this water un-dergoes anisotropic motion, characterized by a fast

uniaxial reorientation around the bond axis (withsingle correlation time ri) and a distribution of re-orientational motions of this axis (characterized by amedian correlation time 7& and a distribution widthA).

The bulk water intrinsic relaxation rate Ric isdetermined by two processes: an isotropic rotationaldiffusion with a single correlation time rw, and atranslational diffusion process with a diffusion con-stant D (D = 1.4x 10~5 cm2/s, according to theexperimental data available). The cross-couplingbetween bound water protons and macromolecu-lar protons is assumed to be due to intermoleculardipole-dipole interactions (93). Its rate a is deter-mined by the average distance between interactingprotons (ri « 2.5A) and the correlation time charac-terizing the reorientation of the proton-proton vec-tor. Since the fastest modulation of this vector ori-entation is caused by the uniaxial mode of boundwater reorientation, a correlation time T\ for thismode was used in cross-coupling formulas.

The chemical exchange rates between bound andbulk water characterize the material exchange be-tween these two reservoirs and are restricted by thecustomary equilibrium flow condition for the firstorder exchange rates:

kcqc = kbqb (19)

so that only one rate (k&) has to be found.This three-site, linear exchange model is de-

scribed by a set of three coupled linear differentialequations:

dma/dt = — ( <rpa)ma + crpamb

= (rspi)ma—(Rib+aspb+kb)mi)-{-kimc (20)

dmc/dt = kcrrib — kc)mc

where

mi = [Mzi(oo) - M«i(*)]/nM«-(oo)

with n = 1 for saturation recovery and n = 2 forinversion recovery sequences. Indices a,b,c repre-sent macromolecular, bound water and bulk waterreservoirs, respectively; pa and p& are the fractions

Vol. 11, No. 1/2 81

MACROMOLECULARRESERVOIR

Distributionof correlation times

BOUND WATERRESERVOIR

anisotropicreorientation

BULK WATERRESERVOIRIsotropic rotation

Trmnslationai diffusion

V

Rib\

\V

LATTI

CE

Figure 8. A three site linear exchange model of the NMR properties of biological materials.

of protons in respective reservoirs, involved in cross-coupling interactions (it was assumed here that po =p& = 1, which means that all protons in both reser-voirs are involved in these interactions); the factor sis the fraction of bound water and macromolecularprotons directly coupled together (s = 1 used heremeans that only single proton-proton interactionswere assumed).

For any given set of parameters, characterizingthe proton dynamics in a biological sample, one cansolve the above set of equations and obtain the ex-perimental (apparent) relaxation characteristics ofthe sample (three apparent relaxation times andnine sizes of different magnetization components).These results can then be directly compared withselected experimental results, thus facilitating thefitting process.

To illustrate this idea, the relaxation data fornormal mouse muscle were modelled. The results ofrelaxation experiments (which used different combi-nations of selective and non-selective pulses to gen-erate different initial conditions for Eqns.20) at 40and 200 MHz (66) were used to select and limit therange of fitted parameters. To facilitate this step,all geometrical factors (proton-proton distances, an-gles etc.) were set according to available literature

data and kept constant during computations. Fi-nally, the fit of the longest relaxation time compo-nent to available frequency dispersion experimentaldata (43) was performed. These results are pre-sented in Figure 9a,b. The fit has been obtainedwith the following values of parameters:

rr = 5 x 10"8s, pT = 2.30; n = 5 x 10"6s,

A = 2.00; n = 5 x lO"7*;-^ = 1.5 x lO"11*;

qa = qb = 0.105; kb = 60s " 1 ;pa =pb = l.

These values are in good agreement with the lit-erature data available for similar, but simpler sys-tems. The modelled true (intrinsic) relaxation timesfor all three reservoirs are plotted versus frequency,in Figure 10.

V. ConclusionsThe MRI technology has opened new and impor-

tant applications of the NMR technique. There hasbeen tremendous progress in technical improvementof this new medical diagnostic tool; new results ofresearch in th\s area are continually being presented

82 Bulletin of Magnetic Resonance

1200,-

9 6 0 -

40 80 120Vo(MHz)

160 200

84.0"T

81.0

78.0J;

: \

M'ob

80 120rO(MHz)

100 200

Figure 9. The results of modelling the NMR prbperties of normal mouse muscle tissue. The lipid componenthas been excluded from this analysis, since it was assumed[that hydrophobie lipid groups interact very littlewith water, a) The frequency dependence of the apparent (measured) relaxation times (1 — long component; m- medium component, seen only in experiments where soft pulses were used; the short component is too smallto be seen). Circles and crosses are experimental data points for long and medium components, respectively(most of the experimental data were taken from ref. (43)). b) The frequency dependence of the apparent(measured) magnetization components. The distinction between a,b, and c is due to different T2s; l,m, and sindicate different Tls.

Vol. U, No. 1/2 83

80 120PQ (MHz)

160 200

Figure 10. The frequency dependence of inherent (true) relaxation times for reservoirs a,b, and c seen inFigure 8.

at scientific meetings. There is a common belief,however, that basic NMR properties of biologicalmaterials are still far from being fully understoodand a great deal of further research is needed toclarify the quantitative information in MR images(94). The resolution of this problem would have animportant influence on the diagnostic efficacy of MRimaging.

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