chapter 3

Chapter 3

Signal Generation andDetection

The magnetic moments ofnuclei in normal matter will result in a nu-clear paramagnetic polarization upon establishment ofequilibrium ina constant magnetic field. It is shown that a radio-frequency field atright angles to the constant field causes a forced precession ofthe to-tal polarization around the constant field with decreasing latitude asthe Larmor frequency approaches adiabatically the frequency of therffield. Thus there results a component of the nuclear polarizationin right angles to both the constant and the rffield and it is shownthat under normal laboratory conditions this component can induceobservable voltages.

Felix Bloch

Having reviewed the mathematical fundamentals, we now begin to discuss theimage formation principles of MRI. This chapter focuses on the signals: whatthey are, and how they are generated and detected from an object. To gain somefundamental understanding, we will start with a description of the nuclear mag-netic resonance (NMR) phenomenon and then gradually arrive at various signalexpressions.

As its name implies, NMR involves nuclei (of an object to be imaged), mag-netic fields (generated by an imager), and the resonance phenomenon (arisingfrom the interactions of the nuclei with the magnetic fields). Therefore, to masterthe mechanism underlying signal generation and detection in MRI and to un-derstand the characteristics of the signals measured, we need, in principle, tostart from the nuclear level. As we know, subatomic particles behave quantum-

57

58 Chapter 3 Signal Generation and Detection

mechanically, but, fortunately, MRI principles can often be accurately describedusing classical vector models because MRI deals with the collective behavior ofan ensemble of a huge number of nuclei present in a macroscopic object. Specif-ically, we will adopt a system approach for our discussions. In this approach, theobject being imaged is viewed as a linear system' (magnetized nuclear spin sys-tem), and the signal detected is a response activated from the system by an inputradio-frequency (RF) excitation that drives the system to a state of resonance. Inthe rest of the chapter, we will first describe what a magnetized spin system is,then discuss the effects ofRF excitations on a spin system, and finally characterizethe observed signals.

3.1 Magnetized Nuclear Spin Systems

To understand the NMR phenomenon, we begin with the object to be imaged. Weknow from basic chemistry that a biological sample or any physical object can bebroken down successively into its constituent molecules, then to atoms, and thento nuclei and their orbiting electrons. Nuclei have a finite radius (rv 10-14 m),a finite mass (rv 10-27 kg), and a net electric charge (rv 10-19 coulomb). Afundamental property of nuclei is that those with odd atomic weights and/or oddatomic numbers, such as the nucleus of the hydrogen atom (which has one pro-ton), possess an angular momentum J, often called spin. Although nuclear spinis a property characterized by quantum mechanics, in the classical vector model,spin is visualized as a physical rotation similar to the rotation of a top about itsaxis. In MRI,. an ensemble of nuclei of the same type present in an object beingimaged is referred to as a (nuclear) spin system. For example, all the protons (at-tached either to water or fat) form one spin system while the nuclei of 31 P formanother spin system. One important property of a nuclear spin system is the so-called nuclear magnetism created by placing it in an external magnetic field. Thismagnetism is the physical basis of MRI. In the rest of this section we will discussits origin and characteristics.

3.1.1 Nuclear Magnetic Moments

Nuclear magnetism ofa nuclear spin system originates from the microscopic mag-netic field associated with a nuclear spin. A classical argument for the existenceof this magnetic field is twofold: (1) a nucleus such as a proton has electricalcharges, and (2) it rotates around its own axis if it has a nonzero spin. Like anyspinning charged object, a nucleus with a nonzero spin creates a magnetic fieldaround it, which is analogous to that surrounding a microscopic bar magnet, asshown in Fig. 3.1. Physically, it is represented by a vector quantity il, which

1While imaging can be treated as a linear process, a spin system behaves nonlinearly during exci-tation, as described later in this chapter.

Section 3.1 Magnetized Nuclear Spin Systems 59

--+J

Figure 3.1 Nucleiwith nonzero ji, are regardedas microscopic magnets.

is called the nuclear magnetic dipole moment or magnetic moment. One funda-mental relationship of particle physics is that the spin angular momentum andmagnetic moment vectors are related to each other by

(3.1)

(3.2)

where 1 is a physical constant known as the gyromagnetic ratio. A related con-stant ~is also widely used, which is defined as

'Y~=-21r

Note that the value of 'Y or Of is nucleus-dependent. For instance, 'Y = 2.675 X

108 rad/s/T (Of = 42.58 MHzff) for IH while 1 = 7.075 X 107 radlsff (Of =11.26 MHzII') for 31 P. The Of values of some diagnostically relevant nuclei arelisted in Table 3.1.

Table 3.1 Properties of SomeNMR-Active Nuclei

Nucleus Spin RelativeSensitivity''

IH 1/2 1.00013C 1/2 0.01619F 1/2 0.87031P 1/2 0.093

Gyromagnetic Ratio"t(MHzJT)

42.5810.7140.0511.26

a Calculated at constant field for an equal number ofnuclei.

Since magnetic moment is a vector quantity, we need to know both its mag-nitude and its orientation to define it uniquely. Based on the theories of quantum


mechanics, the magnitude of il, often denoted as lill or simply It when there is noconfusion, is

It = ,nJI(1 + 1) (3.3)

where nis Planck's constant h (6.6 x 10-34 J-s) divided by 21r and 1is the nuclearspin quantum number. The spin quantum number takes integer or half-integer orzero values such that

1351=0'2,1'2,2'2'·" (3.4)

The value that I assumes for a particular nucleus is governed by the followingthree simple rules:

(a) Nuclei with an odd mass number have half-integral spin.

(b) Nuclei with an even mass number and an even charge number have zero spin.

(c) Nuclei with an even mass number but an odd charge number have integralspin.

For 1H, 13C, 19p, and 31 P nuclei, 1 = !' and such a spin system is called a spin- ~system. A nucleus is NMR-active only if 1 1= O.

Although the magnitude of il is certain under any conditions (with or with-out an external magnetic field), its direction is completely random in the absenceof an external magnetic field due to thermal random motion. This is somewhatanalogous to the situation with a collection of compass needles (analogous to themagnetic moments) sitting on a vibrating table (analogous to thermal motion).Therefore, at thermal equilibrium, no net magnetic field exists around a macro-scopic object.

To activate macroscopic magnetism from an object, it is necessary to line upthe spin vectors. This is accomplished by exposing the object to a strong externalmagnetic field. Following convention, we assume that an external magnetic fieldof strength Bo is applied in the z-direction of the laboratory frame such that

(3.5)

Unlike a compass needle which lines up exactly with an external magnetic field,a magnetic moment vector can assume one of a discrete set of orientations, anessential characteristic of the quantum model. In this model, the z-component ofj1 becomes certain due to the Bo field and is given by

(3.6)

where mI is called the magnetic quantum number. For any nucleus with nonzerospin, mI takes the following set of (21 + 1) values:

mI = -1,-1+1, ... ,1 (3.7)


which corresponds to (21 + 1) possible orientations for j1 with respect to the

direction of the external field. The angle () between j1 and Bo can be calculatedusing the following formula:

cos () = J-Lz = --;:=m=I==J.t JI(I+l)

(3.8)

(3.9)

(3.10)

While the orientation of [i is quantized along the direction of the external field,the direction of its transverse component j1xy remains random. Specifically, let

... ....i1xy = J.tx i + J.tyj

Then, J-Lx and tty can be expressed as

{J-Lx = I~xy' C?S~J-Ly =: IJ-Lxyl SIne

where e is a random variable uniformly distributed over [0, 27r) and liIxyIis givenby

liI",yI== Jp,2 - p,~ = 'YnJI(I + 1) - ml (3.11)

For a spin-! system, I = ~ and mI = ±~. It is easy to show, based onEqs. (3.8) and (3.11), that

(3.12)

(3.14)

(3.16)

and,... I 'Y;"P,,,,y == J2 (3.13)

Equation (3.12) implies that in a spin-! system, any magnetic moment vectortakes one of two possible orientations: pointing up (parallel) and pointing down(antiparallel), as shown in Fig. 3.2.

We next describe the motion of j1 when placed in an external magnetic field.We will use a classical treatment by assuming that iI is a classical magnetic mo-ment vector without mutual interactions. According to classical mechanics, thetorque that j1 experiences from the external magnetic field is given by j1 x Bok,which is equal to the rate of change of its angular momentum. That is,

dJ ...dt = j1 x Bok

Since j1 = ,J, we havedji, ...dt = ,il x Bok (3.15)

which is the equation of motion for isolated spins in the classical treatment. Thesolution to Eq. (3.15) can be expressed by (see derivation in Example 3.1)

{JLxy(t) =: ttXy(O)e-i-,Bot

J-Lz(t) = Jlz(O)

62

Q~.''ri, .~~

!~\~.;... ._{~':..

/':::;

• (~ (I,I --.~. ~ ~

tI:.( •...~......~....

(a)

Chapter 3 Signal Generation and Detection

Parallel

Antiparallel

(b)

Figure 3.2 Nuclearmagnetic momentvectors (a) pointingin randomdirections and (b)alignedin the direction of an external magnetic field.

where J.lxy(O) and J.lz(O) are the initial values and it is understood that

(3.17)

Equation (3.16) describes a precession of j1 about the z-axis (or the B o field),which is called nuclear precession. In the classical vector model, nuclear preces-sion is similar to the wobbling of a spinning top about the gravitational axis, asillustrated in Fig. 3.3.

x

z

.JIl5-----~y

x

t----.... y

Figure 3.3 Precession of a nuclearspin about an external magnetic fieldis similarto thewobbling of a spinningtop in a gravitational field.


Two important points about nuclear precession are evident from Eq. (3.16).First, the angular frequency of nuclear precession is

Wo ="'(Bo (3.18)

which is known as the Larmor frequency . Second, precession of j1 about Eo isclockwise if observed against the direction of the magnetic field. In practice, it iseasy to determine the precession direction using the left-hand rule. That is, if theleft thumb points in the direction of Bo, nuclear precession follows the directionof other fingers.

In addition to Eq. (3.16), nuclear precession can be described by an angularvelocity vector defined as

(3.19)

Another common way to describe the nuclear precession is through the use of arotation matrix. Specifically, let

RAn) = [

Equation (3.25) can be expressed as

cos a-sina

o

sin acos a

o ~] (3.20)

where p. should be interpreted as a column vector p. = [IL"" ILl/ I ILzjT.

(3.21)

64

and '


'(3.25)

d2Jty 2.dt 2 = - WOP,y (3.24)

These decoupled second -order differential equations have .solutions of the ,general f()rmA cos(wot}'+B sin(wot). Setting the initial conditions to Jt~(O).Jty(O), and It~(O) ~ we-get

'{'. " .(1) ~ p. :1: (0) cos(wot) + Jtll(O) Sin(W, ~t). ', Jty(t) ~, ;- Jl~ (O ) sin(wot) + Jty(O) cos(wot) "

it,At) = Jtz(O) , ' ,

which yields the result in Eq. (3.16) immediately when,pht in complex nota"tion.

3.1.2 Bulk Magnetization

To describe the collective behavior of a spin system, a macroscopic magnetizationvector 1\1 is introduced, which is the vector sum of all the microscopic magneticmoments in the object. Specifically, let il« represent the magnetic moment of thenth nuclear spin. Then,

(3.26)

where N. is the total number of spins in the object being imaged . This sectionanalyzes 1\1 for a spino! system .

Recall that 1\1 = 0 in the absence ofan external magnetic field. We shall nowfocus on how i1n behaves collectively when the object is placed in Bo. Based onthe discussion in the previous section, i1n takes one of two possible orientationswith respect to the z-axis at a given time. Spins in different orientations havedifferent energy of interaction with the external magnetic field Bo. Specifically,according to the quantum theory,

Hence, for pointing-up spins (mI = !),

E t = -hliBo

and for pointing-down spins (mI = -!),

E.l- = hliBo

(3.27)

(3.28)

(3.29)

Equations (3.28) and (3.29) indicate that the spin-up state is the lower-energystate, while the spin-down state is the higher-energy state. The energy difference

Section 3.1 MagnetizedNuclear Spin Systems

between the two spin states is given by

65

(3.30)

The nonzero difference in energy level between the two spin states is known asthe Zeeman splitting phenomenonand is illustrated in Fig. 3.4.

E= ~,nBo

E= -!,nBo

Booff Boon

Figure 3.4 Zeeman splitting for a spin-! system.

The spin populationdifferencein the two spin states is related to their energydifference. According to the well-knownBoltzmannrelationship,we have

Nt (b,.E)N.t. =exp KT

s

where

Nt : numberof pointing-up spins

NJ, : number of pointing-downspins

Ts : absolute temperatureof the spin system

K : Boltzmannconstant (1.38 x 10- 23 J/K)

In practice,~E«KTs

Consequently, by first-order approximation,

(dE) ,/iBoexp KT

s~ 1 + KT

s

Therefore,

(3.31)

(3.32)

(3.33)

(3.34)

66

and


Nt - N. ~ N: TnBo (3.35).... s2KTs

Equation (3.35) indicates that there is an excess of a very small fraction(i~~.) of spins in the lower-energy state. This uneven spin distribution betweenthe two spin states occurs because a spin is more likely to take the lower-energystate (with higher stability) than the higher-energy state. Although it is very small,the population difference between the two spin states generates an observablemacroscopic magnetization vector Mfrom a spin system. Such a spin system issaid to be magnetized. The resulting bulk magnetization, according to Eq. (3.26),is

M= Mxi + My) + Mzk

= (~px.n) t+ (~PY,n);+ (~pz.n) k (3.36)

(3.37)

where J-tx,n, J..Ly,n and J-tz,n are the projections of iln along the x-, u-, and z-axes. The first two terms of Eq. (3.36) are zero because the projection of iln ontothe transverse plane has a random phase while it precesses about the z-axis.' asdiscussed in Section 3.1.1. The value of J-tz,n is given, according to Eq. (3.6), by

{+! 'YIi if iln is pointing up

Pz,n = -~')'/i if fin is pointing down

Substituting Eq. (3.37) into Eq. (3.36) gives

(3.38)

(3.39)

Therefore, the bulk magnetization vector points exactly along the positive direc-tion of the z-axis at equilibrium. Its magnitude is

M O = IMI= ,),2/i2B oNs

z 4KTs

Equation (3.39) indicates that the magnitude of M is directly proportionalto the external magnetic field strength Bo and the total number of spins Ns . Thevalue of N; is characteristic of an object being imaged and cannot be changed ingeneral; therefore, Bo and Ts are the only controllable parameters. For a givenspin system, one can increase the magnitude of Mby increasing Boor decreasing

2 According to Eq. (3.10), E~~l J-Lz,n = J-Lzy E~~l cos en = 0 for a random variable enuniformly distributed over [0,211"].


Ts • Since MRI experiments are often carried out with the object being at roomtemperature, one is limited to increasing the magnitude of the applied magneticfield for an increase in the bulk magnetization. The optimal field strength forimaging is dependent on the application.' For most clinical MRI systems, B«ranges from 0.2 to 2 T.

Note that Eq. (3.39) is only valid for a spin-~ system. For a spin-I system,we have

M O = "(2 1i,2 BoNsI(I + 1)z 3KTs

Detailed discussion of this formula can be found in [1].

(3.40)

3Thepenetration depthof an RF field decreases with increasing frequency, but there is a dramaticincrease in detection sensitivity withfieldstrength, roughlyproportional to B~/4

•


3.1.3 More on the Larmor Frequency

Let us restate that the precession frequency of il experiencing a Bo field is givenby

Wo = ,Bo (3.41)

This relation, popularly known as the Larmor equation, is an important equationbecause the Larmor frequency is the natural resonance frequency of a spin system.

Equation (3.41) shows that the resonance frequency of a spin system is lin-early dependent on both the strength of the external magnetic field Bo and thevalue of the gyromagnetic ratio v, This simple relationship is the physical basisfor achieving nucleus specificity. As a case in point, nuclei of 1Hand 31P in anobject resonate at 42.58 MHz and 11.26 MHz, respectively, when the object isplaced in Bo = 1 T; this difference in resonance frequency enables us to selec-tively image one of them without "disturbing" the other.

In practice, a specific spin system (say, protons) may have a range of res-onance frequencies. In this case, we call each group of nuclear spins that sharethe same resonance frequency an isochromat. There are two main reasons for amagnetized spin system to have multiple isochromats: (a) the existence of inho-mogeneities in the B o field, and (b) the chemical shift effect.

It is obvious from Eq. (3.41) that when Bo is not homogeneous, spins withthe same "'( value will have different Larmor frequencies at different spatialloca-tions. It is easy to derive the frequency distribution of a spin system if the inhomo-geneity ofa given Bo is known. The chemical shift effect is due to the fact that nu-clei in a spin system are attached to different chemical environments (molecules)in a chemically heterogeneous object. Since each nucleus of a molecule is sur-rounded by orbiting electrons, these orbiting electrons produce their own weakmagnetic fields, which "shield" the nucleus to varying degrees depending on theposition of the nucleus in the molecule. As a result, the effective magnetic fieldthat a nucleus "sees" is

Bo = Bo(l - (5) (3.42)

where <5 is a shielding constant taking on either positive or negative values. Basedon the Larmor relationship, the resonance frequency for the nucleus is

Wo =Wo - dw = Wo (1 - 8) (3.43)

Equation (3.43) indicates that spins in different chemical environments will haverelative shifts in their resonance frequency even when Bo is perfectly homoge-neous.

Clearly, the frequency shift ~w is dependent on both the strength of theexternal field Bo and the shielding constant J. The value of J is very small,usually on the order of a few parts per million (ppm) and is dependent on the localchemical environment in which the nucleus is situated. A well-known exampleis that "fat" (CH2) protons in biological objects display about a 3.35 ppm shift inLarmor frequency from "water" (H20) protons.

Section 3.2 RF Excitations 69

For biological objects, a large range of 8 values could exist, giving riseto many resonance frequencies. Assuming that the maximum chemical shift iswM /2, the resonance frequency range of a spin system can be expressed as

Iw - wol ::; WM /2 (3.44)

where W M is called the (chemical shift) frequency bandwidth of the spin system.Knowledge of these chemical shift frequencies and the corresponding spin den-sities is of great importance for determining the chemical structures of an object,which is the subject of NMR spectroscopy.

3.2 RF Excitations

We have thus far discussed two aspects of an NMR phenomenon: nuclei and astatic magnetic field. The macroscopic effect of an external magnetic field Bo onan ensemble of nuclei with nonzero spins is the generation of an observable bulkmagnetization vector 1\1 pointing along the direction of Bo. Although there is amicroscopic transverse component for each magnetic moment vector, the trans-verse component of 1\1 is zero at equilibrium because the precessing magneticmoments have random phases, as indicated by Eq. (3.16). A snapshot of an en-semble of a large number of spins (I = !) will be a set of vectors spreading outin the two precessing cones, as illustrated in Fig. 3.5. Establishment of a phasecoherence among these "randomly" precessing spins in a magnetized spin systemis referred to as.resonance.

3.2.1 Resonance Condition

Before we state the resonance condition, let us look at the popular "swing" anal-ogy. Suppose that a row of swings at a children's playground have the same length(thus, each swing-child complex has the same natural frequency). If the childrendo not begin swinging at the same time, a random phase relationship exists amongthe swings; that is, at a given time, the children are at different points of the swing-ing arc. For the swings to reach phase coherence, external forces must be appliedat the natural frequency of the swings. For example, if all the swings are pushedin unison at the natural frequency of the swing by a parent placed behind eachswing, the children will soon swing in phase, exerting a coherent force on thesuspension bar.

For a magnetized spin system, the external force (energy) comes from anoscillating magnetic field denoted as B1(t) in distinction from the static Eo field.

The resonance condition based on classical physics is that .81 (t) rotates in thesame manner as the precessing spins. A more rigorous argument is based on thequantum model. In this model, electromagnetic radiation of frequency Wrf carriesenergy (Planck's law):

Erf = 1ic.vr f (3.45)


Figure 3.5 Distribution of nuclear magnetic moments observedat anarbitrary timeinstant. The excessof vectorspointingalongtheexternalfieldis greatlyexaggerated.

To induce a coherent transition of spins from one energy state to another, theradiation energy must be equal to the energy difference dE between the adjacentspin states. That is,

or

/iu;rf = dE = ,nBo

Wrf = Wo

(3.46)

(3.47)

Equation (3.47) is known as the resonance condition. In the following, we firstdescribe what an RF pulse is and then discuss in detail the effect ofRF excitationsunder on- and off-resonance conditions.

3.2.2 Characteristics of an RF Pulse

RF pulse is a synonym of the B 1 field, so called because the B 1 field is short-lived and oscillates in the radio-frequency range. Specifically, the B1 field isnormally turned on for a few microseconds or milliseconds Also, in contrast tothe static magnetic field Eo, the EI field is much weaker (e.g., B I = 50 mT whileB o = 1.5 T).

Section 3.2 RF Excitations

A typical B1 field takes the following form:

B1(t ) = 2B~(t) cOS(Wrft + tp)i

where

71

(3.48)

Bi(t) : pulse envelope functionWrf : excitation carrier frequency<p : initial phase angle

This field is said to be linearly polarized because it oscillates linearly along thex-axis. Mathematically, it can be decomposed into two circularly polarized fieldsrotating in opposite directions, that is,

B1 (t ) = B~(t)[cos (Wrft+ tp)i- sin (Wrft + <p)J]+ Bf(t) [cos(Wrft + <p)i + sin (Wrft + <p)JJ (3.49)

where the first bracketed term rotates clockwise and the second rotates counter-clockwise, as illustrated in Fig. 3.6. Since the counterclockwise component ro-tates in the opposite direction of the precessing spins, it exerts negligible effectson a spin system if Wrf is near the Larmor frequency.' Therefore, the effectivejj1 ( t) field that needs to be considered here is

B1 (t ) = B~(t)[COS(Wrft + ep)i - sin(wrft + ep)J1 (3.50)

which has an x-component as

B1,x = B~(t) COS(Wrft + <p)

and a y-component as

B1,y = -B~(t) sin(wrft + ep)

(3.51)

(3.52)

Many modem NMR systems use so-called quadrature RF transmitter coilsto generate this circularly polarized field directly, with the advantage of reducedRF power deposition. Unless specified otherwise, the B1(t) field used in theremainder of this book will be assumed to be in this form. For brevity, we willalso adopt the following complex notation:

In summary, an RF pulse generates an oscillating B1(t) field perpendicular

to the Eo field. The main parameters characterizing an RF pulse include (a) theenvelope function Bi(t), (b) the excitation carrier frequency Wrf, and (c) the initial

4The maineffectof this off-resonance component is a very slight shift of the observedresonanceline, whichis known as the Bloch-Siegert shift. This frequency shift disappears when the Bl fieldisturnedon.


z

2BJCoSWI

Figure 3.6 A linearly polarized field represented as two

counter-rotating circularly polarized fields.

phase angle <po The initial phase <p, if it is a constant, has no significant effect onthe excitation result and is assumed to be zero for the present discussion. Theexcitation frequency Wrf is a constant for most RF pulses' and is determined bythe resonance condition. The envelope function Bi (t) is the heart of an RF pulse.It uniquely specifies the shape and duration of an RF pulse, and thus its excitationproperty. In fact, many RF pulses are named based purely on the characteristics ofthis function. For example, the envelope function of the widely used rectangularpulse (shown in Fig. 3.7a), is defined as

o~ t ~ Tp

otherwise(3.54)

where Tp is the pulse width. Another popular pulse, called the sine pulse (Fig. 3.7b),uses the following envelope function:

(3.55)

Before describing the effect of such a pulse on a spin system, we next introducetwo mathematical tools: the rotating reference frame and the Bloch equation.

3.2.3 Rotating Frame of Reference

A rotating frame is a coordinate system whose transverse plane is rotating clock-wise at an angular frequency w. To distinguish it from the conventional stationaryframe, we use x', y', and z' to denote the three orthogonal axes of this frame,and correspondingly, i', I', and k' as their unit directional vectors. Mathemati-cally, this frame is related to the stationary (laboratory) frame by the following

5 For some specialpulses,such as adiabaticpulses,Wrf can be a function of time.


(a) (b)

73

(3.56)

(3.58)

(3.57)

Figure 3.7 RF pulses with (a) a rectangularenvelope function, and (b) asine envelope function.

transformation:

{i' ~ cos(wt)i - sin(wt))l' ~ sin(wt)i+ cos(wt»)k'~k

Two special rotating frames are in use , which correspond to w = Wo andw = Wrf , respectively. In the first case, the (x', y')-plane precesses at the Larmorfrequency of the spin system, and we call it the Larmor-rotating frame. In thesecond case, the (x', y')-plane rotates as the B1(t) field, and we call it the RF-rotating frame. When Wrf = -yBo, both rotating frames are the same. Therefore,when there is no confusion, we use the generic term, rotating frame or w-rotatingframe, to refer to either, depending on the context.

The advantage of introducing the rotating frame lies in the conceptual sim-plicity it affords in describing the excitation effect of an RF pulse, as will becomeevident later in this chapter. In the sequel, we will present several useful relation-ships associated with this transformation.

First, the time derivatives of the unit directional vectors of the rotating frameare given by

di' ~ ~-=wXtdtdJ' ~ ~,-=wxJdt

dk' ~ k~'-=wXdt

where wcan be arbitrary but is equal to -wk for the transformation specified byEq. (3.56).

Second,let

74

and


M-- A M ~ M '"""t, --,rot = x'''' + y'J + Mz,k

Setting M= Mrot yields

(3.59)

(3.60)

Similarly, let(3.61)

and

We then have

[Bl,x' ] = [C?swt - sinwt ] [ Bl,x ]».; Slnwt coswt Bl,y

(3.62)

(3.63)

Equations (3.60) and (3.63) specify how to convert the magnetization vector andB l vector between the laboratory and rotating frames. The transformation rulescan also be succinctly written in complex notation. Specifically,

M - M iwtx'y' - xye

where M xy = M'; + iMy and Mx'y' = M x' + iMy" Similarly,

where B l = Bl,a: + iB1,y and B1,rot = B1,x' + iB1,y"Third, let

dM~

dt

Then,

(3.64)

(3.65)

(3.66a)

(3.66b)

dM _ 8Mrot - M (3.67)dt - at + W X rot

Clearly, based on the definitions in Eq. (3.66), df! is the rate of change of Mas observed in the laboratory frame, while 8~:Qt is the rate of change of M asobserved in the rotating frame. Therefore,

(3.68)


'~.

'ii' Example3;5

Given that. BI(t) ';' BICO~Wrr'ti '-!Ji si'ri w~fi; " .. '

we determin~:the$field as6bserv~din th~wrf-r~t~ti~g fr~~~. , ' ~ : · '.

Since ..": ' [ ~:::J ~ [ -~~=~:n " '. ..... .. the Bi fieldobserV¥i~ ,~~w;f::rot~ting frame, -accordingtOEq; (3:63), is:.given by : -.',.,. . ,.,. '.

~" .In vector notation,

~ , . . ": .,;;',:.' BI,rot(t) =:= Bli _

. Therefore, the given RFfield~6mes a stitir6n~ryfield -pointing aiongthe'<x'-axis in the wrf-rotating frame. . ., . ;''

.- - ';-

3.2.4 The BlochEquation

The time-dependent behavior of if in the presence of an applied magnetic fieldBI (t) is described quantitatively by the Bloch equation. In the context of MRI,the Bloch equation takes the following general form:"

(3.69)

where M2 is the thermal equilibrium value for if in the presence of Bo only,which can be calculated from Eq. (3.39) or (3.40); TI and T2 are time constantscharacterizing the relaxation process of a spin system after it has been disturbedfrom its thermal equilibrium state, a topic to be further discussed in Section 3.3and in Chapter 7. For the present discussion, we drop the last two terms inEq. (3.69) because we are interested only in the behavior of if during the RFexcitation period. This treatment is acceptable if the duration of an RF pulse isshort compared to TI and T2 , as is often the case in practice.

6 A moregeneral formof the Blochequationwas givenby Torrey [256).


Under this assumption, the Bloch equation takes a simpler form:

dM ......-='Y M x Bdt

77

(3.70)

One may recognize that this Bloch equation is identical to the equation of motionfor a free spin in Eq. (3.15) if M is replaced by ji. We next express the equationin the rotating frame. Substituting Eq. (3.67) into Eq. (3.70), we get

We may rewrite Eq. (3.71) as

( ...)... ... w= "YMrot x s.; + :; (3.71)

where

(3.72)

...... ... wBeff = B rot + - (3.73)

'Yis the effective magnetic field that the bulk magnetization vector "experiences"in the rotating frame. The second term in Eq. (3.73) represents a fictitious fieldcomponent for Simplified behavior ofMrot . To see this more clearly, let jj = Bokand W= -'-yBok. Then,

... ... '-yBok ... ...Beff = B rot - -- = Bok - Bok = 0

'Y(3.74)

Therefore, the apparent longitudinal field vanishes and Mrot appears to be sta-tionary in the rotating frame.

Following the same analysis, the general Bloch equation in Eq. (3.69) can beexpressed in the rotating frame as

(3.75)

3.2.5 On-Resonance Excitations

We now look into the effects of an RF pulse on a spin system by examining thetime-dependent behavior of Mduring the excitation period. We first considerthe simple case in which a spin system has a single isochromat resonating atWo = 'YBo. For simplicity, we further assume that the initial phase angle cp is

78 Chapter 3 Signal Generationand Detection

zero for the genericRF pulse defined in Eq. (3.50). Using the transformation rulespecified by Eq. (3.63), we have

B1,rot = B~(t)i'

The effectivefield that the nuclear spins see in the rotating frame is

Beff = Bok' + BHt)f + Wrf'Y

= (Bo - ~rf ) k' + BHt)f

Invoking the on-resonanceexcitationconditionthat

Wrf = Wo = 'YBo

(3.76)

(3.77)

(3.78)

we immediately getBeff = B~(t)i' (3.79)

Substituting the aboveresult into the Bloch equation in Eq. (3.72) yields the fol-lowingequationof motion for the bulk magnetization vector M:

In scalar form, we have

8Mrot M,- B e ( )~~='Y rot X It~ (3.80)

(3.81)

dMx'dt=OdMy ' e( )dt = 'YBI t u;

dMz' e( )--;It = -'YBI t My'

A closed-form solution to Eq. (3.81) under the initial conditions Mx'(O) =My' (0) = 0 and Mz ' (0) = M~ is as follows:

Mx,(t) = 0

My,(t) = M2 sin (it /,BHi)di)

Mz,(t) = M2cos (it /,BHi)di)

(3.82)

These equations indicate that the effect of the on-resonanceexcitation B1 field,as observed in the RF-rotating frame, is a precession of the bulk magnetization


about the x'-axis. This is not surprising since the effective field Beff points alongthe x' -axis. As an example, consider the case where

Then, Eq. (3.82) becomes

BHt) =BIll (t -;;/2) (3.83)

(3.84)

where WI = fBI. It is now apparent that the bulk magnetization vector precessesabout the x'-axis with angular velocity

(3.85)

as shown in Fig. 3.8. The precession of Mabout the B 1 field is called forcedprecession. Equation (3.85) can be derived directly from the Larmor relationship,since the effective field that the spins see in the rotating frame is Bl i'.

x'

--------.. y'

(a)

z

I-------I~y

(b)

Figure 3.8 Motionof thebulk magnetization vectorin the presence of a rotatingRF fieldas observed in (a) the RF-rotating frame, and (b) the laboratory frame.

3.2.5.1 Flip Angle

As a result of the forced precession, the bulk magnetization is tipped away fromthe zl-axis, creating a measurable transverse component MZlyl. The flip angle


a is defined as the smaller angle between M and the z-axis. Clearly, based onEq. (3.82), the value of a at the end of an RF pulse is given by

r: r:a =10

WI (t)dt = 10

"'{Bl(t)dt

In the case of a rectangular pulse,

(3.86)

(3.87)

As a numerical example, let Tp = 0.1 ms and B I = 0.6 G. We then have a = jfor protons.

It is obvious from Eqs. (3.86) and (3.87) that the flip angle depends on boththe magnitude of the .81(t) field and the duration ofexposure. Normally, the pulsewidth is chosen based on the frequency selectivity desired, and we can adjust theexcitation power to vary the flip angle. For example, for a given Tp, increasingthe pulse intensity by a factor of 2 (namely, setting B I to 2B I ) , will double theflip angle according to Eq. (3.87). Another important point to note here is thatthe shape and form of the pulse envelope function are unimportant as long asthe area under Bj(t) is the same. In other words, for different Bj(t), M travelsin different trajectories during the excitation period but will end up in the samespatial location if the area under Bj (t) is the same.

3.2.5.2 Calculation of Mafter an a Pulse

Before we describe how to calculate the effect of an RF pulse through the use ofa rotation operator, it is useful to make clear several notations.

(a) If an RF pulse rotates Mabout the .81 field in the rotating frame by an anglea, we commonly call the pulse an a pulse. Clearly, a 90° or ~ pulse rotates

M by 90°; likewise, a 180° or 1r pulse rotates Mby 180°. Sometimes, itis necessary to make the axis of rotation explicit. Assume that the BI fieldin the rotating frame points in a direction specified by (C{J, 0), as shown in

Fig. 3.9; we call the corresponding .81 field an 0('1',6) pulse. In practice, itis usually assumed that () = 0 and the pulse is simply written as 0'1'. Twopopular choices of C{J are 0 or 90°, corresponding to a ox' pulse and. a 0y'

pulse, respectively.

(b) We use t = 0_ and t = 0+ to represent the time instants immediately beforeand after a pulse, respectively.

(c) We will use --t as a general spin processing operator. For example,

(3.88)


z'

81

y'

(3.89)

(3.90)

Figure 3.9 A tilted RF field with an initial phase angle f{J and a tilt angle 8.

With these clarifications, let us first consider the effect of an a z , pulse. Basedon Eq. (3.82), we have

{

Mz'(O+) =°My'(O+) =M2sinaMz'(O+) = M2cosa

if the spin system was at thermal equilibrium before the pulse. Under a moregeneral prepulse condition, the postpulse magnetization is given by

{

Mz'(O+) = Mx'(O_)My'(O+) = My'(O_) cos a + Mz,(O_) sin aMz'(O+) = -My' (0_) sin a + Mz,(O_) cos a

Similarly to Eq. (3.20), we define a rotation operator about the x'-, y'-, andz'-axis, respectively, as

Rz,(a) = [~ CO~Q Si~Q]o -s sin e cos o

(3.91)

[

cos oRy,(a) = .0

sin a

~ -s~nQ]o cos a

(3.92)

and

[

cos o

-S~Q

sin acos a

o ~ ] (3.93)


As illustrated in Fig. 3.10, R x " R y " and R z ' specify a clockwise rotation asobserved against the x' -, y' -, and z' -axis, respectively.

x'

,z

;.----.. y'

(a) x'

z'

~-t-.....o-.y'

(b) x'

z'

I----.. y'

(c)

Figure 3.10 Graphical representation of (a) Rx " (b) Ry " and (c) R,«

With the matrix operators defined in Eqs. (3.91), (3.92), and (3.93), the effectof an 0 pulse applied along the x'- or y' -axis can be calculated as follows:

Mrot(O+) ~ Rx,(o)Mrot(O_) (3.94a)

Mrot(O+)Q-:l:'

R-x,(o)Mrot(O_) = R x' (-o)Mrot(O_) (3.94b)--t

Mrot(O+) ~ R y ' (0 )Mrot (0_) (3.94c)

Mrot(O+) ~ R_y,(o)Mrot(O_) = R y ' ( -o)Mrot(O_) (3.94d)

where it is understood that Mrot = [Mx' , My" Mz,]T.The effect of an of() pulse can be represented by three cascaded spin rotations:

-epz,ox,epz'. Or in terms of the rotation matrices, we have"

which gives the following postpulse magnetizations:

Mx'(O+) = Mx'(O_)(cosa sin2 cp + cos 2 cp) + My'(O_) sin2 i sin2cp

-MZI(O_) sino sin ep (3.96a)

My' (0+) = u; (0_ ) sin2 i sin 2cp + My' (0_ )(cos a cos 2 cp + sin2 cp)

+Mz,(O_)sinocosep (3.96b)

Mz'(O+) = MXI(O_)sinosinep - My'(O_) sino cos ip

+Mz'(O_) coso (3.96c)

7Notethat the operatorprecedence is fromright to left.


The effect of an o( ep,8) pulse can be calculated in an analogous manner.Specifically, it can be shown that

Mrot (0+) = RZI (ep)Ryl(O)Rxl (0)Ryl(-O)Rzl (-ep)Mrot (0_) (3.97)

where 0= - ~ + 9. The resulting magnetization is given by

MXI(O+) = MXI(O_)[coso(sin 2 ep + cos2 epcos2 9) + cos2 epsin2 8]

+Myl (0- )[sin2 ~ sin 2<p sin2 0 + sin a cos 0]

+MZI (0- )[sin2 ~ sin 20 cos <p - sin a sin <p sin 0] (3.98a)

Myl(0+) = M.,,(O_ )[sin2 ~ sin 2<p sin2 0 - sin a cos 0]

+ Myl(0_) [coso( cos2 ep + sin2 ep cos2 8) + sin2 ep sin2 8]

+Mz /(0_)[sin 2 ~ sin 20 sin <p + sin a cos <p sinO] (3.98b)

MZ/(O+) = Mo:l (O_)[sin a sin <p sin 0 + sin2 ~ cos<psin20]

+Myl(O_)[- sin a cos <psinO + sin2 ~ sin ip sin 20]

+MZI (0_) [cos0 sin2 8 + cos2 9] (3.98c)

The postpulse magnetization can also be expressed in the laboratory frame.According to Eq. (3.60), we have

by noting that

sinWOTp

COSWQTp

°(3.99)

-sinWQTp

COSWQTp

°[

CO~WQTp- slnwQTp

0,

~inWQTp

COSWQTp

° ~ ] (3.100)

where Tp is the pulse duration and it is assumed that the rotating reference framewas set in motion immediately before the pulse is applied.

3.2.5.3 ~xamm))les

Through the next three examples, we show how to use the formulas in the preced-ing section for calculating the excitation effects.


3.2.6 Ott-Resonance Excitations

87

Most excitations are assumed to be on-resonance. However, in practice, if mag-netic field inhomogeneities and chemical-shift effects are not negligible, excita-tions rarely are exactly on-resonance for all the isochromats. When the excitationfield is off-resonance for a certain isochromat, the effective magnetic field that theisochromat sees in the rotating frame is

Beff = (Bo - W;f) k' + BHt)i'

= ~wo k' + BHt)i', (3.105)

where ~wo = Wo - Wrf measures the degree of off-resonance.Equation (3.105) suggests that the effective field has two components: the

usual B 1 component pointing along the x'-axis and a residual component ~wo/'Ypointing along the z'-axis, as shown in Fig. 3.lla. Intuitively, based on the abovediscussion one can predict a precession of Mrot about Beff. A more rigorousanalysis can be obtained by directly solving the following Bloch equation:

(3.106)

which governs the motion of Mduring the RF pulse. Unfortunately, a closed-form solution to the above equations is not available for an arbitrary envelopefunction Bj (t). To illustrate the difference between on-resonance and off-resonanceexcitations, we consider a simple case with a rectangular pulse for which Bi(t) =B1II( t-Tp /2). For this pulse, a closed-form solution to the Bloch equation indeed

T p

exists, which is given by

where

and

{

Mx,(t) = M~ sin9cos9[1 - COS(Wefft)]

My,(t) = M~ sin 9 sin(wefft)

Mz,(t) = M~[cos2 9 + sin2 ocos (Wefft)]

Wef£ = J~w~ + w~

o= arctan (:~J

(3.107)

(3.108)

(3.109)


The magnetization components along each axis immediately after the pulseare given by

{

Mx'(O+) = Mx'(Tp ) = M2sin9cos9(1- cos a)

My'(O+) = My' (Tp ) = M2sin9sina

Mz'(O+) = Mz'(Tp ) = M2(cos2 9 + sin2 9cosa)

(3.110)

where a = WeffTp is now the flip angle about the axis of the effective magneticfield.

Note that the transverse magnetization immediately after the pulse is nolonger along the y' -axis as in the case of on-resonance excitation but has a phaseshift CPo from the y' -axis toward the x' -axis, which is given by

Mx(O+) sinOcos9(1 - cos a)tan 'Po = = --------

My(O+) sin osin a

(1 - cos a) ~wo a ~wo= =tan---

sin a Weff 2 Weft'(3.111)

It is evident from Eq. (3.111) that the phase shift 'Po increases almost linearlywith the frequency shift ~wo. This phase shift can be problematic for some MRIapplications. In addition, the magnitude of the transverse magnetization given by

Mx'y'(O+) = VM;,(O+) + M;,(O+)

=M~ sin Ovsin2 a + (1 - cos a)2 cos2 0

decreases as the frequency offset increases.

3.2.7 Frequency Selectivityof an RF Pulse

(3.112)

From the discussion in Section 3.2.6, we know that for a spin system with morethan one resonance frequency, an RF pulse of the form B}e-iwrft for 0 ~ t ~ "»

will excite not only !VI(Wrf) but other isochromats as well. An important questionthat one often encounters is: How will a pulse of the general form Bi(t)e-iwrrtaffect the various isochromats of a spin system? To give an exact answer to thisquestion, we need to resort to the Bloch equation. However, a closed-form so-lution to the Bloch equation is not available under this general situation. In thissection, we describe an approximate approach based on Fourier analysis.

It is well known that Fourier analysis of a time function reveals its spectralcontent. Specifically, let

(3.113)


z'

.. 'H

e ff•••••

J---- y'

(a) (b)

z'

y'

89

Figure 3.11 Off-resonance excitation: (a) effective field in the rotating frame.and (b) precession of Mrot about the effective field.

We have

(3.114)

(3.115)

and

Bt(t) = -.!... /00 {FBn(w)e-i(W+Wrf)tdw211" -00

In signal processing, {FB~}(w) is interpreted as the frequency spectrumof the pulse envelope function. It is more insightful here to view it as the am-plitude of a clockwise rotating vector specified by e-i(W+Wrf)t. In other words,Eq. (3.115) decomposes B t (t) into a continuum of clockwise-rotating microvec-tors with amplitudes {FB~}(w)dw in the complex plane. Similarly. we can de-compose the bulk magnetization vectors in terms of its isochromats as

M= [: M(w)dw (3.116)

To establish a link between the isochromats and the micro-Bj vectors, wefurther assume that the spin system behaves like a linear system;8 namely, theexcitation effect of B 1(t) is equal to the sum of the excitation effects of indi-vidual component {FB~}(w)dwe-i(W+wr,)t. Since {FB~}(w)dwe-i(w+Wrf)t is

a fictitious pulse of infinite length but of infinitesimal strength. it will, in steady

8The linearsystem assumption is not validfora nuclearspinsystemduring excitation. Therefore.the excitation property of an RF pulsederived fromFourieranalysis is not accurate.


state, excite only the on-resonance isochromat M(w + Wrf) if the relaxation ef-fects can be ignored." Therefore, the frequency excitation property of an RF pulseof the form Bi(t)e- i wrf t is fully characterized by {FBi}(w) under conditions ofFourier analysis. To be more specific, {:FBi} (w)dw is the excitation field that

acts on M(w + Wrf). Since {FBi}(w) is a complex-valued function, we canrewrite it as {FBi}(w) = /{:FBi}(w)/eicp(w). Therefore, the on-resonance mag-netic field that each isochromat sees has a phase shift cp(w) from the x'-axis. 10 Asa result, there is a phase dispersal among the isochromats when they flip down tothe transverse plane, as expected from the discussion in Section 3.2.6. In addition,the flip angle is different for different isochromats. According to Eq. (3.86), theflip angle for the M(Wrf) isochromat is given by

r0:(0) = "I10

BHr)dr (3.117)

Based on the linearity assumption, the flip angle for other isochromats is given by

I{:FBi}(w)Io:(w) = I{FBf}(O) IO:(Wrf) (3.118)

Knowing a:(w) and cp(w), we can calculate the resulting postpulse magnetizationfor each isochromat according to Eq. (3.96).

As an example, let us examine the excitation property of a rectangular pulsewhose envelope function is Bi(t) = B 1II( t- Tp/2). From the results in Exam-

T p

ple 2.3, we have

(3.119)

Therefore, the frequency excitation property is characterized by the followingequations:

a:(w) =0(0)cp(w) =

sine (~WTp) (3. 120a)

(3. 120b)

Note that we allow flip angle a to take negative values instead of advancing ip by180 0

• Clearly, the two ways are equivalent for describing the spin motion.Before concluding this section, an observation from Eq. (3.120a) is in or-

der. If we treat the sine function to be zero beyond the first zero crossing on both

9This is a validassumption if the durationof Br(t) is short relativeto the spin relaxation times.For those pulses, the Fourierprediction is ratheraccurate. For longerpulses, the Fourierprediction isless accuratebecause the relaxationeffectscannot be ignored; consequently, the spin system cannotbe treatedas a linear systemdue to spin interactions.

lOSince <p(0) = 0, M(Wrf) will alwaysfliparound the x'-axis.

Section 3.3 Free Precession and Relaxation 91

sides, a rectangular pulse of the above form will excite nuclear spins resonatingover a frequency range Iw - wrfl < 21r/Tp . Therefore, a short rectangular pulseof T» = 1 us will excite nuclei resonating over a frequency bandwidth of 103 kHzcentered around the excitation frequency Wrf' These short rectangular pulses (ap-proximating a 8-function) are called hard or nonselective pulses, since they aredesigned to excite "everything" in the spin system. On the other hand, a rectan-gular RF pulse of Tp = 10 ms will produce excitation over a narrow frequencybandwidth of 100 Hz. Such long pulses are called selective pulses because theyselectively excite nuclei resonating in the selected frequency range. In practice,based on the asymptotic property of the Fourier transform, better frequency selec-tivity can be achieved by utilizing smoother pulses, such as the Gaussian or sinepulses, rather than the rectangular pulse. For this reason, long selective pulses areoften called soft pulses. The design of an RF pulse with good frequency selectiv-ity is an important subject of MRI and is still an active area of research. We willreturn to this topic when we discuss the signal localization in Chapter 5.

3.3 Free Precession and Relaxation

After a magnetized spin system has been perturbed from its thermal equilibriumstate by an RF pulse, it will, according to the laws of thermodynamics, returnto this state, provided the external force is removed and sufficient time is given.This process is characterized by a precession of 1\1 about the B o field, called freeprecession; a recovery of the longitudinal magnetization M z , called longitudinalrelaxation; and the destruction of the transverse magnetization M x y , called trans-verse relaxation. Both relaxation processes are often ascribed to the existence oftime-dependent microscopic magnetic fields that surround a nucleus as a resultof the random thermal motions present in an object. But the exact mechanismsby which these relaxation events occur for an arbitrary spin system are far too di-verse and complex to be properly covered here. The interested reader is referredto the text by Abragam [1]. In this section, we give only a phenomenological'description of the relaxation process using the Bloch equation. The effect of spinrelaxations on image appearance (contrast) is dependent on the excitation schemeused for data acquisition, an important topic to be discussed in Chapter 7.

Phenomenologically, the transverse and longitudinal relaxations are describedby a first-order process. Specifically, in the Larmor-rotating frame, we have

(3.121)

These equations are directly derived from the rotating frame Bloch equation inEq. (3.75) in which the first term drops out because Beff = (B o- wo/,)k' = o.


Solving Eq. (3.121), we obtain the following time evolution for the transverse andlongitudinal magnetization components:

{MX'y,(t) = M X'y,(0+)e- t / T2

Mz,(t) = M~ (1 - e-t / T1) + Mz,(O+)e-t / T l(3.122)

(3.123)

where Mx'y'(O+) and Mz'(O+) are the magnetizations on the transverse planeand along the z-axis immediately after an RF pulse, and M~ is, as before, thelongitudinal magnetization at thermal equilibrium.

An important point about this phenomenological description is that both thedecay of the transverse magnetization and the recovery of the longitudinal mag-netization after an RF perturbation follow an exponential function. This expo-nential description, especially for the transverse relaxation, applies only to spinsystems with weak spin-spin interactions, as is 'the case with spins residing inliquid state molecules. For solids and macromolecules, the mechanisms for trans-verse relaxation are more complicated. For many biological applications of MRI,however, we deal almost exclusively with "slowly" relaxing spins for which thephenomenological description is often appropriate.

Another point worth noting is that TI and T2 are not defined as the timesat which longitudinal and transverse relaxations are completed. To see this pointmore clearly, consider the TI and T2 relaxations after a 90° pulse, which producesMx'y'(O+) = M2 and Mz'(O+) = 0. By some simple arithmetic, we can easilyverify, based on Eq. (3.122), that '

{Mz,(TI) ~ 63%M2

M X 'y,(T2) ~ 37%Mx 'y' (O+)

Therefore, Mz ' will regain 63% of its thermal equilibrium value after a time in-terval TI , but Mx'y' will lose 63% of its initial value after a time interval T2 , asillustrated in Fig. 3.12. The values of TI and T2 depend on the tissue composi-tion, structure, and surroundings. For a given spin system, T1 is always longerthan T2• As an example, T1 is about 300 to 2000 ms, and T2 is about 30 to 150 msin biological tissues.

The combined effect of free precession and relaxation can be seen by puttingthe magnetization vector back to the laboratory frame. Specifically, applying thetransformation rule in Eq. (3.64) to Eq. (3.122), we obtain

where

Mxy(t) = MXy(0+)e-t/T2e-iwot

Mz(t) = M2 (1 - e-t / T 1) + Mz(O+)e-t/ T l

(3. 124a)

(3. 124b)

(3.125)

Section 3.3 Free Precession and Relaxation

M (t)=M (O)e -tIT2xy xy

93

37°J'oM (0) - - - - - - - -xy

(a)

--------------------------------

(b)

Figure 3.12 Relaxation curves.

is the "initial" transverse magnetization observed in the laboratory frame by thesignal detection system. The time delay term is included because we reset thetime to zero at the end of the pulse (t = Tp ) to describe the relaxation effects.

Equation (3.124) gives an "exact" phenomenological description of howtransverse and longitudinal magnetization evolves after an RF pulse as time pro-gresses. Specifically, it is clear from Eq. (3.124a) that in the laboratory framethe evolution of the transverse magnetization is characterized by an exponentialdecay e-t / T2 and a precession about the Bo field e-iwot . The length of the freeprecession period is dependent on the T2 value. For biological tissues, T2 is onthe order of tens of milliseconds, which enables detection of MR signals duringthis period. It is also worth noting that while 1\1 spirals back to the z-axis, itsmagnitude is not preserved because of the relaxation processes, as illustrated inFig. 3.13. This behavior is different from that of 1\1 during the excitation period,when 1\1 spirals down from the z-axis with a fixed magnitude. II

11 Relaxation is normally ignoredduringthe excitation periodfor mostRFpulses.


z

x y

Figure 3.13 The trajectory of the tip of Mduring the relax-ation period as observed in the laboratory frame.

3.4 Signal Detection

We know from the foregoing discussion that by placing an object in an exter-nal magnetic field Eo and stimulating it with another alternating or oscillatingmagnetic field EI (t). we can induce a macroscopic magnetism in the form of arotating magnetization in the object . This is the so-called NMR phenomenon.The next question is how to detect this magnetism. or more specifically. how toconvert this rotating magnetization to electrical signals. We address this questionin this section. We begin with a brief review of the basic physical principles ofsignal detection and then describe the concept of signal demodulation and quadra-ture detection. Emphasis is placed on deriving mathematical expressions of theactivated signal in terms of various variables characterizing the spin system andthe excitation conditions.

3.4.1 Basic Detection Principles

MR signal detection is based on the well-known Faraday law of electromagneticinduction and the principle of reciprocity. The Faraday law of induction statesthat time-varying magnetic flux through a conducting loop (a receiver coil) willinduce in the coil an electromagnetic force (or voltage) that is equal to the rate atwhich the magnetic flux through the coil is changing. There are many daily-lifeexamples of this law in action. That the power generators convert mechanicalrotation of a permanent magnet into household electricity is one such example.

Section 3.4 Signal Detection 9S

In MRI, the bulk magnetization is precessing at a radio frequency and anyconducting loop resonating at the frequency can be used as a receiver coil. Infact, in many circumstances, the same RF coil used for excitation is also used fordetection. The detection sensitivity of a receiver coil is determined through theprinciple of reciprocity [13]. Specifically, assume that Br (r) is the laboratoryframe magnetic field at location r produced by a hypothetical unit direct currentflowing in the coil. Then, the magnetic flux through the coil by M(r, t) is givenby

4l(t) = r Br(r) • M(r, t)dr (3.126)}object

Then, according to the Faraday law of induction, the voltage V(t) induced in thecoil is

8t1l(t) 81 ~ ~V(t) =--8- = --8 Br(r). M(r,t)drt t object

(3.127)

The voltage V(t) induced in the receiver coil is often regarded as the raw NMRsignal. Therefore, Eq. (3.127) is the most basic formula of MR signal detection,which embodies the Faraday law of induction and the principle of reciprocity.From this formula we can quantitatively determine how various factors in an NMRexperiment affect the received MR signal. In the ensuing section, we use thisformula to derive some commonly used signal expressions.

3.4.2 Signal Expressions

The term signal can mean various things in MRI. It refers sometimes to the trans-verse magnetization, sometimes to the induced voltage signal, and sometimes tothe induced voltage after some processing. This section shows what form thesignal takes at different stages of the signal detection module.

Let us begin by rewriting Eq. (3.127) in scalar form as

V(t) = - ~ r. [Br,:t (r)M:t (r, t) + Br,y(r)My(r, t)JobJect

+ Br,z(r)Mz(r, t)]dr

where the following vector decomposition is assumed:

Br = e. xi+ s, yJ"-t + Br zk, , ,

(3.128)

(3.129)

Since Mz(r, t) is a slowly varying function compared to the free precession ofthe Mx and My components, the last term in Eq. (3.128) can be ignored, yielding

V(t) = -1 [B (r) 8Mx(r, t) B (r)8My(r,t)] dr. r,x 8t + r,y 8t

object(3.130)


Equation (3.130) indicates that the induced voltage is a function of only Mx andMy. This is why it is normally known that MR signals are dependent on thetransverse magnetization.

To develop this expression further, we rewrite Br,x and Br,y as

{Br,x. = IBr,xy(r)1 C.OS<Pr(r)

(3.131)Br,y,,= IBr,xy(r)1 SIn <Pr(r)

where <Pr (r) is the reception phase angle. If the reception field at location r pointsalong the z-axis, then <Pr (r) = o. On the other hand, if the field points along they-axis, <Pr (r) = 1r /2. For other cases, rPr (r) takes a value between 0 and 21r.

To evaluate the time derivative of Mx and My as required by Eq. (3.130), weinvoke the free precession equation, Eq. (3.124), from which we can obtain

Mx{r, t) = IMxy{r,O)le-t/ T2(r ) cos[-w{r)t + <Pe{r)] (3.132a)

= w{r)/Mxy{r, 0)/e-t / T2(r ) sin[-w(r)t + rPe(r)]

1- T

2(r) IM",y(r, O)le-t

/ T2(r ) cos[-w(r)t + ¢e(r)]

(3.133a)

= -w(r)/Mxy(r, 0)le- t / T2(r ) cos[-w{r)t + <Pe{r)]

- T2~r) IM",y(r, O)le-t / T2(r ) sin[-w(r)t + ¢e(r)]

(3.133b)

oMy(r,t)at

My{r, t) = /Mxy(r, O)le-t / T2(r ) sin[-w(r)t + rPe(r)] (3.132b)

where rPe (r) is the initial phase shift introduced by RF excitation. Similarlyto rPr (r ), rPe (r) takes a value between ° and 21r depending on the directionof Mxy{r, 0). Specifically, <Pe(r) = ° if Mxy{r, O) lies along the z-axis, or

rPe(r) = 1r /2 if Mxy(r, 0) lies along the y-axis.From Eq. (3.132), one immediately obtains

8Mx (r ,t )at

For most applications, free precession is at a much faster rate than relaxation,namely,

w{r) ~ 1/T2 {r ) (3.134)

Hence, the second terms in the equations above can be ignored, yielding

8M~~r, t) = w(r)IM",y(r,O)le-t / T2(r ) sin(-w(r)t + ¢e(r)]

(3.135a)

8M~r, t) = -w(r)IM",y(r, O)le-t / T2(r ) cos(-w(r)t + ¢e(r)]

(3.135b)

Section 3.4 Signal Detection 97

Substituting Eqs. (3.131) and (3.135) into Eq. (3.130) with some simplifica-tions, we obtain

or

V(t) = - ( w(r)\Br,xy(r)IIMxy(r,O)le-t/T2(r)Jobject

sin[-w(r)t + cPe(r) - cPr(r)]dr

V(t) = ( w(r)IBr,xy(r)IIMxy(r,O)le-t/T2(r)Jobject

cos [-w(r)t + <Pe(r) - <Pr(r) + iJ dr

(3.136)

(3.137)

Equation (3.136) or (3.137) is a basic signal expression that explicitly shows thedependence of a detected voltage signal on the laboratory frame transverse mag-netization Mxy(r, 0), the free precession frequency w(r), and the detection sen-sitivity of the receiver coil Br,xy(r ).

The voltage signal V (t) is a high-frequency signal because the transversemagnetization vector precesses at the Larmor frequency, as observed at the lab-oratory frame. This can pose unnecessary problems for electronic circuitries inlater processing stages. In practice, V(t) is moved to a low-frequency band usingwhat is known as the phase-sensitive detection (PSD) method, or signal demodu-lation method. Signal demodulation consists of multiplying V(t) by a referencesinusoidal signal and then low-pass-filtering it to remove the high-frequency com-ponent. Referring to Fig. 3.14a and assuming that the reference signal is 2 cos wot,we have

2V(t) coswot = 2 ( w(r)IBr,xy(r)IIMxy(r,O)le-t/T2(r)i-:cos [-w(r)t + <Pe(r) - <Pr(r) + iJ cos wotdr

= ( w(r)IBr,xy(r)IIMxy(r,O)le-t/T2(r)}object

cos [-w(r)t - wot+ <Pe(r) - <Pr(r) + iJ dr

+ ( w(r)IBr,xy(r)IIMxy(r,O)le-t/T2(r)}object

cos [-w(r)t + wot+ <Pe(r) - <Pr(r) + iJ dr

(3.138)

Removing the first component by low-pass filtering will result in a low-frequency signal, which is the output of the PSD system. Denoting this signal


(a)Input

I----~ Low-Pass FilterOutput

PSD

Referencesignal

2coscoot ~S(t)'------~~_-1

V(t)(b)

2sincoot

Figure 3.14 Phase-sensitive detection.

as Vpsd (t), we have

Vpsd(t) =1 w(r)JBr,xy(r)IIMxy(r,O)/e-t/T2(r)object

COS [-w(r)t + wot+ tPe(r) - tPr(r) + iJ dr (3.139)

It is often convenient to express w(r) as

w(r) = Wo + ~w(r) (3.140)

where ~w(r) is the spatially dependent resonance frequency in the rotating frame.Then, we have

(3.142)

Vpsd(t) = r [wo + ~w(r)lIBr,3:y(r)IIMxy(r,O)le-t/T2(r)}object

cos [-~w(r)t + tPe(r) - tPr(r) + iJ dr (3.141)

In practice, ~w(r) «wo, and Eq. (3.141) can be further simplified to

Vpsd(t) = Wo1 /Br,xy(r)IIMxy(r,O)le-t/T2(r)object

cos [-~w(r)t + tPe(r) - tPr(r) + i"J dr

Section 3.4 Signal Detection 99

which is a general expression for signals from a single PSD. Since t1w( r) is theprecessionfrequency in the rotating frame, Vpsd(t) is often regarded as the signaldetected in the rotating frame. A notable drawback of this detection scheme isthat we cannot determine from the signal whether the isochromat is precessingclockwise(t1w > 0) or counterclockwise(t1w < 0). Toovercomethis problem, asecondPSD system is used with referencesignal 2sin wot,which has a 90° phaseshift relative to the first. It is easy to show that the output from this detectionsystem is

Vpsd(t) = Wo1 IBr,xy(r)IIMxy(r,0)le-t/T2(")object

sin [-~w(r)t + 4>e(r) - 4>r(r) + ~] dr (3.143)

In this way,we are detectingthe rotatingmagnetizationwith two "detectors,"which are orthogonal to each other. This detection scheme, known as quadraturedetection, is commonlyused in modernMRIsystems. The two outputs from sucha system are often put in a complex form, as shown in Fig. 3.14b, with one outputbeing treated as the real part and the other as the imaginarypart.

Specifically, letS(t) = SR(t) + iSI(t) (3.144)

with SR(t) being the output from the first PSD given in Eq. (3.142) and SI(t)being the output from the secondPSD given in Eq. (3.143). Then,

S(t) = Wo { IBr,xy(r )IIMxy(r ,O)!e- i [a w(r )t - 4>e(r )+t/>..(r)-1I'/2)dr}object

(3.145)Invokingthe earlier-establishedcomplex notation that

{Br,xy = s.; ~ iBr,yMxy = Mx+'tMy

we have

IB (r)le-i4>r(") = B* (r)r,xy r,xy

IMxy(r ,O)lei 4>e(" } = Mxy(r,O)

(3.146)

(3.147a)

(3.147b)

where B;,xy is the complexconjugateof Br,xy. WithEq. (3.147),Eq. (3.145) canbe written as

S(t) = <"! B;,xy(r)Mxy(r,O)e-idW(,.}tdr (3.148)object

The scaling constant woei1l'/ 2 in Eq. (3.148) is often omitted, resulting in the fol-lowingpopular signal expression:

S(t) = 1 B;,xy(r)Mxy(r,O)e-iAW(,.)tdr (3.149)object


Furthermore, if the receiver coil has a homogeneous reception field over theregion of interest, as is often assumed, the signal expression in Eq. (3.149) can befurther simplified to

S(t) =1 MXy(r,O)e-idw(r)tdr (3.150)object

Note that in the preceding derivation, it is implicitly assumed that the objectsees a static inhomogeneous magnetic field during the free precession period ..Expressing the field distribution as

we have

B(r) = Bo+ ~B(r)

~w(r) = ,LlB(r)

(3.151)

(3.152)

(3.154)

(3.153)

and Eq. (3.150) becomes

S(t) =1 Mxy(r,O)e-i'"YdB(r)tdrobject

If the inhomogeneous field is time-varying, that is, ~B is a function ofboth space and time, then all the foregoing signal expressions need to be mod-ified accordingly. Specifically, denoting the inhomogeneous field componentas ~B(r, t), ~w(r)t should be replaced by , f~ ~B(r, r)dr. For example,Eq. (3.148) should be rewritten as

S(t) = w ei 1r/ 21 B* (r)M (r O)e-i'"Y udB(r,r)dTdro r,xy xy,object

• Example 3.9

We calculate the signal generated by an a pulse in this example.Assume thatthe.object·has.• a thermal. equilibrium magnetizationM~(r) ..

The.transverse magnetization generated. by.thepulse is

Mzy{r,t = 0+) =M2(r)sinaeicPe(r )

Substituting the-result .intoEq. (3~153) .yields

S(t) =sin a 1...• M2(r)eil/>e(r)e-i"(~B(r)tdrobject

whichisthe.desiredexpression for the signal generated. by an arbitrary apulse in thepresenceof an inhomogeneous static field.

Exercises

Exercises

101

3.1 For the following nuclei, does their spin quantum number take an integral,half-integral, or zero value? For each case, discuss whether the nucleus isNMR-active.

(a) IH,2H

(b) 160 , 170

(c) 12C,13C

(d) 31P,23Na

3.2 In the absence of an external magnetic field, a bulk object exhibits no netnuclear magnetism because:

(a) Nuclear magnetic moments for all nuclei are zero.

(b) Nuclear magnetic moments cancel out each other.

(c) The bulk magnetization vector is too small to be detected.

(d) All of the above.

3.3 What are the primary functions of the static magnetic field Eo in MR imag-ing?

3.4 What is the Zeeman splitting phenomenon?

3.5 Why is MRI known as a low-sensitivity imaging technique?

3.6 What is the primary function of the oscillating B1 ( t) field?

3.7 What is the resonance condition?

3.8 Why does a spin system often have more than one resonance frequency?If you place a cup of water in a perfectly homogeneous magnetic field, doyou expect to detect more than one resonance frequency from the protons?Why?

3.9 What is an isochromat?

3.10 Justify the last two equations in Eq. (3.57).

3.11 Given a fixed flip angle, the larger the Mthe stronger the El needed be-cause a stronger force is required to flip a larger M. True or false?


3.12 Briefly discuss how one can selectively elicit the NMR phenomenon fromone spin system of a biological sample (such as protons) without affectingthe others (such as 31P)?

3.13 Justify that the two representations of nuclear precession in Eq. (3.25) andEq. (3.16) are equivalent.

3.14 Prove the following relationships for the rotation matrices RXI (a), Ryl (a),and RZI (a):

(a) R;,1(a) = Rx,(-a) = R_xl(a)

(b) R;/ 1(a) = Ryl (- a) = R_ v'(a )

(c) R~l(a) =Rzl(-a) =R_z,(a)

3.15 In which plane does the receiver coil pick up the activated MR signal?Is the received signal dependent on the time evolution of the longitudinalcomponent after an RF pulse? Why?

3.16 Calculate and sketch B1,rot(t) assuming that

3.17 The bulk magnetization of a proton spin system is flipped 90° by a rectan-gular RF pulse of width 1.0 ms.

(a) What is the magnitude of the B, field required?

(b) How many precession cycles take place in the laboratory frame dur-ing the pulse, assuming B o= 0.5, 1.0, and 1.5 T, respectively.

3.18 Assume that Bl(t) = Bl coswrfti - Bl sinwrftJ is a stationary vector in

the wrf-rotating frame, namely, Bl,rot(t) = Blf. Derive an expression

for .81(t) such that

(a) Bl,rot (t) = BlJ'

(b) Bl,rot(t) = Bli' + BlI'

3.19 Derive the closed-form solution given in Eq. (3.82) for the Bloch equationfor on-resonance excitation with an arbitrary pulse.

3.20 Calculate and depict the bulk magnetization vector of a spin system rel-ative to the prepulse reference frame after a 90~, pulse. Assume that the

Exercises 103

Larmor frequency of the spin system is 10 MHz, the pulse lasts 1.0 ms, andthe prepulse condition is Mx ' (0_) = My' (0_) = 0 and Mz ' (0_) = M~.

3.21 Calculate the resulting magnetization in the laboratory frame immediatelyafter a 90~, pulse with duration of T and 2r, respectively.

3.22 Assume that a spin system with a single resonance component was at ther-mal equilibrium. Calculate the transverse magnetization resulting from thefollowing excitation sequences:

(a) 90~,90;,

(b) 90~, - r - 90;,

(c) 45~,90;,

(d) 30~, (-15~, )80~,15~,

3.23 Calculate the effects of the following excitation sequences on a spin sys-tem with two isochromats at resonance frequencies Wo and Wo - c5wo. It isassumed that the spin system is at thermal equilibrium and T = j.

(a) 90~, - r - 180;,

(b) 45~, - r/2 - 90;,

3.24 Derive the closed-form solution given in Eq. (3.107) for the Bloch equationfor off-resonance excitation with a rectangular pulse.

3.25 Prove the relationships given in Eq. (3.57).

3.26 Prove the result in Eq. (3.60).

3.27 Assume that a known RF pulse B1(t) = B1cos(wot)i - B1sin(wot)]flips the bulk magnetization vector onto the y' -axis (of the rotating frame)immediately after the pulse. Modify this :81 field such that the bulk mag-netization vector ends up in the following positions immediately after thepulse:

(a) Lying along the -y'-axis

(b) Lying along the x'-axis

(c) Lying along the - x'-axis

(d) Lying along a vector 45° away from the y'-axis toward the a' in thetransverse plane


3.28 Specify two pulses that will convert Mx'y' to M;,y' and -M;,y" respec-tively.

3.29 Use a vector model to schematically show the effects of a 90~" 90~x"

90;" 90~y" 180~" 180;, pulse on the bulk magnetization vector originallypointing along the z'-axis.

3.30 The excitation property of an RF pulse is derived from the inverse Fouriertransform of its envelope function. How is it related to the forward trans-form?

3.31 An RF pulse applied along the x'-axis for 100 us flips an "on-resonance"magnetization by 90° onto the y' -axis. How much magnetization is tippedonto the (x', y')-plane if the excitation is "off-resonance" by 10kHz?

3.32 Describe what is meant by "hard" and "soft" pulses.

3.33 The frequency distribution of an RF pulse can presumably be calculatedfrom its Fourier transform. Compare the situation pertaining to Prob-lem 3.31 with the result you expect from the Fourier transform.

3.34 A spin system has three isochromats with resonance frequencies at wo,Wo + ~, and Wo - ~, where Wo =42 MHz and ~ =0.25 kHz. We next as-sume that an RF pulse defined by B 1 (t) = Bi(t)e-iwot, where the Fouriertransform of Bi (t) is given in the following figure, will flip the isochro-mats by 90°, 67.5°, and 67.5°, respectively. Calculate the flip angles of allthe isochromats for the following pulses based on the Fourier theory.

(a) Bi(2t)e-iwot

(b) 2Bi(2t)e-iwot

(c) B~ (t /2)e- iwot

(d) !Bi(t/2)e-iwot

(e) 2Bi (2t)e- i (Wo+A)t

(f) 2B~ (2t)e- i (Wo-A)t

(g) !Bi(t/2)e-i (wo+A)t

(h) !Bi (t / 2)e- i (wo- A)t

Exercises

{FBi}(f)1

__ "-- ---L ~_~ f

lOS

-1 kHz 1 kHz

3.35 Design an RF pulse that will selectively excite a 10 kHz wide frequencyband centered at 42 MHz with 450 flip angle for a spin system of protons.

3.36 After an RF pulse, M x y decays to zero and M z returns to M2. During thisrelaxation process, the amount of M x y lost is equal to the amount of M z

that is gained. True or false?

3.37 During the excitation period, the magnitude of M(t) stays constant while

M(t) spirals down. Give an example to demonstrate that this statement isnot true during the relaxation period when M(t) spirals up.

3.38 How long does it take for the longitudinal magnetization M z of a spinsystem with longitudinal relaxation time constant T1 to recover 63% of itsthermal equilibrium value after (a) a 900 pulse and (b) a 750 pulse?

3.39 A spin system is excited by a 180~, - r - 90~, sequence with r rv 2T1 .

(a) Plot the time evolution of the MZIcomponent in the r time interval.

(b) Calculate the magnitude of Mx'y' immediately after the 90~, pulseand plot its time evolution after this pulse.

3.40 An imaging sequence often involves a series of excitation pulses to gen-erate signals to cover k-space. Since a 900 pulse completely rotates anyavailable Mz component onto the transverse plane, magnetization alongthe z-axis is always zero immediately after a 900 pulse in any imagingsequence with 90 0 excitation pulses. True or false?

3.41 Why is forced precession much slower than free precession?

chapter 3

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