Magnetic Resonance Imaging - Tutorial I

H. Panepucci And A. Tannús

Description

11994Magnetic Resonance ImagingSão Carlos - SP - BrasilDepartamento de Física e InformáticaInstituto de Física de São Carlos - Universidade de São Paulo - USPH. Panepucci and A. TannúsTutorial Idonupnnn2nunNature of the MR imageThe MR image is actually a map of the very weak magnetization, originated from some ofthe atomic nuclei in the body tissue, in the presence of an external magnetic field. Since thismagnetization is proportional to the density of those nuclei, the MR image shows the distributionof the selected atoms. Because of the large hydrogen concentration, soft tissue is easily seen inproton MR images, that show even the small differences due to their different chemicalcomposition. The resulting soft tissue contrast is thus better that in X- ray techniques and can befurther improved by methods that use the difference in dynamic behavior of the tissueProperties of the atomic nucleiThe atomic nuclei useful for MRI have two properties that are fundamental for the- an intrinsic angular momentum or spin, Ih - a permanent magnetic moment, m = g I hwhere I is the nuclear spin operator, is Planck's constant and g g is called the nuclearhThe above two properties make these nuclei resemble a small magnet spinning around itsom nFigure 1As shown in Figure 1 , the magnetic field B produces a torque, m m x B , on the magneticdipole moment m of the nucleus. This torque, being normal to the angular momentum vector,m to precess ...

Informations

• Publié par: Shiem
• Nombre de lectures: 26
• Langue: English

Extrait

Magnetic Resonance Imaging

Tutorial I

H. Panepucci and A. Tannús

Instituto de Física de São Carlos - Universidade de São Paulo - USP Departamento de Física e Informática São Carlos - SP - Brasil 1994

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Nature of the MR image

The MR image is actually a map of the very weak magnetization, originated from some of the atomic nuclei in the body tissue, in the presence of an external magnetic field. Since this magnetization is proportional to the density of those nuclei, the MR image shows the distribution of the selected atoms. Because of the large hydrogen concentration, soft tissue is easily seen in proton MR images, that show even the small differences due to their different chemical composition. The resulting soft tissue contrast is thus better that in X- ray techniques and can be further improved by methods that use the difference in dynamic behavior of the tissue magnetization.

Properties of the atomic nuclei

The atomic nuclei useful for MRI have two properties that are fundamental for the occurrence of the magnetic resonance phenomena: - anintrinsic angular momentumorspin,I h   and - apermanent magnetic moment,m =Ig  h   whereI  is the nuclear spin operator, h  is Planck's constant andgi s called the nuclear gyromagnetic ratio whose value depends on the nuclear species. The above two properties make these nuclei resemble a small magnet spinning around its axis. Therefore they behave in a magnetic field, like a top spinning on a table in the earth's gravity field.

B o

m n : m agnetic om ent m

spinning nucleus

Figure 1

As shown in Figure 1, the magnetic fieldB produces a torque, mx B , on the magnetic dipole moment m the nucleus. This torque, being normal to the angular momentum vector, of continuously changes the spinning direction making the nuclear momentm to precess aroundB. Using the definitions above, this result can be written as v d d t m r = - gB´ r m[1] which means that the component of the nuclear magnetic momentmperpendicular to the magnetic fieldBrotates around it with an angular frequency

w L =gB ,

called the nuclear Larmor frequency. This relation is of central importance in MRI.

2

[2]

Voxel magnetization

We shall use the termvoxelto refer to an elementary tissue volume, ideally homogeneous in composition, whose proton magnetization density will be represented by the pixel brightness in the MR image. It is composed of a large number of molecules comprising some N ~ 10 20 hydrogen nuclei each having a magnetic momentm. If no field is present these moments will be r randomly oriented and the net magnetizationM= å r m will be zero. In a fieldB, according to the above discussion, the transverse component of each moment is rotating aroundBwith the same frequencywwith random phases in thermal equilibrium., but Thus, at equilibrium, there is no net magnetization component transverse to the field , i.e.

M T = 0

[3]

The component of m alongB be either parallel or anti parallel to it. At absolute could zero complete alignment would occur. At temperatureT, due to thermal agitation, both orientations are present with a small excess fraction, of the order of mB/kT, in the parallel lower energy direction. This results in a net equilibrium voxel magnetization, longitudinal to the field, given by

Figure 2 illustrates this.

x

M L = M 0 = N m. mB / kT

z

M o

y

[4]

Figure 2 As explained bellow, the nuclear spin system can be excited, changing the values ofM T andM L  those given above. In that case it is generally observed that fromM L  andM T  return exponentially to their thermal equilibrium values with characteristic time constantsT 1  andT 2  , known as longitudinal and transverse relaxation times respectively. Since the microscopic mechanisms that relax the longitudinal component ofm also are effective in relaxing the transverse one,M T relaxes faster thanM L , that isT 2 is shorter or equal tT han 1.

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then

and

Assuming that at t = 0 the voxel magnetization has values M L (0) = 0 and M T (0) = M 0 ,

M L  M(t) = 0 (1 - exp(-t/T1) )

M T (t) = M 0 exp(-t/T2) .

This is illustrated in Figure 3 below.

M o

a)

M o ( 1 - exp (- t / T 1  ))

t

M T

exp (- t / T 2  )

b)

t

[5]

[6]

Figure 3 It must be noticed that if there is a transverse component of the voxel magnetization present, it will rotate at the Larmor frequencyw L ,as pictured in Figure 4.

B o z

M L

M

M T w o =

g B o

y

x Figure 4 WithM = M L  + M T the total pixel vector magnetization is the overall behavior of  , described by the following vector equation due to Felix Bloch r r r d r =g .M r ´ r B - M 2 T - (M L T - M 0 ) [7] M dt T 1

The values of T 1 and T 2 depend not only on the nuclear species, but also on the chemical composition of the tissue. Typical values are 300 to 600 ms for T 1 and 30 to 80 ms for T 2 .

4

RF excitation of the voxel magnetization

The basic phenomenon that allows the measurement of the voxel magnetization giving the name to this imaging technique, is Nuclear Magnetic Resonance or NMR, discovered simultaneously by F. Bloch and E. M. Purcell in 1946. It consists of the RF resonant excitation of the nuclear spins, precessing in a magnetic field, and the subsequent observation of the spin system response. The resonant excitation occurs when the applied magnetic RF field,B 1  , has approximately the same frequency as the precession of the nuclear moments in the external fixed fieldB 0 this happens consider the simple apparatus shown in Figure 5.. To understand how z B o

x

Figure 5

B 1 (t)

y

A RF generator that can deliver short pulses of lengtht, is connected to a coil where an oscillating magnetic field , normal toB 0 is produced and thus applied on the nuclei. Let the frequency of the oscillating field be equal to the Larmor frequency and its amplitude 2B 1 . As shown in Figure 6, it can be thought as the superposition of two circularly polarized counter-rotating fields in the plane perpendicular to B 0 , each of amplitude B 1 . y

B 1

B + 1

2 B 1 cos(wt)

Figure 6

x

To the nuclear moments precessing atw L , one of the rotating fields will look stationary, while the other will be changing direction at a rate too fast. We can then ignore this last, that will have no first order effect, and look at the situation from a coordinate system rotating around B 0 with angular frequencyw L . In thisrotating framethe individual nuclear momentsm i  ,will only see a fieldB 1 at fixed angles to them as illustrated in Figure 7-a.

5

x

a)

z

M o

y

b)

M o

Figure 7 They will therefore precess around it at the corresponding frequency w 1 =gB 1

[8]

Now, since the total voxel magnetization is justM =S m i , it will also precess aroundB 1 at the same angular frequency w, ure 7-b. Going back to the experiment, if the RF pulse 1    Fig lastst then at the end of it, the original equilibrium magnetization,seconds, M 0   , initially along B 0 , would have rotated aroundB 1  by an angle

q = wt=gB 1  t .

This implies in the creation of a non zero transverse magnetization

M T = M 0 sinq

[9]

[10]

as shown in Figure 8. This is a non equilibrium situation, associated with the increase of the internal energy of the nuclear spins system due to the resonant RF excitation. Two very important cases, also illustrated in the figure are those for q = p/a2n d   q = p.R F pulses that produce this rotations are calledp/2a n d   ppulses and are respectively used toexciteorinvertthe nuclear magnetization.

x

B 1

z

q=g B 1 t

M

y M T = M o se n q

B 1

Figure 8

6

q = p /2

M T = M o

B 1

q = p /2

M T = M o

Measurement of the voxel magnetization: the FID

We just saw that, immediately following ap/2p ulse, a magnetization normal toB 0  and equal in size to the equilibrium value M 0 is present in every excited voxel. It will then start precession in the transverse plane atw L . Denoting this initial value by M T (0) and using complex notation to represent a vector in the plane, we can write for the value at a later time t

M T (t) = M T (0) exp( iw L t) exp( - t / T 2 ) ,

[11]

where the last exponential factor represents the relaxation of this non equilibrium, and therefore transient transverse component, towards its thermal null value . We can now go again back to our experiment and place a second coil also normal toB 0 and connected to a receiving device such as an oscilloscope as in Figure 9-a.

x

a)

B o

z

M

e . m .f .

y

I n d u c e d e m f

F I D

b)

t

Figure 9 Each voxel of volumedvwill have a momentM T (t) dvand will thus be equivalent to a small revolving magnet that will induce in the receiving coil a small oscillating e.m.f. of frequency w L   and decreasing amplitude. This is called theFree Induction Decaysignal or FID whose amplitude is a measure of the voxel nuclear magnetization and thus of proportional to the local proton density. The overall signal from all nuclei inside the coil will be the sum of those coming from the individual voxels, if all experience the same fieldB 0 gets an FID as in Figure 9-, then one b of amplitude proportional to the total number of the selected nuclei.

The relaxation mechanisms

at the longitudinal magnetizatio results fro er of We saw before th nM L  m an excess numb individual momentsm i  precessing with its fix component parallel toB 0 . Similarly, in order to have a finite magnetization transverse to the field,M T  , an excess number of nuclear moments must be precessing in phase , that is, their transverse components must be pointing simultaneously in the same direction. Thisphasingof the spins is accomplished by the RF pulse during the resonant excitation described above.

7

The return to equilibrium after excitation say, by ap/2p ulse involves two different processes that take place at the same time although at different rates; these are:

- relaxation of the transverse component M T , calledtransverseor T 2  axatreloin   -relaxation of the longitudinal component M L, , calledlongitudinalor T 1 relaxation

The microscopic mechanisms behind this two processes involve the random fluctuation of the nuclear spin interactions. For the case of protons in tissue, the most relevant of this is the dipole-dipole interaction between the nuclear momentsm i  ,t he fluctuation in this case being mostly due to the random reorientation of the tissue molecules. To see how these works, consider thelocal field produced by a dipolem at the site of another placed at a distancer. Its module is

|B loc | =m (c3o s2q - 1)r 3/, 

[12]

whereqi s the angle between the vectorrand the external fieldB 0 , as illustrated in Figure 10.  

m i

B o

q ij

r ij

m j

B loc

Figure 10 This local field has components parallel and transverse toB 0  both will undergo rapid and as well as slow fluctuations due to changes inqwith molecular reorientation. Considering first the transverse component of the randomly time dependent fieldB loc (t), some of the fast fluctuations may introduce high frequencies nearw L ,   in its frequency spectrum. Thisresonant harmoniccomponents of the  random rotations producetransverse local field will of the local moments m i . It has two effects: First, it randomly changes the phases of the individual spins precession, destroying phase coherence between them and thus reducingM T . This contributes to theT 2  relaxation.

8

Second, by changing the component of m i  parallel toB 0 , it allows the exchange between the nuclei magnetic energy, given by the dot productm i . B 0 , and the thermally excited mechanical energy of the moving molecules. This is a thermalization process that takes the longitudinal magnetization to its equilibrium valueM 0 it is therefore responsible for the, T 1  relaxation . Let us now look at the component ofB loc (t)that is parallel to the external field. This cannot produce spin rotations, no matter its frequency spectrum, and therefore does not contribute to T 1 other hand the slow fluctuating parallel local field, adds to. On the B 0 producing local random changes in the nuclear Larmor frequency. This makes some nuclei precess faster than others thus destroying their phase coherence. This mechanism also contributes to T 2 . This is the reason why the transverse magnetizationM T  relaxes faster thanM L , that is, T 2 is always shorter than T 1 . z M L = M o

x

z M L

x

x

y

z

M L

y

M T =

T 2

y

M

0 e - t/ T 2

<

T 1

x

M L = M 0 ( 1 - e - t/T 1 )

x

t

M L

z

M T = M o y

x

y

z

M T

y

Figure 11 Figure 11 is an attempt to illustrate the above discussion showing the behavior of the nuclear momentsm i and the resulting net longitudinal and transverse magnetizations,M L  andM T  , in the rotating frame, following ap/2e xcitation pulse.

Non uniform magnetic field: Spin Echoes

9

All the previous discussions assumed a spatially uniform fieldB 0 .This is not the real case because real magnets do not produce perfectly homogeneous fields. Furthermore in MRI, as we shall see, magnetic field gradients are used to encode the nuclear signals. Let us consider the case of an inhomogeneousB 0 field and let  <   DB   >   be the average root mean square value of the field deviations from the mean B 0 over a voxel.    Then one has an average spread in precession frequencies given byDw =g <     DB   >   that will speed up the dephasing of the nuclear spins. Usually, in tissue, this effect is larger than that of the internal local fields and the spins will be spreaded in phase over a complete 2pcycle after a time T2*,  that suchDwT2*   <   DB  >     ~ 2p  .Since T2*<< T 2  , then, calling f o  the resonant frequency and < dB> t   he field inhomogeneity in ppm,

= T2*~ 2p/ <     gD B  >  1/ f o  < dB>

[13]

is an effective transverse relaxation time, giving the experimentally observed decay time constant of the FID signal. For an MRI magnet with B 0 = 0.5 Tesla and 10 ppm field homogeneity T 2* ~ 5 ms. There is a very important difference between this dephasing, due to the external field static inhomogeneity, and the true, intrinsic, T 2  relaxation due to internal fields fluctuations. In fact, while T 2  is associated with random interactions, and therefore involves a spontaneous thermodynamically irreversible process, T 2 *results from the evolution of the spins phases under unknown but well defined conditions, given by the intensity of the external field at the individual nuclear positions. This T2*then reversible, at least in principle.process is This makes possible the following experiment, that leads to the observation of the phenomenon known asspin echo. A [p/2 - t -s epqu e]n ce, composed of ap/2e xcitation pulse followed by appulse at a timetla ter, is applied to the transmitter coil as shown in Figure 12-a. The oscilloscope connected to the receiver coil will then show the usual decaying FID signal following thep/2p ulse, and anadditionalsignal, that first grows and then decays, peaking at time 2tt,h at ists econds after theppulse, Figure 12-b.

RF

pulses

Signal

p/2

0

t

exp(-t/T * ) 2

FID

exp(-t/T 2 )

Figure 12

10

2t

ECHO

a)

b)

This extra signal that marks the reappearance of a transverse magnetization, after it had apparently died, at a time well beyond T 2*  , was named a spin echo by Erwin Hahn who first observed it in 1951. The origin of the spin echo is explained by the following sequence of events illustrated in Figure 13. z ro t z ro t z ro t a ) b ) c )

3

2

4

x ro t

1

x ro t

t=0

z ro t

M T

y ro t

d )

y ro t

x ro t

x ro t

z ro t

1

4

t=2 t

3

2

y ro t

e )

y ro t

2

3

B 1 x ro t

1

4

x ro t

t= t

z ro t

y ro t

f)

y ro t

Figure 13 It shows the evolution of the spinatomscorhsi, (groups of spins that see identical local fields) in a frame rotating at the average resonant frequency. After being phased by the excitation pulse (a) the various isochromats begin to precess at slightly different frequencies given by the local values of the external field. In the rotating frame the “slow” isochromats will be seen rotating in opposite sense to that of the “fast” ones. This starts the dephasing process (b). The   isochromats continue to fan out in this way for a timetuntil theprotation pulse is applied (c). At this time every nuclear moment m i  is rotated by 180° around B 1 (d). Now the difference in precession frequencies makes the isochromats tofan inat the same rate, so that the transverse magnetization is almost completely desucoferts econds later (e), except for the small fraction irreversibly lost by the intrinsic T 2 relaxation, Figure 12-b. From this point the spins fan out making the echo signal disappear e).

Field gradient encoding of position: one-dimensional image. In this section we will study the principles of image generation using Nuclear Magnetic Resonance. v In the presence of a magnetic field the macroscopic magnetization M(r v  proportional to) is the proton density,r H  ( v  We saw in the last section how to measure the macroscopicr ). magnetization and the main purpose here is to show how to use the NMR signals to build an

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