Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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Magnetic Resonance Imaging

C L Epstein, University of Pennsylvania,

Philadelphia, PA, USA

F W Wehrli, University of Pennsylvania,

Philadelphia, PA, USA

Introduction

Nuclear magnetic resonance (NMR) is a subtle

quantum-mechanical phenomenon that, through

magnetic resonance imaging (MRI), has played a

major role in the revolution in medical imaging over

the last 30 years. Before being conceived for use in

imaging, NMR was employed by chemists to do

spectroscopy, and it remains a very important tech-

nique for determining the structure of complex

chemical compounds like proteins. In this article we

explain how NMR is used to create an image of a

three-dimensional object. Scant attention is paid to

both NMR spectroscopy, and the quantum descrip-

tion of NMR. Those seeking a more complete

introduction to these subjects should consult the

article Nuclear Magnetic Resonance in this Encyclo-

pedia, as well as the monographs of Abragam (1983)

or Ernst et al. (1987), for spectroscopy, and that of

Callaghan (1993) for imaging. All three books

consider the quantum-mechanical description of

these phenomena. Comprehensive discussions of

MRI can be found in Bernstein et al. (2004) and

Haacke et al. (1999), and a historical apprecia tion of

the development of MRI is given in Wehrli (1995).

The Bloch Equation

We begin with the Bloch phenomenological equa-

tion, which provides a model for the interactions

between applied magnetic fields and the nuclear

spins in the objects under consideration. This is a

macroscopic averaged model that describes the

interaction of aggregates of spins, called isochro-

mats, with applied magnetic fields. An isoc hromat is

a collection of ‘‘like’’ spins, which is spatially large

on the atomic scale, but very small on the scale of

the variations present in the applied magnetic fields.

Spins are alike if they belong to the same species and

are in the same chemical environment . There may be

several different classes of spins , but, in this article,

it is assum ed that they are noninteracting and so it

suffices to consider each separately. Heretofore, we

suppose that there is a single class of like spins. The

distribution of isochromats for these spins is

described macroscopically by the spin density

function, which we denote by (x, y, z). In most

medical applications, one is imaging the distribution

of spins arising from hydrogen protons in water

molecules.

The state of the isochromat at spatial location

(x, y, z) is given by a 3-vector:

Mðx; y; zÞ¼ðm

ðx; y; zÞ; m

ðx; y; zÞÞ

which is interpreted as the magnetic moment per

unit volume. It is an ensemble mean of the quantum

dipoles caused by the spins within the isochromat. In

most applications of NMR to imaging, the applied

magnetic field is descr ibed as the sum of a large,

time-independent field, B

(x, y, z), and smaller time-

dependent fields, B

(x, y, z; t). In the presence of a

static field, thermal fluctuations cause the nuclear

spins to slightly prefer an orien tation aligned with

the field. Using the Boltzmann distribution, one

obtains that the nuclear paramagnetic susceptibility

of water protons is given by

 ¼

h



½1

here h is Planck’s constant, k

the Boltzmann’s

constant, and T the absolute temperature, (see Levitt

(2001)). The constant  is called the gyromagnetic

(or magnetogyric) ratio. For a proton ,

  2  42:5764  10

rad s

1

½2

For water molecules at room temperature,

  3.6  10

9

If the sample is held stationar y in the field B

for a

sufficiently long time, then the spins become

polarized and a bulk magnetic moment appears;

this is called the equilibrium magnetization:

ðx; y; zÞ¼ðx; y; zÞB

ðx; y; zÞ½3

The Bloch equation describes the evolution of M

under the influence of the applied field B = B

þ B

dMðx; y; z; tÞ

¼Mðx; y; z; tÞBðx; y; z; tÞ



ðx; y; z; tÞþ

ðM

ðx; y; zÞ

 M

ðx; y; z; tÞÞ ½4

Here  is the vector cross-product, M

(x, y, z; t)

the compone nt of M(x, y, z; t) perpendicular to

(x, y, z) (called the transverse component), and

the component of M parallel to B

(called the

longitudinal component). For hydrogen protons in

other molecules, the gyromagnetic ratio is expressed

in the form (1  ). The coefficient  is called the

Magnetic Resonance Imaging 367

nuclear shielding; it is typically between 10

4

and

þ10

4

. The difference in the nuc lear shielding causes

a shift in the resonance frequency by .

The second and third terms in eqn [4] are

relaxation terms. They provide a phenomenologi-

cal model for the averaged i nteractions of the spins

with one another and their environment. The

coefficient 1=T

(x, y, z) is the spin lattice relaxation

rate;itdescribestherateatwhichthemagnetiza-

tion returns to equilibrium. The c oeffi cient

1=T

(x, y, z) is the spin–spin relaxation rate; it

describes the rate at which the transverse compo-

nents of M decay. The physical processes causing

these relaxation phenomena are different and so

are the rates themselves, with T

less than T

.The

relaxation rates largely depend on the localized

thermal fluctuations of the molecules and provide

a useful contrast mechanism in MR imaging.

Spin–spin relaxation occurs very rapidly in s olids

(<1 ms) and, therefore, we usually assume that we

are imaging liquid-like materials such as water

protons in soft mammalian tissues. In this case, T

takes values in the 40 ms to 4 s range. Notice that

this model does not include any explicit interac-

tion betwee n i sochromats at diff erent s patial

locations. A variety of such interactions exist,

but, at least in liquid-like materials, they lead only

to small c orrections in the Bloch equation model.

A derivation of the Bloch equation from the

Schro¨ dinger equation can be found in Abragam

(1983) and Slichter (1990). For coupled systems,

the Bloch equation formalism breaks down and a

full quantum-mechanical treatment is necessary

(see Nuclear Magnetic Resonance and Ernst et al.

(1983)).

Much of the analysis in NMR imaging amounts to

understanding the behavior of solutions to eqn [4]

with different choices of B. We now consider some

important special cases. The simplest case occurs if

B has no time-dependent component; then this

equation predicts that the sample becomes polarized

with the transverse part of M decaying as e

t=T

and the longitudinal component app roaching the

equilibrium magnetization, M

,as1 e

t=T

.To

simplify the subsequent discussion, we assume that

the field B

is homogeneous with B

= (0, 0, b

). If

B = B

and we omit the relaxation terms (set

= T

= 1 in [4]), then an initial magnetization

M(x, y, z; 0) simply precesses about B

at angular

frequency !

= b

: M(x, y, z; t) = U (t) M(x, y, z; 0),

with

UðtÞ¼

cos !

t sin !

t 0

sin !

t cos !

t 0

001

½5

The frequency !

is called the Larmor frequency;

this precession of M about the axis of B

is the

resonance phenomenon referred to as NMR. In

typical medical imaging systems, b

is between 1

and 3 T and the corresponding resonance frequency

is between 40 and 120 MHz.

Typically, the field B takes the form

B ¼ B

G þ B

½6

where

G is a gradient field and B

is a radio-

frequency (RF) field. Usually, the gradient fields are

‘‘piecewise time-independent’’ fields, small relative

to B

. By piecewise time-independent field, we

mean a collection of static fields that, in the course

of the experiment, are turned on and off. The B

component is a time-dependent RF field, nominally

at right angles to B

. It is usu ally taken to be

spatially homogeneous, with time dependence of

the form

ðtÞ¼UðtÞ

ðtÞ

ðtÞ

½7

The functions  and  define an envelope that

modulates the time-harmonic field, [ cos !

sin !

t , 0]. They are supported in a finite interval

, t

], that is, the B

field is ‘‘turned on’’ for a finite

period of time. The change in the state of the

magnetization between t

and t

is called the RF

excitation. It may be spatially dependent.

In light of [5] it is convenient to introduce the

rotating reference frame. We replace M with m,

where m(x, y, z; t) = U(t)

1

M(x, y, z; t). It is a classi-

cal result of Larmor, that if M satisfies [4], then m

satisfies

dmðx; y; z; tÞ

¼mðx; y; z; tÞB

eff

ðx; y; z; tÞ



ðx; y; z; tÞþ

ðM

ðx; y; zÞ

 m

ðx; y; z; tÞÞ ½8

where

eff

¼UðtÞ

1

B  0; 0;





G is much smaller than B and quasistatic, it turns

out that one can ignore the components of

orthogonal to B

. Indeed, in imaging applications,

one usually assumes that the components of

depend linearly on (x, y, z) with the

z-component

given by h( x, y, z), (g

, g

)i. The constant vector

G = (g

, g

) is called the gradient vector. With

= (0, 0,b

)andB

given by [7], we see that B

eff

can be taken to equal (0, 0, h(x, y, z), Gi) þ (, , 0).

368 Magnetic Resonance Imaging

In the remainder of this article, we assume that B

eff

takes this form.

If G = 0 and   0, then the solution operator for

Bloch’s equation, without relaxation terms, is

VðtÞ¼

10 0

0 cos ðtÞ sin ðtÞ

0 sin ðtÞ cos  ðtÞ

½9

where

ðtÞ¼

ðsÞds ½10

This is simply a rotation about the x-axis through

the angle (t). If B

6¼ 0 for t 2 [0, ], then the

magnetization is rotated through the angle ().

Thus, RF excitation can be used to move the

magnetization out of its equilibrium state. As we

shall soon see, this is crucial for obtaining a

measurable signal. Note that the equilibrium mag-

netization is a tiny perturbation of the very large

field B

and is, therefore, in practice not directly

measurable. Only the precessional motion of the

transverse components of M produces a measurable

signal. More general B

fields, that is, with both 

and  nonzero, have more complicated effects on the

magnetization. In general, the angle between M and

at the con clusion of the RF excitation is called

the flip angle.

If, on the other hand, B

= 0andG

= (0, 0,

l(x, y, z)), where l() is a function, then V depends on

(x, y, z), and is given by

Vðx; y; z; tÞ

cos lðx; y; zÞt sin lðx; y; zÞt 0

sin lðx; y; zÞt cos lðx; y; zÞt 0

001

½11

This is precession about B

at an angular

frequency that depends on the local field strength

þ l(x, y, z). If both B

and

G are simultaneously

nonzero, then, starting from equilibrium, the

solution of the Bloch equation, at the conclusion

of the RF pulse, has a nontrivial spatial depen-

dence. In other words, the flip angle becomes a

function of the spatial variables. We return to this

in a later section.

A Basic Imaging Experiment

With these preliminaries, we can describe the basic

measurements in magnetic resonance imaging. When

exposed to B

, the sample becomes polarized at a

rate determined by T

. Once the sample is polarized,

a B

-field, of the form given in [7] (with   0), is

turned on for a finite time . This is called an RF

excitation. For the purposes of this discussion, we

suppose that the time is chosen so that () = 90



,see

eqn [10].AsB

and B

are spatially homogeneous,

the magnetization vectors within the object remain

parallel throughout the RF excitation. At the conclu-

sion of the RF excitation, M is orthogonal to B

After the RF is turned off, the vector field

M(x, y, z; t) precesses abou t B

, in phase with the

angular velocity !

. The transverse component of M

decays exponentially. If we normalize the time so

that t = 0 corresponds to the conclusion of the RF

pulse, then, in the laboratory frame,

Mðx; y; z; tÞ¼

!

ðx; y; zÞ



t=T

cos !

t=T

sin !

t; ð1  e

t=T

½12

Recall Faraday’s law: a changing magne tic field

induces an electromotive force (EMF) in a loop of

wire according to the relation

EMF

loop

d

loop

½13

Here 

loop

denotes the flux of the field through the

loop of wire (see Introductory Articles: Electromag-

netism). The transverse components of M are a

rapidly varying magnetic field, which, according to

Faraday’s law, induce a current in a loop of wire. In

fact, by placing several such loops clos e to the sample

we can measure a signal of the form

ðtÞ¼

!



sample

ðx; y; zÞe

t=T

ðx;y;zÞ

 b

1rec

ðx; y; zÞdx dy dz ½14

Here b

1rec

(x, y, z) quantifies the sensitivity of the

detector to the precessing magnetization located at

(x, y, z). From S

(t) we easily obtain a measurement

of the integral of the function b

1rec

. By using a

carefully designed detector, b

1rec

can be taken to be

a constant, and therefore we can determine the total

spin den sity within the object of interest. For the rest

of this article, we assume that b

1rec

is a constant.

Note that the size of the measured signal is

proportional to !

, which is, in turn, proportional

to kB

. This explains, in part, why it is so useful to

have a very strong B

-field. Though even with a

1.5 T magnet, the measured signal is only in the

microwatt range (see Hoult and Lauterbur (1979)

and Edelstein et al. (2004)).

Suppose that, at the end of the RF excitation, we

turn on the gradient

G. As the magnetic field

B = B

G now has a nontrivial spatial dependence,

the precessional frequency of the spins, which equals

Magnetic Resonance Imaging 369

kBk, also has a spatial dependence. In fact,

assuming that T

is spatially indepe ndent, it follows

from [11] that the measured signal would now be

given by

ðtÞ

b

1rec

t=T



sample

ðx; y; zÞe

2ihðx;y;zÞ;ki

dx dy dz ½15

Up to a constant, e

i!

t=T

(t) is simply the

Fourier transform of  at k =  tG=2. By sam-

pling in time and using a variety of different gradient

vectors, we can sample the three-dimensional Fourier

transform of  in a neighborhood of 0. This suffices

to reconstruct an approximation to . In medical

applications, T

is spatially dependent, which, as

described later in the section ‘‘Contrast and resolu-

tion,’’ provides a useful contrast mechanism.

Imagine that we collect samples of

(k)ona

rectangular grid



ðj

k

; j

k

; j

k

Þ:



 j



; 

 j



;



 j





Since we are sampling in the Fourier domain, the

Nyquist sampling theorem implies that the sample

spacing determines the spatial field of view from which

we can reconstruct an artifact-free image: in order to

avoid aliasing artifacts, the support of  must lie in a

rectangular region with side lengths [k

1

, k

1

k

1

], see Haacke et al. (1999), Epstein (2003),and

Barrett and Myers (2004). In typical medical applica-

tions, the support of  is much larger in one dimension

than the others, and so it turns out to be impractical to

use the simple data collection technique described

above. Instead, the RF excitation takes place in the

presence of nontrivial gradient fields, which allows for

a spatially selective excitation: the magnetization in

one region of space obtains a transverse component,

while that in the complementary region is left in the

equilibrium state. In this way, we can collect data from

an essentially two-dimensional slice. This is described

in the next section.

Selective Excitation

As remarked above, practical imaging techniques do

not excite all the spins in an object and directly

measure samples of the three-dimensional Four ier

transform. Rather, the spins lying in a slice are

excited and samples of the two-dimensional Fourier

transform are then measured. This process is called

selective excitation and may be accomplished by

applying the RF excitation with a gradient field

turned on. With this arrangement, the strength of

the static field, B

G, varies with spatial position,

hence the response to the RF excitation does as

well. Suppose that

G = (0, 0, h(x, y, z),Gi) and set

f = [2]

1

h(x, y, z), Gi. This is called the offset

frequency, as it is the amount by which the local

resonance frequency differs from the resonance

frequency !

of the B

-field. The result of a selective

RF excitation is described by a magnetization profile

(f ), which is a unit 3-vector-valued function of

the offset frequency. A typical case would be

ðf Þ¼

½0; 0; 1 for f =2½f

; f



½sin ; 0; cos  for f 2½f

; f





½16

The magnetization is flipped through an angle ,in

regions of space where the offset frequency lies in

the inte rval [ f

, f

] and is left in the equilibrium state

otherwise.

Typically, the excitation step takes a few milli-

seconds and is much shorter than either T

or T

;

therefore, one generally uses the Bloch equation,

without relaxation, in the discussion of selective

excitation. In the rotating reference frame, the Bloch

equation, without relaxation , takes the form

dmðf ; tÞ

02f 

2f 0 

  0

mðf ; tÞ½17

The problem of designing a selective pulse is

nonlinear. Indeed, the selective excitation problem

can be rephrased as a classical inverse-scattering

problem: one seeks a function (t ) þ i(t) with

support in an interval [t

, t

] so that, if m(f ; t)is

the solution to (17) with m(f ; t

) = [0, 0, 1], then

m(f ; t

) = m

(f ). If one restricts attention to flip

angles close to 0, then there is a simple linear model

that can be used to find approximate solutions.

If the flip angle is close to zero, then m

 1

throughout the excitation. Using this approxima-

tion, we derive the low -flip-angle approximation to

the Bloch equation, without relaxation:

dðm

þ im

¼2if ðm

þ im

Þþið þ iÞ½18

From this approximation, we see that

ðtÞþiðtÞ

F ðm

þ im

ÞðtÞ

i

where F ðhÞðtÞ¼

1

hðf Þe

2ift

df ½19

370 Magnetic Resonance Imaging

For an example such as in [16],  close to zero, and

= f

, we obtain

 þ i 

i sin  sin f

t

½20

A pulse of this sort is called a sinc-pulse. A

sinc-pulse is shown in Figure 1a, the result of

applying it in Figure 1b. A more accurate pulse can

be de signed using the Shinnar–Le Roux algorithm

(see Pauly et al. (1991) and Shinnar and Leigh

(1989)), or the inverse scattering approach (see

Epstein (2004)). An inverse-scattering 90



-pulse is

shown in Figure 2a and the response in Figure 2b.

Spin-Warp Imaging

In an earlier section we showed how NMR

measurements could be used to measure the three-

dimensional Fourier transform of . In this section,

we consider a more practical technique, that of

measuring the two-dimensional Fourier transform of

a ‘‘slice’’ of  . Applying a selective RF pulse, as

described in the previous section, we can flip the

magnetization in a region of space z

 z < z <

þ z, while leaving it in the equilibrium state

outside a slightly larger region. Observing that a

signal near the resonance frequency is only produced

by isochromats whose magnetization has a nonzero

transverse component, we can now measure samples

of the two-dimensional Fourier transform of the

function



ðx; y Þ¼

2z

þz

z

ðx; y; zÞdz ½21

If z is sufficiently small then



(x, y)  (x, y, z

0 0.5

(a) (b)

1 1.5 2 2.5 3 3.5 4 4.5 5

–0.2

–0.1

0.1

0.2

0.3

0.4

0.5

RF pulse KHz

–5 –4 –3 –2 –1 0 1 2 3 4 5

–0.2

0.2

0.4

0.6

0.8

KHz

magnetization

Figure 1 A selective 90



pulse and profile designed using the linear approximation. (a) Profile of a 90



sinc-pulse. (b) The

magnetization profile produced by the pulse in (a).

(a) (b)

1 2 3 4 5 6

–0.2

–0.1

0.1

0.2

0.3

0.4

0.5

0.6

RF pulse KHz

–5

–4 –3 –2 –1 0 1 2 3 4 5

–0.2

0.2

0.4

0.6

0.8

KHz

magnetization

Figure 2 A selective 90



pulse and profile designed using the inverse scattering approach. (a) Profile of a 90



inverse-scattering

pulse. (b) The magnetization profile produced by the pulse in (a).

Magnetic Resonance Imaging 371

In order to be able to use the fast Fourier

transform (FFT) algorithm to do the recons truction,

it is very useful to sample



on a uniform grid. To

that end, we use the gradient fields as follows: after

the RF excitation we apply a gradient field of the

form G

= (0, 0, g

y þ g

x) for a certain peri od of

time T

. This is called a phase encoding gradient.

At the conclusion of the phase encoding gradient,

the transverse components of the magnetization

from the excited spins has the form

ðx; yÞ/e

2iðk

yk

xÞ



ðx; yÞ½22

where (k

, k

) = [2]

1

T

(g

, g

). At time T

we turn off the y-component of G

and reverse the

polarity of the x-component. At this point, we begin

to measure the signal. We get samples of



(k, k

)

where k varies from k

x max

to k

x max

. By repeating

this process with the strength of the y-phase

encoding gradient being stepped through a sequence

of uniformly spaced values, g

2 {ng

}, and col-

lecting samples at a uniformly spaced set of times,

we collect the set of samples





ðmk

; nk

Þ:



 m 

; 

 n 



½23

The gradient G

= (0, 0, g

x), left ‘‘on’’ during

signal acquisition, is called a frequency encoding

gradient. While there is no difference, mathemati-

cally, between the phase encoding and frequency

encoding steps, there are significant practical differ-

ences. This app roach to sampling is known as spin-

warp imaging; it was introduced in Edelstein et al.

(1980). The steps of this experiment are summarized

in a pulse sequence timing diagram, shown in

Figure 3. This graphical representation for the

steps followed in a magnetic resonance imaging

experiment is ubiquitous in the literature.

To avoid aliasing artifacts, the sample spacings

k

and k

must be chosen so that the excited

portion of the sample is contained in a region of size

k

1

 k

1

. This is called the field of view or

FOV. Since we can only collect the signal for a finite

period of time, the Fourier transform



(k

, k

)is

sampled at frequencies lying in a rectangle with

vertices (k

x max

, k

y max

), where

x max

k

; k

y max

k

½24

The maximum frequencies sampled effectively deter-

mine the resolution available in the reconstructed

image. Heuristically, this resolution limit equals half

the shortest measured wavelength:

x 

x max

FOV

y 

y max

FOV

½25

Whether one can actually resolve objects of this size in

the reconstructed image depends on other factors such

as the available contrast and the signal-to-noise ratio

(SNR). We consider these factors in the final sections.

Signal-to-Noise Ratio

At a given spatial resolution, image quality is largely

determined by SNR and the contrast between the

different materials making up the imaging object. SNR

in MRI is defined as the voxel signal amplitude divided

by the noise standard deviation. The noise in the NMR

signal, in general, is Gaussian distributed with zero

mean. Ignoring contributions from quantization, for

example, due to limitations of the analog-to-digital

converter, the noise voltage of the signal can be

ascribed to random thermal fluctuations in the receive

circuit (see Edelstein (1986)). The variance is given by



thermal

¼ 4k

TR ½26

where k

is Boltzmann’s constant, T the absolute

temperature, R the effective resistance (resulting from

both receive coil, R

and object, R

), and  the

receive bandwidth. Both R

and R

are frequency

dependent, with R

/ !

1=2

,andR

/ !. Their relative

contributions to overall circuit resistance depend in

a complicated manner on coil geometry, and

the imaging object’s shape, size, and conductivity

ADC

Slice selection gradient

Phase encoding gradient

Frequency encoding gradient

Signal acquisition

Δt

Figure 3 Pulse timing diagram for spin-warp imaging. During

the positive lobe of the frequency encoding gradient, the analog-

to-digital converter (ADC) collects samples of the signal

produced by the rotating transverse magnetization.

372 Magnetic Resonance Imaging

(see Chen and Hoult (1989)). Hence, at high magnetic

field, and for large objects, as in most medical

applications, the resistance from the object dominates

and the noise scales linearly with frequency. Since the

signal is proportional to !

,inMRI,theSNRincreases

in proportion to the field strength.

As the reconstructed image is complex valued, it is

customary to display the magnitude rather than the

real component. This, however, has some conse-

quences on the noise properties. In regions where the

signal is much larger than the noise, the Gaussian

approximation is valid. However, in regions where the

signal is low, rectification causes the noise to assume a

Raleigh distribution. Mean and standard deviation can

be calculated from the joint probability distribution:

PðN

; N

Þ¼

2

ðN

þN

Þ=2

½27

where N

and N

are the noise in the real and

imaginary channels, respectively. When the signal is

large compared to noise, one finds that the variance



= 

. In the other extreme of nearly zero signal,

one obtains for the mean:

S ¼

ﬃﬃﬃﬃﬃﬃﬃﬃ

=2

ﬃ 1:253 ½28

and, for the variance:



¼2

ð1  =4Þﬃ0:655

½29

Of particular practical significance is the SNR

dependence on the imaging parameters. The voxel

noise variance is reduced by the total number of

samples collected during the data acquisition pro-

cess, that is,



¼

thermal

=N ½30

where N = N

in a two-dimensional spin-warp

experiment. Incorporating the contributions to

thermal noise variance, other than bandwidth, into

a constant

u ¼ 4k

TR ½31

we obtain for the noise variance:



u

avg

½32

Here N

avg

is the number of signal averages collected

at each phase enco ding step. We obtain a simple

formula for SNR per voxel of volume V:

SNR ¼C

 V

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

avg

u

¼C

 x yd

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

avg

u

½33

where x, y are defined in [25], d

is the thickness

of the slab selected by the slice-selective RF pulse,

and

 denotes the spin density weighted by effects

determined by the (spatially varying) relaxation

times T

and T

and the pulse sequence timing

parameters. Figure 4 shows two images of the

human brain obtained from the same anatomic

location but differing in SNR.

Contrast and Resolution

The single most distinc tive feature of MRI is its

extraordinarily large innate contrast. For two soft

tissues, it can be on the order of several hundred

percent. By comparison, contrast in X-ray imaging is

a consequence of differences in the attenuation

coefficients for two adjacent structures and is

typically on the order of a few percent.

We have seen in the prece ding sections that the

physical principles underlying MRI are radically

different from those of X-ray computed tomogra-

phy, in that the signal elicited is generated by the

spins themselves in response to an external pertur-

bation. The contrast between two regions, A and B,

with signals S

and S

, respectively, is defined as

 S

½34

If the only contrast mechanism were differences in

the proton spin density of various tissues, then

contrast would be on the order of 5–20%. In reality,

it can be several hundred percent. The reason for

this discrepancy is that the MR signal is acquired

under nonequilibrium conditions. At the time of

excitation, the spins have typically not recovered

from the effect of the previous cycle’s RF pulses, nor

(a) (b)

Figure 4 T

-weighted sagittal images through the midline of

the brain: Image (b) has twice the SNR of image (a), showing

improved conspicuity of small anatomic and low-contrast detail.

The two images were acquired at 1.5 T field strength using two-

dimensional spin-warp acquisition and identical scan para-

meters, except for N

avg

, which was 1 in (a) and 4 in (b).

Magnetic Resonance Imaging 373

is the signal usually detected immedi ately after its

creation.

Typically, in spin-warp imaging, a spin-echo is

detected as a means to alleviate spin coherence

losses from static field inhomogeneity. A spin-echo

is the result of applying an RF pulse that has the

effect of taking (m

, m

)to(m

, m

). As

such a pulse effects a 180



rotation of the

z-axis, it is

also called a -pulse. If, after such a pulse, the spins

continue to evolve in the same environment then,

following a certain peri od of time, the transverse

components of the magnetization vectors through-

out the sample become aligned. Hence a pulse of

this type is also called a refocusing pulse. The time

when all the transverse components are rephased is

called the echo time, T

The spin-echo signal amplitude for an RF pulse

sequence =2      , repeated every T

sec-

onds, is approximately given by

Sðt ¼ 2Þð1  e

T

Þe

T

½35

This is a good approximation as long as T

<< T

and T

<< T

, in which case the transverse magne-

tization decays essentially to zero between successive

pulse sequ ence cycles. In eqn [35],  is voxel spin

density and the echo time T

= 2. Empirically, it

is known that tissues differ in at least one of

the intrinsic quantities, T

, T

,or. It, therefore,

suffices to acquire images in such a manner that

contrast is sensitive to one particular parameter. For

example, a ‘‘T

-weighted’’ image would be acquired

with T

 T

and T

>> T

and, similarly, a

‘‘T

-weighted’’ image with T

< T

and T

<< T

with T

, T

representing typical tissue proton relaxa-

tion times. Figure 5 shows two images obtained with

the same scan parameters except for T

and T

illustrating the fundamentally different image con-

trasts that are achievable.

It is noteworthy that object visibility is not just

determined by the contrast between adjacent

structures but is also a function of the noise. It is,

therefore, useful to define the contrast-to-noise ratio as

CNR



eff

½36

where 

eff

is the effective standard deviation of the

signal. Finally, it may be useful to reconstruct

parametric images in which the pixel signal values

represent any one of the intrinsic parameters. A

-image can be computed from eqn [35],for

example, either analytically from two image data

sets acquired with two different echo times, or from a

series of T

values, obtained from a Carr–Purcell spin-

echo train, using regression techniques (see Nuclear

Magnetic Resonance and Haacke et al. (1999)).

We have previously shown that the limiting

resolution is given by k

max

, the largest spatial

frequency sampled, see [25]. In reality, howev er,

the actual resolution is always lower. For example,

spin–spin (T

) relaxation causes the signal to decay

during the acquisition. In spin-warp imaging, this

causes the high spatial frequencies to be further

attenuated.

A further consequence of finite sampling is a

ringing or Gibbs artifact that is most prominent at

sharp intensity discontinuities. In practice, these

artifacts are mitigated by applying an appropriate

apodizing filter to the data. Figure 6 shows a portion

of a brain image obtained at two different resolu-

tions. In Figure 6b , the total k-space area covered

was 16 times larger than for the acquisition of the

image in a). Artifacts from finite sampling and

blurring of fine detail such as cortical blood vessels

are clearly visible in the low-resolution image. SNR,

according to eqn [33], is reduced in the latter image

by a factor of 4.

(a) (b)

Figure 5 Dependence of image contrast on pulse sequence

timing parameters: (a) T

-weighted; (b) proton density-weighted.

(a) (b)

Figure 6 Effect of k-space coverage on spatial resolution in

axial image of the brain: the field of view in both images was

20 cm and all scan parameters were the same except that (a)

was acquired with N

= N

= 128 and (b) with N

= N

= 512.

374 Magnetic Resonance Imaging

See also: Nuclear Magnetic Resonance; Stochastic

Resonance.

Further Reading

Abragam A (1983) Principles of Nuclear Magnetism. Oxford:

Clarendon.

Barrett HH and Myers KJ (2004) Foundations of Image Science.

Hoboken: Wiley.

Bernstein MA, King KF, and Zhou XJ (2004) Handbook of MRI

Pulse Sequences. London: Elsevier Academic Press.

Callaghan PT (1993) Principles of Nuclear Magnetic Resonance

Microscopy. Oxford: Clarendon.

Chen C-N and Hoult DI (1989) Biomedical Magnetic Resonance

Technology. Bristol: Adam Hilger.

Edelstein WA, Glover GH, Hardy C, and Redington R (1986)

The intrinsic signal-to-noise ratio in NMR imaging. Magnetic

Resonance in Medicine 3: 604–618.

Edelstein WA, Hutchinson JM, Johnson JM, and Redpath T (1980)

Spin warp NMR imaging and applications to human whole-

body imaging. Physics in Medicine and Biology 25: 751–756.

Epstein CL (2003) Introduction to the Mathematics of Medical

Imaging. Upper Saddle River, NJ: Prentice-Hall.

Epstein CL (2004) Minimum power pulse synthesis via the inverse

scattering transform. Journal of Magnetic Resonance 167:

185–210.

Ernst R, Bodenhausen G, and Wokaun A (1987) Principles of

Nuclear Magnetic Resonance in One and Two Dimensions.

Oxford: Clarendon.

Haacke EM, Brown RW, Thompson MR, and Venkatesan R

(1999) Magnetic Resonance Imaging. New York: Wiley-Liss.

Hoult D and Lauterbur PC (1979) The sensitivity of the

zeugmatographic experiment involving human samples.

Journal of Magnetic Resonance 34: 425–433.

Levitt MH (2001) Spin Dynamics, Basics of Nuclear Magnetic

Resonance. Chichester: Wiley.

Pauly J, Le Roux P, Nishimura D, and Macovski A (1991)

Parameter relations for the Shinnar–Le Roux selective excita-

tion pulse design algorithm. IEEE Transactions on Medical

Imaging 10: 53–65.

Shinnar M and Leigh J (1989) The application of spinors to pulse

synthesis and analysis. Magnetic Resonance in Medicine 12:

93–98.

Slichter CP (1990) Principles of Magnetic Resonance, 3rd enl. and

upd. ed., Springer Series in Solid-State Sciences, vol. 1. Berlin–

New York: Springer.

Wehrli FW (1995) From NMR diffraction and zeugmatography

to modern imaging and beyond. In: Progress in Nuclear

Magnetic Resonance Spectroscopy 28: 87–135.

Magnetohydrodynamics

C Le Bris, CERMICS – ENPC, Champs Sur Marne,

France

The Basic Modeling

Magnetohydrodynamics (MHD) is the study of the

interaction of (electro-) magne tic fields and con-

ducting fluids. When a conducting fluid (e.g., a

liquid metal, a weakly ionized gas, or a plasma) is

placed within a magnetic field, two coupling

phenomena appear: the electric currents modify the

magnetic field, and the Lorentz forces due to the

magnetic field modify the motion of the fluid. At the

mathematical level, two sets of equations, very

different in nature, are involved. The usual descrip-

tion of the hydrodynamics phenomena is most often

that provided by the continuum mechanics for

fluids, while the description of electromagnetic

phenomena essentially proceeds from the Maxwell

equations.

Either category of equations can be declined in a

variety of models. The coupling between the two

categories might also be accounted for at different

levels of accuracy. For the sake of conciseness in

such an expository survey, it is neither desirable nor

doable to present all the possible set of equations

and their possi ble coupling. The difficulty stems

from the incredibly large spectrum of physical

phenomena where MHD plays a role. A list of

such phenomena includes



astrophysical and geophysical applications (mod-

eling of stars in the galactic field, of pulsar s, of

solar spots, of the flows in the earth’s core, ...),



advanced ‘‘terrestrial’’ applications such as the

magnetic confinement of plasmas in controlled

fusion, MHD propulsion engines for rockets, and



industrial applications in the engineering world

(electromagnetic pumping, metal forming, alumi-

num electrolysis, and many other metallurgical

applications).

Due to this variety of physical situations, no

unified setting can be presented with a satisfactory

degree of details. We therefore mostly concentrate

throughout this article on the MHD of conducting

fluids that are homogeneous, incompressible, vis-

cous, and Newtonian. This is often the case of

liquid metals in many industrial processes. The

equations manipulated will first be given in their

most general f orm and then immediately adapted to

the above context. For other contexts, the modeling

follows the same pattern, but other variants of the

general equations must be employed. The biblio-

graphy of this article contains such general

information.

Magnetohydrodynamics 375

The Hydrodynamics Description

The usual description for fluids follows from

continuum mechanics. In this setting, the governing

equation is the equation for the conservation of

momentum

@ðuÞ

þ divðu uÞdiv ¼f ½1

where  denotes the densi ty of the fluid, u its

velocity,  the stress tensor, and f the density of

volumic (or per unit volume) body forces applied to

the fluid. For incompressible viscous Newtonian

fluids, the stress/velocity relation reads

 ¼ðru þðruÞ

ÞpId ½2

together with the constraint

div u ¼0 ½3

on the velocity. Here,  denotes the viscosity of the

fluid, p the pressure, and A

denotes the transpose

matrix of the matrix A. A third usual assumption is

that the incompressible fluid is in addition homo-

geneous, that is,

 ¼

 ¼constant ½4

Equations [1]–[4] lead to the equations for

conservation of momentum in the case of incom-

pressible homogeneous viscous Newtonian fluid,

that is, the incompressible Navier–Stokes equations



u ru  u þrp ¼f

div u ¼0

½5

These equations are supplied with initial and

boundary conditions on the velocity u. At initial

time, the velocity is assumed to be known

u(t = 0,) = u

on the whole domain occupied by

the fluid , a domain that is supposed here not to

vary in time (see , neverthel ess, the sect ion ‘‘The

indust rial pr oduction of alum inum’’ for a differe nt

setting). On the other hand, the boundary conditions

on the boundary @ of  can be of various forms.

For simplicity, the boundary is supposed regular, so

that its unitary outward normal n

@

can be

unambiguously defined. The standard choice is to

set Dirichlet conditions on the velocity u = u

given

.In

the following, we will assume for simplicity that the

boundary condition is the homogeneous Dirichlet

boundary condition u = 0, as a superposition of the

nonpenetration condition u  n

@

= 0 and the no-slip

boundary condition u n

@

= 0. One can also

impose alternative boundary conditions, for exam-

ple, involving the pressure.

The Electromagnetic Description

Classical electromagnetism is described by the

Maxwell equations. For the sake of consistency, we

recall here that these are:

The Maxwell–Ampe`re equation



þ curl H ¼ j ½6

The Maxwell–Coulomb equation

divD ¼ 

½7

The Maxwell–Faraday equation

þ curl E ¼ 0 ½8

The Maxwell–Gauss equation

divB ¼ 0 ½9

In the above equations, the three-dimensional vector

fields D, B, E, H denote the electric and magnetic

inductions, and the electric and magnetic fields,

respectively. On the other hand, the three-dimensional

vector field j denotes the current density, and the scalar

field 

denotes the charge density. Inside an elec-

trically conducting medium, the standard assumption

of perfect medium consists in assuming the following

relations:

D ¼"E

H ¼



½10

often called ‘‘constitutive laws,’’ where " and ,

respectively, denote the (electric) permittivity and

the (ma gnetic) permeability of the medium. In the

simple isotropic homogeneous case, both these

parameters are scalar and constant. They are often

expressed as

" ¼"

 ¼



½11

where "

, 

are the permittivity and the perme-

ability of the vaccum (that satisfy "



= 1=c

, with

c denoting the speed of light), and "

, 

are the

permittivity and the permeability relative to vaccum,

or relative permittivity and relative permeability.

When collecting [6]–[9], together with [10], [11],

one obtains the following general system of

376 Magnetohydrodynamics