Available at: http://publications.ictp.it IC/2008/085

United Nations Educational, Scientific and Cultural Organization and

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

APPLICATION OF BLOCH NMR FLOW EQUATION ANALYTICAL MODELS FOR MAGNETIC RESONANCE IMAGING

O.B. Awojoyogbe1

Department of Physics, Federal University of Technology, Minna, Niger-State, Nigeria and

The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

Abstract

In this study, solutions to the Bloch NMR flow equations in the form of polynomials are presented. The polynomials are obtained in terms of trigonometric, algebraic, ordinary and special functions. The polynomials represent the T2 weighted NMR transverse magnetizations and signals obtained in terms of Chebyshev polynomials, which can be an attractive mathematical tool for simplified analysis of hemodynamic functions of blood flow system. By means of these polynomials, appropriate mathematical algorithms are developed to understand several physical characteristics that form the link between the magnetic resonance image and the tissue characteristics (proton density, T1 and T2 relaxation parameters). The results obtained from this mathematical formulation will enhance our ability to appropriately adjust the imaging process to be especially sensitive to each of the characteristic being evaluated.

MIRAMARE – TRIESTE

December 2008

1 Regular Associate of ICTP. awojoyogbe@yahoo.com

2

Introduction

The magnetic resonance image is a display of rF signals that are emitted by the tissue

or fluid molecules during the image acquisition process. The source of the signals is a

condition of magnetization that is produced in the tissue when the patient is placed in

the strong magnetic field. The tissue magnetization depends on the presence of

magnetic nuclei. The specific physical characteristic of fluid or tissue that is visible in

the image depends on how the magnetic field is being changed during the acquisition

process.

Traditionally, the first thing we see in an image is rF signal intensity emitted by the

tissues, bright areas in the image correspond to tissues that emit high signal intensity.

There are also areas in an image that appear as dark voids because no signals are

produced. Between these two extremes there will be a range of signal intensities and

shades of gray that show contrast or differences among the various tissues [1-9]. It is

the level of magnetization of specific “picture snapping” times during the imaging

procedure that determines the intensity of the resulting rF signal and image brightness

[9]. Tissues or other materials that are not adequately magnetized during the imaging

procedure will not be visible in the image.

In this study, two basic conditions are required for transverse magnetization: (i) the

magnetic moments of the nuclei must be oriented in the transverse direction, or plane

and (ii) a majority of the magnetic moments (protons) must be in the same direction,

or in phase, within the transverse plane. When a nucleus has a transverse orientation,

it is actually spinning around an axis that is parallel to the magnetic field. This

rotation or spin is a result of the normal precession discussed in earlier studies [9-14].

One important effect we consider is the exchange of energy among the spinning

nuclei (spin-spin interactions), which results in relatively slow dephasing and loss of

magnetization. The rate at which this occurs is determined by the characteristics of

the blood molecule. It is this dephasing activity that is characterized by the T2 values.

3

Mathematical Method

We study the magnetic resonance imaging properties of the modified time

independent Bloch NMR flow equation which describes the dynamics of magnetic

resonance imaging of fluid flow under the influence of rF magnetic field as derived in

the earlier studies [10-14].

For steady flow

(1)

(2)

Two reasonable initial boundary conditions which may conform to the real-time

experimental arrangements were chosen. These are:

1. Mo≠ Mz a situation which holds good in general and in particular when the rF B1(x)

field is strong say of the order of 1.0G or more.

2. Before entering signal detector coil, fluid particles has magnetization

Mx =0, My = 0 (3)

If B1(x) is large; B1(x) >> 1G or more so that My of the blood bolus changes appreciably

from Mo.

Resonance condition exists at Larmor frequency

(4)

based on the following condition

(5)

where γ denotes the gyromagnetic ratio of fluid spins; ω/2π is the rF excitation

frequency; fo/γ is the off- resonance field in the rotating frame of reference. V is the

instantaneous velocity of the fluid flow; T1 and T2 are the spin-lattice and spin-spin

relaxation times respectively. Mo is the equilibrium magnetization and My is the

transverse magnetization. When a tissue containing magnetic nuclei, i.e., proton, is

placed in a strong magnetic field, the tissue becomes magnetized. When a 90 rF pulse

is applied to the longitudinal magnetization, it produces two effects: (i) it temporarily

destroys the longitudinal magnetization, a condition known as saturation, and Mo = 0;

(ii) it produces transverse magnetization My, a condition known as excitation because

4

transverse magnetization is an unstable excited state. At the point where the

equilibrium magnetization Mo is zero and the transverse magnetization My is

maximum, equation (2) can be written as

(6a)

We solve equation (6a) when the fluid velocity V(x), is not a constant (V=V(x) ≠

constant). Equation (6a) can be written as

(6b)

where

(7)

(8)

€

n2 =T1T2

(9)

The solutions of equation (6) are obtained by power series:

(10)

The general recurrence relation can be written as

(11)

From equation (11), we obtain for the even coefficients n = 2p

(12)

and the odd coefficients n =2p-1 as

5

(13)

The general solution is obtained in closed form as

(14)

Performing a change of variables gives the equivalent form of the solution

(15)

where

€

Mn (x) =n2

(−1)k (n − k −1)!k!(n − 2k)!

(2x)n−2kk= 0

n / 2

∑ (16)

and

€

M*n (x) = (−1)k (n − k)!

k!(n − 2k)!(2x)n−2k

k= 0

n / 2

∑ (17)

Generally, whether an NMR transverse magnetization model polynomial is an even or

odd function depends on its degree n.

(18)

Mn(x) is an even function, when n is even and odd function, when n is odd.

Mn(x) is the NMR transverse magnetization model polynomial of the first kind and

M*n(x) is the NMR transverse magnetization model polynomial of the second kind.

Another equivalent form of the solution is given by

€

My = c5 cosh[(nIn x + x 2 −1( )]+ ic6 sinh[nIn( x + x 2 −1)( )] (19)

6

Generating Functions of the NMR Transverse Magnetization Model

The NMR transverse magnetization model polynomial of the first kind are a set of

orthogonal polynomials defined as the solutions to the modified Bloch NMR flow

equation and denoted by Mn(x). They are used as an approximation to a least squares

fit, and are intimately connected with trigonometric multiple-angle formulas. They are

normalized such that Mn(x) = 1. The first few polynomials are illustrated in table 1

and figure 1 respectively for -1 < x < 1 and n = 1, 2, 3, 4, 5.

The NMR transverse magnetization model polynomial of the first kind Mn(x) can be

defined by the contour integral

€

Mn (x) =14πi

1− t 2( )t−n−11− 2xt + t 2( )

dt∫ (20)

Table 1. The first few NMR transverse magnetization model polynomial of the first kind.

(n) Mn(x)

1 x

2

3

4

5

6

Fig.1. The first few NMR transverse magnetization model polynomial of the first kind.

7

The NMR transverse magnetization model polynomials of the second kind are defined

as

(21)

Table 2. The first few NMR transverse magnetization model polynomials of the second kind.

N M*n(x)

2

3

The NMR transverse magnetization model polynomials of the first kind are defined

through the identity

(22)

The NMR transverse magnetization model polynomials of the first kind can be

obtained from the generating functions

(23)

and

(24)

For x≤ 1 and t< 1

A closely related generating function is the basis for the definition of the NMR

transverse magnetization model polynomials of the second kind.

The polynomials can also be defined in terms of the sums

€

Mn (x) = cos(ncos−1 x) =2q

n

xn−2q x 2 −1( )

q

q= 0

n / 2

∑ (25a)

8

where is a binomial coefficient and [x] is the floor function or the product

€

Mn (x) = 2n−1 x − cos (2k −1)π2n

k=1

n

∏ (26)

Mn(x) also satisfy the curious determinant equation

(25b)

The NMR transverse magnetization model polynomials are orthogonal polynomials

with respect to the velocity weighting function (1-x2)-1/2

(27)

where δnm is the Kronecker delta. The NMR transverse magnetization model

polynomials of the first kind satisfy the additional discrete identity

(28)

where xk for k =1, ..., m are the m zeros of Mn(x).

By using this orthogonality, a piecewise continuous function f(x) in -1 ≤ x ≤ 1 can be

expressed in terms of NMR transverse magnetization model Polynomials:

9

€

CmPm (x) =f (x − )+ f (x + )

2at discontinuous po int s

f (x ), f (x ) continuous

m= 0

∞

∑ (29)

where

(30)

This orthogonal series expansion is the Fourier-NMR series expansion or a

generalized Fourier series expansion.

The NMR transverse magnetization model polynomials also satisfy the recurrence

relations

(31)

€

Mn+1(x) = xMn (x) − V 1− [Mn (x)]2{ } (32)

For n ≥ 1, as well as

(33)

(34)

They have a complex integral representation

€

Mn (x) =14πi

1− t 2( )t−n−11− 2xt + t 2( )

dtc∫ (35)

and a Rodrigues representation

€

Mn (x) =(−1)nV π

2n (n − 12)!

dn

dxn[(1− x 2)n−1/ 2] (36)

10

The NMR transverse magnetization model polynomial of the first kind is related to

the Bessel function of the first kind Jn(x) and modified Bessel function of the first

kind In(x) by the relations

(37)

(38)

Putting x = cosφ allows the NMR transverse magnetization model polynomials of the

first kind to be written as

(39)

The second linearly dependent solution to the transformed differential equation

(40)

is then given by

(41)

which can also be written

(42)

where Gn(x) is not a polynomial.

The Signal Property of the NMR Transverse Magnetization Model Polynomials

The signal property of the NMR transverse magnetization model polynomials is the

trigonometric representation on [-1,1]. These celebrated NMR transverse

11

magnetization approximation polynomial Sm(x) of degree ≤ m for f(x) over [-

1,1] can be written as a sum of {Mn(x)}:

€

f (x) ≈ Sm (x) = dnMn (x)n= 0

m

∑ (43)

The coefficients {dn} are computed with the formulas

€

dn =2

m +1f (xk )Mn (xk ) =

2m +1

f (xk )k= 0

m

∑k= 0

m

∑ cos n 2k +12m + 2

π

(44)

for n = 2,3,……m where

€

xk = cos 2k +12m + 2

π

for k = 0,1,2,3,......m (45)

For illustration, several NMR transverse magnetization approximation polynomials of

degree n = 1,2, 3, 4, and 5 and their error analysis for trigonometric, algebraic,

ordinary and special functions have been presented as Chebyshev polynomials [15].

Dynamics of the NMR Transverse Magnetization

Based on equation (6), when the equilibrium magnetization Mo along the z –axis

becomes zero, the transverse constant magnetization My with amplitude is

maximum and there is no precession around z-axis. My(t) can be written as,

where

(46)

The frequency fo is called the Rabi frequency which describes transition between the

states 0〉 and 1〉, under the action of resonant rF B1 field. At t = 0, the spin is in the

ground state,

12

(47)

From equation (46), we can write,

(48)

If the rF B1 field is applied for a duration of τ such that

(49)

then we can write from equation (48),

(50a)

and

(50b)

This mathematical analysis indicates that a pulse of a resonance rF field with duration

given in equation (49) drives the spin system from the ground state to excited state.

Such a pulse is called π-pulse. If the spin is already in the excited state,

(50c)

After the action of π-pulse we have

(50d)

Thus, a π-pulse drives the spin into the ground state. The π-pulse changes the state of

the NMR system from 0〉 to 1〉 or from 1〉 to 0〉. If a pulse of different duration is

applied, we can drive the NMR system into a superpositional state, creating a one-

cubit rotation. Following the same procedure, a π/2-pulse drives the system into a

superposition with equal weights of the ground and the excited states. The

measurement of the state of the system gives the state 0〉 or the state 1〉 with equal

probability, 1/2. The same result is obtained when a π/2-pulse drives the system from

a pure excited state as given in the initial conditions of equation (50c).

Considering the change of the average value of magnetization components under the

action of resonant rF B1 field when the system is initially in the equilibrium (ground)

13

state and its dynamics is described by equation (48), the evolution of the average

values of the magnetization components is given by

(51)

(52)

Equation (51) describes the precession of the average magnetization around the x-

axis, in the rotating system of coordinates. At t = 0, the average magnetization points

in the positive z- direction. The z-component of the average spin decreases, and the y-

component increases. At any moment we have

(53)

After the action of π/2-pulse, we have

(54)

This shows that the average magnetization points in the positive y-direction. A π-

pulse, we obtain

(negative z-direction) (55)

Qualitative Description of the Position and Potential Energy of Blood Particles

Multiplying both sides of equations (7) and (8) by m/2, where m is the mass of the

fluid particles gives

(56)

(57a)

(57b)

The solution of equation (57b) can be written as

€

x(t) = xoe−4T1T2(T1 +T2 )

t

(57c)

14

In equation (56) the quantity on the right depends only on the initial conditions and is

therefore constant during the motion. It is called the total energy , and we have

the law of conservation of kinetic , plus potential energy ,

which holds, as we can see, only when the force is a function of x alone:

+ = = E. (58)

where . Solving for V, we obtain

V = =

€

2m

E − E x( )[ ]1/ 2 (59)

The function x(t) is to be found by solving for x in the equation

€

m2

E − E x( )[ ]x0

x∫

−1/ 2dx = (60)

In this case, the initial conditions are expressed in terms of the constants E and xo.

In applying equation (60), and in taking the indicated square root in the integrand, we

must carefully use the proper sign, depending on whether the velocity V given by

equation (59) is positive or negative. In cases where V is positive during some parts

of the motion and negative during other parts, it may be necessary to carry out the

integration in equation (60) separately for each part of the motion.

Equation (60) becomes, for this case, with to = 0,

€

m2

E − 12kx 2

x0

x∫

−1/ 2

= (61)

Making the substitutions

= (62)

=1 (63)

so that

€

k2

E − 12kx 2

x0

x∫

−1/ 2

€

dx = dθ =1

θ −θ0( )θ 0

θ

∫

15

by equation (61),

From equation (62):

=

€

2Eksinθ = A

€

sin t + θ0( ) (64)

where

(65)

Thus the coordinate x oscillates harmonically in time, with amplitude A and frequency

ω/2. The initial conditions are determined by the constants A which are related to E

and xo by

(66)

(67a)

We can now determine the position function x(t) from equation (57c) as

€

x(t) = sinθ0e−4T1T2T1 +T2 )

t

(67b)

It may be interesting to note that there is the sign difficulty in taking the square root in

equation (61) by replacing (1-sin2 θ)-1/2 by (cos θ)-1, a quantity which can be made

either positive or negative as required by choosing θ in the proper quadrant.

The function in equations (56) and (58) is called the energy integral. The equation of

motion is generally defined as

€

d2xdt 2

+ kx = 0 (68)

An integral of the equations of motion of a mechanical system is called a constant of

the motion. In general, any mechanical problem can be solved if we can find enough

first integral, or constants of the motion. The general solution of equation (68) is

given in equation (64).

16

Even in cases where the integral in equation (60) cannot easily be evaluated or the

resulting equation solved to give an explicit solution x (t), the energy integral in

equation (58), gives us useful information about the solution. For a given energy E,

we see from equation (59) that the blood molecule is confined to those regions on the

x-axis where E(x) E. furthermore, the velocity is propositional to the square root of

the difference between E and E(x). Hence, if we plot E(x) versus x, we can give a

good qualitative description of the kinds of motion that are possible. For the potential-

energy function shown in Fig.2 we note that the least energy possible is x4Eo. At this

energy, the blood molecule can only be at rest in xo. With a slight higher energy E1,

the molecule can move between x1 and x2; its velocity decreases as it approaches x1 or

x2 , it stops and reverses its direction when it reaches either x1 or x2, which are called

turning points of the motion. With energy E2, the blood molecule may oscillate

between turning points x3 and x4, or remain at rest at x5. With energy E3, there are four

turning points and the blood molecule may oscillate in either of the two potential

valleys. With energy E4, there is only one turning point; if the molecule is initially

flowing to the left, it will turn at x6 and return to the right, speeding up over the

valleys at x0 and x5, and slowing down over the hill between. At energies above E5,

there are no turning points and the blood molecule will move in one direction only,

varying its speed according to the depth of the potential at each point.

Fig.2. A potential energy function for one-dimensional motion of blood particle when a blood particle

is oscillating near a point of stable equilibrium we can find an approximate solution for its motion.

17

Theoretical Analysis of Magnetic ResonanceT2-weighted Image of Blood

Molecules

Based on equation (6), Transverse magnetization is produced by applying a pulse of

energy to the magnetized blood molecule. Transverse magnetization My is used

during the image formation process to develop image contrast based on differences in

T2 value and to generate the rF signal emitted by the blood molecule. The longitudinal

magnetization is an rF silent condition and does not produce any signal. However,

transverse magnetization is a spinning magnetic condition within each voxel, and that

generates an rF signal used to form the image. The intensity of the rF signal is

proportional to the level of transverse magnetization. The difference in T2 values of

fluid molecules is the source of contrast in T2- weighted images. This is shown in

figure 1, here we watch magnetizations from five different blood molecules with the

same T1 relaxation time (of T1 =1.00s typical of blood) n = 1, T2 =0.57735; n = 2, T2

=0.5; n = 3, T2 =0.44721; n = 4, T2 =0.40825; n = 5, T2 =0.37796. We see that the

transverse magnetizations and hence the signals are getting weaker with the relaxation

time index n. However, they are not getting weaker at the same rate. The fluid

molecule with shorter T2 becomes weaker and thus will become darker in a T2-

weighted image leaving the blood molecule with the longer T2 and stronger signals to

be brighter during the relaxation time. What we will actually see in a T2-weighted

image, as shown in figure 1, depends on the level of magnetization at the time when

we snap the picture. The important thing to observe here is that the fluid particle with

long T2 values will appear bright inT2-weighted images.

Generally in MRI, a T2-weighted image appears to be a reversal of a T1-weighted

image. Tissues that are bright in one image are dark in the other image because T1 and

T2 values are generally related as expressed in equation (9), which indicates that T1 is

proportional to T2 with

€

1n2

as the reversal constant of proportionality defined as the

relaxation time index, a property of a fluid molecule or tissue as shown in table 3.

18

Table 3. Relaxation time index n, T1 and T2 values for various tissues

Serial No. Tissue T2 (sec) T1 (0.5T)

(sec)

n(0.5T)

T1(1.5T)

(sec)

n(1.5T)

1 Adipose(Fat) 0.080 0.210 1.62 0.26 1.80

2 Liver 0.042 0.350 2.89 0.50 3.45

3 Muscle 0.045 0.550 3.50 0.87 4.40

4 White matter 0.090 0.500 2.36 0.78 2.79

5 Gray matter 0.100 0.650 2.55 0.92 3.03

6 CSF 0.160 1.800 3.35 2.40 3.87

Conclusion

We have presented a theoretical model for the dynamics of blood molecules flowing

in one dimension only. We take this as the x direction and that the trajectory is a

function of x(t) which depends on T1 and T2 relaxation parameters of blood flow

according to equation (67b). Solutions to the Bloch NMR flow equations in the form

of polynomials allow us to express the blood flow velocity of the blood molecule as

the differential of position with respect to time, and the acceleration as the differential

of the velocity. In this way we are able to specify the position and velocity of the

blood molecule at a given time. Another important quantity derived as a result of the

solution is momentum, p, defined as the product of the mass and the velocity in

equation (57a).

The key to MRI is that the signal from hydrogen nuclei varies in strength depending

on the surroundings. NMR relaxation is a consequence of local fluctuating magnetic

fields within a molecule. Local fluctuating magnetic fields are generated by molecular

motions. In this way measurement of the position, velocity, acceleration, momentum

of blood molecules in terms of relaxation times can provide functional information of

motions within a molecule at the atomic level.

A very important quantity derived in this study is the energy, E of the blood molecule.

We distinguish between kinetic energy, Ek, which comes from the motion of the blood

19

molecule, and the potential energy, Ep(x), which depends on its position. A point

where Ep(x) has a minimum is called a point of stable equilibrium. A molecule at rest

at such a point will remain at rest. If displaced a slight distance, it will experience a

restoring force tending to return it, and it will oscillate about the equilibrium point.

For a blood molecule moving without a force, both potential energy and kinetic

energies are constant. Generally, they will vary during the motion, but in such a way

that the total energy given by equation (58) remains constant.

A point where E(x) has a minimum is called a point of unstable equilibrium. In

theory, a molecule at rest there can remain at rest, since the force is zero, but if it is

displaced the slightest distance, the force acting on it will push it farther away from

the unstable equilibrium position. A region where Ep(x) is consistent is called a region

of neutral equilibrium, since a molecule can be displaced slightly without suffering

either a restoring or a repelling force.

This kind of qualitative discussion, based on the energy integral, is simple and very

useful. It can be very interesting to study this model and understand it well enough to

be able to see at a glance, for any potential energy curve, the types of motion that are

possible. However, it may be only part of the force on a blood molecule is derivable

from a potential function Ep(x). If the remainder of the force is represent by FR, we

can write

= (69)

In this case the energy [Ek+Ep(x)] is no longer constant. Since the motion of the blood

molecule is governed by

(70)

If we substitute F from equation (69) in equation (70), and multiply by we

have, after rearranging terms, the time rate of change of kinetic plus potential energy

is equal to the power delivered by the additional force .

Since , we have , which means that, classically, the blood molecule

can be found only in the range . At the end of the interval, where ,

its kinetic energy vanishes; the point are called turning points.

20

It can be very interesting to note that appropriate application of classical and quantum

mechanics to equations (6)-(9), (58) and (68) can give valuable information about the

physical quantities such as the relaxation index n, position , velocity, energy and

momentum of a blood molecule in terms of NMR relaxation parameters. Specifically,

the relaxation index n, developed in this study, is a very important physical

characteristic that forms the link between the magnetic resonance image, tissue

characteristics (proton density, T1 and T2 relaxation parameters), the magnetic

characteristic of fluid (usually blood) movement (vascular flow, perfusion and

diffusion) and the spectroscopy effects related to molecular structure.

Generally, the magnetic resonance imaging process consists of the acquisition of rF

signals from the patient’s body and the mathematical reconstruction of an image from

the acquired signals. It is significant to note that the signals generated in figure 1, can

be digitized, and stored in computer memory in a configuration known as k space

(equation 25b). The k space is divided into lines of data that are filled one at a time.

One of the general requirements is that the k space as shown in equation (25b) must

be completely filled before the image reconstruction can be completed. The size of k

space is determined by the requirements for image detail. The relaxation index n, may

have tremendous control over the characteristics and the quality of magnetic

resonance images that are produced. This will be investigated separately.

Acknowledgments

This work was supported in part, by Federal University of Technology, Minna,

Nigeria and the Abdus Salam International Centre for Theoretical Physics (ICTP),

Trieste, Italy. This work was done within the framework of the Associateship Scheme

of ICTP.

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