APPLICATION OF BLOCH NMR FLOW EQUATION ANALYTICAL...
Transcript of APPLICATION OF BLOCH NMR FLOW EQUATION ANALYTICAL...
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. [email protected]
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.
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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
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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,
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(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)
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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.
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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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