Journal of Applied Mathematics and Physics, 2013, 1, 71-78
Published Online November 2013 (http://www.scirp.org/journal/jamp)
http://dx.doi.org/10.4236/jamp.2013.15011
Open Access JAMP
Flow Dynamics in Restricted Geometries: A Mathematical
Concept Based on Bloch NMR Flow Equation and
Boubaker Polynomial Expansion Scheme
Omotayo Bamidele Awojoyogbe1*, Oluwaseun Michael Dada1, Ka r e m Bou b ak e r2,
Omoniyi Adewale Adesola1
1Department of Physics, Federal University of Technology, Minna, Nigeria
2UPDS/ESSTT/63 Rue Sidi Jabeur 5100, Mahdia, Tunisia
Received August 4, 2013; revised September 25, 2013; accepted October 2, 2013
Copyright © 2013 Omotayo Bamidele Awojoyogbe et al. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
ABSTRACT
Computational techniques are invaluable to the continued success and development of Magnetic Resonance Imaging
(MRI) and to its widespread applications. New processing methods are essential for addressing issues at each stage of
MRI techniques. In this study, we present new sets of non-exponential generating functions representing the NMR
transverse magnetizations and signals which are mathematically designed based on the theory and dynamics of the
Bloch NMR flow equations. These signals are functions of many spinning nuclei of materials and can be used to obtain
information observed in all flow systems. The Bloch NMR flow equations are solved using the Boubaker polynomial
expansion scheme (BPES) and analytically connect most of the experimentally valuable NMR parameters in a simpli-
fied way for general analyses of magnetic resonance imaging with adiabatic condition.
Keywords: Bloch NMR Flow Equations; Boubaker Polynomial Expansion Scheme (BPES); Magnetic Resonance
Imaging (MRI); Adiabatic Condition
1. Introduction
Flow through porous media represents a vast field of
study with many scientific and engineering applications
[1-7]. A great number of experimental and theoretical
studies on flow in restricted motion using NMR are avail-
able in the literature [1-18]. Most of these studies are based
on either numerical or approximation solutions of Bloch
NMR equations. However, it will be fundamental and ideal
if the theoretical and experimental application of MRI for
flow analysis in restricted geometry is based on the ana-
lytical solutions of Bloch NMR equations. This has been
claimed over the years to be the best approach for ob-
taining fundamental information to accurately access fluid
dynamical properties in porous media/restricted geome-
try. It is possible to derive necessary relationships ana-
lytically for free motion. However, in the case of re-
stricted motion for which porous media are defined, the
macroscopic approach becomes mathematically intracta-
ble. Thus, in general case, one is forced to use different
method to find mathematical relation for the MRI signal
in terms of NMR experimental parameters [19-21].
In this investigation, we solved the Bloch NMR flow
equation which is transformable to Bessel equation of
order zero using the Boubaker Polynomial Expansion
Scheme to obtain the NMR transverse magnetization for
the analysis of flow in anisotropic fluid flow. The rela-
tionships between fluid velocity, the NMR relaxation
rates and the path length x for cerebrospinal fluid, white
and gray matter of human cerebrum are demonstrated.
The Boubaker Polynomials Expansion Scheme BPES is a
resolution protocol which has been successfully applied
to several applied-physics and mathematics problems.
Solutions have been proposed through the BPES in many
fields such as numerical analysis [22-27], theoretical
physics [24-29], mathematical algorithms [26], heat
transfer [30,31], homodynamics [28,29], material char-
acterization [32], fuzzy systems modeling [31] and boil-
ogy [32,33].
*Corresponding author.

O. B. AWOJOYOGBE ET AL. 72
2. Mathematical Analysis
The BPES protocol ensures the validity of the related
boundary conditions regardless of the main features of
the equation. The BPES is mainly based on Boubaker
polynomials first derivatives properties


4
10
4
1
20
0;
q
N
q
qx
N
q
qxr
Bx N
Bx


;










(1)
and


 


4
10
4
11
22
4
13
4
41
d0
d
d
d
42
with: 4
q
Nq
qx
NN
q
q
qq
xr
n
nn qn
q
nnn n
n n
Bx
x
Bx H
x
rr Br
H
Br r
Br



















 

 









(2)
In this investigation, the Boubaker Polynomials Ex-
pansion Scheme BPES has been applied to the bound-
ary-valued second order Bloch NMR flow differential
equation through setting the expression:
 
0
4
1
0
1
2
N
yk
k

kk
M
xB
N



xr
(3)
where My(x) are the time independent NMR transverse
magnetizations, 4k are the 4k-order Boubaker poly-
nomials,
B


0,x1 is the normalized variable, k are
minimal positive roots, is a prefixed integer
r
4k
B0 N
and 0
1, ,
kkN

 are unknown pondering real coefficients.
Based on the conditions which may conform to the
real-time experimental arrangements, we obtained a sec-
ond order non homogeneous differential equation from
the Bloch NMR flow equation [34,35] at Larmor fre-
quency:
0
o
fB

 
The x, y, z components (in the rotating frame) of the
magnetization of a fluid moving with spatially varying
velocity v is given by the Bloch equations which may be
written as follows:
2
d
d
x
x
x

1
2
d
d
yy
yz
y
M
MM
vgradMMB x
tt


 
T
(5)


0
1
1
d
d
z
zz
zy
MM
MM
v gradMM Bx
tt



 
T

(6)
Subject to the following conditions:
1) Mo  Mz a situation which holds well in general and
in particular when the RF B1(x) field is strong say of the
order of 1.0 G or more.
2) Before entering signal detector coil, fluid particles
has magnetization.
Mx = 0, My = 0.
3) If B1(x) is large;


11GBx or more so that My
of the fluid bolus changes appreciably from the equilib-
rium magnetization Mo.
 denotes the gyromagnetic ratio of fluid spins; 2


is the RF excitation frequency; 0
f

is the off-reso-
nance field in the rotating frame of reference. T1 and T2
are the spin-lattice and spin-spin relaxation times respec-
tively, the reciprocals of T1 and T2 are defined as relaxa-
tion rates. Mo is the equilibrium magnetization and RF B1
is the spatially varying magnetic field [35] which may be
designed as


1
Bx gx

  (7)
where g is the field gradient. Equations (5) and (6) give a
second order non-homogenous differential equation called
the Bloch NMR flow equation:
 
2
00
1
222
1
dd
d d
yy
y
MM Sx
TM
M
Bx
vx xvvT
   (8)
where
 
22
01
12 12
11 1
,TSxBx
TT TT
 
In NMR systems, when the RF B1 field is applied, My
has a maximum value when RF B1 has maximum ampli-
tude and Mo ≈ 0. In biological systems especially at the
molecular level we need to solve Equation (8) to provide
velocity profiles for different tissues materials such that
o
v
x
vnTR
T

 (9)

o
TTR
n

 (10)
where n is the number of pulses, TR is the repetition time.
If  is the time between two pulses, we write:
0
TR
nT

  (11)
x
M
MM
v gradM
tt

 
T
(4)
For adiabatic condition, Equation (8) becomes:
Open Access JAMP

O. B. AWOJOYOGBE ET AL. 73
2
2
22
2
dd 0
d d
yy
y
o
MM
x
xT x






g xM  (12)
where

22
1
12
1
Bx TT  
Equations (13) and (14) can be solved using the Bou-
baker polynomial expansion scheme [21-23] with bound-
ary conditions based on traditional NMR procedures.
 
2
2
2;1;2,3,4,5
dd
0
d d
d0
01; 0;
d
yy n
y
n
y
y
MM
xxM
x x
M
Mx









 (13)
where β is a constant which is unique to the NMR system
being described. For this system, the gradient field is
chosen (under the condition 2


) such that
2
o
T
g


We define n as a dimensionless variable
2
o
n
fTx


 (14)
where  is a special flow property of the fluid (for this
analysis 1

), α is dimenssioless constant and f is a pro-
perty of the medium. Equations (13) and (14) reduce to:




00
0
2
4
,,
2
11
00
2
,4
1
0
d
11
2 d
1
NN
kk kk
kn kn
kk
N
knkk
k
Bxr Bxr
NxN x
Bxr
N






 




4
d
dx
(15)
The BPES solution is obtained by determining the
non-null set of coefficients
0
1, ,
kkN


 that minimizes
the absolute difference :
0
N

00
0,,
11
00
11
22
NN
Nknkkn
kk
NN



k


 






(16)
with:

 
0
2
1
24
2
0
2
1
4
4
1
00
dd
d
d 21
d
d
k
kk k
N
k
kkkk k
k
B
rxrx
x
B
rxrBxr
xx N

 


 
 






x






(17)
The final solution is:
 
0
,,
2,3,4,5 1
0
1
2
N
ynknk k
nk
3. Analysis of Results
From Equations (7), (9)-(13), we obtain for the value of α
= 2, the following
2
o
v
xT  (19)
2
2
o
o
nT n
fTx v
2


 (20)
Tables 1-4 show how the fluid velocity and relaxation
parameters changes with x for different human tissues
materials at 1.5 T.
The tables show the usefulness of BPES to different
tissues on MRI scan. They can also be used to observe
the same tissue materials at different locations.
Figure 1 shows the NMR transverse magnetization
when the value of x is small, high and very high for the
Boubaker polynomial expansion scheme (BPES). The
number of pulses n have more influence on the NMR
signal when the value of x is small that when it is high.
This can be useful to determine the number of pulses
needed for a particular NMR experiment.
Figures 2 and 3 show velocity profiles for different
tissues materials. The color bands represent the different
magnitude of the fluid velocity for different tissue. For
example in Figure 2, the velocity profile is 0.030 m/s in
cerebrospinal fluid while it is 0.25 m/s in gray matter for
the same color band with the T1 and T2 relaxation rates
providing tissue contrast.
Table 1. Values of the path length, velocity and the relaxa-
tion rate for cerebrospinal fluid at 1.5 T.

4
M
xB
N



xr
(18)
Cerebrospinal Fluid
x x2 τ = α/T0 T0 v
0.046114 0.002127 0.332266 6.019278 0.138786
0.044374 0.001969 0.307664 6.500591 0.144229
0.042634 0.001818 0.284009 7.042029 0.150115
0.040894 0.001672 0.261300 7.654041 0.156502
0.039154 0.001533 0.239537 8.349447 0.163457
0.037414 0.001400 0.218720 9.144115 0.171059
0.035674 0.001273 0.198849 10.05788 0.179402
0.033934 0.001152 0.179924 11.11578 0.188601
0.032194 0.001036 0.161946 12.34980 0.198795
0.030454 0.000927 0.144913 13.80134 0.210153
0.028714 0.000824 0.128827 15.52468 0.222888
0.026974 0.000728 0.113687 17.59216 0.237266
0.025234 0.000637 0.099493 20.10193 0.253626
0.023494 0.000552 0.086245 23.18975 0.272410
0.021754 0.000473 0.073943 27.04779 0.294199
0.000000 0.000000 0.000000 ∞ ∞
Open Access JAMP

O. B. AWOJOYOGBE ET AL. 74
Table 2. Values of the path length, velocity and the relaxa-
tion rate for gray matter of the cerebrum at 1.5 T.
Gray Matter
x x2 τ = α/T0 T0 v
0.046114 0.002127 1.329063 1.504819 0.034697
0.044374 0.001969 1.230657 1.625148 0.036057
0.042634 0.001818 1.136036 1.760507 0.037529
0.040894 0.001672 1.045200 1.913510 0.039126
0.039154 0.001533 0.958147 2.087362 0.040864
0.037414 0.001400 0.874880 2.286029 0.042765
0.035674 0.001273 0.795396 2.514469 0.044851
0.033934 0.001152 0.719698 2.778944 0.047150
0.032194 0.001036 0.647784 3.087451 0.049699
0.030454 0.000927 0.579654 3.450335 0.052538
0.028714 0.000824 0.515309 3.881169 0.055722
0.026974 0.000728 0.454748 4.398041 0.059316
0.025234 0.000637 0.397972 5.025483 0.063407
0.023494 0.000552 0.344980 5.797437 0.068102
0.021754 0.000473 0.295773 6.761946 0.159695
0.000000 0.000000 0.000000 ∞ ∞
Table 3. Values of the path length, velocity and the relaxa-
tion rate for white matter of the cerebrum at 1.5 T.
White Matter
x x2 τ = α/T0 T0 v
0.046114 0.002127 1.772084 1.128615 0.026022
0.044374 0.001969 1.640877 1.218861 0.027043
0.042634 0.001818 1.514715 1.320380 0.028147
0.040894 0.001672 1.393599 1.435133 0.029344
0.039154 0.001533 1.277530 1.565521 0.030648
0.037414 0.001400 1.166506 1.714522 0.032074
0.035674 0.001273 1.060529 1.885852 0.033638
0.033934 0.001152 0.959597 2.084208 0.035363
0.032194 0.001036 0.863711 2.315588 0.037274
0.030454 0.000927 0.772872 2.587751 0.039404
0.028714 0.000824 0.687078 2.910877 0.041791
0.026974 0.000728 0.606331 3.298531 0.044487
0.025234 0.000637 0.530629 3.769112 0.047555
0.023494 0.000552 0.459973 4.348078 0.051077
0.021754 0.000473 0.394364 5.07146 0.055162
0.000000 0.000000 0.000000 ∞ ∞
Table 4. Values of the path length, velocity and the relaxa-
tion rate for white matter of cystic tumor at 1.5 T.
Cystic tumor
x x2 τ = α/T0 T0 v
0.046114 0.002127 0.189866 10.53374 0.242876
0.044374 0.001969 0.175808 11.37603 0.252400
0.042634 0.001818 0.162291 12.32355 0.262701
0.040894 0.001672 0.149314 13.39457 0.273879
0.039154 0.001533 0.136878 14.61153 0.286050
0.037414 0.001400 0.124983 16.0022 0.299353
0.035674 0.001273 0.113628 17.60129 0.313954
0.033934 0.001152 0.102814 19.45261 0.330052
0.032194 0.001036 0.092541 21.61216 0.347891
0.030454 0.000927 0.082808 24.15235 0.367768
0.028714 0.000824 0.073616 27.16819 0.390054
0.026974 0.000728 0.064964 30.78629 0.415215
0.025234 0.000637 0.056853 35.17838 0.443846
0.023494 0.000552 0.049283 40.58206 0.476717
0.021754 0.000473 0.042253 47.33363 0.514848
0.000000 0.000000 0.000000 ∞ ∞
Based on Equation (19) and Tables 1-4, the analysis
of fluid velocity, relaxation rates and the path length x
can be described within the following three limits:
21
o
n
fTx

 (21)
21
o
n
fTx


 (22)
21
o
n
fTx

 (23)
Equation (21) is the short time limit where the particle
does not flow far enough during time
o
nT

to feel the
effect of parameter  . When f  1, as shown in equation
(22) some of the particles feel the effects of restriction
and the value of  measured within this time scale will be
a function of
o
nT

. Equation (23) shows that the time
is long enough for all the particles to feel the effects of
restriction and the displacement of the particle depends
not on time
o
nT

, but only on path length x. This indi-
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O. B. AWOJOYOGBE ET AL.
Open Access JAMP
75
Figure 1. Plots of the NMR transverse magnetization against (a) Small values of x; (b) Higher values of x; (c) Much higher
values of x for the Boubaker polynomial expansion scheme (BP ES).
(a) (b)
Figure 2. Plots of the fluid velocity against the relaxation rate and the path length x for cerebrospinal fluid and gray matter
within the human brain at a static magnetic field of 1.5 T.

O. B. AWOJOYOGBE ET AL. 76
(a) (b)
Figure 3. Plots of the fluid velocity against the relaxation rate and the path length x for white matter of human cerebrum and
cystic tumor at a static magnetic field of 1.5 T.
cates that the value of n and not  , in Equations (13) and
(14) as solved by the Boubaker polynomial expansion
scheme is very significant for the analysis of flow in re-
stricted geometry where the measured fluid velocity de-
pends of the relaxation parameters as shown in Figures 2
and 3. We may conclude that f, has a memory of the
chemical differences within the spin’s immediate envi-
ronment or the magnitude of the static magnetic field Bo.
Therefore, selecting a particular value of x may corre-
spond to selecting certain magnitude of Bo field or the
molecular imprints of the tissue containing a flowing
spin. The values of  used in this study, is for computa-
tional purposes.
4. Conclusion
A mathematical concept of magnetic resonance imaging
for flow analyses in restricted geometries has been pre-
sented by solving the Bloch flow equation using the
Boubaker polynomial expansion scheme (BPES). These
demonstrate the usefulness of Bloch NMR flow equation
and the Boubaker polynomial expansion scheme for
studying fluid flow in restricted geometries to obtain the
NMR transverse magnetization for the analyses of flow
in anisotropic fluid flow. The relationship between fluid
velocity, the NMR relaxation rates and the path length x
for cerebrospinal fluid, white and gray matter of human
cerebrum as demonstrated provides tissue contrast for
different tissues materials. This can prove to be a very
good starting point for building more sensitive and less
expensive magnetic resonance imaging sequences.
5. Acknowledgements
The authors acknowledge the support from Federal Uni-
versity of Technology, Minna, Nigeria through the STEP
B research programme of the World Bank.
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