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Chapter 3
Basic Principles of MRI
3.1
Nuclear Magnetic Moment
In clinical MRI the image is formed by the signals from protons in water and lipid. At
the atomic level, since a proton is a charged particle which spins around an internal axis
of rotation with a given value of angular momentum P , it also has a magnetic moment
µ, and therefore can be thought of as a very small magnet with a north and south
pole, as shown in Fig.3.1(a). The magnetic field generated by the spin of the particle
are collinear with the direction of the spin axes and normally it is termed magnetic
moment. The strength of the magnetic moment determines the sensitivity of detection
in magnetic resonance and it is dependent on the type of nucleus. Most frequently, the
hydrogen nucleus with one proton is the nucleus of choice in MRI because it possesses
the strongest magnetic moment and its abundance in organic tissues. The variations in
spin angular momentum result from interactions with an applied static magnetic field
and electromagnetic radiation. These particles have mass and thus generate angular
momentum as they rotate. Positively and negatively charged particles can be regarded
as spheres of distributed positive or negative charges, while neutral electrical particles
such as the neutron can be thought of as a combination of distributed positive and
negative charges. Relevant to MRI, the magnitude of tha angular momentum of the
proton is quantized and has a single, fixed value. The magnitude of the proton’s
magnetic moment is proportional to the magnitude of the angular momentum:
|~
µ| = γ|P~ |
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(3.1)
3. BASIC PRINCIPLES OF MRI
(a)
(b)
(c)
Figure 3.1: (a) The internal rotation of a proton creates a magnetic moment, and so
the proton acts as a magnet with north and south pole. (b) In the absence of a strong
magnetic field, the orientation of the magnetic moments are completely random. (c) When
there is a strong magnetic field present the magnetic moment must be align at the angle
θ = ±54.7◦ with respect to the direction of B0 .
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3.1 Nuclear Magnetic Moment
where γ is a constant called the gyromagnetic ratio, and has a value of 267.54
MHz/Tesla for protons. As a result, the magnitude of the magnetic moment has a
single, fixed value. In absence of an external magnetic field, as shown in Fig.3.1(b),
the magnitude of the magnetic moment of every proton in our bodies is fixed, but the
orientation is completely random. Therefore, the net magnetization, i.e. the sum of all
the individual magnetic moments in our bodies, is zero.
The situation changes with the application of an external magnetic field B0 . From
quantum mechanics, the component of the magnetic moment in the direction of B0
can have only two possible discrete values, which results in the magnetic moments
being aligned at an angle of 54.7◦ with respect to the direction of B0 , aligned either in
the same direction, as shown in Fig.3.1(c). The former configuration is termed as the
parallel, and the latter as the anti-parallel configuration: note however that the terms
parallel and anti-parallel only refer to the z-component of µ, and that µ is actually
aligned at an angle with respect to B0 . The relative number of protons in the parallel
and anti-parallel configurations depends upon the value of B0 . Protons in the parallel
configuration are preferred because it guarantees the lowest energy state. The energy
difference ∆E between the two states is shown in Fig.3.2 and given by:
γhB0
(3.2)
2π
where h is Plank’s constant (6.63 × 10−34 Js). To calculate the relative number of
∆E =
protons in each of the two configurations the Boltzmann equation can be used:
γhB0
∆E
Nanti−parallel
= e− kT = e− 2πkT
Nparallel
(3.3)
where k is Boltzmann’s constant with the value of 1.38 × 10−23 J/K, and T is the
temperature measured in Kelvin. Since the value of the exponent is very small, a first
order approximation, e−x ≈ 1 − x , can be made:
Nanti−parallel
γhB0
=1−
Nparallel
2πkT
(3.4)
The MRI signal depends upon the difference in populations between the two energy
levels:
Nparallel − Nanti−parallel = Ntotal
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γhB0
2πkT
(3.5)
3. BASIC PRINCIPLES OF MRI
Figure 3.2: Proton configurations. (left) in the absence of a strong magnetic field, the
energies of all the random orientations of the magnetic moments are the same. (right)
When a strong magnetic field is applied, the single energy level splits into two levels, on
corresponding to the magnetic moments being in the parallel state, and the other the anti
parallel state. The energy difference between the two states depends upon the value of B0 .
where Ntotal is the total number of protons. It is important to note that MRI can
detect only the difference Nparalllel −Nanti−parallel , and not the total number of protons.
As shows equations 3.3 and 3.4, thermal energy causes the energy difference between
the two orientations to be minimal, with the two orientations almost equally populated
resulting in a net bulk magnetization M. Naturally, the protons can change from one
orientation to another by absorbing or emiting photons with energy equal to the energy
difference, as shown in Fig.3.2.
3.1.1
Classical Precession
Having determined that the proton magnetic moments are all aligned at an angle of
54.7◦ with respect to the direction of B0 . The motion of these magnetic moments can
most easily be described using classical mechanics. The B0 field attempts to aling the
proton magnetic moment with itself, and this action create a torque, C, given by the
cross product of the two magnetic fields:
~ =µ
C
~ × B~0 = iN |µ||B0 |sinθ
(3.6)
where iN is a unit vector normal to both µ
~ and B~0 . The direction of the torque,
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3.1 Nuclear Magnetic Moment
Figure 3.3: A proton in a magnetic field. Using classical mechanics, the torque C acting
on the magnetic moment, spinning about an internal axis, causes it to precess about the
vertical axis, B0 .
shown in Fig.3.3, is tangential to the direction of µ
~ and so causes the proton to ”precess”
around the axis of the magnetic field, while keeping a constant angle of 54.7◦ between
µ
~ and B~0 .
To calculate how fast a proton precesses, we use the fact that the torque is defined
as the rate of change of the proton’s angular momentum:
~
~ = dP = µ
~ × B~0
C
dt
(3.7)
From Fig.3.3, the magnitude of the component of the angular momentum which
precesses in the plane perpendicular to B0 is given by |P~ |sinθ. In a short time dt,
µ precesses through an angle dϕ resulting in a change dP~ in the angular momentum.
Simple trigonometry gives the relationship that:
sin(dϕ) =
dP~
|P~ |sinθ
=
~
Cdt
|P~ |sinθ
(3.8)
if dϕ is small then we can make the approximation that sin(dϕ) = dϕ. The angular
precession frequency, ω, is given by dϕ/dt and so has a value:
ω=
~
dϕ
C
mu
~ × B~0
γ P~ × B~0
γ|P~ ||B~0 |sinθ
=
=
=
=
= γB0
dt
|P~ |sinθ
|P~ |sinθ
|P~ |sinθ
|P~ |sinθ
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(3.9)
3. BASIC PRINCIPLES OF MRI
The effect of placing a proton in a magnetic field, therefore, is to cause it to precess
around B0 at a frequency directly proportional to the streght of the magnetic field. This
frequency, termed ω0 , is termed the Larmor frequency after renowned Irish physicist
Joseph Larmor. For hydrogen protons, γ is given as 4257 Hz/Gauss. Thus, in a field
stregth of 7 T, the hydrogen proton will precess with a frequency of 297.99 MHz≈ 300
MHz.
3.1.2
Total Magnetization
By superimposing several proton magnetic moments we can represent the net magnetization in a simple vector form. Figure 3.4 shows on the left a representation of several
proton magnetic moments, each aligned at 54.7◦ to B0 , each precessing at a frequency
ω0 , with slightly more protons in the parallel than anti-parallel state. The total magnetization can be calculated by a simple vector sum of the individual components, and
is shown on the right of Fig.3.4. It can be seen that the net magnetization has only
a z-component since the vector sum of the components has only a z-component, since
the vector sum of the components on the x- and y-axes is zero. The net magnetization
of the sample is defined as M0 :
M0 =
NX
total
n=1
3.2
µz,n =
γ 2 h2 B0 Ntotal
γh
(Nparallel − Nanti−parallel ) =
4π
16π 2 kT
(3.10)
Effects of Radio Frequency Pulses on Magnetization
The detection of an NMR signal is facilitated by the establishment of a resonance
condition. The resonance condition represents a state of alternating absorption and
dissipation of energy. Energy absorption is achieved through the application of RF
pulses, while energy dissipation is caused by relaxation processes. The energy levels for
protons in a magnetic field, shown in Fig.3.2, are analogous to energy levels in semiconductors. As with all such a multi-level systems, to obtain an MR signal, energy must
be supplied with a specific value ∆E, given by Equation 3.2, to stimulate transitions
between the energy levels. The energy is supplied as an electromagnetic (EM), usually
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3.2 Effects of Radio Frequency Pulses on Magnetization
Figure 3.4: Magnetization represented by vectors. (left) individual magnetization vectors
are randomly distributed around a cone which subtends an angle of 54.7◦ with respect to
the B0 (z) axis. The vector sum of all the individual magnetization vectors (right) is simply
a static component in the direction of B0 .
as a rediofrequency (RF) field, the frequency (f) of which can be calculated from the
Broglie’s relationship ∆E = hf :
γhB0
2π
γB0
⇒f =
or ω = γB0
2π
hf = ∆E =
(3.11)
By comparing equations 3.11 and 3.9 it can be seen that the frequency of the RF
field is identical to the precession frequency. Consider the application of RF radiation
at Larmor frequency to a bulk sample of non-magnetic material in an applied static
magnetic field. In MRI, the energy is applied as a short RF pulse and it is composed
by a coupled electric and magnetic field components. The magnetic field component
is denoted by B1 , and it resides in a plane perpendicular to B0 , as shown in Fig.3.5.
Applying the same classical analysis as for proton precession, B1 produces a torque
which causes the net magnetization to rotate towards the xy plane as shown in Fig.3.5.
The consequence of the application of B1 is to rotate M by a certain angle away from
the B0 axis. This angle is called the flip angle (α) is defined as the angle through which
the net magnetization is rotated. This angle is proportional to both the streght of the
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3. BASIC PRINCIPLES OF MRI
Figure 3.5: on the left, application of an RF pulse about the x-axis rotated the magnetization from z-direction towards the y-axis. If the RF pulse streght and duration are chosen
to produce 90◦ pulse, then the magnetization lies directly along the y-axis. When the RF
pulse is switched off (right), the magnetization precesses around the z-axis at the Larmor
frequency ω0 .
applied RF field (measured in Tesla) and the time τB1 , for which it is applied:
α = γB1 τB1
(3.12)
Hence, if B1 persists for the appropriate duration of time, M can be made to rotate
onto the transverse plane. While in the transverse plane and rotating at the Larmor
frequency, M will induce an NMR signal in the RF receiver coil which is oriented
in the transverse plane as shown in Fig.3.6. This signal can be used to observe the
characteristics of M in the transverse plane and constitute the basis of MR signal
detection, this process constitutes the basis of MR signal detection. The RF pulse that
brings M into the transverse plane is usually referred to as the 90◦ pulse. A flip angle of
90◦ results in the maximum value of the My component of magnetization, whereas one
of 180◦ produces no My magnetization but rotates the net magnetization M0 form +z
to the -z axis. The 90◦ flip angle is very important because the strongest NMR signal
is obtained when M rotates in the transverse plane. The 180◦ flip angle is primary
important in spin-echo imaging techniques where it is used to reverse the direction of
M once it is on the transverse plane.
3.3
The Basis of MR signal detection
In the most simple case, the MR detector consists of a pair of conductive loops (of
copper wire for example) placed close to the patient at an angle of 90◦ with respect to
each other. Faraday’s law of induction states that the voltage (V) is induced in each of
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3.3 The Basis of MR signal detection
Figure 3.6: The MR signal is measured via Faraday induction. Either one or two RF
coils can be used, with a voltage being induced across the ends of the conductor loops by
the precessing magnetization.
these loops with a value proportional to the time rate of change of the magnetic flux
dϕ:
V ∝−
dϕ
dt
(3.13)
Figure 3.6 shows the situation a short time after 90◦ pulse has been applied about
the x-axis: in this case the respective voltages induced in the two coils are given by:
Vy ∝ M0 ω0 sinω0 t
(3.14)
Vx ∝ −M0 ω0 cosω0 t
It is important to note that the requirement for a time − varying magnetic flux to
induce an MR signal is the reason why only magnetization precessing in the xy-plane
gives rise to an MR signal. Any z-component of magnetization does not precess and
therefore does not induce any voltage.
3.3.1
MR signal intensity
The intensity of the recived MR signal is determined by three different factors. First
the signal is proportional to the number of protons in the object, from Equation.3.5. In
terms of MRI, as will be seen later, this corresponds to the number of protons in each
voxel of the image. The other two factors depend upon the value of the B0 field. From
Equation 3.10, the value of M0 is proportional to B0 . Therefore, a 3 Tesla MRI system
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3. BASIC PRINCIPLES OF MRI
Figure 3.7: (left) magnetization vector after a 90◦ RF pulse about the x-axis. (centre)
T1 and T2 relaxation of the magnetization a certain time after the pulse has been applied
results in an increased Mz component and reduced My component, respectivley. (right)
After further time, the Mz and My components have almost returned to their equilibrium
values of M0 and zero, respectively.
has twice the M0 of a 1.5 Tesla system. Aditionally, from Equation 3.14, the induced
voltage is proportional to the precession frequency, which in turn is proportional to B0 .
Overall, therefore the MR signal is porportional to the square of the B0 field, one of
the reason of why there is a such a strong drive towards higher field MRI systems.
3.4
Relaxation
In a presence of a strong magnetic field B0 , as seen in sub-section 3.1.2, the equilibrion
magnetization state corresponds to a z-component, Mz , equal to M0 and transverse
components, Mx and My , equal to zero. Application of an RF pulse creates a nonequilibrium state by adding energy to the system. After the pulse has been switched off,
the system must relax back to termal equilibrium. The phenomenon of MRI relaxation
is similar to the application of an impulse voltage pulse to an RC electrical circuit, where
the circuit produces time-varying voltages across the lumped elements, the values of
which return in time to their values prior the pulse being applied, this process being
characterized by certain time-constants. After the application of a 90◦ RF pulse, M
rotates in the transverse plane at the Larmor frequency and gradually decays to zero
as shown in Fig.3.7.
There are two relaxation times which govern the return to equilibrium of the zcomponent, and the x- and y- components, respectively. These are referred to as T1 relaxation (which affects only z-magnetization) and T2 -relaxation (which affects only x-
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3.5 Magnetic Gradients Coils
and y-magnetization). These are also called spin-lattice (T1 ) and spin-spin (T2 ) relaxation. MR relaxation is described mathematically by first order differential equations
known as the Bloch equations. Solutions of these equations yields the relation of the
Mz component at a time t with the flip angle α of an RF pulse after being applied,
given by:
Mz (t) = M0 cosα + (M0 − M0 cosα)(1 − e
− Tt
1
)
(3.15)
For example, after a 90◦ pulse the value of Mz is given by:
Mz (t) = M0 (1 − e
− Tt
)
1
(3.16)
Different tissues have different values of T1 , and deseased tissues often have substantially altered T1 relaxation time compared to healthy tissue, and these differences
form the basis for introducing contrast into tha MR image.
The second relaxation time, T2 , governs the return of the Mx and My components
of magnetization to their thermal equilibrium values of zero. If an RF pulse of arbitrary
flip angle α is applied along the x-axis, the value of My at time t after the RF pulse is
given by:
My (t) = M0 sinα e
− Tt
2
(3.17)
As is the case for T1 relaxation times, different tissues in the body have different
values of T2 , and these can also be used to differentiate between healthy and diseased
tissues in clinical images
3.5
Magnetic Gradients Coils
The concept of making MR an imaging modality originated with the realization that
if the magnetic field could be made to vary spatially within the subject, this would in
turn impose a spatial variation in resonant frequencies that could be exploited to form
an image. Such spatial variations have to be varied dynamically. This is performed by
incorporating three separate ’gradient coils’ into the design of an MRI scanner. These
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3. BASIC PRINCIPLES OF MRI
Figure 3.8: The resulting magnetic field, Bz , is a function of position in z. The slope of
the graph is Gz , the z-gradient.
gradient coils are designed so that the spatial variation in magnetic field is linear with
respect to spatial location, i,e.
δBz
δBz
δBz
= Gz ,
= Gx ,
= Gy
δz
δx
δy
(3.18)
where G represents the gradient measured in T/m. The three separate magnetic
field gradients are produced by passing a DC current through separate coils of wire.
The current in each set of gradient coils comes from high-power gradient amplifiers
which supply hundreds of amps and where the current can be turned on and off very
quickly under computer control. The gradient coils are also designed such that there
is no additional contribution to the magnetic field at the isocenter (z=0,y=0,x=0) of
the gradients, which means that the magnetic field at this position is simply B0 . By
convention, the y-axis corresponds to the anterior/posterior direction, and the x-axis
to the left/right direction of the patient lying in the magnet.
The magnetic field, Bz expereinced by protons with a common z-coordinate is given
by:
Bz = B0 + zGz
(3.19)
From the graph shown in Fig.3.8, at position z=0, Bz = B0 ; for all positions z > 0,
Bz > B0 , and for the position z < 0, Bz < B0 . The precession frequencies (ωz ) of the
protons, as a function of their position in z, are given by:
ωz = γBz = γ(B0 + zGz )
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(3.20)
3.6 Image Acquisition
Figure 3.9: Pulse sequence diagrams for imaging sequences. An RF pulse is applied, various gradients are turned on and off. Individual steps in image formation can be considered
independently in terms of slice selection(RF and Gz), phase-encoding (Gy) and frequency
encoding (Gx).
Analogous expressions can be obtained for the spatial dependence of the resonant
frequencies in the presence of the x- and y-gradients.
3.6
Image Acquisition
The process of image formation can be broken down into three separate, independent
components, slice selection, phase-encoding and frequency-encoding. An overall imaging ’pulse sequence’ is shown in Fig.3.9. The transmitter line indicates when an RF
pulse is applied, and the lenght and power of the pulse are adjusted to give an indicated flip angle For each gradient line, the height of the gradient pulse indicated its
stregth, and the polarity(positive or negative) indicates which direction current is flowing through the particular gradient coil. The entire sequence of RF pulse and three
gradients has to be repeated a number of times (Np tipically is between 128 and 512)
to build-up a two dimensional data-set, with the arrow next to the phase encoding
gradient indicating that different values are used for each repetition of the sequence.
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3. BASIC PRINCIPLES OF MRI
Figure 3.10: An RF sinc pulse and its fourier transform. The RF pulse excites a band
of frequencies of width ∆ωs centered around the frequency ω0
3.6.1
Slice Selection
Selective slice excitation refers to the process of restricting the signal response in the
third spatial dimension in order to create a 2D image of the sample. This is achieved by
selectively exciting only a well defined slice of the sample within the ROI. If a magnetic
field gradient is applied along an axis normal to the chosen slice plane. In fact, MRI
can acquire the image in any given orienation. For example, coronal, axial or sagital
images, corresponding to slice selection in the y-,z-, or x-directions, respectively, can
be chosen. Equation 3.9 dictates that there will be a linear variation in resonance
frequencies along that axis. This forms the basis of the selective slice exciation process
in MRI imaging. The relationship between the thickness of the excited slice T , the RF
pulse bandwidth ∆ωs and the applied field gradient amplitude Gslice is given by
T =
2∆ωs
γGslice
(3.21)
Slice selection uses a frequency-selective RF pulse applied simultaneously with one
of the magnetic field gradients (Gx , Gy or Gz ), denoted Gslice . The RF pulse is generally
a sinc pulse because of the desirable properties of its fourier transform as illustrated
in Fig.3.10. The RF pulse is applied at a specific frequency ωs , with an excitation
bandwidth of ±∆ωs . Thus, only the desired band of frequencies will be generated for
slice excitation. The location of the slice z along the axis is given by 3.20. The slice
position can therefore be moved to different parts of the patient by changing the value
of ωs while from 3.21, it can be seen that the slice thickness can be made thinner either
by increasing the strength of Gslice or decreasing the frequency bandwidth of the RF
Pulse.
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3.6 Image Acquisition
After the selective excitation process, the next step is to encode the image information within the desired slice. The image information sought is made up of the amplitude
of the NMR signal generated within the various locations in the excited slice. The two
spatial axes of the image plane are encoded using two distinct processes referred to as
frequency encoding and phase encoding.
3.6.2
Phase Encoding
Continuing with the example of an axial slice, in which the slice select gradient was
applied in the z-direction, the x and y direction are now encoded via the phase and
frequency of the MR singnal. As shown if Fig. 3.9, a phase encoding phase gradient
(Gy ) is turned on for a period τpe and then switched off. During the interval τpe the
proton precess at a frequency ωy = γGy y. The net effect is to introduce spatially
dependent phase shift, ϕpe (Gy , τpe ), with a value given by:
ϕpe (Gy , τpe ) = ωy τpe = γGy yτpe
3.6.3
(3.22)
Frequency Encoding
The x-dimension is encoded by applying a frequency-encoding gradient (Gx ) while the
receiver is gated on and the data are being acquired. During this time t, protons precess
at a frequency given by ωx = γGx x determined only by their x-location. A total of Nf
data points are acquired while the receiver is on.
Overall, this means that for each face encoding step each voxel in the image is
characterized by a specific phase which depends upon its position in y, and specific
freqeuncy which depend upon its position in x.
To form, for example, a 256 × 256 image, Nf = 256 and the sequence must be repeated 256 times, each time with different value of the phase encoding gradient, ranging
from its maximum negative to maximum positive value in equal increments of ∆Gpe .
In order for sufficient T1 relaxation to occur so that a significant fraction of Mz magnetization recovers between successive RF excitations, there is a delay between successive
RF pulses, called the TR (time of repetition) time. Overall the total data acquisition
time is therefore given by the TR multiplied by the number of phase encoding steps
applied, T R ∗ Npe .
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3. BASIC PRINCIPLES OF MRI
114
Chapter 4
Head-Brain Modeling
As previously mentioned, obtaining a head model is also significant for the simulation
of the RF coil-head interaction. A realistic head model cosists in a tetrahedreal volume
mesh discretization of the entire head considering the different constitutive parts of the
brain.
Name
Label
Background
CSF
Grey Matter
White Matter
Fat
Muscle
Skin
Skull
Vessels
Around Fat
Dura matter
Bone Marrow
Bone Marrow
0
1
2
3
4
5
6
7
8
9
10
11
12
Table 4.1: Labels associated with each anatomical structure
Few methods have been proposed on tetrahedreal mesh generation for head-brain
models. In this chapter the sphere carving algorithm (27) and iso2mesh (28) toolbox
were used to obtain the definitive head-brain model. The highlights of the method
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4. HEAD-BRAIN MODELING
and the resulting 3D head-brain tetrahedral mesh are presented in sections 4.2 and 4.3,
respectively.
Figure 4.1: Simulated MRI -
4.1
3D MRI Image
We start with a 3D grey scale MRI generated by (Kwan et al 1999). Each MRI
image has been produced taking into consideration a repetition time τ , equal to 20ms,
and a T1 scanning modality with a thickness equal to 1 mm along the z direction.
The images are obtained using a simulator (Kwan et al 1999). Once we have the
MRI in grey scale to obtained a segmented version of the different brain’s anatomical
structures. According to (Kwan et al 1999), the digital MRI is classified by different
constitutive parts of the brain, which include: cerebrospinal fluid (CSF), grey matter
(GM), white matter (WM), fat, muscle, skin, skull, vessels, dura matter (DM) and bone
marrow. The segmentation of the single regions are obtained from http://mouldy.
bic.mni.mcgill.ca/brainweb/. In that model each anatomical tissue is classified
membership volumes, one for each class listed in table 4.1. The voxel value in these
volumes reflects the proportion of that tissue type, present in that voxel, in the range
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4.1 3D MRI Image
(a) CSF
(b) Grey matter
(c) White matter
(d) Fat
(e) Muscle
(f) Skin
(g) Skull
(h) Vessels
(i) Around fat
(j) Dura matter
(k) Bone marrow
Figure 4.2: Brain’s anatomical structures
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4. HEAD-BRAIN MODELING
[0,1]. The segmentation yields a particular label for different anatomical structures as
indicated in table 1. The whole assembled head-brain model is shown in Figure 4.1,
while all the different constitutuve Brain’s parts are shown in Figure 4.3.
4.2
The Sphere Carving algorithm
Methods to tetrahedralize volume data are well studied for FEM research. However,
the methods to tetrahedralize volume with complicated geometric structure such as
the head-brain are somewhat rare in the literature, although they are used occasionally
for surgical simulation, or mapping intraoperative brain change. In this dissertation,
we apply first a sphere carving algorithm (27) to volume data tetrahedralization of
the head-brain model. At the beginning, the algorithm constructs a large sphere that
contains all the volume data. Then the algorithm keeps removing the outside tetrahedra
while maintaining the surface genus number http://en.wikipedia.org/wiki/Genus_
(mathematics). The input image to this algorithm is a binary 3D volumetric image
that in the case of brain volume construction are the brain MRI images shown in Fig.4.1.
The final goal is to build a tetrahedral volume while maintaining a surface with desired
genus number. First it builds a large sphere tetrahedral mesh which totally encloses
the brain 3D volume Fig. 4.3(a). Then it keeps removing the tetrahedra outside of
the brain volume while maintaining a genus zero surface. When a tetrahedra is inside
the brain 3D volume its baricenter is mapped into one of the slices in the set of the
MRI immages. The mapped point turns into a voxel of the immage, where the voxel’s
value represents the tissue label, one for each class listed in table 4.1. However, if the
point is mapped in between two immage slices. The neighbors voxels of the top and
the botton slices are considered to carry out a label counting list. The first place is
assigned to the label with more occurrences which is finally the one assigned to the
tetrahedron. The output of the algorithm is a head-brain tetrahedral model shown in
Fig. 4.3(b). Finally, a refinement process of the output mesh was done by first, taking
only the skin’s tetrahedrons and second, by moving only the nodes on the surface of
the head face looking to meet perfect coherence between the mesh and the 3D MRI
Immages, the final mesh is shown in Fig.4.3(d). Coronal, sagital and transversal cuts
of the mesh are shown in Fig. 4.4.
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4.2 The Sphere Carving algorithm
(a)
(b)
(c)
(d)
Figure 4.3: Sphere Carving Algorithm work flow: (a) Initial large sphere tetrahedral
mesh (radius = 15cm) , (b) and (c) Head tetrahedral mesh before refinement process, (d)
Head Tetrahedral mesh after refinement process
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4. HEAD-BRAIN MODELING
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4.4: Cross-cut views of the final head Tetrahedral mesh produced with the Sphere
Carving Algorithm: (a) Head coronal MRI view, (b) Head tetrahedral mesh coronal view,
(c) Head sagital MRI view, (d) Head tetrahedral mesh sagital view, (e) Head trasversal
MRI view, (f) Head tetrahedral mesh transversal view.
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4.3 A 3D Surface and Volumetric Mesh Generator for MATLAB/Octave:
iso2mesh
The algorithm is given below.
Sphere Carving Algorithm
Input (a sequence of volume images and a desired surface genus number),
Output (a tetrahedral mesh whose surface has the desired genus number).
1. Build a solid sphere tetrahedral mesh consisted of tetrahedra, such that the sphere
totally enclose the 3D data. Let the boundary of the solid sphere be S.
2. Put all the tetrahedra which share faces with S in a queue Q, the elements in Q is
sorted by the sharing face number, the first in the queue is the one which shares
the most number of faces with S.
3. Exit if Q is empty, otherwise pop out the first tetrahedron, t, from Q.
4. If t is not inside the object and the new surface after removing it still has genus
0, then remove the tetrahedron from the mesh and go to Step 2, otherwise go to
step 3.
Despite of the the fast approach and easy implementation of this method, the output
mesh after the refinement process presents some topological errors due to the complexity
of the model. Additionally, there is still the issue that the boundaries between tissues
are not well defined.
4.3
A 3D Surface and Volumetric Mesh Generator for
MATLAB/Octave: iso2mesh
The iso2mesh toolkit (28) is a very powerful and free 3D mesh generation tool for
creating finite-element surface or volumetric mesh. It incorporates a number of free
mesh processing utilities, and is capable of producing quality 3D tetrahedral mesh or
triangular surface directly from binary, segmented or grayscale medical images. The
structure of the software is highly modularized, optimized for processing efficiency. In
addition, iso2mesh was written in Matlab making it easy to use and very versatile.
Unlike, other commercial software packages with similar functionalities, iso2mesh is
accesible and scalable for general use.
In iso2mesh, the process of mesh generation can be roughly divided into two
subsequent steps. In the first step, triangular iso-surfaces with the specified density are
extracted from the input 3D image. In the second step, they fill tetrahedral elements
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4. HEAD-BRAIN MODELING
for the sub-volumes bounded by the previously extracted iso-surfaces. Holes or subregional labels can be supplied so that the resulting FEM mesh may carry hollow
structures or sub-domains that correspond to different tissue types. A more detailed
description of the mesh generation work-flow and the functuonalities are porvided in
(28).
One of the examples for the applications of iso2mesh given in (28) is a human
brain MRI image with segmented gray and white matters. In that case the segmentation was performed using FreeSurfer (61). The authors highlight the fact that a
thresholded brain image contains a large number of disconected regions within the
domain. So first, a hole-filing process has to be applied to produce single-connected
region. Then, the brain tissue segmentation has to be overlapped to produce a multiregional volume. And finally, by using the ”vol2mesh” utility, a volumetric mesh is
produced where the tetrahedra in each tissue type are tagged correspondingly. In this
dissertation, we followed the same procedure aforementioned. However, the segmentation was performed using some functions provided by the image toolkit of Matlab
(MathWorks, MA, USA) instead of using FreeSurfer. The reason of this choice was
because the segmented surfaces produced by FreeSurfer presents multiple issues:
• they are extremely dense: each surface alone carries over 350000 nodes which is
overkilling in many modeling tasks;
• they typically contain self-intersecting triangular elements; these must be fixed
before feeding to a mesh generator;
• you can only get gray-/white-matter surfaces, no CSF and skin/skull;
• the surfaces only cover the cortex domain, they do not include cerebellum, brain
stem and ventricles;
In addition, the computational time to produce the surfaces are extremely long which
is not desirable in our case. The next section explains how segmentation was performed using a preprocessing of the MRI immages using the Matlab image toolkit and
iso2mesh utilities.
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4.3 A 3D Surface and Volumetric Mesh Generator for MATLAB/Octave:
iso2mesh
4.3.1
Image Preprocessing
The preprocessing of the MRI images was done by following the same work-flow shown
in http://iso2mesh.sourceforge.net/cgi-bin/index.cgi?Doc/Workflow. Since the
thresholded brain MRI image contains a large number of disconnected regions. The
first step is to perform a hole-filling process for each single-connected region. Then,
the images obtained from the brain tissue segmentation have to be overlapped to produce a multi regional-volume. However, special attention should be given in this last
process to the fact that many times intersections appears between overlapped regions;
indeed, this is indesirable because after building the triangular iso-surfaces, iso2mesh
starts a tetrahedral filling process for the sub-volumes bounded by the iso-surfaces, but
since there are intersections, the space in between the two bourders are not suitable
to correctly create the tetrahedral mesh. Finally, once a topologically correct image is
produced, a volumetric mesh is generated by using ”vol2mesh” utility.
The hole-filling process consists in filling up completely the region enclosed by the
tissue bourder in order to connect the disconnected regions. The steps of the hole-filling
process are given below.
1. From the initial MRI immage, pick up the tissue to fill up and all the tissues
inside.
2. Use the directive ”fillholes3d” of iso2mesh to fill the biggest holes.
3. For each sagital, coronal and transversal slice of the 3D image apply ”imfill”
directive of Matlab with the option ’holes’ for further hole-filling.
4. For each sagital, coronal and transversal slice of the 3D image apply ”medfilt2”
directive of Matlab to remove the isolated islands, this process can be also be
performed by ”deislands3d” directive of iso2mesh.
5. Finally, a ’disk’ filter is created using ”fspecial” function of Matlab. The filter is
then applied to the 3D image yielding an smoother border.
The hole-filling process is performed on all the tissues. For our porpose we simplified
the model compared to the one produced in section 4.2. In this case, we only considered
cerebrospinal fluid (CSF), grey matter (GM), white matter (WM) , skull and skin parts.
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4. HEAD-BRAIN MODELING
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
Figure 4.5: Hole-filling process, sagital views: (a) and (b) Skin, (c) and (d) Skull, (e)
and (f) CSF, (g) and (h) Gray matter, (i) and (j) White matter
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4.3 A 3D Surface and Volumetric Mesh Generator for MATLAB/Octave:
iso2mesh
In Fig.4.5 it is shown the sagital view of the input and output images of the hole-filling
process.
Once the hole-filling process is done, the next step in the list is to overlap the
tissues to produce a multi-regional 3D volume. However, as it was mentioned before,
special attention should be given to possible intersections between consequtive tissues.
Therefore to solve this issue, one possible approach is to enlarge the outer tissues util the
inner tissues are completely enclosed. Nevertheless, this approach yields a deformation
of the anatomical shape of the head. Other possible approach, is to find the intersections
by means of logical operations and then cutting the intersecting region. This approach
was done for each pair of consequtive tissues, starting first with the skin and the skull
Fig.4.6(a), following by the skull and the CSF Fig.4.6(c), the CSF and the gray matter
Fig.4.6(e) and finally the gray matter and the white matter Fig.4.6(g). The cross-cut
views of the final multi-regional 3D volume are shown in Fig. 4.8(a),4.8(c) and 4.8(e).
At this point the 3D image is ready to be processed by the ”vol2mesh” directive of
iso2mesh. In ”vol2mesh”, parameter opt sets the maximum edge length of the surface
triangles, and maxvol sets the maximum tetrahedron volume. This parameters allow
us to control the surface and volume mesh density. The final mesh is shown in Fig.4.7.
Coronal, sagital and transversal cuts of the mesh are shown in Fig.4.8(b), 4.8(d) and
4.8(f) respectively.
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4. HEAD-BRAIN MODELING
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 4.6: Intersection-removing process, sagital views: (a) and (b) Skin-Skull before
and after process, (c) and (d) Skull-CSF before and after process, (e) and (f) CSF-GM
before and after process, (g) and (h) GM-WM before and after process
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4.3 A 3D Surface and Volumetric Mesh Generator for MATLAB/Octave:
iso2mesh
(a)
(b)
Figure 4.7: Final head-brain Tetrahedral mesh produced by iso2mesh
127
4. HEAD-BRAIN MODELING
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4.8: Cross-cut views of the final head-brain Tetrahedral mesh produced by
iso2mesh: (a) Head coronal MRI view, (b) Head tetrahedral mesh coronal view, (c) Head
sagital MRI view, (d) Head tetrahedral mesh sagital view, (e) Head trasversal MRI view,
(f) Head tetrahedral mesh transversal view.
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Chapter 5
Model Implementation and
Validation
In this chapter, we will discuss the implementation and validation of our Parallel fast
Method of Moments Model for dielectric bodies and Perfect Electric Conductors (PEC).
The modeling of dielectric and metal structures consists in a numerical implementation
of the MoM formulation for the Volume Surface Integral Equation (VSIE) derived in
chapter 2. Our goal is to use the full wave solutions obtained by the Parallel fast Method
of Moments in the determination of the electrical and magnetic properties of structures
with dielectric materials and metals. We will show particularly the case for modeling
a Magnetic Resonance Imaging (MRI) RF coil. Knowledge of the electromagnetic
properties of the RF coils can subsequently be used not only to influence their design for
improved and optimum performance but also to test new shimming methods (29, 62).
5.1
Software Implementation
The development of efficient software implementations for the numerical solutions of
MoM based integral equations is of paramount importance in the design and development of appropriate electromagnetic models. Several critical factors come together to
influence the software development process. However anong them, the most important
to reach a good performance are program structure, algorithmic implementation, and
the software development language of choice. The Fortran programming language (63)
was chosen as the software development language of choice. This is because the Fortran
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5. MODEL IMPLEMENTATION AND VALIDATION
programming language facilitates software portability across several different operating
systems as well as various hardware-dependent machine architectures. Aditionally, it
is widely used for the scientific community for numerical methods due to its versatility
with matrices .
With the choice of programming language and hardware platforms, the next step is
the implementation of an efficient program structure. For this reason, we have divided
our program structure into three software stages: Pre-processing stage, the Kerneprocesing stage and Post-processing stage.
In the Pre-processing stage, the configuration parameters (e.g simulation frequency,
losses in the free space, the excitation type, number of gaussian integration points for
test-source and far-near interactions, ACA precission ,etc...) are set it. The parameters
are readed from configuration files. In addition, the input mesh model are loaded
into the software at this point. This model approximates the conducting surface of
metals and the inhomogeneous dielectric material into surface triangles and volume
tetrahedra. These mesh models are obtained using any meshing software that is capable
of generating triangular and tetrahedral meshes of surface and volume discretization.
The one we have used is named GID pre and postprocessor (64), which is widely used
because of its user-friendly enviroment. GID allows users the generation of large meshes
for surface and volumes. The mesh is exported in a single .msh file which is readed and
loaded in this stage. Additionally, during this stage all the geometrical variables used to
define the volume and surface basis function are defined where there is included the near
and far interaction list of the groups resulting from the hirarchical subdivision in the
octree. The outcome of the Pre-processing stage will feed the kernel Processing stage.
A simplified diagram depicting the input and output relationship of the Pre-processing
stage is shown in Fig. 5.1.
The Kernel-processing stage forms the core fondation of the software implementation of the parallel fast MoM numerial model. It takes as an input, the outputs of the
Pre-processor stage and compute the set of Znear and Zf ar matrices. Finally, this stage
produces a numerical solution for the discretized VSIE. The solution of the resulting
discretized VSIE is obtained through a Conjugate Gradient Square (CGS) iterative
method (2). Hence, the output of the kernel Processor stage is essentially the surface
current density coefficients as defined in 2.41, as well as the unknown total volume
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5.1 Software Implementation
Figure 5.1: Input-Output relationship of the Pre-processing stage
current density coefficients defined in 2.67. The input-output relationship of this stage
is shown in Fig. 5.2.
Figure 5.2: Input-Output relationship of the Kernel-processing stage
The Post-processing stage uses the currents solution from the Kernel-processing
stage and the user-defined output file to determine the electromagnetic fields of interest. A simplified schematic block of the Post-Processing stage is shown in Fig. 5.3.
The required electromagnetic field are computed and saved into output files with an
appropriate format for visualization.
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5. MODEL IMPLEMENTATION AND VALIDATION
Figure 5.3: Input-Output relationship of the Post-processing stage
5.2
Simulation Models and Validation
In this section, we discuss the validation of our MoM implementation based on four
different simulation models. These simulation models have been developed explicitly to
validate the three different integral equations that form the foundation of our MoM implementation. The base equations are the Surface Integral Equation (SIE) the Volume
Integral Equation (VIE) and the hybrid Volume-Surface Integral Equation (VSIE) as
discussed in section 2.4.3. For the first simulation model, we consider the computation
of the electric near field radiation of a small dipole antenna. The second simulation
model is based on the classic electromagnetic problem of scattering of an incident electromagnetic wave by a dielectric sphere.
5.2.1
Small Dipole Antenna Model
We now consider the implementation and validation of a simulation model based on the
determination of the electric near field of a small dipole antenna. Our goal is to simulate
a small dipole antenna of dimension (l = λ/10) to obtain the current distribution on
the antenna and the electric and magnetic near field in the longitudinal planes, and
compare them with those obtained from a comercial software. The mesh model of the
small dipole is shown in Fig. 5.4. As seen, the lenght of the dipole is denote by l,
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5.2 Simulation Models and Validation
while the width of the strips is denoted by w. The source of excitation was chosen to
be a delta-gap voltage source of 1V with a corresponding frequency of 300MHz. The
solution for the surface currents on the thin strips of the antenna are then obtained
using our parallel fast MoM implementation. Since the problem setup does not involve
any inhomogeneous bodies, the surface current solutions were all obtained by solving
the SIE discretized the the MoM for the unknown current distributions. Once the
surface current solutions have been obtained using the MoM, the electric and magnetic
field is computed in the logitudinal plane at a distance of λ. For verification purposes,
the near field was then determined using a comercial software and later compared with
those obtained using the MoM formulation. The results of the comparison for the
electric and magnetic field are presented in Fig.5.5(a) and 5.5(b), respectively. There
is certainly good agreement between the fields obtained using the parallel fast MoM
solver and the commercial solver. This undoubtedly validates our implementation of
the parallel fast MoM using SIE. The magnitude of the surface current distribution
is shown in Fig.5.6. The current distribution agrees with the trinagular distribution
typical from small dipole antennas shown in (31).
5.2.2
Incident Plane Wave Dielectric Sphere Scattering Model
In this part, we discuss the implementation of an incident wave scattering model to
validate our VIE implementation. The incident wave scattering model completely embodies the classical problem of scattering of an incident wave by a dielectric sphere.
Analytical solutions to this classic problem are readily available in the form of the socalled Mie Series. For the implementation of the scattering model, we consider a sphere
of radius r = 0.1 m. The volumetric discretization of the sphere resulted in a volume
mesh with around 6000 tetrahedra. A mesh of the discretized spherical region is as
shown in Fig. 5.7. The sphere was assigned a dielectric constant of r = 43.8, while its
conductance was set it as σ = 0.0 S/m. The precission of the Adaptive cross Approximation (ACA) method for compression and the iterative solver precission was of 1e-6.
The incident electromagnetic wave was chosen to be a plane wave traveling along the
positive z-axis with an amplitude of 1.0V /m and a frequency of 300MHz. The polarization of the electric field component of the plane wave is along the positive x-axis.
The scattered electric field was obtained in the longitudinal plane (θ = [0◦ , 180◦ ] and
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5. MODEL IMPLEMENTATION AND VALIDATION
Figure 5.4: Small dipole antenna model
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5.2 Simulation Models and Validation
(a)
(b)
Figure 5.5: Near fields of the small dipole antenna at 300 MHz in the longitudinal plane:
(a) Electric field, (b) Magnetic field.
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5. MODEL IMPLEMENTATION AND VALIDATION
Figure 5.6: Current distribution for the small dipole antenna
φ = 90◦ ) at a distance of R = 100 m. Results from our parallel fast MoM implementation with the configuration given above, were compared against both, those obtained
using the Mie series implementation and a commercial software with the same input
configuration. The results of the comparison are as depicted in Fig. 5.8. We observe
that our numerical MoM solution is well behaved, and hence provide us with values of
electric and magnetic fields that agree very well with exact values.
5.2.3
Incident Plane Wave Coated Sphere Scattering Model
In this arrangement, we also consider the scattering of an incident wave by a dielectric
sphere. However, in this case, half of the sphere is coated with a perfect electric conductor. This model was developed to help validate our complete MoM implementation
of the hybrid MoM VSIE since it contains both surface and volume discretizations.
There are no analytical solutions available for this setup, so we will base our validation
only in the comparison with the commercial software results. The mesh model setup is
shown in Fig.5.9. The sphere is of radius r = 0.1 m with dielectric constant r = 43.8 as
in the incident wave dielectric sphere scattering model. The conductance of the sphere
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5.2 Simulation Models and Validation
Figure 5.7: Volumetric discretization of a dilectric sphere of radius r = 0.1 m and permittivity and conductivity of r = 43.8, σ = 0.0
is set as σ = 0.4377 S/m. In this case, the sphere was inserted in a lossy medium of
permittivity and conductivity of b = 1.0 and σb = 0.01 S/m, respectively. After discretization, the volume mesh yields around 6000 tetrahedra while the surface mesh gave
600 triangles for the PEC. The precission of the Adaptive cross Approximation (ACA)
method for compression and the iterative solver precission was of 1e-6. The incident
electromagnetic wave was chosen to be a plane wave traveling along the positive z-axis
with an amplitude of 1.0V /m and a frequency of 300MHz. The polarization of the
electric field component of the plane wave is along the positive x-axis. The scattered
electric field was obtained in the longitudinal plane (θ = [0◦ , 180◦ ] and φ = 90◦ ) at a
distance of R = 0.4 m. Results from our parallel fast MoM implementation with the
configuration given above, were compared against those obtained using the commercial
software with the same input configuration. The results of the comparison are as depicted in Fig. 5.10. Also for this case, we observe that our numerical MoM solutions
are well behaved.
5.2.4
Head-Brain Model irradiated by a Small Dipole Source
In this part, we consider the scattering of the head brain model discussed in section
4.3, when a small dipole source is irradiating. This model was developed to validate the
MoM VIE when a dipole source is present. There are no analytical solution available for
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5. MODEL IMPLEMENTATION AND VALIDATION
Figure 5.8: Scattered electric field of a dielectric sphere in the longitudinal plane to the
plane wave, results comparison with MIE series and a commercial solver
this setup, so we will base our validation on deductions about the wave behaviour in the
dielectric body. The mesh model is shown in Fig.5.11. For simplicity, the constitutive
parts considered in the head model were only the CSF (r = 69.3, σ = 2.32), skull
(r = 12.7, σ = 0.1153), skin (r = 43.1, σ = 0.78), white matter (r = 39.9, σ = 0.51)
and grey matter (r = 54.2, σ = 0.83). All the dielectric porperties were set at the
simulation frequency of 650 MHz. The head model was inserted in lyquid glycerinwater with permittivity and conductivity of b = 40 and σb = 0.01, respectively. The
dipole is distant 10 cm from the head and is fed with a current of 1A and is oriented
along the z-axis. The entire domain is discretized with around 53000 tetrahedra. This
kind of configuration is set on microwave imaging for brain stroke monitoring (65).
Where, it is necessary to improve the coupling between the probing wave and the
inspected tissues. The results obtained for the electric field in the head are shown in
Fig.5.12(a) and 5.12(b). From these results, we observe that the wave propagation of
the dipole source radiation penetrates the head and the field is more intense in the side
closer to the source and weaker in the opposite side. This make sense considering the
losses inside the dielectric body.
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5.2 Simulation Models and Validation
Figure 5.9: Volumetric discretization of a dielectric sphere half-coated of radius r = 0.1 m
and permittivity and conductivity of r = 43.8, σ = 0.4377 inserted in a lossy medium of
permittivity and conductivity of b = 1.0 and σb = 0.01 S/m, respectively.
Figure 5.10: Scattered electric field of a dielectric sphere half-coated in the longitudinal
plane to the plane wave, results comparison a commercial solver
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5. MODEL IMPLEMENTATION AND VALIDATION
Figure 5.11: Volumetric discretization of a head model (CSF, skull, skin, white matter,
grey matter) and a dipole source set beside.
(a)
(b)
Figure 5.12: Electric field in the head model made of 53K tetrahedra, when a dipole
source is irradiating from one side
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5.2 Simulation Models and Validation
5.2.5
Unloaded RF Shielded Birdcage Coil Model
The bird cage RF coil is the most common volume RF coil used in MRI imaging (20).
It finds particular use as an RF transmit coil where it establishes the requires uniform
magnetic field to excite the sample in the region of interest. In this section, we intent
to simulate a spetial configuration of bird cage named Inverted-Microstrip resonator
(66). Unlike the original model, we have introduced some modifications that give some
advantages and improve the performance of the resonator. The aim of the simulation
is to determine the magnetic field in the region of interest and compare its behaviour
with the one found in literature. A mesh model of the bird cage was generated as shown
in Fig. 5.13 with an outer diameter of 40 cm and a height of 21 cm. The separation
between the metallic shield and the microstrips is of 20 cm. The reason of this choice
is because by increasing the separation, the coupled dipole effect due to the closeness
between shield and microstrips is attenuated and hence there is more field penetration
in the region of interest. Additionally, our model do not have dielectric substrate
which made the model simplier. The mesh model has 16 rungs each of width 3 cm.
The bird cage model was discretized into a surface mesh of 2300 truangular elements.
The birdcage model was treated as an 80-port system where 64 ports corresponds to a
capacitor location, placed on the microstrips. And the other 16 of them acting as input
ports that drive the coil. Our MoM implementation is first used to solve the system at
300 MHz and determine its admittance matrix Y shown later in section 5.2.5.1. From
the Y-matrix and the values of the capacitors, we determine the values of voltage gaps
needed to achieve resonance at 300 MHz, this is shown later in section 5.2.5.2. The
simulated electric and magnetic field of the unloaded birdcage coil are shown in Fig.
5.15(a) and 5.15(b), respectively. From Fig.5.15(b), we observe than in the central
portion of the plane ( −10 ≤ x ≤ 10,−10 ≤ y ≤ 10 ) the gap between the maximum
and the minumim value is around 0.1x10−3 A/m so we can conclude that in this region
the magnetic field is highly uniform which is normally associated with the birdcage coil.
Additionally, the surface currents distribution on the birdcage is depicted in Fig.5.14,
we observe that the effect of the capacitances placed on the microstrips is to guarantee
uniformity of the current’s magnitude and since the amplitude of the magnetic field
along the microstrips resonators depends upon the magnitud of the currents flowing
through them. Therefore, we expect that the magnetic field pattern along the z-axis
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5. MODEL IMPLEMENTATION AND VALIDATION
Figure 5.13: The bird cage model: An Inverted-Microstrip resonator at 300 MHz for high
field MRI B0 = 7T
will be highly uniform as shown in Fig. 5.15(c). However, since all the feeding ports
are located at the top of the birdcage instead of both top and botton, there is a slightly
inhomogeneity of the ∼ 10% at the top, which is acceptable for further analysis.
5.2.5.1
RF Coil Equivalent Circuit Model
In order to determine an equivalent circut model of an RF coil from the solution of the
MoM matrix equation, we use the system arrrangement shown in Fig.5.16. The linear
equivalent circuit model will provide a complete network description of the RF coil.
Such a description can be obtained by computing the admittance matrix Y. if the RF
coil system. For the N −port RF coil system of Fig. 5.16 the admittance matrix Y is
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5.2 Simulation Models and Validation
Figure 5.14: Surface currents distribution in the bird cage model
defined as

 
I1
Y11
 I2   Y21
  
 ..  =  ..
 .   .
IN
YN 1
 
Y11 . . . Y1N
V1
 V2 
Y22 . . . Y2N 
 
..
..   .. 
..
.
.
.  . 
. . . . . . YN N
VN
I=Y∗V
(5.1)
(5.2)
Looking at the N port coil system of Fig.5.16, a voltage source V with source
impedance is applied across port i while all other ports, including port j are terminated
by short circuts. For this circuit arragement, the currents Iii and Iji are determined by
solving the underlying MoM matrix equations for the case of the loaded RF coil. Once
a solution for the port current Iii is obtained, the input admittance Yii at port i can
be determined as
Yii =
Iii
Iii
=
Vii
V
(5.3)
Similarly, the transmission coefficients Yji from port i to port j when all other ports
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5. MODEL IMPLEMENTATION AND VALIDATION
(a)
(b)
(c)
Figure 5.15: Simulation results of a unloaded birdcage coil: magnitude of (a) Electric
(V/m) and (b) magnetic field (A/m) in the x − y plane (z = 10.5 cm). (c) Magnetic field
pattern (A/m) in the z−plane
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5.2 Simulation Models and Validation
Figure 5.16: Determination of an equivalent circuit of RF coil from its MoM solution
all terminated by short circuits can subsequantly be found by doing
Yji =
Iji
Iji
=
Vii
V
(5.4)
Once, we have obtained the admittance matrix, we can find the impendance matrix
by doing
Z = Y−1
(5.5)
While the scattering matrix can be computed as
Sii =
Zii − Z0
Zii + Z0
Sji = Yji Z0 (1 + Sii )
(5.6)
(5.7)
With the aid of 5.3-5.7, we have been able to realize an equivalent circuit model in
terms of the Y, Z and S matrices of an RF coil. Aditionally, this matrices completely
characterizes the electrical circuit properties of the RF coil. This will allow us to
determine pertinent circuit parameters such as decoupling capacitors, as well as tuning
and matching capacitors required to resonate the RF coil at its Larmor resonance
frequency.
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5. MODEL IMPLEMENTATION AND VALIDATION
5.2.5.2
Tuning and Matching Requirement
The process of tuning and matching RF coils to their associated coupling circuits can
be achieved by using lumped element capacitors, the same that are used to keep the
current constant across the microstrip. All RF coils possess an inherent inductance
and capacitance due to the spatial distribution of their conductors around the region of
interest, and also because of the size of their current-carrying conductors. As such, RF
coils will be highly efficient when they are operated at their resonance frequencies. The
tuning process establishes a resonance condition in the RF coil at the desired Larmor
frequency which in our case is 300 MHz for High field MRI of 7T. The resonance phenomenon guarantees some form of rudimentary signal amplification as well as frequency
selectivity.
Let’s start recalling the equation for the impedance for narrow microstrip lines
(w/h ≤ 3.3) descibed in (67), which is:

119.9
h
Z0 = p
ln 4 +
w
2(r + 1)

2
h
16
+ 2
w
s
for w/h ≤ 3.3
(5.8)
where h is the thickness of the substrate, w is the width of the microstrip and r is
the effective dielectric constant which is this case is equal to 1, since there is not any
dielectric substrate.
The value of the inductance and the capacitance per unit length for the microstrip
line at a working frequency ω are given by
L0 =
C0 = −2
Z0
vp
cos( Kπa π) − 1
ω 2 L0
(5.9)
(5.10)
where vp stand for the phase velocity of the wave traveling along the line. As
in this case, when the substrate of the microstrip line is removed we have an airfilled line along which the wave will travel at c, the velocity of light in free space
(vp = c = 2.99793 × 108 m/s). Ka represents the phase coefficient normalized to
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5.2 Simulation Models and Validation
Figure 5.17: Angular velocity dispersion
π. Now we can rewrite equation 5.10 to find the expression for the angular velocity
dispersion, given by
ω
1
=
ω0
ω0
s
2(1 − cos( Kπa ∗ π))
L0 C0
(5.11)
where the resonance frequency ω0 is given by
ω0 = √
1
L0 C0
(5.12)
By doing ka = [0, 1], we can plot the normalized angular velocity ω/ω0 in terms
of Ka shown in Fig.5.17. From this figure we can have some insight of the values for
Ka and the working frequency. The critaria to pick a working point regards to the
fact that we need to obtain a constant current along the strip which guarantee a good
magnetic fild homogeneity along the z-axis inside the birdcage. One of the ways of
keep the current almost constant is to slow down the traveling wave through the strip.
Therefore, we need to pick one point over the resonant frequency which will allow us
to slow down the wave. In Fig.5.17 it is shown that the working point chosen is when
Ka /π = 0.4 and ω/ω0 = 1.2. From this values we can obtain L0 and C0 and then
compute new resonant frequency.
The tuning part is to find the capacitance per lenght to be placed on the microstrips
which keep the current constant and set the frequency to 300 MHz. Let us rewrite the
147
5. MODEL IMPLEMENTATION AND VALIDATION
Figure 5.18: Angular velocity dispersion with a tuning capacitor
equation 5.11 when the tuning capacitance Cr is placed
ω
1
=
ω0
ω0
s
2(1 − cos( Kπa ∗ π))
L0 (C0 + Cr )
(5.13)
By changing the value of Cr we obtain different curves shown in Fig. 5.18. From the
figure we can set again the working point near to the one chosen before (Ka /π = 0.44
and ω/ω0 = 1.2) as well the value for the tuning capacitor Cr = C0 /5 can be chosen.
Since Cr is the tuning capacitance per lenght, now we need to specify in how many
ports we will pllace it. In our birdcage model we have set this capacitance in 4 ports
distributed along of each rung. The tuning capacitor value to be placed in each port is
given by:
Cs = Cr
l
4
(5.14)
where l is the height of the birdcage which is the same lenght of the microstrips.
Once we know the values of the tuning capacitors for the lumped elements placed
in the ports on each rung of the microstrip, we can now compute the volatge gap values
to be placed in our parallel fast MoM solver. The voltage gap vector which contains
the values to be set in all the ports can be expressed as
V = −ZL ∗ I + Vs
148
(5.15)
5.2 Simulation Models and Validation
Combining equation 5.2 and 5.15, we obtain
−1
V = Y−1 ∗ = + Y ∗ ZL
∗ Y ∗ Vs
(5.16)
Equation 5.16 gives the voltage gap values to be set in the 80 ports of the birdcage
necessary to have a resonant frequency of ∼ 300MHz and to keep the currents constant
along the strips to guarantee a high homogeneity of the magnetic field in z-axis.
5.2.6
Loaded RF Shielded Birdcage Coil Model
In this section, we intent to simulate the birdcage coil model used in section 5.2.5 when
is loaded with a dielectric body. The aim of the simulation is to determine the magnetic
field in the region of interest, in this case inside the load and compare its behaviour
with the one found in literature. The mesh model shown in Fig.5.19 has 16 rungs as in
the one in section 5.2.5. The biological load is a simple sphere of diameter 20 cm with
electrical properties corresponding to white matter tissue at 300 MHz. The birdcage
model was dicretized into a surface mesh of around 23000 triangular elements, while
the spherical load was discretized into a volume mesh with around 18000 tetrahedra.
Our MoM implementation is first used to solve the system at 300 MHz and determine
the admittance matrix Y by means of the procedure explained in section 5.2.5.1. From
the Y-matrix and the values of the capacitors, we determine the values of voltage gaps
needed to achieve resonance at 300 MHz as shown in section 5.2.5.2. The simulated
magnetic field of the spherical load placed inside the birdcage coil is shown in Fig.5.20(a)
and 5.20(b). We observe that there is inhomogeneity of ∼ 50% which is common in
birdcage with heights over 20 cm (68). Since the inhomogeneity represents a critical
factor that affects the quality and resolution of the MRI, this can be corrected by
means of shimming techniques for enhancing the uniformity (29). We then obtained
the S-matrix parameters (S11,S21) shown in Fig. 5.21(a) and 5.21(b). Inmediatly upon
viewing, we observe that the S11 profile presents a resonance at 300 MHz which was
expected. In addition, the S21 profile describes that at the frequency of interest, there
is not coupling between ports 1 and 2 which are the case of closest ports.
149
5. MODEL IMPLEMENTATION AND VALIDATION
Figure 5.19: The bird cage model: An Inverted-Microstrip resonator at 300 MHz for high
field MRI B0 = 7T with a volume load mesh of dielectric material
(a)
(b)
Figure 5.20: Magnetic Field Intensity in the spherical biological load: (a) Transversal
plane, (b) longitudinal plane
150
5.2 Simulation Models and Validation
(a)
(b)
Figure 5.21: Simulated S parameter plots: (a) S11 (b) S21
5.2.7
RF Shielded Birdcage Coil Model Loaded with the Brain Model
In this section, we intent to simulate the birdcage coil model used in section 5.2.5 when
is loaded with a realistic brain model. The aim of the simulation is to determine the
electric and magnetic fields and the specific energy absortion rate (SAR) (W/Kg) for
determining RF exposure. The mesh model is shown in Fig. 5.22(a). The Brain model
employed is composed by CSF, Skin, white and grey matter. The frequency simulation
is 300 MHz for high field MRI at 7T. The birdcage model was dicretized into a surface
mesh of around 2300 triangular elements, while the brain model load was discretized
into a volume mesh with around 53000 tetrahedra. The tuning approach at 300 MHz
of the coil was the same used in the previous sections. The specific energy absortion
rate (SAR) in the biological load is computed and shown in figures 5.23(a),5.23(b)
and 5.23(c). We can observe at the three planes that the greatest absortion of energy
happens in the CSF tissue, this fact agrees with SAR results found in literature (29).
The Figures 5.24(a) and 5.25(a) report the magnetic and electric field computation
inside the brain, respectively. When the system coil is loaded with the brain asymmetric
model, the field loses its symmetry both in radial and asimuthal directions as expected.
5.2.8
Parallel Performance: Speedup and Efficiency
There are various methods that are used to measure the performance of a certain
parallel program. No single method is usually preferred over another. In this section,
151
5. MODEL IMPLEMENTATION AND VALIDATION
(a)
(b)
(c)
Figure 5.22: RF shielded Birdcage coil model loaded with the brain model
(a)
(b)
(c)
Figure 5.23: Specific energy absortion rate (SAR) of the load brain model: (a) Coronal
plane, (b) Sagital plane, (c) Transversal plane.
152
5.2 Simulation Models and Validation
(a)
(b)
(c)
Figure 5.24: Normalized Magnetic Field inside the load brain model: (a) Coronal plane,
(b) Sagital plane, (c) Transversal plane.
(a)
(b)
(c)
Figure 5.25: Normalized Electric Field inside the load brain model: (a) Coronal plane,
(b) Sagital plane, (c) Transversal plane.
153
5. MODEL IMPLEMENTATION AND VALIDATION
we intent to test the performance of the parallel code so that we need to determine the
speedup and the efficiency.
In the simplest of terms, the most obvious benefit of using a parallel computer is the
reduction in the running time of the code. Therefore, a straightforward measure of the
parallel performance would be the ratio of the execution time on a single processor (the
sequential version) to that on a multicomputer. This ratio is defined as the speedup
factor and is given as
S(n) =
Execution time usign one processor
ts
=
Execution time using N processors
tn
(5.17)
where ts is the execution time on a single processor and tn is the execution time on a
parallel computer. S(n) therefore describes the scalability of the system as the number of processors is increased. The ideal speedup is n when using n processors, i.e.
when the computations can be divided into equal duration processes with each process
running on one processor (with no communication overhead). Ironically, this is called
embarrassingly parallel computing.
In some cases, superlinear speedup (S(n) > n) may be encountered. Usually this is
caused by either using a suboptimal sequential algorithm or some unique specification
of the hardware architecture that favors the parallel computation. For example, one
common reason for superlinear speedup is the extra memory in the multiprocessor
system.
The speedup of any parallel computing environment obeys the Amdahl’s Law. Amdahl’s law states that if F is the fraction of a calculation that is sequential (i.e. cannot
benefit from parallelisation), and (1 − F ) is the fraction that can be parallelised, then
the maximum speedup that can be achieved by using N processors is
1
F + (1 − F )/N
(5.18)
In the limit, as N tends to infinity, the maximum speedup tends to 1/F . In practice,
price/performance ratio falls rapidly as N is increased once (1−F )/N is small compared
to F . As an example, if F is only 10%, the problem can be speed up by only a maximum
of a factor of 10, no matter how large the value of N used. For this reason, parallel
computing is only useful for either small numbers of processors, or problems with very
low values of F : so-called embarrassingly parallel problems. A great part of the craft
154
5.2 Simulation Models and Validation
of parallel programming consists of attempting to reduce F to the smallest possible
value.
In the other hand, the efficiency of a parallel system describes the fraction of the
time that is being used by the processors for a given computation. It is defined as
E(n) =
Execution time using one processor
ts
=
Execution time using N processors x N
N tn
(5.19)
which yields the following
E(n) =
S(n)
N
(5.20)
For example, if E = 50%, the processors are being used half of the time to perform
the actual computation. Program that scales linearly has parallel efficiency of 1. However, efficiencies over the 70% are acceptable and do not need of further optimizations
of the code.
Speedup ratio, S, and parallel efficiency, E, may be used:
1. to provide an estimate for how well a code speed up if it was parallelized.
2. to generate a plot of time vs. workers to understand the behavior of the parallelized code.
3. to see how the parallel efficiency tends toward the point of diminishing returns.
With this information, you would be able to determine, for a fixed problem size,
what is the optimal number of workers to use.
To test the parallel performance we carry on simulations for a metalic and dielectric
sphere (r = 40, σ = 0.6) model of radius r = 1m when is radiated by a plane wave
at a frequency of 10 MHz. The mesh discretization in both cases was continuously
increased for every simulation from 1 to 20 processors. Thus, in the case of the metalic
sphere the discretization was of 1000, 10000 and 100000 triangular elements while the
dielectric sphere considered discretizations of 6000, 9000 and 14000. In all the cases
the speedup and the efficiency were computed and are depicted from figure 5.26(a)
to 5.27(f) . The red line stands for the ideal performance and the blue one for the
actual performance. From the figures we can conclude that there is a good behaviour
of our code. The speedup in most of the cases is linear and in some cases superlinear.
155
5. MODEL IMPLEMENTATION AND VALIDATION
From the speedup results we can conclude aswell that the curve settles down around
10 and 15 processors. In other words, this means that our code is from 10% to 15%
sequential which is reasonable since the pre-processing stage has to be performed by
every processor involved in the computation. The efficiency so far is around 1 when
we are simulating with less than 10 processors and then it starts to decrease. This fact
gives us some idea of how many processors should we use for the computation, a good
tradeoff is to mantain the efficiency over 70 %. We can observe also that small models
allow us to simulate with more processors since the value of the efficiency keeps high
values. The reason of this is because there is not communication overhead, the opposite
occurs when the mesh model is bigger and hence there is necessary to communicate
more information among the processors for computing.
156
5.2 Simulation Models and Validation
(a)
(b)
(c)
(d)
(e)
(f)
Figure 5.26: Parallel performance test over 20 processor units: (a) Sppedup and (b)
efficiency results of a metallic sphere discretized with 1000 triangles, (c) Sppedup and (d)
efficiency results of a metallic sphere discretized with 10000 triangles, (e) Sppedup and (f)
efficiency results of a metallic sphere discretized with 100000 triangles
157
5. MODEL IMPLEMENTATION AND VALIDATION
(a)
(b)
(c)
(d)
(e)
(f)
Figure 5.27: Parallel performance test over 20 processor units: (a) Sppedup and (b)
efficiency results of a dielectric sphere discretized with 6000 tetrahedra, (c) Sppedup and
(d) efficiency results of a dielectric sphere discretized with 9000 tetrahedra, (e) Sppedup
and (f) efficiency results of a dielectric sphere discretized with 14000 tetrahedra
158
Chapter 6
Conclusion and Future Work
6.1
Summary
In this dissertation, we presented a novel approach to modeling dielectric materials and
metals using the Method of Moments combined with fast and parallel techniques to
enhance the possibilities to simulate bigger problems. The method combined a surface
integral formulation with a volume integral formulation in order to describe the interaction between dielectric materials and the electromagnetic fields produced by metals.
Our approach is based on formulating an electromagnetic scattering problem and providing a solution of the electric and magnetic field in terms of surface and volume
current densities. As such, two distinct basis function definitions were used in order to
describe the surface current density on metals and the volume current density inside the
dielectric materials. We discussed the development and implementation of a multilevel
kernel free method to reduce the overall complexity of the method to O(N LogN ). Additionally, we discussed the message passing programming paradigm used to implement
a parallel version of the multilevel kernel free method yielding a further reduction of the
complexity to O(N LogN/Nprocs ). The parallel fast MoM formulation for the hybrid
volume-surface integral equation was validated against several analytical models and
results obtained with commercial solvers. The results showed good agreement and help
establish the reliability of our method.
Among the several possibilities that our parallel fast MoM solver has. The main
application for our solver was the modeling and design of a 16-chanel birdcage coil
for clinical brain imaging at 7T. The coil featured one resonant mode at 300 MHz
159
6. CONCLUSION AND FUTURE WORK
necessary for high field magnetic resonance immaging (7T). For simulation puposes of
the loaded 16-chanel RF coil, we were able to produce a realistic head model considering
the main tissues inside. The resutls from this simulations gave us insights of field
distribution and the Specific Absortion Rate (SAR) of the tissues. In addition, we
were able to accurately determine its frequency response as well as optimum values of
tuning, matching capacitors.
6.2
Future Research
Based on issues encountered during the development and implementation of this work,
suggestions for further research should be focus towards:
• The construction of physical prototypes of the 16-channel birdcage model for
validation of our simulation results against real measurements to check if they
show a good agreement.
• Test the scalability of the parallel code for large structures in a massively parallel
computer system.
• Expand the application of the code and the brain model for simulations of biomedical applications in diagnosis and monitoring of deseases when there is involved
any kind of electromagnetic device, such is the case of Brain stroke detection.
160
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