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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~ | 99 (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 . 100 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 101 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, 102 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 103 (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 104 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 105 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 106 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 107 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- 108 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 109 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 ) 110 (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. 111 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. 112 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 . 113 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 115 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 116 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 117 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. 118 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 119 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. 120 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 121 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. 122 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. 123 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 124 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. 125 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 126 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. 128 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 129 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 130 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. 131 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, 132 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 133 5. MODEL IMPLEMENTATION AND VALIDATION Figure 5.4: Small dipole antenna model 134 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. 135 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 136 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 137 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. 138 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 139 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 140 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 141 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 142 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 143 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 144 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. 145 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 146 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. 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