Coil
Riccardo Lattanzi
Laurea, Electronic Engineering
University of Bologna (Italy), 2000
Submitted to the Department of Electrical Engineering and Computer Science in Partial Fulfillment of the Requirements for the Degree of
Master of Science at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2006
@ 2006 Massachusetts Institute of Technology
All rights reserved
Signature of Author.
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Department of Electrical Engineering and Computer Science
January 26,2006
Certified by..
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Daniel K.
Accepted by..
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Arthur C.
Chairman, Department Committee on Graduate Students
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
1
JUL 1 0 2006
1

Riccardo Lattanzi
Submitted to the Department of Electrical Engineering and Computer Science on January 26,2006 in Partial Fulfillment of the
Requirements for the Degree of
Master of Science
ABSTRACT
Parallel MRI techniques allow acceleration of MR imaging beyond traditional speed limits. In parallel MRI, arrays of radiofiequency (RF) detector coil arrays are used to perform some degree of spatial encoding which complements traditional encoding using magnetic field gradients. As the acceleration factor increases, coil design becomes critical to the overall image quality. The quality of a design is commonly judged on how it compares with other coil configurations. A procedure to evaluate the absolute performance of RF coil arrays is proposed. Electromagnetic calculations to compute the ultimate intrinsic signal-to-noise ratio (SNR) available for any physically realizable coil array are shown, and coil performance maps are generated based on the ratio of experimentally measured SNR to this ultimate intrinsic SNR
Thesis Supervisor: Daniel K. Sodickson
Title: Assistant Professor of Health Sciences and Technologies

If somebody had told me when I graduated from the University of Bologna that in a few years I was going to study at MIT, I would not have believed it. If today I am submitting this thesis, it is thanks to my wife Marta. Back in 2002, she convinced me to send an application and since then she has supported and motivated me, without ever complaining for the weekends we stayed home because I had to study for an exam. She is the greatest!
I want to thank also the rest of my family, because, although in another continent, they have been always at my side with their love and support. Besides, I mostly owe to them what I am today.
Daniel Sodickson guided me in the realm of parallel imaging, after showing me the wonders and mysteries of MRI. He has been much more than a supervisor, with his precious academic and personal advice.
I feel very fortunate for having joined the Laboratory for Biomedical Imaging Research. I have met such great people in this group! Aaron Grant taught me a lot of physics and his help was crucial to the outcome of my project. Michael Ohliger shared with me his deep understanding of SNR and showed me how to build coils. Charlie McKenzie has always had an answer to any question about MRI. Ernie Yeh, also an electrical engineer, has comforted me anytime I got lost in quantum physics discussions and has been a great lab buddy. Connie Cheng has given me programming advice and helped me with any issue regarding the internal network. Steve Suddarth taught me how to measure the noise figure of the MR receiving system. Shmuel Cohen helped me to uncover some secrets, at least for me, of Windows Server 2003. A special thank to Lois Gilden and Yemi Talabi, who are the unseen hands that keep the wheels of the lab in motion.
This research has been supported through the generosity of the Harvard-MIT Division of
Health Sciences and Technology and the National Institutes of Health.

Acknowledgements
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Tableofcontents
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ListofFigures
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1 Introduction
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1.1 General Overview
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1.2 MRI Signal Detection
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1.3 Spatial Encoding Using Magnetic Field Gradients
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1.4 Introduction to Parallel MRI
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1.4.1 Detector Coil Arrays
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1.4.2 General Formulation of Parallel MRI
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1.5 Analysis of Coil Performance in Parallel MRI
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2 Electrodynamic Determination of Ultimate Intrinsic SNR
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2.1 Signal
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2.2 Noise
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2.3 Intrinsic SNR
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2.4 Ultimate Intrinsic SNR
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3 Ultimate Intrinsic SNR for a Parallelepiped Geometry
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3.1 Parallelepiped Phantom Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Ultimate SNR Simulation Algorithm for a Parallelepiped Geometry . . . . . . . 36
3.3 Numerical Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Overall SNR behavior
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4 Ultimate Intrinsic SNR for a Cylindrical Geometry
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4.1 Cylindrical Phantom Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Ultimate SNR Simulation Algorithm for a Cylindrical Geometry . . . . . . . . . 44
4.3 Numerical Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4 Overall SNR behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5 Coil Performance Maps
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5.1 Experimental SNR Measurement
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5.2 Coil Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6 Discussion and Future Work
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Appendix A
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Dispersion Relation in a Source-Free Region
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Appendix B
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Derivation of an Expression for the Electromagnetic Field Inside a Cylinder Filled with a Uniform Medium
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References
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[l-11 On the left (a) a single loop detector coil and a plot of its sensitivity to the signal, simulated using the Biot-Savart law. The combined sensitivities of the elements of a coil array are shown on the right (b), together with a picture of a six-element array of surface detector coils.
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[l-21
Schematic diagram of a 4-element linear array of surface detector coils.
Adjacent coils are overlapped to force the mutual inductance to zero. All coils are connected to low input impedance preamplifiers to reduce coupling between coils that are not overlapped. A passive blocking circuit employing crossed diodes is placed on each coil to eliminate induced RF currents during transmit.
[l-31
[3-11
[3-21
Three-fold accelerated image acquisition of a head phantom using a 4-element cardiac array. The undersampled data from the four coils is used to reconstruct a full-FOV image. The "folding" effect is clearly visible in the aliased images at the top of the figure.
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Rectangular parallelepiped phantom with uniform electrical properties modeled for the simulation of the ultimate intrinsic SNR.
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Schematic diagram of the image plane and the FOV used to calculate the ultimate intrinsic SNR for the rectangular parallelepiped phantom.
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[3-31 Convergence of the ultimate SNR calculation for a reduction factor of one, as a function of the dimension of the plane wave basis set. The plot in the top left quadrant

shows the convergence at the center of a rectangular parallelepiped object, whereas the other plots refer to positions selected on an arbitrary coronal plane (see Fig. [3-21) at various distances (d = 5 cm, 9.5 cm, 14 cm) from the center of the object. The values of the SNR in each plot are normalized to the respective maxima.
. . . . . . . . . . . . . . . 39
Convergence of the ultimate SNR calculations as a function of pixel location for an arbitrary coronal slice passing through the center of a parallelepiped object. Using 1000 plane waves mode and 200 digits of numerical precision, only for about 113 of the FOV is the simulated value of the optimum SNR accurate.
. . . . . . . . . . . . . . . . . . . . . .
. 4 0
Ultimate intrinsic SNR computed at each pixel of a 120x48 coronal slice, passing through the center of the rectangular parallelepiped object. Values are normalized to the largest number. SNR is smaller along the z-direction because of the effect of the main magneticfield
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41
Normalized spatial variation of the ultimate intrinsic SNR along the y-axis (top) and the z-axis (bottom) of the coronal plane. In both cases values are larger near the edges of the object and become smaller when approaching the center. . . . . . . . . . . . . . . . 42
Cylindrical phantom with uniform electrical properties modeled for the simulation of the ultimate intrinsic SNR.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Schematic diagram of the image plane and the FOV used to calculate the ultimate intrinsic SNR for the cylindrical phantom.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Convergence behavior of the ultimate SNR calculation at a position in the center of the cylindrical object, for different reduction factors. Each plot shows the normalized ultimate SNR for as a function of the number of modes in the basis function (see Eq.
[2-101 and Eq. [4-61)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Convergence behavior of the ultimate SNR calculation at a position on the surface of the cylindrical object, for different reduction factors. Each plot shows the normalized ultimate SNR for as a function of the number of modes in the basis function (see Eq.
[2-101 and Eq. [4-61).
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Ultimate intrinsic SNR computed at each pixel of a 108x48 coronal slice, passing through the center of the cylindrical object. Values are normalized to the largest number
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50

[4-61 Behavior of the ultimate intrinsic SNR as the reduction factor along the z direction is increased, for a point in the center (top) and on the surface (bottom) of the cylindrical object. Values are normalized with respect to the unaccelerated case. Note that the intrinsic SNR values plotted here do not include the inverse proportionality to the square root of reduction factor, which is treated as an extrinsic sequence- dependent effect.
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[5-11 Eight-element linear array of surface coils used to acquire images of the cylindrical phantom
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[5-21 An MR image displaying the SNR value at each pixel was computed from a set of replicated coronal slices, acquired along the same plane used in the simulation of the ultimate intrinsic SNR. The image was then cropped to exactly match its pixels to the simulated positions. The SNR values are low because of the choice of using a small flip angle for the acquisitions.
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[5-31 Coil performance maps showing the fraction of ultimate achievable SNR for unaccelerated (top) and 2-fold accelerated (bottom) studies with the eight-element coil array atop the cylindrical phantom.
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1.1 General Overview
Magnetic Resonance Imaging (MRI) [l, 21 has become a widely accepted diagnostic tool for its ability to produce high quality internal anatomical images. In a magnetic resonance experiment, the signal, which relates to the physical and biochemical properties of the sample, is detected in the form of a radiofrequency (RF) voltage induced in a detector coil in response to the application of alternate magnetic fields to the object of interest. In order to reconstruct an image, the signal is spatially encoded by imposing magnetic field gradients during the acquisition.
The downside of such a method is that the gradients have to be re-applied many times to encode the entire field of view (FOV) and thus the acquisition time becomes longer as the resolution increases. Long acquisitions, besides being unpleasant for the patient and increasing the cost of the examination, can promote motion artifacts, which may limit the diagnostic value of some examinations. In recent years scientific advances in imaging hardware, as well as the development of fast imaging sequences [3, 4, 5, 61, have contributed to substantial increases in the speed of MRI scans. However, the nature of the
MR signal, together with physical and safety issues regarding the rate of application of radio frequency pulses and the switching rate of the magnetic field gradients used in the acquisition, imposes complex constraints on the maximum achievable speed.

In the late nineties these limits were partially circumvented as parallel magnetic resonance imaging was successfully applied to clinical scanning protocols [7, 81. Parallel
MRI is a revolutionary technique that uses arrays of RF coils to increase imaging speed, without increasing gradient switching rate or RF power deposition.
Detector arrays are commonly used in MM because they provide images with a high signal-to-noise ratio (SNR) across a large FOV [9]. The sensitivity of a detector coil to signal is high in the region directly below it and falls off with distance from the center of such area (Fig. 1-la). Thus, in the case of a coil array, each detector captures a strong signal from a local region and the contribution, to both signal and noise, from the rest of the sample is negligible (Fig. 1-lb). That clearly shows the SNR advantage in using an array of coil to cover an extended region, rather than a large single coil with the same total area.
Parallel MRI methods similarly take advantage of the local nature of the reception pattern of each array element, but exploit it to extract spatial information about the detected magnetization. Complementing such additional data with the spatial information obtained from the externally applied magnetic field gradients, images can be reconstructed from undersampled datasets. The omission of phase-encoding gradient steps enables scanning of the same FOV in less time.

Figure 1-1. On the left (a) a single loop detector coil and a plot of its sensitivity to the signal, simulated using the Biot-Savart law. The combined sensitivities of the elements of a coil array are shown on the right (b), together with a picture of a six-element array of surface detector coils.
Although the concept of parallel acquisition in MRI was introduced relatively early after the development of MR multicoil arrays [ 10, 1 1, 12, 131, it wasn't until 1997 that parallel
MRI was successfully applied in vivo with the simultaneous acquisition of spatial harmonics (SMASH) technique [7]. SMASH emulates the effect of the omitted phase encoding gradients, shaping linear combinations of coil sensitivity functions into spatial harmonics. The method was made more robust and generally applicable in successive improved implementations [14, 151 and its original version is no longer used. In 1999, the sensitivity encoding (SENSE) technique for parallel MRI was introduced and also employed to acquire accelerated in vivo images [8]. SENSE is based on a different approach, which was first proposed in the early nineties [12], that involves using measured information about coil sensitivities to reconstruct an image from a set of aliased images, acquired with a reduced FOV.
Numerous other approaches have been proposed since the introduction of SMASH and
SENSE [16, 17, 18, 19, 20, 21, 22, 23, 241 and parallel MRI is now a well-established

technique in many areas of clinical medicine. Among those applications that strongly benefit from accelerated image acquisition, are dynamic contrast enhanced angiography
[25, 26, 27, 28, 291, real-time cardiac imaging [30], large volumetric abdominal scans
[31, 321.
Despite the many advantages of parallel imaging, there are some downsides, including both practical issues (such as hardware compatibility and increasing system complexity) and intrinsic tradeoffs associated with fast scanning. In fact, the penalty for using fewer k-space points in the reconstruction is a reduced time averaging of noise, which leads to a loss in the SNR of the reconstructed image. Furthermore, parallel MRI reconstructions require non-orthogonal transformations, which lead to spatially dependent noise amplifications. This additional source of SNR loss has been referred to as the geometry factor (g) and has been used to assess the performance of coil arrays for parallel MRI, since it depends on the geometry of the coil sensitivity functions [8].
That dependency suggests that the SNR of the reconstructed image can be improved by optimizing the design of the receiver coil array. Nowadays it has in fact become common practice to include simulations of g in the design process. First, sensitivities are calculated for an initial coil array geometry and next g is simulated for given sample shapes at different acceleration rates. Design parameters, such as spacing between array elements, dimensions of each coil, etc, are then adjusted until an acceptable SNR is reached. Although effective in improving existing array configurations, such a method doesn't help to discern innovative designs and, furthermore, it doesn't say how close to the optimum SNR is the resulting enhanced coil performance.
The empirical optimization approach also raises the question of whether we can in principle model any desired sensitivity pattern by increasing the number and decreasing the size of coil array elements, in order to obtain an adequate g at any acceleration rate.
An affirmative answer would mean that in optimizing an array of coils for parallel MRI we are only limited by the size of electronic components and by the capacity to decouple an ever larger number of elements. However, recent studies have showed that optimized coil designs yield g factors that are favorable up to acceleration factors of 3-4 in one dimension, but dramatically worsen beyond that threshold [33, 34, 35, 361. This evidence

suggests that design imperfection is not the only constraint on the speed of parallel MRI, and that there may be other fundamental limitations.
Inherent limits of parallel MRI performance were the subject of two recent articles, in which the ultimate intrinsic SNR was calculated for objects with uniform electrical properties, and its dependence on acceleration rate, main field strength and position in the sample was analyzed [37, 381. Both investigations proved that the largest achievable undersampling in parallel MRI is intrinsically limited by electrodynamics, as the ultimate
SNR performance exponentially decays after a threshold reduction factor, which varies slightly depending on the main field strength, the electrical properties and the size of the sample.
The existence of an upper bound, independent of coil array design, on the performance of parallel MRI, may be very useful for coil optimization. The comparison of the ultimate
SNR with the SNR of a coil array under development can tell whether there is room for further improvement and can help to choose the best design for given field strengths and sample properties. Although extremely valuable to theoretically inspire future coil designs, the results of the two papers cited above cannot be directly applied to assess the performance of actual coil array configurations. Ocali et al. showed a method to compare the experimentally measured SNR of a coil with ultimate SNR values in the case of standard MM [39]. At the present time, there are no published studies attempting to extend such work to parallel MM.
This thesis present a method for constructing coil performance maps for sequential and parallel MRI applications. For a given phantom geometry the ultimate SNR is computed at each point of an arbitrary image plane and then these positions are mapped to the pixels locations on an MRI image acquired along the same plane. Adjusting for multiplicative factors related to the particular MRI scanning sequence, it is possible to compare the SNR of each pixel of the acquired image to the respective ultimate value and use the results as a measure of the performance of the specific coil array used. In the studies reported here, this whole procedure was repeated for different acceleration factors and tested on a particular coil array configuration.

The rest of this chapter illustrates the general principles of spatial encoding in parallel
MRI. Section 1.2 is a brief review of the physics underlying MR signal generation and detection, whereas Section 1.3 describes how spatial information is encoded by means of magnetic field gradients. Section 1.4 explains how additional spatial encoding can be obtained using coil arrays and introduces to the formalism of parallel MRI. In the section that follows, an overview of the current method to evaluate coil array performance is presented.
In Chapter 2, an expression for the SNR of a parallel MRI experiment is determined using electrodynamics. The calculations involved in the simulation of the ultimate SNR and the generation of coil performance maps are described in detail. In Chapter 3 and
Chapter 4 the theory is applied to assess the performance of a particular coil array, using a rectangular parallelepiped phantom and a cylindrical phantom, respectively. The final chapter discusses the results of this work and its implications for coil array design.
1.2 MRI Signal Detection
In order to understand how an image is formed in MRI, it is necessary to start with a brief introduction to nuclear magnetization. Only the basic principles will be described in this section
- a more rigorous description of the physics of MRI can be found in reference
1401
The term NMR describes a resonance phenomenon, &st observed in 1946 [4l, 421, which involves magnetic fields and nuclei, more specifically nuclei with an odd number of protons andlor odd number of neutrons. Atoms with this characteristic possess an angular momentum, or spin. In the absence of an external magnetic field, the spins inside any object are randomly oriented and the net macroscopic magnetic moment is zero. In the presence of an external magnetic field, the magnetic moment vectors of the protons tend to align in the direction of the magnetic field and create a net magnetization.
Furthermore, the nuclear spins have a resonance at a well-defined frequency called the
Lannor frequency. The Lannor frequency
(o) y Bo) Bo of the magnetic field and to a constant, the gyromagnetic ratio y, which is unique for each type of atom and describes the ratio of its magnetic moment to its angular momentum.

Placing an object inside a static magnetic field Bog, will induce a net magnetization vector M that points in the same direction 2 of the main field (i.e. longitudinal direction) and that at thermal equilibrium will be equal to: where A is the Planck's constant divided by 2n, sample, I is the nuclear spin quantum number, ka is the Boltzmann's constant and TS the absolute temperature of the sample.
If the equilibrium of the proton system is perturbed by an externally applied magnetic field Bi, oscillating at the same frequency of the spins, the magnetization vector is rotated by a certain angle and starts precessing with angular frequency about the main magnetic field Bo. After the external excitation is removed, the system returns to its thermal equilibrium state. This process is characterized by a precession of M about Bo (free precession), the recovery of the longitudinal component of M (longitudinal relaxation) and the decay of the transverse component of M due to dephasing among the ensemble of precessing spins (transverse relaxation).
The time-varying magnetization resulting from the just described NMR phenomenon can be detected using the well-known Faraday's law of induction. Any loop of wires resonating at the radio frequency (RF) of the precessing magnetization can be used as a receiver coil. Its complex-valued detection sensitivity C(r) can be determined recalling the principle of reciprocity [43, 441. According to Faraday's law M is responsible for a magnetic flux 3> through the surface of the detector coil that induce an electromotive force given by:
In the case of biological specimens, the MR system is commonly tuned to the Larrnor frequency of the hydrogen nucleus (H), with a single proton, as it is the most abundant nuclear species in biological tissue and gives rise to the largest signals.

The sample is composed of many atoms and the nucleus of each atom can be seen as a small oscillator inducing a RF signal in the receiver coil. The signal described by Eq.
[1.2] is then a sum of local signal from all these components. The aim of MRI is to spatially differentiate the various contributions to the signal and map the distribution of their amplitudes, in order to generate an image. In conventional MRI, this is done by superimposing magnetic field gradients on the main magnetic field.
1.3 Spatial Encoding Using Magnetic Field Gradients
If a linear magnetic field gradient G with a general orientation r is applied together with
Bo, the Lamor frequency at position r becomes:
As a consequence, the net magnetization density acquires a spatially varying phase and can be expressed as:
-iy(Gr)t
M'(r,t)
=
M(r,t)e
[ la4]
Introducing the wave vector k
= t (y123i) G and using the symbol S for the voltage signal,
Eq. [1.2] can be rewritten as:
The expression shows that, when linear gradients are used, the signal corresponds to the
Fourier transform of the magnetization density. The spin density distribution is simply obtained by performing the inverse Fourier transform of the signal and it is then used to generate an image.
For Cartesian sampling schemes, linear combinations of magnetic field gradients can be applied to acquire images in any specific image plane. The strategy is the same for any arbitrary plane. From Eq. [1.5] it is possible to map each gradient encoding step to a point in the discrete space defined by k. Points of k-space along a single axis (i.e. frequency- encoding axis) are acquired applying gradients at the same time the signal is being received (frequency-encoding readout). In order to change the k-space coordinates in the axes perpendicular to the frequency-encoding axis (i.e. phase-encoding axes), magnetic

field gradients are applied before the frequency-encoding readout. For each increment in the phase-encoding direction, a complete frequency-encoding readout is necessary.
Volumetric (3D) acquisitions use two phase-encoding directions, whereas 2D acquisitions use only one. Many MR imaging protocols sample k-space following non-
Cartesian schemes, but a description of any of these approaches is not relevant for understanding the rest of this thesis, as the whole work employs only Cartesian sampling strategies.
1.4 Introduction to Parallel MRI
The description above highlights the advantages associated with the use of magnetic field gradients for encoding spatial information, but also the intrinsic limits of such a method.
If, on one hand, the technique enables one to use any arbitrary image plane or slice thickness and it is suitable for any kind of application, on the other hand it can be very slow, especially when high-resolution images are needed. In fact, for each k-space position in the phase-encoding direction the complete gradient readout in the frequency- encoding direction must be re-applied.
One approach to this problem would be to apply the magnetic field gradients at a higher rate and strength to speed up the acquisitions. This approach results in reduced SNR, but
SNR may in principle be increased by moving to higher-field MR systems. However, the switching-rate of the magnetic field gradients is limited by physiological constraints, as rapidly varying magnetic fields have the potential to induce currents in the human body.
The use of high-field MR scanners, meanwhile, raises issue of human tissue heating.
The limits on imaging speed were in part overcome only after the development in the late nineties of parallel MRI, a new technique for spatial encoding, which enables to generate an image combining partial information acquired in parallel by the different elements of an array of detector coils. In order to understand the general principles of parallel MRI, a brief introduction to coil arrays is necessary.

1.4.1 Detector Coil Arrays
The first method for generating an image from simultaneously acquired data from multiple RF detector coils was described in 1990 [9]. number of overlapping coils, closely positioned and each associated with an independent preamplifier and receiver chain (Fig. 1-2). Each element receives a separate signal for every k-space position and further information is needed to combine the various portions into a single signal. The sensitivity profile, which accounts for the spatial inhomogeneity of the coil response to the precessing spins, can be measured for each coil of the array and used to weight the individual signal contributions. For example, the total signal for each k-space point can be obtained as the sum of the signals from the single coils, each multiplied by the complex conjugate of the value of the respective sensitivity at the same position.
Figure 1-2. Schematic diagram of a 4-element linear array of surface detector coils. Adjacent coils are overlapped to force the mutual inductance to zero. All coils are connected to low input impedance preamplifiers to reduce coupling between coils that are not overlapped. A passive blocking circuit employing crossed diodes is placed on each coil to eliminate induced RF currents during transmit.

Coil arrays were initially used to acquire large FOVs as they offer the SNR and the resolution of a small coil over an extended region. The advantages carried by this development were immediately evident in the field of body imaging, for which a large coil was previously used to cover the region of interest, resulting in non-uniform signal, low SNR and long acquisitions. However, it was with the introduction of parallel MRI that coil arrays reached their greatest popularity.
The basic idea of parallel MM is to exploit the different reception patterns of the array elements to perform spatial encoding. Using the spatial information provided by coil sensitivities to substitute for time-consuming gradient encoding steps enables one to accelerate the image acquisition process. The next section describes the general principles of parallel MRI.
1.4.2 General Formulation of Parallel MRI
The discretized MR signal received by the 1-th element of the detector coil array at k- space point k_ , pixel- j where Cl(rj) is the sensitivity function for coil 1, M(rj) is the transverse magnetization and
nrfkm) is the time-dependent Gaussian white noise. If the coil sensitivities and the gradient-induced modulation are combined in a single encoding matrix B , can be re-written in the simplified matrix notation:
S = B M + n 1 -71
The encoding matrix has a number of rows equal to the number of array elements times the total acquired k-space points and a number of columns equal to the pixels in the image:
In order to reconstruct an image is sufficient to find the matrix inverse of B. Let us say there are N pixels in the image and L elements in the coil array. Assuming the coil sensitivities are known (e.g. from a calibration measurement), if a full set of N k-space

points is acquired, then Eq. [1.7] is an over-determined system of L*N equations and N unknowns. It is therefore possible to speed up the acquisition by reducing the sampled positions in k-space by a factor R, so long as L*N/R 3 N, i.e. the matrix B is still invertible.
Typically R is smaller than the number of coils L andso there are different choices for the inverse matrix B"". It has been shown that the solution that leads to the lowest noise amplification in the reconstructed image is the modified Moore-Penrose pseudoinverse
P I
: where properties of the noise received by the coils and B* is the conjugate transpose of B.
Undersampling by a factor R is equivalent to reducing the FOV by the same factor. That causes aliasing, which results in a fold-over effect in the image. Practically, parallel MRI enables reconstruction of a full-FOV image by combining the aliased images detected by each element of the coil array. The value of each pixel in the aliased images corresponds to the superposition of the signals from R aliased positions. Applying the matrix in Eq.
[1.8] to the set of folded images, the R signal contributions are separated so that each pixel of the aliased images resolves into R pixel values that contribute to form the full-
FOV image (Fig. 1-3).

Coil I Coil 2 Coil 3 Coil 4
Reconstructed Image
Figure 1-3. Three-fold accelerated image acquisition of a head phantom using a 4-element cardiac array. The undersampled data from the four coils is used to reconstruct a full-FOV image. The "folding" effect is clearly visible in the aliased images at the top of the figure.
The general formulation presented in this section represents a brief introduction to the concepts of parallel MRI, showing with a simple formalism how the sensitivity functions of the coil array elements are exploited to achieve additional spatial encoding. There are many other parallel MRI techniques, which originated from the same basic idea, but a description of all of them would deviate from the purpose of this thesis.
1.5 Analysis of Coil Performance in Parallel MRI
As mentioned earlier, the reduction in scan time provided by parallel MRI comes at a cost in terms of the SNR of the final image. The SNR of the images reconstructed from an undersampled dataset can therefore be used to assess the performance of a particular coil array.
The first cause of SNR loss associated with parallel MRI is intrinsically related to the mathematical operators involved in the reconstruction. As an effect of using the Fourier transform, the noise associated with each acquired k-space line is averaged and so, in the

case of a fully sampled dataset, each pixel reconstructed will have a noise variance per unit signal equal to: where
2 o is the noise received by the coil from each k-space point and N is the total number of acquired k-space points. In parallel MRI, in order to accelerate image acquisition by a factor of R, the number of acquired k-space points is reduced by an equal factor. As a consequence, from Eq [1.8], each pixel suffers a proportional loss in SNR equal to the square root of R.
The second source of noise derives from the employment of non-orthogonal transformations in parallel MRI reconstructions. The resulting reconstruction matrix is non-unitary and, as a consequence, the noise level varies from pixel to pixel and there is noise correlation between pixels.
The previous observations can be expressed in mathematical language and used to derive an expression to quantify the SNR performance of a coil array in a parallel MRI experiment. The derivation is particularly intuitive for the SENSE approach with a uniform Cartesian sampling pattern, and for this reason the next paragraphs assume a
Cartesian SENSE reconstruction [8].
The noise power associated with each pixel can be expressed in terms of the encoding matrix B as
It can be shown that the noise level [8] at each pixel in the case of an undersampling factor of R is related to the correspondent noise level in the fully sampled case by the simple equation: where the subscripts R and full indicate respectively the undersampled and the fully sampled case. Eq [ l . 101 can be manipulated to write a compact expression for the SNR of the accelerated image in terms of the SNR of the correspondent unaccelerated case:

g is the geometry factor, or "g-factor", which describes for each pixel the spatial amplification of noise introduced by parallel MRI:
The g-factor is by definition always greater than or equal to one and it depends on the shape of the coil sensitivity functions, as well as on the pattern of acquisition in k-space.
Because of its dependence on the geometry of the coil array, the g-factor has been used to evaluate the performance of coil arrays for parallel MRI.
Given the geometry of a coil array and the k-space acquisition pattern, the g-factor can be computed using the laws of electrodynamics. This type of simulation is extensively used in coil array design to optimize the arrangement of the detectors in an array. Although such an optimization approach is very useful to improve existing designs, it is important to understand that this method provides a relative measure of the performance of the coil array and therefore doesn't tell if the chosen coil array configuration is optimal for a particular task. The next chapter shows how to derive an expression for ultimate intrinsic
SNR, which is independent from coil array design and represents the best possible SNR achievable for a particular parallel imaging experiment. The ultimate SNR can be used as a reference against which to test any coil array configuration, providing an absolute measure of the performance.

2.1 Signal
As previously mentioned, the precessing magnetization creates a time-varying magnetic flux that intersects the surface of the MR detector coils, inducing a voltage in the circuit.
The total signal received by a coil is given by the integral, over the whole sample, of the time-derivative of the component of the magnetization orthogonal to the surface of the coil. Such an integral is not easy to perform, but the computation can be simplified by utilizing the principle of reciprocity [43, 441, which states that the electromotive force induced around a closed loop by a nuclear magnetization, present somewhere outside the detector, is proportional to the magnetic field that a unit current flowing through the loop would generate at the same position.
With this simplification, the signal voltage per unit volume, assuming the main magnetic field is in the z direction, can be written as [37, 391:
V'
=
M ~ ~ L I H , ; +
= & M ~ B 6 ~ ~ ~ ~ ~ ~ ~ where <o is the Lannor frequency, & is nuclear magnetization at equilibrium, p is the magnetic permeability of the sample, Hx , Hy field transmitted by the unit current and 6- (2
- i
?)I&
RF magnetic polarized unit vector.

The net sensitivity, which is the weighted combination of the coil sensitivities of an array of detectors, is given by [37] :
2.2 Noise
There are various mechanisms causing power loss in an MRI system. These dissipation factors include ohmic losses in the conductor, radiation losses, as well as thermal noise coming directly from the sample. In a well-designed system, the only component that limits SNR performance is the dissipation caused by thermal agitation of particles in the sample, as all the other sources of noise can be diminished by improving the technology.
Fluctuations of charged molecules in the sample induce voltage fluctuations in the receiver coil. Since the sample and the coil are in thermal equilibrium, for the fluctuation- dissipation theorem, there must be a correspondent dissipation, which is described as resistive power loss in the coil circuit. For this reason, from here on, "noise" will be taken to refer to the power loss associated with the sample.
Since the sample and the coil are in equilibrium, thermal fluctuations in the sample induce a noise voltage in the detector coil, which has the characteristics of white noise and whose power per unit bandwidth is [45]: where ke is the Boltzmann's constant, T is the temperature of the sample and Q is the series resistance that represents the sample in the equivalent circuit of the coil. The noise equivalent resistance can be computed from the formula of ohmic losses in a coil driven by a unit current [46] :
Q
=
$ o(r)I'~(r)~d'r sample where o is the conductivity of the sample and E is the RF electric field generated by a unit current flowing in the coil.

2.3 Intrinsic SNR
The SNR of an MR experiment is the ratio between the signal voltage amplitude and the root-mean-square noise voltage at the output of the detector. Using Eq. [2.1], Eq. [2.3] and Eq. [2.4] it is possible to derive a general expression for the intrinsic SNR per unit volume of sample:
V sample
Eq. [2.5] represents the intrinsic SNR because it is independent from image acquisition parameters and it accounts only for the noise intrinsically associated with the sample. The
SNR actually observed in the image can be calculated adding the factors associated with the particular pulse sequence: where Vsnamle number of phase encoding steps, NEX is the number of signal averages, 0 is the flip angle, NF is the noise figure of the preamplifiers connected to the coils and Af is the receiver bandwidth.
2.4 Ultimate Intrinsic SNR
In Eq. [2.5] the only parameters depending on the coil are the net sensitivity and the noise equivalent resistance. Since we are interested ultimately in the ratio of signal to noise, we are free to scale one of these quantities by an arbitrary amount at any given location, so long as we adjust the other accordingly. For convenience we can specify that the net signal sensitivity is always equal to one at the reconstructed point. This allows us to eliminate the net sensitivity from the numerator of the SNR expression, reducing the problem of finding the ultimate intrinsic SNR to the minimization of the noise loss term for fixed unit sensitivity.
It is not possible to simply assume that the detector has sensitivity to signal but no sensitivity to the noise coming from the sample -

be infinite - since the physical electromagnetic (EM) field originating in the sample is coupled to the coil sensitivities by the laws of electrodynamics and the principle of reciprocity used to derive the SNR expression (see Eq. [2.2]). The minimization problem is constrained by the fact that both the EM field and the coil sensitivities must obey
Maxwell's equations for a source-free region [46]: where p and e are the permeability and the dielectric constant of the sample (assumed constant for our purposes here). For the frequencies used in MRI, the permeability can be assumed equal to that of free space, p,. Assuming an harmonic time dependence for the
EM fields: and using the fact that the current density J in a homogeneous sample having conductivity o is equal to oE, it is possible to derive the following wave equation from the set of Eq. [2.7]:
The magnetic field can be expressed as a linear combination of a basis set of waves that span the whole solution space of the wave equation:
The solution is also characterized by a set of electric fields:
Eq. [2.10] specifies a complete basis of coil sensitivities, which can be used to derive any arbitrary sensitivity:

= x a , C , ( r ) 2;a&(r) 6-
I 1
Any actual coil can thus be hypothetically constructed as a linear combination of such a basis set of virtual coils. For an SNR-optimal image reconstruction algorithm, contributions from additional coils are included only to the extent that they improve the resultant SNR, therefore a net coil derived from a complete basis set guarantees optimal performance.
The noise resistance term can also be expressed in terms of the net field:
Q = f*rfd3r = sample Ill' f o i ~ ~ ( r ) ~ ~ r ) d ~ r sample where Y fuEl(r)~;(r)d3r sample
The problem of finding the ultimate intrinsic SNR has thus been reduced to finding the optimal set of weights a,, Eq. [2.13], while assuring that the net sensitivity is one at the reconstructed points and zero at all potential aliasing positions.
It has been demonstrated that the SNR-optimal weighting matrix is given by the Moore-
Penrose pseudoinverse of the encoding matrix [8, 381. Substituting the right-hand side of
Eq. [ 1.81 in Eq. [2.13], the noise resistance becomes:
So in the case of uniform Cartesian sampling of k-space, a basis set of coil sensitivities can be optimally combined to ideally achieve the ultimate intrinsic SNR for each reconstructed point: where the H matrix is evaluated at the position of the voxel of interest.

The ultimate intrinsic SNR, appropriately scaled as in Eq. [2.6] to account for sequence related parameters, can be compared with correspondent SNR values from actual MRI acquired data in order to evaluate the performance of a specific coil. When such evaluation is performed on every pixel of an arbitrary image, the result can be effectively displayed as a coil performance map, which is defined as the ratio of the actual SNR image to the ultimate SNR image, as a function of position inside the sample of interest
[39]. The next chapters describe two experiments aimed at generating coil performance maps for a coil array using two different sample geometries.

3.1 Parallelepiped Phantom Description
Ultimate intrinsic SNR was simulated at different positions inside a rectangular parallelepiped object (Fig. 3-I), modeled on an existing nickel-sodium phantom specifically designed for SNR measurements (USA Instruments Inc., Aurora, OH, USA).
Figure 3-1. Rectangular parallelepiped phantom with uniform electrical properties modeled for the simulation of the ultimate intrinsic SNR.
The rectangular parallelepiped phantom was 37.2 cm long, 14.8 cm high and 15 cm wide.
It contained 8.02 liters of a water solution with 3.3685 g/L of Nick-6H20 and 2.4 g/L

NaCl. The conductivity and permittivity, assumed uniform inside the object, were equal to 0.447795 c2 magnetization.
3.2 Ultimate SNR Simulation Algorithm for a Parallelepiped Geometry
For this geometry, the net EM field inside the object was expanded into a complete set of
EM plane waves, following the general approach of [37]. Although plane waves enable one to compute analytically the noise correlation matrix, they change very slowly at the
MRI wavelength causing the rows of such a matrix to be highly linearly dependent. The inversion of the noise correlation matrix thus becomes an ill-conditioned problem, which requires high numerical precision for the computer operations in order to avoid numerical instability. For that reason the simulation was implemented using a public C++ library for arbitrary precision computation [48].
The first step was to define a basis set of plane waves. The general expression of every complex wave vector in the basis set is:
The value of the coefficients a and b is specified by the dispersionrelation in Eq. [2.9], as k1 must satisfy the condition:
The unit vectors kd and selected to be evenly distributed on the unit sphere. kd and kimg that were close to be orthogonal to each other were discarded to improve the numerical stability of the computation. For each wave vector, the following procedure was repeated to generate the real and imaginary unit vectors. Two random numbers were chosen:
Then, defining B cos-I (z), the components of a unit vector were derived as:

k,
= cos (<^)sin (6) k,,
= sin (#)sin (f+) k,
= cos(8)
Next, each wave vector k, was used to specify two orthogonal polarizations of an EM field:
In this way, N wave vectors were sufficient to generate 2N different modes. The electric and magnetic fields inside the sample were then expressed as a linear combination of all these modes:
The plane wave basis functions in Eq. [3.6] were utilized to compute the noise correlation matrix for a finite uniform parallelepiped of length 21, width 2w, and height 2h.
Substituting in Eq. r2.14 1, the analytical expression becomes:
Y,,, = $u2Zl(r)~;(r)d3r sample
=
The inverse matrix o f ? was then computed. The algorithm up to this point needs to be executed only once for a given sample geometry and composition. The last part of the algorithm depends on the reduction factor and the position r inside the sample. The points of interest were chosen on the section formed by a coronal plane passing through the center of the parallelepiped. The rectangular section was divided in a grid of 5760 squares of side 3.125 mm, arranged in a 120 by 48 mesh (Fig. 3-2). The positions of the squares centers were used in the equations.

37.2 cm 120 pixels
Figure 3-2. Schematic diagram of the image plane and the FOV used to calculate the ultimate intrinsic SNR for the rectangular parallelepiped phantom.
For each position, the encoding matrix was constructed combining Eq. [1.8], Eq. [2.2] and Eq. [3.6] and then substituted in Eq. [2.15] to obtain the corresponding noise resistance. The ultimate intrinsic SNR was calculated as in Eq. [2.16] at each point. The algorithm was implemented in C++ (Visual Studio .NET, Microsoft Corporation,
Redmond, WA, USA) using two open-source libraries [48, 491. The results of the simulation were exported into MATLAB (The Mathworks, Natick, MA, USA), which was used to display them in the form of a 120-by-48 pixel ultimate SNR image, both for the fully sampled and decimated cases.
3.3 Numerical Convergence
It was shown in a previous study using an elliptical geometry that about 800 plane waves are needed for the ultimate SNR algorithm to converge at a point in the center of a phantom and more at positions closer to the edge of the object [37]. In the case of the rectangular parallelepiped phantom, the analytical expression of the noise correlation matrix leads to even worse numerical instabilities, most likely due to the multiplication of

three sine functions (see Eq. [3.7]). The convergence of the algorithm depends on the number of modes and the number of numerical precision digits used, both of which are limited by the available computer memory. The outcome of a simulation was taken to be convergent if it changed less than 5% after adding 100 modes. Figure 3-3 illustrates the convergence of the calculations, using 200 digits of precision, as the number of plane wave modes is increased, at different distances from the center of the phantom, for the fully sampled (unaccelerated) reconstruction.
Figure 3-3. Convergence of the ultimate SNR calculation for a reduction factor of one, as a fimction of the dimension of the plane wave basis set.
The plot in the top left quadrant shows the convergence at the center of a rectangular parallelepiped object, whereas the other plots refer to positions selected on an arbitrary coronal plane (see Fig. 3-2) at various distances (d
=
5 cm, 9.5 cm, 14 cm) from the center of the object. The values of the
SNR in each plot are normalized to the respective maxima.

Note that close to the center of the object the simulation convergences rapidly, whereas for positions closer to the edges of the object the ultimate SNR doesn't come near to a converging value even using 1000 plane waves. Figure 3-4 displays for which pixels of the coronal slice passing through the center of the parallelepiped phantom (see Fig. 3.2) the ultimate SNR calculation converged, using 1000 modes and 200 digits of precision on a computer with 2 Gigabyte of RAM.
Converging Region for Ultimate SNR calculations
Figure 3-4. Convergence of the ultimate SNR calculations as a function of pixel location for an arbitrary slice passing through the center of a parallelepiped object. Using 1000 plane waves mode and 200 digits of numerical precision, only for about 113 of the FOV is the simulated value of the optimum SNR accurate.
As the convergence behavior is expected to degrade for reduction factors greater than one
[37, 381, the calculations for the parallelepiped phantom were not repeated for any accelerated case. Although it is possible to run the simulation on a more powerful machine, such a decision should come after exploring other possible implementations of the algorithm, in particular a different choice of basis functions (see Chapter 4).

3.4 Overall SNR
Although the convergence of the calculations was not sufficient to allow confidence in
SNR values near any edge of the phantom, the qualitative spatial variation of the ultimate intrinsic SNR agrees with the results of other published research papers [37,39], in which
SNR also rapidly decreases from the edge toward the center of the object (Fig. 3-5). This behavior is expected as the EM waves are more and more attenuated while traveling inside the phantom.
Normalized Ultimate Intrinsic SNR (parallelepiped phantom)
.
.
. . . . .
1 " .
.
Figure 3-5. Ultimate intrinsic SNR computed at each pixel of a 120x48 coronal slice, passing through the center of the rectangular parallelepiped object. Values are normalized to the largest number. SNR is smaller along the z-direction because of the effect of the main magnetic field.
The described spatial variation is less accentuated, but nevertheless present (see Fig. 3-6), in the z-axis than in the other direction, as any EM wave corresponding to a coil with the sensitivity along the direction of the main magnetic field would not contribute to the total signal.

1
Central Profile
- y Axis
Figure 3-6. Normalized spatial variation of the ultimate intrinsic SNR along the y-axis (top) and the z-axis (bottom) of the coronal plane. In both cases values are larger near the edges of the object and become smaller when approaching the center.

SNR
4.1 Cylindrical Phantom Description
Ultimate intrinsic SNR was simulated at different positions inside a cylindrical object
(Fig. 4- l), modeled on an existing nickel-sodium phantom specifically designed for SNR measurements (USA Instruments Inc., Aurora, OH, USA).
Figure 4-1. Cylindrical phantom with uniform electrical properties modeled for the simulation of the ultimate intrinsic SNR.

The cylindrical phantom was 34.1 cm long, with a diameter of 15.3 cm. It was filled with
6.28 liters of the same water solution contained in the rectangular phantom (see section
3.1). Also in this case, the conductivity, permittivity and magnetic susceptibility were assumed to be uniform inside the phantom and were, respectively, equal to 0.447795
4.2 Ultimate SNR Simulation Algorithm for a Cylindrical Geometry
For the cylindrical geometry, the net EM field inside the cylinder was expressed as a linear combination of cylindrical harmonic functions, in order to match as much as possible the phantom geometry.
Because of the cylindrical geometry, it is possible to single out the spatial variation of the modes in the direction of the longitudinal axis (i.e. z axis) and to assume plane wave dependence in the z-direction [46] :
Each mode can then be separated into components parallel and transverse to the z axis:
When the components in the z direction are known, the transverse components can be computed (see appendix A) as: where V, is the transverse part of the Laplacian operator V:
=
V
- a2/az2. Transforming
Eq. [4-21 and Eq. [4-31 in cylindrical coordinates, the following equations hold for every
EM mode in the cylindrical sample:

The EM field in Eq. [4-41 and Eq. [4-51 is completely known once the z components of the electric and magnetic field are specified. In this work, Ez and B; were selected to be solutions of the Helmholtz wave equation in cylindrical coordinates [50]: where Jm is the first type Bessel function of integer order m and k is k = i / p m 2 - + i p w In order to find a complete basis to express the net EM field inside the sample, m was varied among positive integers and kz was expressed as 2;r n/L, with L equal to the length of the cylinder and n equal to the integers 0, 1, 2,
. . .
For each couple of parameters, the correspondent EM field was generated as the sum of two components, one with Ez
=
0 and the other with Bz
=
0. This approach simplifies the expression of the noise correlation matrix, which from Eq. [2-141 can be computed as: where R is the radius of the cylindrical phantom and L its length.
The resulting noise correlation matrix has only few non-zero off-diagonals elements and its inversion is not troublesome. As for the other phantom, the points of interest were chosen on a coronal section passing through the center of the object. The rectangular

section was divided in a grid of 5 184 squares of side 3.125 mm, arranged in a 108 by 48 mesh (Fig. 4-2) and for each point the ultimate SNR was calculated. The algorithm in this case was implemented using Mathematica (Wolfram Research, USA), which enabled accurate computation of the integral in Eq. [4-71.
Figure 4-2. Schematic diagram of the image plane and the FOV used to calculate the ultimate intrinsic SNR for the cylindrical phantom.
4.3 Numerical Convergence
The choice of using cylindrical harmonics functions to expand the EM field for the cylindrical object improved dramatically the numerical convergence of the ultimate SNR calculations. The convergence was tested for different reduction factors, repeating the computation for an ever greater set of parameters m and n. The total number of basis functions used to expand the EM fields is given by the product of the number of different m and n added to the simulation. Since the work with the rectangular parallelepiped phantom showed that the convergence behavior is worse at positions far from the center of the object (see Fig. 3-3), numerical convergence was investigated for a point at the center of the cylindrical object (Fig. 4-3) as well as for a point on its surface (Fig. 4-4).

The tests were performed for different acceleration factors along the largest dimension of the cylinder (z axis).
Figure 4-3. Convergence behavior of the ultimate SNR calculation at a position in the center of the cylindrical object, for different reduction factors. Each plot shows the normalized ultimate SNR the number of modes in the basis function (see Eq. [2-101 and Eq. [4-61)

at
1.00050 i
--..
,
Figure 4-4. Convergence behavior of the ultimate SNR calculation at a position on the surface of the cylindrical object, for different reduction factors. Each plot shows the normalized ultimate SNR for as a function of the number of modes in the basis function (see Eq. [2-101 and Eq. [4-61).

A calculation was assumed to be convergent when its output increased by less than 0.1% after adding more modes to the basis, varying the values of either m or n by 2. The previous figures show that it takes a larger number of modes for the computation to convergence at the position far from the center of the phantom. Thus, the smallest number of different m and n for which the calculation converged at a position on the surface of the cylinder was used to define the minimum size of the basis set that enables one to accurately compute the ultimate intrinsic SNR at every position of the coronal slice. Table 4- 1 summarizes the results of these convergence tests, for different reduction factors along the z direction.
Table 4-1. Minimum EM Basis Size as a Function of the Reduction Factor
4.4 Overall SNR
The spatial variation of the ultimate SNR on the coronal slice was similar for each reduction factor and it is shown in Fig. 4-5. As expected, the SNR values are smaller near the center of the object and larger at the surface. The SNR variation is symmetrical around the longitudinal axis and is diminished along the y direction, because, as for the other geometry, any contribution to the net coil sensitivity along the z direction is negated by the effect of the main magnetic field.

Normalized Ultimate Intrinsic SNR (cylindrical phantom)
Figure 4-5. Ultimate intrinsic SNR computed at each pixel of a 108x48 coronal slice, passing through the center of the cylindrical object. Values are normalized to the largest number.
Although its qualitative spatial variation remains the same, the ultimate intrinsic SNR assumes smaller values as the reduction factor is increased. Fig. 4-6 shows how the value of the ultimate intrinsic SNR decreases as the reduction factor becomes larger. This is in agreement with the theory of parallel MRI (see section [IS]) and it was shown also in previous work [37, 381. The optimal SNR falls off dramatically as acceleration is increased beyond a comparatively low level and such behavior is more evident at the center of the cylinder, where a for a reduction factor of three its value is already one third of the unaccelerated case.

0.80
0.60
0.40
0.20
0 .OO
1 2 3 4 5
Reduction Factor
6 7 8
Figure 4-6. Behavior of the ultimate intrinsic SNR as the reduction factor along the z direction is increased, for a point in the center (top) and on the surface (bottom) of the cylindrical object. Values are normalized with respect to the unaccelerated case. Note that the intrinsic SNR values plotted here do not include the inverse proportionality to the square root of reduction factor, which is treated as an extrinsic sequence-dependent effect.

The accuracy of the simulated ultimate SNR values is fundamental in order to derive an absolute benchmark to evaluate coil array performance. For this reason, only the results for the cylindrical object can be used to compare quantitatively the optimum SNR to the actual SNR of an MR image. Making such a comparison is the principal purpose of this work, and so the next step was to measure the SNR at each pixel of an image for a geometry exactly matched to the simulated case.
5.1 Experimental SNR Measurement
The cylindrical phantom modeled in the ultimate SNR simulation (see Fig. 3-1) was scanned on a GE 1.5T Twinspeed system (GE Medical Systems, Milwaukee, WI, USA), using an eight-element linear array (Fig. 5-1) along the same plane used in the simulations. The array was positioned on top of the phantom with its longest side along the cylinder's longitudinal axis. The total area of the array was 13.3 cm x 27.3 cm, providing full coverage of the chosen FOV. The array is described more fully in reference [5 11.

'F-
Figure 5-1. Eight-element linear array of surface coils used to acquire images of the cylindrical phantom.
A gradient echo pulse sequence with a TR of 1 s and a TE of 5.1 ms was chosen to avoid dealing with the effects of relaxation. The pixel size was the same as in the simulations
(3.125 x 3.125 mm), the slice thickness was 1 mm and the flip angle was 10'. A small flip angle was chosen deliberately in order to avoid systematic signal fluctuations causing artifacts in regions of high SNR. The receiver bandwidth was ±15.6 KHz. For each study, the same slice was sequentially acquired 112 times and reconstructed off-line using a Cartesian SENSE algorithm [20]. An SNR image was then generated, computing the SNR at each pixel as its mean value along the 112 replicas, divided by the standard deviation in the same interval.
The SNR image was then cropped around the object contours (Fig. 5-2). Since the pixel size was the same of that used in the simulation, such reshaping resulted in a final 108- by-48 SNR image, with each pixel corresponding to a pixel in the ultimate SNR map.

48 x 108 pixels
Figure 5-2. An MR image displaying the SNR value at each pixel was computed from a set of replicated coronal slices, acquired along the same plane used in the simulation of the ultimate intrinsic SNR. The image was then cropped to exactly match its pixels to the simulated positions. The
SNR values are low because of the choice of using a small flip angle for the acquisitions.
5.2 Coil Performance Evaluation
In order to compare the experimental SNR data with the simulated data, the ultimate SNR values need to be properly scaled to account for the specific sequence parameters (see Eq.
[2-61). The noise figure of the RF receiving chain, which must be taken into account, was measured experimentally using a liquid nitrogen method and its value was 1.1 1 dB.
The ratio of the actual SNR to the scaled ultimate SNR can be used to measure the performance of the coil array for the specific task. Taking the ratio for each pixel provide quantitative information as a function of position inside the sample of interest and such data can be visualized by means of a coil performance map. In this work, coil performance maps were generated in the case of a fully sampled and a 2-fold undersampled image reconstruction (Fig. 5-3).

Figure 5-3. Coil performance maps showing the fraction of ultimate achievable SNR for unaccelerated (top) and 2-fold accelerated (bottom) studies with the eight-element coil array atop the cylindrical phantom.
In the unaccelerated case the SNR was always less than 45% of the optimum, whereas in the case of 2-fold acceleration, although the overall value of the SNR was lower, performance of the coil array reached 60% of the ultimate in the region of high signal.
The lateral elements of the array performed better than the central elements. Otherwise, as expected, the performance was generally higher in the center, where the ultimate SNR is smaller.

As the number of available receiver channels on modem MR systems increases, increasing attention will be paid to the design and performance of many-element RF coil arrays. Questions regarding the balance of coil-noise and sample-noise, or the suitability of any particular array design for parallel imaging, promise to take on new significance as the number of elements increases. In the present work a method has been described to evaluate the absolute performance of any particular coil array. Although originally conceived to improve the design of receiver coils for parallel MRI applications,' the procedure can be applied to sequential imaging as well.
The performance of any coil array is strictly constrained by the behavior of electromagnetic fields within the sample [37, 38, 391. For a chosen imaging task, the best possible SNR achievable for any coil configuration can be computed using a complete set of coil sensitivity basis functions. In Chapter 3 and 4, an algorithm to compute the ultimate intrinsic SNR for parallelepiped and cylindrical geometries was presented.
Ultimate SNR was calculated for each pixel position on specific image planes, modeled on existing phantom geometries. The numerical complexity of the calculations was found to be strongly affected by the choice of EM basis functions. Plane waves proved not to be the best choice to express the EM field inside a parallelepiped object, as the resulting calculations suffered from serious convergence issues. Cylindrical harmonic functions

enabled computing the ultimate SNR with high accuracy in the case of a cylindrical sample.
For this reason, only the cylindrical phantom was employed for the remaining part of the project, aimed to produce a quantitative measure of the performance of a coil array. A linear 8-element array was used to acquire exactly the same image section modeled in the simulation. The SNR at each pixel was compared with the correspondent optimal value, scaled to account for the specific acquisition sequence. The results were presented in the form of performance maps for the specific coil array.
Maps were generated for unaccelerated and 2-fold accelerated parallel imaging reconstructions. Interestingly, the performance of the 8-element coil array was higher with respect to ultimate intrinsic values in the 2-fold accelerated case, even though both the ultimate and the actual SNR with acceleration are reduced as compared to the unaccelerated case. That could be explained recalling that an array configuration which is optimized for parallel imaging is not necessarily the best also for unaccelerated acquisitions [35]. This result will be further investigated in the future, by building performance maps for higher acceleration factors and for arrays with larger numbers of elements.
In defining the set of basis functions that describes the EM inside the cylindrical phantom, for computational convenience, a plane-wave dependence was assumed in the direction of the longitudinal axis. That should capture the dominant behavior for long cylinders, but requires future investigations on evanescent-wave effects in the z-direction.
Cylindrical harmonics may be a suitable choice of basis functions also for parallelepiped geometries, even if the computation of the elements of the noise correlation matrix, as well as its inversion, could be more troublesome.
In this work, sensitivity functions were continually added as input of a simulation until a complete basis set was found to describe the EM field inside the sample, completely disregarding the field outside the sample. Another possible way to completely characterize the EM field inside the sample is to accurately determine boundary conditions at the object surface. The computational advantages of this approach will be investigated in the future.

Although further improvements are needed, the method proposed in this thesis is a good candidate to become a valuable tool for the evaluation of coil designs. Ultimate intrinsic
SNR defines an absolute performance target for coil designers, and coil performance maps provide useful and immediate feedback on how far a particular array configuration is from such expectations. The fact that the ultimate SNR needs to be computed only once for each particular geometry and imaging section will eventually facilitate its employment as an additional instrument for coil engineers.
Given the comparatively large set of basis functions required for convergence of the ultimate intrinsic SNR at multiple positions, it is likely that the simulated optimum cannot be practically realized with an actual coil array. However, if further investigation shows that a smaller subset of the larger basis can capture the dominant SNR behavior, then it may be possible to build an array targeted to that subset in order to approach the simulated optimal sensitivity patterns.
Furthermore, since such an array should in principle perform very close to the optimum, any discordance could be linked to noise other than that coming from the sample. That type of insight would represent a valid contribution to the field of parallel MRI. In fact, the development in recent years of arrays with very many elements [52, 531 has raised to practical priority the question of what is the smallest size for array elements before the final SNR begins to be dominated by the noise coming from the electronic components.
In conclusion, the capability of parallel MRI to accelerate image acquisitions is fundamentally limited by electrodynamic constraints, but the knowledge of such limitations can be exploited to improve current coil design and to eventually develop innovative receivers that operate close to the optimum.

Assuming a sinusoidal dependence e""* for the fields inside a uniform medium,
Maxwell's equations in a source-free region (see Eq. [2.7]) can be written as: where p and e are the permeability and the dielectric constant of the medium. Applying the identity: to the curl of the magnetic field, the equation becomes:
Substituting the curl of the electric field from Eq. [A.l], it's possible to derive the wave equation: which specifies the dispersion relation:
=

This appendix describes how to derive an expression for the electromagnetic field inside a cylindrical object filled with a medium having uniform electrical properties. It is based on the approach described in reference [46].
Because of the cylindrical geometry it is useful to single out the spatial variation of the field in the z direction of the cylinder axis and to assume:
With the assume z dependence, Eq. [A.4] reduces to: where the transverse part of the Laplacian operator is defined as:
If the fields are separated into components parallel and orthogonal to the z axis,
Faraday's and Ampere's equations in Eq. [A. 11 become:

x(E, +EZ)=iw((Bt
+(Bz) x (B,
+@)
=
( p - i m p e ) ( E t + E z ) which can be separated in four different equations:
V, x E z + z - = i & ,
Bi
V, x <Bt
= ([iff - i m p e ) ~ ,
Using the identities:
2 x (V, x E z )
=
V,Ez
2 x ( V , x q = - 5 5 , on both Eq. [BS] and Eq. [B.6], they reduce to:
[B. 101
Taking the partial derivative in z of all members in Eq. [B. 101 and using Eq. [BS] and
Eq. [B.6] to substitute in for the right hand sides, results in:
It comes from Eq. [B. I] that:
[B. 131

Substituting Eq. [B. 131 in both Eq. [B. 1 1] and Eq. [B. 121 and solving for the transverse components of the fields:
[B. 141
The last two equations describe how to derive the transverse components of the electric and magnetic fields once their z components are known. Using Eq. [B.13] they can be reduced in the form shown in Eq. [4.3]

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