Signal-to-noise ratio improvement in NMR via receiver hardware optimization

MISSING IMAGE

Material Information

Title:
Signal-to-noise ratio improvement in NMR via receiver hardware optimization
Physical Description:
vi, 136 leaves : ill. ; 29 cm.
Language:
English
Creator:
Duensing, George Randall, 1964-
Publication Date:

Subjects

Subjects / Keywords:
Physics thesis, Ph. D
Dissertations, Academic -- Physics -- UF
Genre:
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1994.
Bibliography:
Includes bibliographical references (leaves 130-135).
Additional Physical Form:
Also available online.
Statement of Responsibility:
George Randall Duensing.
General Note:
Typescript.
General Note:
Vita.

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
oclc - 33343990
System ID:
AA00022372:00001

Full Text




SIGNAL-TO-NOISE RATIO IMPROVEMENT IN NMR
VIA RECEIVER HARDWARE OPTIMIZATION











BY

GEORGE RANDALL DUENSING


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


1994










ACKNOWLEDGEMENTS


The author acknowledges the enormous significance of the love

and support of his wife Tracey Kryslak Duensing. He is also grateful
for all that his parents, Clyde and Winnie Duensing, have done to

enable this accomplishment. In addition, the NMR resource group of
students and faculty have been of great assistance and a source of

academic pleasure. Dr. Jeffrey Fitzsimmons has been an excellent
mentor, allowing freedom while providing direction, and has shown

great concern for the goals of the author. The author is also thankful
to Dr. H. Ralph Brooker and Dr. William Brey for many fruitful
discussions.








TABLE OF CONTENTS


page


ACKN OW LED GEM ENTS................................................................................................ ii
ABSTRA CT ........................................................................ ....... ............. ........................ v

CH APTER 1: INTRODU CTIO N ................................................................................ 1...

The Phenomenon of Nuclear Magnetic Resonance (NMR).............. 1

M agnetic Resonance Im aging..................................................................... 5
The N M R Instrum ent..................................................................................... 7

The NM R Receiver........................................................................................... 8

Original W ork.................................................................................................. 13

CHAPTER 2: REVIEW OF STATE OF THE ART................................................ 15
Signal-to-N oise Ratio.............................................................................. 15

Radio-frequency Coils.................................................... ..........................19

Independent Signal A cquisition............................................................. 23

CHAPTER 3: VARIABLE FIELD OF VIEW COILS........................................... 27
Introduction................................................................................................. 27

M ethods...................................................................................... ................... 29
Results ................................................................................................ ...... 31

Conclusion.................................................................................................... 34

CHAPTER 4: COUPLING BETWEEN NMR COILS....................................... 37
Introduction ............................................................................... .................... 37
Discussion ..................................................................................................41

Conclusion................................................................................................ ..55

CHAPTER 5: GENERALIZED QUADRATURE ...................................................... 56
Introduction....................................................................................................56

iii








Theory ................................................................................................................61
Applications and Exam ples....................................................................... 72

Summ ary of Attainable Gains.................................................................. 89

Conclusion......................................................................................................... 91

CHAPTER 6: INDEPENDENT SIGNAL ACQUISITION.................................... 94
Introduction.................................................................................................... 94
Tim e Division M ultiplexing...................................................................... 95

Frequency D om ain M ultiplexing.......................................................... 100

RF Prefiltering.............................................................................................. 102

Conclusion...................................................................................................... 111l

CHAPTER 7: MISCELLANEOUS HARDWARE IMPLEMENTATIONS........ 114
Im pedance M atching................................................................................ 114

Im pedance Preserving Com biners...................................................... 118

Transm mission Synchronized Shielding............................................... 120

Birdcage Surface Coil................................................................................. 123

CHAPTER 8: CONCLUSION ................................................................................... 127
REFERENCES............................................................................................................... 130

BIOGRAPHICAL SKETCH....................................................................................... 136


















iv








Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

SIGNAL-TO-NOISE RATIO IMPROVEMENT IN NMR
VIA RECEIVER HARDWARE OPTIMIZATION

By
George Randall Duensing

December 1994

Chairman: Professor E. Raymond Andrew
Major Department: Physics
The goal of this research was to increase available signal-to-
noise ratio (SNR) in magnetic resonance imaging (MRI) by applying

specific knowledge of the imaging system to improve receiver probes
(coils) and receiving hardware. A brief history of improvements in
MRI receiver and coil design is presented, including the transition

from large linear volume coils to local surface coils and quadrature
volume coils. Then quadrature surface coils are introduced and
finally multi-coil arrays with independent acquisition systems. The

research covers improvements in these areas and begins with a
surface coil which is adjustable in size to optimize performance given
the region of interest. By careful design of trombone-like coil
elements, physical adjustment can be made without electrical
adjustment. Second, new understanding of noise correlation and
crosstalk between coils is developed and applied to muli-coil arrays.
This provides the ability to increase available SNR for such systems.
Third, a method for optimally combining multiple coils in a
transverse (extending perpendicular to the static magnetic field)
array into a single channel by proper signal combination is








presented. This method is termed generalized quadrature because of

the similarity of the method to standard quadrature combination, but
with freedom in weighting and phasing in the combination process.

Fourth, several methods of manipulating the multiple signals from an

array to allow separation after acquisition are presented. These
methods require new hardware demands but allow significant
improvements in SNR for either transverse or longitudinal arrays.
Fifth, several novel design methods are demonstrated, including an

algorithm for impedance matching, a generalized quadrature

combination method, transmission synchronized rf shielding and a
bird-cage surface coil. Finally, the potential future applications and
benefits of this research are presented.















CHAPTER 1
INTRODUCTION


The Phenomenon of Nuclear Magnetic Resonance (NMR)


The properties of the nuclei of atoms are described by

quantum mechanics. The significant property for this work is the
property of spin. Some nuclei possess an intrinsic angular
momentum or spin. Denoting spin S, we can describe its quantized

values as S = 2 s(s+l) where h is Planck's constant and s is the
2I1
spin quantum number and is integral or half integral. A second
quantum number is introduced to indicate as much as allowable
about the direction of the spin in a magnetic field. The convention is
h
to define a magnetic field in the z-direction with S, =- -m, being the
271

projection of the spin onto the z-axis. The quantum number ms may

take on the values of {s,s-l,s-2 ..... -s). A magnetic moment can be
thought of as being the result of an electric charge distribution
within a spinning nucleus.
Since the spin magnetic moment is quantized in a magnetic
field it can be identified with certain energy states. Hydrogen nuclei
are almost always the nuclei of interest in magnetic resonance
imaging (MRI) and their spin properties are characterized as spin 1/2.








These nuclei have 2 allowable quantum states in a magnetic field.
One is higher energy, with the magnetic moment anti-parallel with
the field and the other of lower energy which is parallel. The energy

difference is AE= h-yB where y is the gyromagnetic ratio and is
21c
defined empirically from this expression and thus relates the angular
frequency of radiation for a transition between states, to Bo, the

static magnetic induction. The population difference of the two
energy states defines the amplitude of the bulk magnetic moment

which is available for perturbation and is described by Boltzman's
equation as n -n, = N where N is the number of nuclei, k is
2kT

Boltzman's constant and T is the absolute temperature of the sample.
The population difference, n. -n, is typically on the order of 10-5
N

for typical MRI situations which is the root of the problem of
sensitivity in MRI.
In quantum mechanics, a Hamiltonian which describes the

energy states of the system is created in analogy with classical
mechanics. The energy states of a nuclei are in general extremely

complex. In the following description all terms except that of the
energy of a magnetic moment in a magnetic field will be ignored.
This term is
H=-M-H,

where H is the spin Hamiltonian, M is the magnetic moment of the
spin (proportional to the spin operator S) and H is the magnetic field.
By neglecting orbital angular momentum, the behavior of the spin in
a magnetic field can be readily observed. In the Heisenberg
represention the equation of motion is









i S,()=[S,(t),H].

This can be rewritten using the commutation relations for the
components of the spin operator and the definition of the spin
Hamiltonian above as
S(t) = coS(t) x H(t).

This equation is an operator description of the classical expression
which equates the time derivative of angular momentum and the
applied torque. In NMR this expression is called the Bloch equation
and describes the macroscopic motion of the bulk magnetic moment
in a material. If the magnetic field is constant in time, it is easily
shown that the spin moment will process or rotate about the axis of
the applied field. The rate at which this occurs is called the Larmor
frequency. As with classical systems which have a natural
frequency, excitation of the system is most easily produced by
coupling to the system at that frequency. In this case a magnetic
field which varies at the Larmor frequency will strongly couple to
the system. It should also be clear that for a given amount of power
used in producing a time-varying magnetic field at the frequency of
interest, a rotating magnetic field will most strongly interact with the
spin magnetic moment. Therefore, if other parameters are held
constant a magnetic field described by an exponential, with exponent
ito0t, will couple most efficiently and optimally produce the dynamics

of interest. This will be further expounded upon in the following
sections. Figure 1-1 demonstrates the relationship of the magnetic
moment precession, the perturbing rf magnetic induction BI1 and the
main fixed magnetic induction Bo.








M
















NMR




Figure 1-1. Pictorial representation of the precession of the spin
magnetic moment and the spatial relationship with the static field
Bo and the RF field Bl

In reality, as with any oscillator (other than a perfect
superconductor), the oscillations will decrease with time. This has
been termed relaxation and occurs due to a number of effects. The
energy in the excited state nuclei which form the net magnetization
will be dissipated in some way. One method is by local interactions
of the spin with its surroundings or lattice. This form of dissipation
is termed spin-lattice relaxation and the time constant of this process
is called T1. In general terms each spin is effected by other magnetic
nuclei which fluctuate to some degree at the Larmor frequency. At
typical MR field strengths, the rotational rates of medium sized
molecules are approximately the inverse of the Larmor frequency








thus protons in this environment have the most efficient relaxation

and therefore the shortest TI (1).
The other common relaxation method, termed T2 relaxation

describes the case where energy is transferred from an excited

nucleus to a ground state nucleus. If all nuclei were processing at
the same rate, the magnetization would not decay (ignoring spin-
lattice relaxation). However, in real magnets and with real samples

there is always some inhomogeneity in the magnetic field, which in
turn implies a difference in the precessional frequencies of various

spins. The net effect is a loss of coherence which eventually leads to
no net transverse magnetization. Again the local magnetic fields of
other nuclei are most significant in this process. However, here the

perturbation of frequency is most pronounced for large, slow moving
molecules which therefore produce short T2. For typical samples
used in MRI, too little signal is absorbed by the receiver to damp

oscillations, although this phenomenon has been observed. (2)


Magnetic Resonance Imaging


The phenomenon of nuclear magnetism was well known long

before the development of MRI as a tool for the examination of
morphological structure and was applied to the more basic physical

and chemical analysis problems. The local magnetic field of a
nucleus is responsible for the exact Larmor frequency of the
particular spin moment. This is described by the chemical shift
which is the frequency difference of a particular nucleus compared
to some standard. This property along with many other more








complicated interactions involving steric hindrance and thus

relaxation time differences, and coupling between nuclei through a

number of different mechanisms, allow precise characterization of
local chemical environments and thus chemical structure (3).
The basis of MRI is to eliminate the fine spectral details and

observe only the nuclear distribution without regard to small
environmental differences between nuclei. The common
experiments produce contrast through differences in proton density
and TI and T2 differences, although sequences for many other
contrasts have been developed (4). This is accomplished in the

standard imaging sequences by producing a term in the spin
Hamiltonian which dominates the dynamic behavior. To produce a
one-dimensional picture of the nuclear density in a sample, a strong

linear gradient of magnetic field is added to the homogeneous main
field. This serves to spread the frequency to such an extent that

chemical shift differences are normally unnoticable. Furthermore

the frequency differences encode the spatial dimension and allow an
image of the material to be produced. Application of a radio
frequency (rf) pulse followed by acquisition of the signal induced by

the moment precession during application of a linear magnetic
gradient results in a one-dimensional image which is a projection of
the sample onto one line. This is the simplest imaging pulse
sequence, but is representative of one dimension in almost all state-
of-the-art imaging pulse sequences.
The most basic imaging sequence currently used clinically

involves the use of the gradient described above for frequency
encoding in one dimension and then a repetitive gradient which








varies incrementally with an acquisition at each step. This gradient

set produces phase encoding and allows an inversion via Fourier

transform to the two spatial dimensions. An important point to note

for future reference is that one dimension is a frequency dimension,
the bandwidth of which is dictated by the strength of the frequency
encode gradient.


The NMR Instrument


The introductions to NMR and MRI above refer indirectly to

many of the basic sections of the NMR instrument. The primary

requirement is a magnetic field. Although experiments have been

carried out in the earth's magnetic field (5) and fringe fields (6), the

standard instrument includes a carefully designed magnet with a
strong, homogeneous magnetic field. The magnet may be a
permanent magnet, an actively maintained "resistive" electro-magnet
or a superconducting electromagnet. An accessory to the magnet is
the shim winding set, which is a group of coils which can be driven to

correct for inhomogeneities in some particular region of the magnetic
field. Closely associated are gradient windings which are used to
produce the linear magnetic fields used in imaging. Typically, the
computer which controls the pulse sequence, sends the voltage shape
required to produce the current (and thus gradient field) of interest.
These three components, the magnet, the shim coils and the gradient
coils along with their associated power supplies complete the static/
low frequency magnetic field requirements.








The next basic requirement is the ability to perturb the spins
from equilibrium. This is normally implemented with the use of a
large excitation coil tuned to resonance at the Larmor frequency and

a high power rf amplifier, although local transceive coils are
sometimes employed at lower power levels. For modern pulse
sequences, it is necessary to produce shaped rf bursts. As with the
gradient waveform, the computer sends information to a digital-to-

analog converter whose voltage modulates the Larmor frequency
sine wave. The rf coil may receive as well as transmit, or separate
coils may be employed. In either case, the spin magnetic moment

induces a voltage in the coil which is delivered to the receiver.


The NMR Receiver


Since NMR and thus MRI inherently generate low level signals,

it is crucial to produce as little extra noise as possible during the
process of detection and recording of the signal. The signal-to-noise
ratio (SNR) is the standard quantitative measure of the quality of a
signal in the presence of noise and will be discussed below and in

chapter 2. The block diagram of the standard MRI receiver is shown
in figure 1-2. The first item in the chain is the probe which is
sensitive to a magnetic field. Typically this is a loop antenna
designed to be sensitive to near field signals. The system is typical
of communications receivers, which deal with an audio frequency
signal modulated on a higher frequency carrier. The relaxation
phenomenon has approximately audio frequencies, while the Larmor








preamplifier
phase B. sample C
I I I detector :- : ]
~\ ^ 0 ~--~ aIdio and hold A
receiver sampled
I Isignal signal
bandpass
filter Q

coil A. modulated
rf signal




Figure 1-2. Block Diagram of NMR receiver



frequency is typically tens of megahertz, so this type of receiver is
well suited to the purpose. It is important to note that the
modulation contains both amplitude and phase information. The

incoming signal is preamplified and eventually mixed down to audio
for digitization.

The signal-to-noise ratio is the measure of the quality of some

information in the presence of noise. Chapter 2 contains specific
discussions of formulations of SNR, but qualitatively the signal is the

voltage induced in a receiving prove by the rotating magnetic
moment. Similarly, the noise is a random voltage induced in the

same probe. There are many possible sources of noise, but in the
context of these discussions only white thermal noise will typically
be considered. This is related to any dissipative process by the
fluctuation-dissipation theorem of statistical mechanics. As will be

discussed, the most straightforward quantitation of the noise is in







terms of an equivalent electrical resistance of the probe which is the
addition of resistances from a variety of sources.
SNR is typically measured in a MRI image (which is a
magnitude Fourier spectrum) using the algorithm depicted in figure
1-3.




Sinland noise
... mean here
Noise mean and
13 std. dev here


Figure 1-3 A depiction of a simple image of a round phantom with
indications for SNR measurement


The SNR for the image depicted is SNR=( ) The means and
std.dev(N)
the standard deviation are with respect to a large number of pixels.
The area of the noise which is used for the measurement is arbitrary
in most cases, since the noise is uniformly distributed in the image
space, except for a few specific cases discussed in chapter 6.
As is well known, noise figure (7) is a standard means of
characterizing the extra noise inserted into the source noise and
signal by system electronics. An expression for noise figure for a
multi-stage system is

NF = NF, + NF2 1 + NF3 1
G, GIG2







where NF stands for noise figure of a particular stage and G stands
for gain of that stage. It is clear from this expression that the first
stage is critical to the system performance. The combination of low
noise figure and high gain in the preamplifier is most desirable.
Currently, the state-of-the art preamplifier has approximately 0.5 dB
noise figure and 30 dB gain.
Figure 1-4 shows the typical method for detection of the
narrowband signal modulated on the Larmor frequency carrier.







^ amplifieri er\/ ^ ^. "
-- ifm a low ps toA/D

90 (o
(o -- -| spl i tter |

Figure 1-4. Quadrature phase detection receiver

The signal is first down-converted to an intermediate frequency.
This prevents the existence of large signals at the Larmor frequency
which could affect the signal. This is split into two equal signals.
Each signal is multiplied by a reference signal at approximately the
intermediate frequency with the phase of the two reference signals
maintained 90 different. The low frequency component of the
outputs describe a vector which rotates and decays with time.
Mathematically only a single down-conversion is equivalent to the








method depicted and the equations for direct mixing are shown
below. A particular frequency component of the incoming signal is
ce-'/1T cos((0ot + )). The process of quadrature phase detection

produces
Low passlce-'/T cos(mot + 0)) cos(ot)}
and Low pass{ce-'T/ cos(o0t+0))sin((0ot). Using
trigonometric identities these can be rewritten as
Low pass{ce-'`T [cos((o0 + co, )t + )) + cos((o. o,)t + ))]}
and Lowpass{ce-'/T [sin((o0.+ co)t + ))-sin((Oo. -co,)t +0)]}.
Typical T2 decay has no influence on the separation of the sum and
difference frequencies and the resultant signals are
ce"'7'1 cos(( o0 -co, )t + )

and -ce-'11T sin((O0o o, )t + 0). If these signals are

assigned to the x and y axes respectively, a spiral pattern is
produced which traces out a direct representation of the end of the
magnetic moment vector as it decays. The term quadrature
detection refers to the use of two reference signals and detectors.
Single detection produces only a projection of the spiral on to one
axis. This results in ambiguity as to whether the signal is greater or
less than the reference signal. Besides incomplete information of the
frequency of the signal, this also results in noise which is both above
and below the signal added together in the resultant acquisition.
Typically the noise in NMR is due to the source probe/sample system
and receiver losses. This implies that the noise is characterized as
white noise and the resultant standard deviation is thus V2 higher
than if quadrature detection is utilized. This improvement in SNR by








-V2 due to the use of quadrature detection is of interest for future
discussion of quadrature coil technology.


Original Work


The above descriptions serve as basic introductions to the
concepts to follow. Chapter 2 gives more detailed background on the
topics necessary to the original analyses in the remaining chapters.
The bulk of this dissertation, chapters 3-8, describes the results

of applied physics aimed at the central theme of improving
sensitivity in NMR and MRI. This is of crucial importance in the field,
because the source signals are so small as to be comparable to the
thermal noise of a probe. The results of almost every interesting
experiment are improved by an increase in the effective SNR. MRI
continues to be a highly significant clinical tool for diagnosis of
disease (8-10) and promises new utility in the form of diagnosis of
breast cancer (11,12) and functional imaging of the brain (13,14) to
name a few. Even when increasing SNR does not reveal new detail of
structure (morphological or spectral), it always has the capacity to
improve time resolution.
Specifically, this work details the attempt to improve the SNR
of images acquired using MRI systems in the range of 0.5T to 2.0T
and with fields of view approximately 5-40cm. The general
philosophy was to design the optimum receiver equipment with
knowledge of the signals to be obtained. Though similar in intent to
optimal receivers for digital systems, the information available about
the analog signals to be received is much less complete. However,





14


some improvements were made over the state of the art for general
systems and further improvements were made for specific situations.
The system from the coil to the first receiver stage were considered

for improvement. Sometimes these improvements were identified
with a particular type of experiment. Again, this allowed the ability
to increase specificity and thus make an improvement in the system.
Because this work has been in an extremely active research

area, some of the topics have been reported in the literature

following the authors independent developments. Specifically
sections of chapter 6 have been reported by other workers in the
field. Unless otherwise noted, it can be assumed that all of chapters
3-8 describe the independent work of the author, with invaluable
assistance and support from the UF NMR research group.












CHAPTER 2
REVIEW OF STATE OF THE ART

Signal-to-Noise Ratio

Since the fundamental theme of this work is enhancement of
the signal-to-noise ratio, a model is needed which describes the
parameters of interest in SNR. A standard formula for the SNR (1,15)
following a 90 pulse is

SNR- = Kq o '0 V [2.1]
!4;FTkT.AJ;

K is a numerical factor dependent on coil geometry; il is the filling
factor (the ratio of sample volume to coil volume); Mo is the nuclear
magnetization; go is the permeabilty of free space; Q is the quality
factor of the coil; Coo is the Larmor angular frequency; Vc is the
volume of the coil; F is the noise figure of the preamplifier; k is
Boltzmann's constant; Tc is the temperature of the probe; and Af is
the bandwidth of the receiver.
The description of SNR in NMR was simplified by Hoult and
Richards in 1976 (15). The improvements over the older formula,
shown above, are primarily based on the use of reciprocity (a
theorem which relates transmission performance of a probe to its
receive performance) and the identification of the important
variables which can be most easily controlled or at least measured.
The formula is presented as









J {B,.MJ}dV,
sample --,T [2.2]
4kTAJR

in which the numerator describes the signal induced by the spin and
the denominator describes the thermal noise generated by a resistor
(the coil). The most essential part of the equation for our purposes is
the Mo, the magnetic moment, is equivalent to the field of a dipole

rotating in a plane at the Larmor frequency, and the magnetic field
appears in the numerator, while the square root of the product of coil
resistance and bandwidth appear in the denominator. These are the
only parameters that will be used in the following discussions.
In 1979 Hoult and Lauterbur extended their discussion of SNR
to biological tissue (16). The expression above is still fundamental;
however, the resistance in the denominator is shown to come from a
number of sources of relevance in MRI. Reciprocity is here applied
to NMR probes; however, the basic concept is well known from
general electromagnetics. The primary result is that a receiving
probe may be described in terms of its transmission properties.
Using reciprocity, one finds that the power dissipated in a coil can be
lost in 4 fundamental ways. First, the coil may be made of a non-
superconducting material which implies ohmic losses in the
conductor. This is worsened by the skin-effect conduction pattern of
current at radio frequencies. The skin depth decreases as f-1/2
(where f stands for frequency) which implies that the resistance
increases as f1/2. Since the signal induction voltage increases as f2
(Mo increases linearly with Bo and thus frequency, and the time

derivative introduces another linear dependence of frequency), the








SNR increases as f7/4 for ohmic losses only. Second, there will be non-

conservative electric fields associated with the coil components. If

the sample is conductive these electric fields will deposit power in

the sample. The frequency dependence for this effect is complicated

and may involve a resonance (17). Third, the magnetic field which is
the primary property of the coil will also generate eddy currents in a

conducting sample which represent power loss. The inductive losses

are such that the resistance increases as f2 (16) thus providing a

linear increase of SNR with frequency for inductive losses alone.

Fourth, the coil has a certain propensity to radiate energy as an

antenna. This is directly related to the size of the structure

compared to a wavelength of the frequency of interest and the

geometry of the structure. Radiation resistance of a simple Hertzian
loop increases as f4 (18), and it would appear that there is no

improvement in SNR with frequency; however, the signal induction

formula must be modified as well if the radiation is an important

term.

The first effect can be addressed by maximizing the cross
section of the conductor as far as is reasonable, using the best

available material, and choosing high quality discrete components.

For mid to high field MRI, this loss is not normally dominant. The
second effect can be reduced by use of balanced matching (19),

reducing the effective inductance of the coil (20), and the use of

Faraday shields (17). All of these methods reduce the effective

electric field in the lossy sample. An example of the process is
discussed below which demonstrates the effect of a particular

implementation of reducing the inductive reactance of the coil.








The power lost due to the electric field in the sample is
P= fffo"EI dV, where o is the conductivity of the volume and E is the
v

electric field. Figure 2-1 shows the voltages on the components
which are responsible for the non-conservative E-field losses.
C 2C




L L/2 2,C L/2


V + V/2 +








Figure 2-1. A comparison of two tuned circuits with different
placement of components, which produce different power losses


Since V=JE.dx, it is clear that 1/4 as much power is lost to the

sample if the inductive reactance is reduced to 1/2 its previous
value.

The third effect remains if one couples to the spin system. The
ideal would be to have the magnetic field only in the region of the
sample of interest and only in the transverse direction. (The rf
magnetic field parallel to the static field produces loss but does not
couple to the transverse magnetization.)
The fourth effect can be moderated by use of geometries which
have lower radiation resistances, for example quadrupoles.








Distributing capacitors around the inductance may also lower the

radiation resistance by prohibiting a buildup of phase of current

around the loop.

There is one potential complication in the equation for SNR
given above. The calculation of the rf magnetic field may not be

accurate if done in the absence of the sample. Normally, however,

the permeability of biological tissue is approximately that of free

space and the conductivity is not so high as to induce currents which

are on the order of the coil currents. If these conditions apply and
the sample dimensions are not close to a wavelength, then a free

space calculation of magnetic field is sufficiently close. (19)

Radio-frequency Coils

The first level of application of knowledge of the problem to
improving the SNR for an image is to identify a typical sample and

characterize it in terms of size and electrical properties. For optimal

performance a receiving probe should be designed so as to pick up

only noise which is inseparable from the signal of interest, i.e.
inductive losses only in the region of interest with as little field in

the static field direction as possible.

The first NMR based systems were small bore magnets for

characterization of physical properties and chemical analysis. The

magnetic probe was a volume coil such as a saddle coil (15) or

solenoid. This was appropriate to match the typically non-electrically
conducting small samples in test tubes.

In the early 1980's the process of matching the coils to specific
purposes began to be applied to clinical imaging. Surface coils, which
were basic Hertzian loops, were used for obtaining inhomogeneous







but high SNR images of structure near the surface of the human
body. The first use of a surface coil in vivo was described by
Ackerman et al in 1980 (21). This work dealt specifically with in-
vivo spectroscopy; however, the result that higher local SNR was
obtained with surface coils was soon applied by those interested in
MRI. Ackerman et al. showed the well known magnetic field
produced by an electrically small circular loop with reference to
figure 2-2.
x














Figure 2-2. A circular current loop with radius a, and unit current


The two components of the magnetic field are
BPx [K K +P2 +x2 E] and
aap)x-]I- (a-p)+2 I
B, =2P 1p1o2. + -2 where K and E are complete
elliptic integral of the first and second kind.
elliptic integrals of the first and second kind.








By 1984 several papers dealing with region-specific surface

coils had reached the MRI literature (22.23,24). These papers

quickly showed that images of many anatomical regions could be

greatly improved by this procedure. From the description of the
signal-to-noise ratio above, it was clear that the more local the field
to the region of interest, the lower the overall resistance of the coil

and therefore the better the SNR. An analytical solution for the
optimum diameter of a circular surface coil was described in 1985

(25), and was later taken to derive the ultimately acheivable SNR for
a given depth (26).
Before this time the only coils available were volume coils for
the whole body or perhaps the head. These coils include the saddle
coil (16) and various versions of the birdcage resonator (27). The
saddle coil is derived by finding the positions of four currents lying

on a cylinder which flow parallel to its axis and provide the best
homogeneity of the field in the region of the center of the coil (28).

The birdcage coils are a general class of distributed phase coils which
are analogous to a full wavelength of transmission line connected in a

continuous loop. One of the early versions of this was exactly this,
where the transmission line was coiled into a toroid shape and the
center conductor was exposed on each loop near the sample (29).
The Alderman-Grant resonator is a two-leg version of the discrete

element implementation (30). The general version of this coil
described first by Hayes et al. (27), may have any number of legs

and the current in these legs follows a sinusoidal distribution which
has been shown to be the optimum coil for cylindrical geometry at a
point in the center (31).









About this time quadrature volume coils also became useful for

clinical imaging (32). Quadrature excitation uses half the power for

the same rotating magnetic field. The method of production of the

quadrature rotating field is exactly analogous to using Euler's
identity to produce an imaginary exponential with the sum of two
sinusoids. One sinusoid produces a rotation and a counter-rotation

each with an amplitude of one half. Putting all input power into the
rotation which couples to the spin is most efficient as can be
observed from eq. 2.2. The arguments demonstrate lower power for
excitation and reciprocally higher SNR in reception. Thus quadrature,

like the quadrature detection receiver, was designed to receive two
equal magnitude signals which differed in phase by it/2.

The birdcage resonator naturally has two modes related to the

direction a traveling wave propagates around the structure. Driving

the coil from some location of the coil will non-preferentially activate

both modes with the result that the two traveling waves add and
form a standing wave. This results in a cosinusoidal distribution of
currents. At a position 90 degrees from the drive point (where a

maximum in current appears) the current should correspond to
cos(900) = 0. Therefore another drive point can be located
independently at this location. The relative phase of the second
drive allows cancellation of one or the other rotational mode
produced by the first drive point and thus will determine which
rotational direction remains. Any other volume coil is made
quadrature simply by placing another coil physically perpendicular
to the first coil and driving the two with a 90 phase difference.








The next improvement of coils was the application of
quadrature principles to inhomogeneous coils, i.e. surface coils. The

first reference to non volume quadrature was by Arakawa (33) who

described the principle applied to slightly curved surface coils. The

first entirely planar quadrature coil was described by Hyde et al. in
1987 (34). Quadrature surface coils have since become quite
common with routine use on clinical MRI systems. The

generalization of quadrature to more coils and other phases and
magnitudes is discussed in chapter 5.

Independent Signal Acquisition
In 1986 Hyde et al. (35) first published a discussion of the
possibility of making images using independent coils. In this paper a
claim is made for the equivalence of independence and the condition

of zero mutual inductance for coils. Much controversy has arisen
concerning this contention. Hyde et al. presented a theorem relating
noise correlation and mutual inductance in 1988 (36) and refined in

1991(37, p. 36): "If two coils exhibit zero mutual inductance, there
can be no correlation of noise." This position was refuted in 1989

(38, p. 402): the claim that "two coils with no mutual inductance will
always have uncorrelated noise....is not correct in most instances" and
also in 1990 (39, p. 208): "these calculations....contradict assertions

that....coils with zero mutual inductance have no correlated noise."

The authors described the noise correlation coefficient as an electric
coupling coefficient. In 1992 (40, p. 85), a slightly different result
was given: "if....there is no voltage cross talk between the coils, then
noise, either from coils or sample, will not be correlated." Chapter 4








contains a concise refutation of the claims summarized above from

references 36, 37 and 40.

The fundamental concept is to receive signals from separate

and somewhat independent coils and preserve the signals and noise

to allow pixel-by-pixel combination of the signals. The degree of

independence in the noise determines how much the SNR improves

by addition of the two signals.
This work led to the phased array which was described by

Roemer et al. in 1990 (39). This is a general discussion of multi-

channel coil systems. Each coil has its own receiver. A linear array

of loops is a simple model to consider. The first coil receives signal
from a region primarily beneath it. The second coil receives some

signal from the region of the first coil, but mostly from sample

immediately adjacent to it. If one considers having made two images

separately it should be clear that the optimum image of the two
signals will have SNR's higher than either of the single images in

every location. The advantage this brings over quadrature is the

ability to arrange the coils in the z direction along the main field and
to combine each pixel with different combination weighting to obtain

optimum performance everywhere. It should be pointed out that

coils which are coupled to the same lossy sample will not generally

be entirely independent. This is because noise generated in the lossy

medium can couple to both coils producing a correlation which

prevents complete independence.
One of the problems solved by Roemer et al. (39), is the

apparent necessity to decouple the receiver coils as much as possible.

Voltage crosstalk between coils introduces a noise correlation and








non-optimum signal combination (in general). The method employed

is to mismatch the coil to the preamplifier in such a way that the

current induced in each loop during reception is much lower than it

would be in the matched condition. In a signal-to-noise sense, this
process is non-lossy. The reduction in current means that the

receiving coil, as a source, loses efficiency, resulting in less transfer

of signal and noise to another receiving coil. The intrinsic noise
correlation induced by the sample is not influenced by this method.

It is shown in chapter 4 that isolation is not actually necessary for

optimization as long as the coupling is measured under experimental
conditions.

Another topic discussed in the work of Roemer et al. (39) is the

problem of image reconstruction utilizing the multiple images

received. The general conclusion is that the optimum treatment of

the images requires knowledge of field characteristics of the coils

used as receivers and noise correlations between each pair. A
complex addition utilizing this information gives the optimum SNR

for each pixel in the final image. As a matter of practical importance,

a much simpler algorithm, i.e. the square root of the sum of the

squares of the individual image pixels, give a nearly optimum result-

- at worst about 10% inferior.

There are many ways in which any intrinsic independence is

preserved while still getting all of the signals to the computer. The

most general is described above--using independent receivers all the
way to the a-to-d converters. Another method which will be

described in chapter 6 is by time multiplexing the data (41). The





26


requirements for this will be discussed later but the result is exactly

the same as for the multiple receivers.

Another method is a more specific procedure in which narrow

band filters at the Larmor frequency are matched to the coil

positions and field characteristics (42). The great benefit of this

method is that the output can be fed to a standard single channel
system and no software modifications are necessary. This will be

further discussed in chapter 6.













CHAPTER 3
VARIABLE FIELD OF VIEW COILS

Introduction
A variable field of view surface coil was designed to allow
variation of the field of view to be imaged while maintaining the
tuning of the coil to a substantial degree. This is an attempt to
improve imaging by allowing a significant size region to be imaged

but still enable the user to obtain optimal performance on a smaller
region. This work was completed prior to the appearance of multi-

channel systems.

Radio frequency (rf) surface coils have become a necessity for
high resolution MRI of the human body. This is because optimization

of the SNR primarily requires that the size and shape of the receiver
coil provide an optimum filling factor given the region of interest
(43). Small rf coils provide larger BI fields per unit current while

presenting a smaller surface area to the load. The reduction in the
surface area of the load results in less degradation of the coil circuit

Q, reducing coil losses (17,44,45).

Popular regions of interest for surface coil designs include the
orbits, TMJ, neck, spine, heart and extremities. A number of
different geometries with varying degrees of performance (46-50)
have been proposed to optimize SNR for each of these regions of
interest. Due to a large demand for MRI of the nervous system, this





28

work has focused on the spine. This area is especially challenging

because the clinician would like to survey the entire spine in a single

acquisition and at the same time obtain the highest possible SNR for

high resolution from thin slices. The result is that surface coils for

the spine may be a compromise between field of view and high

performance.

Several workers have attempted to solve this problem by

constructing coils which either can be repositioned (51) or have sets

of elements which can be switched into place to change the field of

view (52-54). Repositioning devices can be very effective and easy

to use because there is no need to reposition the patient: the coil
simply moves in a space created between the patient and the table.

However, larger fields of view require multiple acquisitions. The use

of multi-element coils, however, is limited by the number of sections
which can be effectively switched. In addition, such switching

requires PIN diodes at each junction which may reduce the efficiency

of the design by reducing coil Q. Another approach to the problem is

to construct an array of coils which have no mutual inductance and

to connect each coil to a separate preamplifier, phase detector and an

analog-to-digital converter (39). This approach is effective; however,

it requires much additional electronic modification of the receiver

system which is quite expensive.

The objective of this work was to provide a flexible, efficient.
low-cost solution to the problem of varying the field of view. The

general strategy was to design NMR coils that would be variable in

size but remain electrically stable during adjustment. This kind of
probe would be applicable to clinical imaging, where tuning and








matching for individual patients is highly inefficient and undesirable.
A common method of adjusting dimensions of electrical systems is
the use of the trombone (55). This adjustment of size coincides,
however, with a drastic change in the impedance of the system. In
fact, trombones are typically used to tune to a particular impedance
and have even been utilized for this purpose in NMR (56). The goal
was to make a self compensating trombone which adjusts with no net
change of impedance (tuning in the NMR probe).


Methods
A prototype flat rectangular coil was constructed that could be
adjusted from about 12.5 cm x 17.5 cm to about 12.5 cm x 32.5 cm
and was made from 1/2" and 3/8" tubular copper. This coil is shown

in figure 3-1.
Inner
copper tubing



Outer
copper tubing


Figure 3-1. A schematic representation of the adjustable coil







Using this coil as a model, an analysis can be performed which will
lead to an approximately constant resonant frequency over a
considerable range of size adjustment. This analysis assumes that
coil dimensions are small enough to ignore wavelength effects and
negligible resistance in the coil. Figure 3-2 shows a circuit model of
the coil and figure 3-3 shows a simplified version.

LI C(x) L2z+ L(x)


cTrUNET I ~

L1 C(x) L2+ L(x)

Figure 3-2. Circuit model of the adjustable coil

LT(x)

0 L / 'ICT(X)

SCTUNET T

Figure 3-3. Simplified circuit model of the adjustable coil


For constant tuning, Leq must be a constant, where
.1 1
jotLq = jLT(x)-X ji I and o= I
(OCT (X) LeQne-
For the case of a simple trombone, schematically shown in
figure 3-4, the following equations approximately apply (57):
L(x) = 1.97E- 7x[ln(4x/d2) + d2/2x- 0.75] H








C()- 2EE(1- x) F
ln(d, d2)
where linear dimensions are in meters and SI units are used for E.,
the permittivity of free space and c,, the relative permittivity for the
dielectric.

C(x)
I L(x)
II





j44 1i

x

Figure 3-4. A schematic of a trombone section


Also CTW(x)- CC(x) and LT(x) = 2[L, +L, +2L(x)].
C(x) + 2C,

In order to satisfy the constraint that Leq be constant, C(x) and/or

L(x) must be variable. As can be seen from the equations above, if
the trombone is used, there is only one way to make L a constant.
This is to make c, vary as a function of x to give LT(x) and
1/c02CT(.X) equal derivatives in x. By slightly modifying the
trombone, the goal of constant L can be attained in several

different ways.
Results
The approach presented in the following section leaves the
inductive change nearly the same as in a standard trombone to






simplify analysis. For the prototype discussed, the capacitance of the
system decreases too rapidly (the resonant frequency increases) as
the coil is enlarged. Two methods were utilized to compensate for
this situation.
The first method utilizes a set of telescoping conductors similar
to the radio antenna of an automobile. From figure 3-5, it can be
observed that the rate of decrease in capacitance is slowed with this
method.


.c C




Figure 3-5. A set of telescoping conductors

One can vary the diameter and length of each segment to make the
resonant frequency approximately constant (within the Q bandwidth)
over the range of size variation desired.
The second method, shown in fig. 3-6, involves the use of a
layer of copper foil between layers of dielectric. This foil can be
easily trimmed to compensate for the fast decrease in capacitance.
The two methods described can be used separately or
combined to obtain the desired result. Figure 3-7 shows empirical
results for the prototype coil described using a normal trombone, one
sliding inner conductor, and an inner conductor plus a piece of foil








Copper
Dielelectric Tubing

Dielelectric TFoil















Figure 3-6. A trombone modified with extra layers of conductor and
dielectric


tape of a particular shape. The results demonstrate that the change

in resonance frequency with change in coil size can be made
negligible with appropriate modifications. Similar results could be
obtained for coils made with foil strips for the conducting surfaces.
To evaluate the coil's effectiveness in a clinical setting, a

volunteer allowed imaging of his spine. It is frequently of interest to

obtain a large field of view of the spine and then to concentrate on

one or two vertebral bodies in particular. Figures 3-8a and 3-8b

show the results of using the coil in an extended and a fully
contracted position respectively. The images were obtained at 1.5T
with a GE Signa using a spin echo sequence with 256 x 128 pixel
resolution, 500 ms repetition time, 30 ms echo time, four averages, 5
mm thick slices and a 40 cm field of view.













80




C
(-
C
0
0)

U
0 70- 13


c x


C) X




60.
0 2 4 6 8 10 12

1 trombone
cm extended 3 conductors
x 3 cond. + foil

Figure 3-7. Empirical results of resonant frequency vs. coil extension



Conclusion

The possible applications of the basic idea are numerous. Any

probe which has a linear or circular segment can be made in this

way. Also, if one does not wish to attempt to modify the trombone, a

varactor diode can be used to maintain a constant resonant

frequency. The size could also be varied remotely with hydraulic or

pneumatic devices or motors.








































b
Figure 3-8. Spine coil images a. coil
extended b. coil contracted








Another slightly different application involves any case in

which two loops are coupled, producing splitting of the resonant peak

dependent upon the separation of the two loops. By utilizing the

above modified trombone approach, it should be possible to vary the
coupling while forcing the peak of interest to remain at constant

frequency. An example of this is an adjustable cardiac coil.
It should be noted that the resonant frequency does depend on

coil resistance as well as reactance. If the Q is fairly low this may not
be negligible. Therefore, both the resonant frequency and match
cannot be exactly corrected because the extension makes the coil

have greater resistance.

The work presented here was previously published in Magnetic
Resonance in Medicine 13, 378-384 (1990) and forms the basis for
US patent # 5,049,821 assigned to the University of Florida.













CHAPTER 4

COUPLING BETWEEN NMR COILS
Introduction

Quadrature combination, as introduced in chapter 2, was

implemented primarily for use with volume coils and provides a root
two improvement in SNR and half the power required for excitation.
Hyde (34), and Arakawa (33) initiated the use of non-volumetric
coils with quadrature combination and substantial improvements can
be produced. The real difference is that the magnetic fields of these

coils are not homogeneous, which implies that the optimal
combination occurs only for those regions where the magnetic fields

of the two coils to be used in the combination are of equal
magnitudes and are perpendicular.

The theory of quadrature is insufficient to describe all
situations of different field magnitudes and arbitrary crossing angles.

The following two chapters develop a theory to cover all situations.

This will be denoted generalized quadrature and is also accurately

described as optimal signal combination which, however, also
includes consideration of coupling. The general topic of optimal
combination will be broken into two sections. In this chapter
compensation of signal and noise coupling will be considered. Chapter
5 deals primarily with the generalization of phasing to the array
with inter-coil noise correlation.








Since the introduction of the quadrature coil (32). the first

receiver coil array in MRI, the issue of isolation (32. 58) has

frequently been an area of concern and controversy. Because of

existing disagreement, especially on the relationship of noise

correlation and coupling (37,38,40) as discussed in chapter 2, the

general problem will be examined by reducing it to more specific

areas of interest. A semantic description of coupling and noise

correlation needs to be carefully given.

cross talk--this can also be referred to as coupling--cross talk

between two coils is non-zero if and only if a current originating in
coil 1 induces a current in coil 2.
noise correlation (nc)--is zero if and only if the noise

voltages delivered to each coil's load (preamplifier) can be added

with arbitrary phase with the result that the total noise power is the

same as the sum of the individual noise powers.

intrinsic nc --noise voltage originating from a shared lossy

load. This is independent of crosstalk.
extrinsic nc --noise voltage originating from the coupling of

one coil's noise voltage to the other coil

It is shown here that cross-talk and noise correlation are, in

principle, separable. Consider the circuit model in figure 4-1. The

coil matching is done as described in the work of Roemer et al. (39).

The coils are positioned so as to have zero mutual inductance in the

absence of a load and it is assumed that the load does not
substantially change the field pattern. Therefore the coupling path

between coils is confined to the path through the load. The model
shown includes each coil's isolated noise sources into the resistance






































Figure 4-1. Equivalent circuit for two coils loaded by the same
lossy sample and thus exhibiting intrinsic noise correlation










in series with the coil inductance. Since each coil is coupled equally
to the model's inductive load, this resistance produces purely

correlated noise.

To examine the process of cross-talk, assume a voltage source
va in series with the inductance of coil 1 (vb shorted for this case).

It is of interest to know how much the induction of signal and noise

into coil 1 is transferred to coil 2. The cross-talk constant will be
defined as i2/i l. It is easily shown that lim = 0 This was

demonstrated by Roemer (39) for the case of no intrinsic noise
correlation.

Now consider the noise which is generated by the resistor of
the lossy inductive load (vb active, va shorted). It is clear without

any calculation that the induction of the noise voltages into coil 1 and
coil 2 is identical and not a function of Ramp. The noise delivered to

amplifier 1 comes from the inductive load resistor and the local
resistor of coil 1. The relationship between these voltages is also not
a function of Ramp. The conclusion is that intrinsic noise correlation

and cross-talk are entirely separable whereas cross-talk is entirely

responsible for extrinsic noise correlation. Any lossless method of
increasing isolation should result in reduction of extrinsic noise
correlation. Intrinsic noise correlation is impossible to change in a
lossless way. With these factors in mind it is logical and useful to
treat noise correlation and coupling as distinct phenomena.

The remainder of this chapter discusses the investigation of
coupling between MRI coils and the effect on the attainable signal-







to-noise ratio (SNR). It will be shown that if the signals are
combined properly, there is no loss in SNR due to coupling. Long
wavelength approximations will be used and it will be assumed that
isolaimon (dBl
the degree of effective coupling (k 10 20 ) is not equal to one.
In chapter 5 the issue of noise correlation and the generalization of
quadrature gain will be examined. The technique of reciprocity (15)
will be used to examine SNR by assuming the coils are driven, and
the principle will be confirmed with a comparison to signal reception.
Coupling with no shared resistance will first be examined and several
examples will be given before describing the general situation.
Discussion
Relative SNR is the parameter of interest and this is
represented via reciprocity as the rotating component of magnetic
field divided by the square root of power input to the system. This
method is clarified by example. Consider the single saddle type coil
with arbitrary arc length shown in figure 4-2.


(~B)






Figure 4-2. A saddle type coil seen from an end perspective


A point of interest (POI) is defined at the center of the coil The coil
has some resistance R which in general, is due to coil electronics and








local sample loading. When the coil is driven with power P, the
current in the loop must be of magnitude \'P/R. It is assumed that

any matching circuit loss can be lumped into this loop resistance.
The magnetic field is a linear function of current and we define B as
the amplitude of the rotating field produced by unit current at our
POI. B/VP is independent of current for a single coil and is a point-
wise measure of efficiency exactly analogous to SNR, thus

T -P- [4.1]


The next step is to consider two coils which by assumption
have no intrinsic noise correlation. This implies that whatever the
mechanism of coupling, there is no resistor other than the individual
isolated coil resistances. To see the effects of coupling, the case of
completely independent coils are first considered. Figure 4-3 depicts
two identical reduced arc saddle coils which have been oriented for
zero mutual inductance.


Figure 4-3. Two reduced arc saddle coils, oriented for zero mutual
inductance









There is indeed a rotational position which corresponds to zero
mutual inductance aside from the usual position of a 7t/2 rotation. To

observe that this is so, assume that the coils are positioned directly
on top of one another. They will be strongly coupled with all of the
flux from one coil passing with the same sense through the other. If
one is rotated until it does not overlap at all, the flux of the first coil
now passes through the other loop with the opposite sense. It is
expected, in practice, that the flux linkage would be a continuous
function of angle which must have a zero between a positive and a
negative region. Therefore, it is possible to overlap the coils for zero
mutual inductance and the angle 0 is defined as the physical angle
(between bisection axes of the coils) at which this occurs. This angle
also gives the angle at which the field lines at the POI cross.
Assume the loops have the same resistance R and coil 1 is
driven with current ii and coil 2 with i2. The fields cross at angle

and correspondence of the x- axis with real numbers and the y-axis
with imaginary values gives the total rotating magnetic field
efficiency:
BI1, + i2ej i1 +i2e
-2 [4.21
Ii,12R+ i2I2R i, r +1i2r[

Since the fields are rotating we note that a positional delay is
equivalent to a time delay. Suppose the currents are electrically
varied in an equal power split so that one of the currents is delayed
with respect to the other by the angle 0. The equal power split will
result in equal magnitude currents and thus the efficiency







B I1+ebeeJ
A It is clear that the choice of 0 = produces the
VR V2
result that the total rotating magnetic field efficiency is -/2 times
greater than for the linear coil. This action simply time delays the
field which is positionally advanced so that the fields magnitudes
add. This is the generalization of quadrature gain (59) to any
physical angle given the important assumption that the coils are
truly independent. This concept is more fully developed in chapter
5. It should be noted that this principle directly applies for low
frequency coils, micro imaging coils, coils with nonconducting
samples and any other application where the resistance of the coil
conductor is the dominant source of resistance, and there is no
coupling between the coils.
Since the discussion above shows that /2 gain can be obtained
for any physical angle if the coils are independent, the case of
coupled circuits is next examined. The description will remain
general by assuming the change due to coupling can be described as
shown in Table 4-1.


Table 4-1. The forms of coupled vs. uncoupled coils currents


coil 1 coil 2


uncoupled il i2
coupled II = il + ki2eJi3 12 = i2 + kileJ 3








The parameter 3 describes the phase angle with which the coupling
occurs, k is the magnitude of the coupling and 0, as before, is the
electrical phase angle imposed on one of the coils by the
splitter/combiner. Note that the currents which are seen as
originating with a given coil are not the same as the uncoupled coil
currents but they converge for small k. In the coupled case il is the
current produced in coil one when driven and with the other coil
present but not driven. Figure 4-4 shows an electrical block diagram
of the coils in figure 4-3. The total efficiency of the pair of coils with
e+BI
the net currents I, and 12 after coupling is [4.3]
-,112 R + I,2 R

If these net currents can be manipulated as freely as the uncoupled
currents by variation of the splitter circuits the condition 1, = le-"

may be satisfied and the maximum gain remains 2. Note that
complete freedom of manipulation is impossible if k = 1. The splitter

solution is given by i = e' 1- ke--- [4.4]
i. 1- ke'oe-'*
This implies generally different magnitudes of the driving currents
for equal amplitude net currents after coupling. Note that the
imbalance in i, and i, does not correspond to amplification but

rather a shifting of power from one loop to the other. No active
devices are required. An example of this process is impedance
matching to some arbitrary impedance. This scales the noise and
signal equally if the circuit is lossless.
Because the ability to obtain lossless performance in the
presence of arbitrary coupling may be unexpected, the gain in
efficiency with coupling will be examined in detail for several











rA


keJO
C- r


Figure 4-4. Electrical block diagram of the coils of figure 4-3








specific cases. The single coil field value, B/ R, will be separated
and the remainder designated as the gain G to which we will refer
hereafter. The total gain in efficiency is then

1 + Ake..'. + (Ae' + ke')e [4.5
FIl + Ake j'o2 +I(Ae' + ke'P)e' 1

where i' = Aei,.


Case I deals with parallel fields and case II addresses perpendicular
fields.

]1 + Ake^' + (Aej + keJ)]
1. = 0 G(O = 0)= + Ake" + (Aef + kej5)j [4.6]
1+ Ake"O'" + Ae' + kel'


Selection of 0 = 0 and A = 1 results in gain

1 + ke" + (1 + ke') .
G( = 0,0 = 0) = 2+1 = 2. [4.7]
V1 + k5 +|1 +k I


Note that this particular condition leads to equal currents in
the loops which add in phase. Figure 4-5 shows vectors
representing the rotating fields (at some instant in time)
from the two coils at the POI which are added under the
conditions described above. The same arguments apply for
0 =nt.









0 =0

Independent loops Coupling, arbitrary 13 coupling, 0 = 0











Figure 4-5. Vectors representing the additions of the rotating
fields described in case I


II. S = it/2
A. Consider the special case 13 = 0 with A = 1:
|1 + ke'9 + (el' + k)e"2
G( 0 = 7/2,p3= 0) = [4.8]
^211+2kcosO + k2}
If one chooses the splitter angle to be 6 = -7c/2, the

resultant gain is ,-k2 However, the currents in loop one
v+ k'
and loop two after coupling are of equal magnitude but are
not out of phase by 7t/2. The angle 0 should be chosen with

full knowledge of the problem, including the description of
coupling. It is easily shown that the maximum gain of
occurss at the angle where cosO = --2k. Fig. 4-6a

graphically demonstrates the field addition under these
circumstances.









O=7C/2
coupling, [=0


'N,


S= Cos'r -2k2







Figure 4-6a. Field additions for Case II, part A


It is important to note that the splitter angle is simply the
angle which produces a net current phase difference of -7r/2
when coupling is considered. Again the same argument
applies for [ = ic.


B. Consider the special case c = 7t/2:
/2.
For A = 1 and 0 = -It/2, a gain of ,l+2 is obtained. This
l+k2
result is the same as above and has been previously obtained
for standard quadrature coils (2). In this case there is no


coupling, O=-it/2


Independent lc








phase which produces the full gain. Another degree of

freedom must be exploited -the combiner power split. On
1lk
optimization one finds that the gain is -2 when A = ---

SIt is no coincidence that this equalizes the currents in the

loops. Figure 4-6b shows the field additions for these cases.


Independent loops


u i=7t/2

coupling, P=iT/2


NO
N


coupling, 0=-ir/2


N


1+k
coupling, A = -k
1-k


Figure 4-6b. Field additions for Case II, part B


Reciprocity predicts the above results, but care must be taken

in proper consideration of the situation. The special cases above are

illustrations that if one drives two equivalent loops, such as shown

in figure 4-3, which have only internal resistance (no shared

resistance) and there is no other power sink, then forcing equal

magnitude currents with electrical phase difference which is the







negative of the physical field crossing angle will result in a gain of
2 at a symmetry point, regardless of coupling.
Next the reciprocal model of SNR is compared with a
description of reception. As indicated above, the general expression
for efficiency of an equivalent coil pair with coupling is
B 1 + kAe""" + (Aej + kefl)eo" [4.9]
R 11l + kAeeO +|+(Ae + kte)e'2

Suppose the following situation applies for two equivalent
independent loops:
S, +N, ; S, +N2 where S, = S,e",NN,N = 0 and (N, 12=(IN212).
Note that 0 here has the inverse of its value in transmission for the
same spin system. The phased addition of the two signals give the
SNR

S. + S, e S
\2 if 0 [4.10]
(FN, + NJr ) y,~

If the coils are coupled and then the signals are phased and scaled,
as done for the currents the result is

IS, + kS,e"'+O' + A(S,e' + kSe'")ele
'N, +kN,e' +A(N, +kN,e"p)e'o

S 11 + kAe"8+' + (Ae" + ke')e' 1 [4.11]
(N2) [l + kAe^'2 + jAeO + ke' ]'

The expression for gain obtained from reception and transmission
are exactly the same by identification of B/,Rwith N The
phasing and weighting which equalize current levels actually scale2)
phasing and weighting which equalize current levels actually scale








the signals according to their levels. The effect of a particular
splitter/combiner is different in transmission and reception because
the coupling occurs after the power split in transmission but before
the combination in reception. This makes sense when one observes
that the noise in each loop is equivalent, which means the combiner
adds SNR values which are different. It is easy to see that optimal
treatment implies weighting them according to their value. The
coefficient A does this in the signal reception equation as it equalizes
current in the transmission equation.
So far only the case of equivalent loops at a symmetry point
has been considered. It can, however, be shown that the conclusion
that coupling does not effect attainable SNR is general when there is
no shared resistance.
Consider the following situation for two coils and fields at a
particular POI:
coil 1 has resistance R, and produces Bi, with input power
I,' 12R ,.
coil 2 has resistance R2 and produces cBi2 e" with input
power li,2 R2.
+ i,+ci9
For this case the efficiency is Ai +ci2e 1 [4.12]
12i [R, +1ji, I'R2"

If the power splitter is defined so that i2 = Ae' i, the proportionality
Bil +cAe~ieJQ 4.3
can rewritten as +cAee14.13
V/R +A2 R,
Since the denominator has no dependence on 0, the expression can be
maximized by separately maximizing the numerator by choosing 0 =








1 + cA
- Thus the scaled rotating field is proportional to -
R +A' 2R,

Through a maximization procedure one finds that A =- and the
R,
B
corresponding efficiency ,- I -+c'R/R, When coupling is
R,
A1i, + ke'0i, + c( i + ke'0i, )e"
considered, the efficiency is I __ [4.14]
1, + ke'i2 2R, + i, + kei'iR2,

If i2 = Aeli, is again defined and for simplicity of notation

K = ke'5 and a = Ae'O are defined as well, this can rewritten as

i +C( a + K ,
B~l+Ka+c(a+K)e^ ~ ____________
fil=___________I___ + Ka [4.15]
S'l+Kal2R +la+KI2R, R, + a+K 2R
VI I 1+Ka
The form of this expression is identical to the expression without
an y a+K R,.
coupling and by setting a+--- =c e-j or equivalently
1 + Ka R2
c Re-" K
R,
a = -R the optimum gain is once again obtained.
1-c e- K
R,


Finally, the general case with noise correlation is considered, in
which the fields at the POI and resistances for two coils are generally
unequal and some of the resistance is shared. The details of the
derivation of noise correlation parameters will be investigated in
chapter 5. The proof that coupling need not be detrimental follows
that above. First the expression for rotating magnetic field divided
by the square root of power is examined for the case of no coupling.







B i, + cie [4.16
Ifp I'1R +i-i, '2 R ir +( i,'i,)R,

The extrinsic correlation which would typically be associated with
the shared resistance is not included; instead a hypothetical ideal
case in which coupling has been totally eliminated is defined.
If i, = ai, is defined as above, the following expression is
obtained.
B 1+ cae'17
P" VR, +a2 R, + (a+a)R,2 [4.17]
This can be maximized with respect to a to provide the best SNR (as
B
in Part II). When coupling is included it is clear that -

ji, + K;i + c(i, + Ki, )el"'
I', + Ki2I R) +i, + Ki, IR2 +R, 2{(i, + Ki2)(i +Ki)' + (, +Ki2, )' (i, +Ki,)}

[4.18]
B
If i, = ai, is defined it can be observed that =
TP
11 + Ka + c(a + K)e" I
[11+ Ka12R, +la+ K2R, 2+R,2{(I+ Ka)(a+ K)'+(l+ Ka)'(a+ K)}

[4.19]
and the expression can be rewritten to show that
( a+iK e ,
B l+cda+- e'*[
B K l+Ka) [4.20]
IRD a+K +R a+K a+K (
I l+Ka [l+Ka ll+KaJ J
a+K
Identification of a and -- demonstrates that the form of the
1+Ka
expression and therefore the maximum is the same as the case of no
coupling.








Conclusion
It has been shown that noise correlation and coupling are
separable and distinct processes. Furthermore, it has been
demonstrated that coupling alone does not represent any loss in the
attainable SNR if one accounts for the parameters of the coupling and
uses the two degrees of freedom, weighting and phasing, in the

splitter/combiner. Coupling is not intrinsically lossy, rather it is the

effect on the current distribution which is of significance. Reciprocity
implies that the current distribution gives all of the information to
describe the coil. From reciprocity the desired end point is known
and the combiner and or the coils are adjusted to achieve this end.
The above treatment implies that multiple receiver systems can be
designed so that all of the effects of coupling are eliminated after

acquisition providing the coupling constants are measured with the
load in place. In practice the smaller the coupling the easier the

compensation, because if the coupling is strong, a change in the
characteristics of one coil affects all other coupled coils substantially.

This results in a multivariable solution, instead of a single variable
solution with small perturbation effects on other coils which applies

for small coupling. In theory, however any degree of coupling other

than unity can be overcome.













CHAPTER 5
GENERALIZED QUADRATURE
Introduction
The first use of an array of coils to improve signal-to-noise ratio

(SNR) in NMR appears to have been by Chen et al. (32) with the use
of a two element radial array. This was a so-called "quadrature"
volume coil in which the individual saddle coils were oriented with
their fields perpendicular. Direct combination through a 90 degree
combiner gives close to a -2 gain in SNR for this case. Several other
implementations of these orthogonal two element arrays have since
been introduced to the literature (60,61). The concept was applied to
non-volume geometries by Arakawa et al. (33) with the quadrature

half saddle surface coil. The concept is the same as for the volume
case except that the gain is approximately -i2 in a smaller region
where the field lines are perpendicular and equal in magnitude.

Slight modification of the geometry then led to the completely flat
quadrature surface coil (a two element array) (34). As a result of the
existing literature, it is commonly assumed that the benefits of
quadrature reception exist only for perpendicular fields and with 90

degree electrical phase angles. The validity of this assumption will
be examined and the gain mechanism will be generalized to other
conditions.
Recently arrays of several coils have been used in conjunction
with parallel (39) and time-multiplexed (41, 62) receivers. In these








applications the initial aim was to extend the field of view while

attempting to maintain the same SNR as that obtainable from a single

element of optimal size and position. The spine array (39) is used

primarily to extend the field of view (FOV) in the z-direction,
however, even signals from coils far away from a given pixel are
used in forming the best composite image. The volume abdominal

array (63) consists of four planar surface coils, two above and two

below the body, which all, to a great extent, "see" the same FOV. The

signals from all of the elements are used to obtain optimum SNR gain
for a given pixel. Performance can be greatly increased with this

approach if the coils are nearly independent. If two coils have noise
voltages which are not exactly the same (within multiplication by a

complex constant) then the coils are at least partially independent.

Each of the signals may be separately acquired and after
reconstruction the signals from each pixel may be phased, weighted,
and summed to maximize SNR. It can be concluded from this work

that, in theory, the best performance for a given point of interest
(POI) would be obtained by receiving with as many separate coils as
possible and combining their signals using knowledge of the coil

characteristics. This principle applies to fixed combiners as well as

independent receiver acquisition.
It is important to distinguish between transverse and

longitudinal arrays which are fundamentally different and for
optimum performance should be treated differently. Consider, for

example, a planar array of two circular loops as shown below in
figure 5-1.












00


z____

Figure 5-1. A depiction of two circular surface coils assumed to be
independent, arranged longitudinally in the direction of the static
magnetic field

In the long wavelength approximation the signals induced from
spins along the bisection plane, which includes the z-axis, in the two
coils will always be either in phase or out of phase. The noise
correlation between the two coils is independent of the orientation
with respect to the main field. Optimum combination of the signals
will always be with either 00 or 180.
Consider a rotation of this array by 90. When the coils are
oriented perpendicular to the main static field, the phases between
signals originating from different pixels will differ over a continuous
range. The noise correlation is the same as in the previous case and
optimization for each pixel in this case will produce arbitrary phases.
In general, this will provide an increase in SNR over the previous
case. In fact, every pixel will have a SNR which is equal to or
superior to the previous case. For the case where optimal
combination procedures aren't used, but instead the simple algorithm
of the square root of the sum of the squares of the signals is








employed, there is no difference due to the above effects. There is

however another difference.

Consider the two cases below in figure 5-2.

Z---- ZO0






Figure 5-2. A single coil in the first case (on the left) with voxel of
signal below and displaced down the static field direction, with the
spin orientation shown. On the right is the same system rotated in
the static field.


In the first case, the magnetic field is never aligned with the

rotation of the spin and the signal will be much smaller than in the
second case where perfect alignment with the field will occur as the
spin rotates. This also implies that the a single coil has an induction

pattern which is weaker in the z- direction than in the x direction.

This stems from the fact that the spins rotate in a plane
perpendicular to the static field. In conclusion, the transverse array

will be superior to the longitudinal array, assuming the coils in the

array are the same.

If the optimum phasing and weighting of signals from an array

is known a priori for a particular POI, then simple hardware can be
used to directly combine the signals. For this POI the result would in
theory be equivalent to a totally parallel acquisition with optimum

phasing and weighting. For example, in the case of an ideal

quadrature volume coil, acquisition of the signals from the two








perpendicular coils through independent receivers followed by

optimum recombination would result in -,' gain in the center of the

coil and would fall to a gain of 1 very near the coil elements. If the
coils are combined with the standard quadrature combiner the result
is F2 gain at the center but with a more rapid fall-off near the

periphery. This is because both coils contribute equal noise for all

pixels but unequal signals in a standard quadrature combiner. If the

signals are independently acquired the appropriate weighting can be

imposed to scale the noise according to the signal level for each pixel.
In chapter 4 it was demonstrated that coupling is not

necessarily detrimental, but coupling can reduce performance by
imbalancing currents and changing the phases of the currents. By
compensating for these effects with the splitter/combiner one can

prevent all loss for a given POI. Reciprocity suggests that the current
distribution and the characteristics of the sample determine

everything there is to know about the system in terms of SNR and/or
magnetic field. In this chapter the effects of coupling on currents

will be ignored. Noise correlation will be defined as the intrinsic
presence of a shared resistance and the coupling which may be

produced by this shared current path will be ignored.

It seems, then, that all future coil designs should consider these
findings. This work is concentrated on applying transverse array

technology to single receiver systems. Arrays provide the basis for
the best possible performance under conditions currently existing in
state of the art MRI. The use of arrays with fixed phase combiner
networks will be referred to as generalized quadrature (GQ)
combination. In the following the importance of these considerations








will be investigated and some particular new coil geometries will be

discussed. The SNR of phased coil sets under various conditions will

be examined. First, the SNR of coil sets will be calculated for the case

of no noise correlation. Since the outcome of this is favorable, the

more common case where the noise in all coils in a set is correlated
must also be examined. In the last section examples and new coil

designs are described.

Theory
It will be assumed in the following, unless otherwise noted,

that long wavelength approximations apply. This implies that

radiation resistance is not part of the central development and that
the use of the concept of mutual inductance is justified. It is then

assumed that all noise comes from the sample, since any other source

is, in principle, removable. The best possible performance for a

given POI will be obtained by the use of as many independent coils

as can reasonably interact with spins at that POI. To utilize this

approach, the degree of independence of the coils and the relative
signals for the coils induced from that POI must be known. The

approach to the calculations is straightforward. The SNR will be

represented via reciprocity (15) by rotating magnetic field per

square root of input power as in chapter 4. In the following the

attainable gains will be observed by examining the magnetic field
produced by several coils when total independence is assumed. The

gain will be defined as the ratio of the efficiency of a combination of

similar coils to the efficiency of one of the single coils. Next the noise

correlation issue will be examined by calculating the change in power








deposition when the coils are placed on the same loading sample

(38).


Field Analysis for no Noise Correlation


The case of no intrinsic noise correlation in multiple coils around

the same sample is first considered. This is physically realistic in

many cases. For example, in low field MRI and in micro-imaging at
higher fields, the coil components account for most of the coil
resistance. Alternatively, one can imagine a sample which is lossy

(noise producing) only in distinct areas where the total field is
primarily due to a single coil but the signal POI is in a region where
all coils can interact strongly with a spin. The motivation for this

analysis is that gains greater than V2 can be obtained in theory for
certain physically realizable situations.
If there is no correlation of noise between coils, then the coils do

not have a shared resistance. This implies total independence of coils
which are oriented for zero mutual inductance. In each of the

following three cases, power will be supplied to the coil system and
the resulting field which couples to a spin will be computed. Recall
that magnetic field is a linear function of source current. B is the
rotating field at the POI for unit current in a particular coil.
Consider first the linear saddle type coil with resistance R. This

coil is depicted in Fig. 5-3 from an end view with a vector
representing the magnetic field at the center of the coil.











( )

Figure 5-3. A saddle type coil seen from an end perspective

The input power produces a current i in the loops which produces at
the center the field

B = Bie'"'+ Aie'J [5.1]
where the Cartesian coordinates have been represented by real and
complex numbers and the rotating and anti-rotating components
have been explicitly written. The efficiency for a single coil is then
BA< A
R- R [5.2]
,'1', R R-

Now consider the case depicted in Fig. 5-4 in which a second saddle-
type coil is positioned so as to have no mutual inductance, resulting
in an angle 0 between the field lines at the center of the coil. As
expressed in chapter 4 it is a matter of practical convenience to
minimize coupling. The efficiency for this case is
Bi, +Bi2e'* i, +iee'j
RI ,2R+. 2R + '12 [5.31
,,' R [2 I R w ", 2+i1

























Figure 5-4. Two reduced arc saddle coils, oriented for zero
mutual inductance



It should be noted at this point that the condition for zero mutual
inductance can be expressed as


I BI dS2 = 0. [5.4]


This is the integral of the field of coil one which passes through the
loop defined by coil two. The individual elements are assumed to
have the same impedance in this configuration as they have
separately which implies each of the loops will have equal magnitude
currents given an equal power split. If one of the currents is delayed
by an angle 6 we observe a gain of efficiency over the linear coil of
1 + eje' [5.5]
V-2-5.5






which is clearly equal to -/2 if6 =-0.
Note that no dependence on the value of 0 alone is shown. In
other words, changing the physical angle of the field crossing does
not change the resulting generalized quadrature gain. For the case of
no noise correlation quadrature gain occurs even when the fields are
not perpendicular and can occur even if the field lines are parallel.
Note that 0 is variable under the constraint of zero mutual
inductance, by changing the arc length of each coil.
Now consider three coils rotated 60 0 from one another as
shown in Fig. 5-5. For complete independence the power will have






( )"




Figure 5-5. Three saddle type coils oriented for zero mutual
inductance (at 60 0 rotations) seen from an end view with vectors
representing the fields at the center of the coils
increased by a factor of 3 given constant current. If the second and
third coils have their signals delayed by electrical angles 0 and Yi
respectively, the gain in efficiency is -l+ -e-e-' + -e' .e'2 [5.6]
y3








Again by delaying the currents so as to exactly cancel the

advancement due to the relative position we can force all rotating
components to add in phase. Thus selection of 0 = -7i/3 and W = -27/3

results in a gain of V-3.

The case of 4 coils is a straight forward extension and leads to a
gain of 2. This hypothesis has been tested, as will be discussed in the

final section, and gains greater than 2 for multiple coils have been
found This is an enticing result since in the general case of n coils

we see that as long as independence is assumed we obtain ,r n gain in
SNR at the center. It should be recalled, however, that the field

(which is the basis for the gain) decreases in strength as n increases
because of dwindling size of each coil, resulting in a finite SNR. It is

not actually necessary to decrease the size of the coils, but since the
current distribution is the only important parameter, it is logical to

do this. It should be pointed out that, in this regime, gain can come
from increase in the effective cross-sectional area of the conductor
when the resistance originates from ohmic conductor losses. For two

coils the attainable gain is always V2 for equal strength fields
regardless of the physical angle of the field crossing. To see if such
gains are practical, the gains for the case of correlated noise are next
calculated.


Field Analysis given Noise Correlation
To consider the case where complete independence in the noise
is not obtained we derive a relative expression for the total power
deposited by the set of coils. If there is no correlation, the power is

simply the sum of the individual powers. The cases are often








specialized to that of equal powers delivered to each coil. The degree
of correlation will be left as a parameter in the following discussion,
since the correlation is essentially unpredictable for a typical sample
(i.e. a human body) but can be found through direct measurement
(39) (or measurement of the shared impedance (40)). The special
case discussed is the case of equal correlations for all coil pairs.
As above, a certain time-average power is expended in driving
the coils. For the identical coil sets above, the powers added to form
the total power. To derive the shared power for a pair of coils, we
consider all the power expended in driving the coil as power
deposited in the sample. Several expressions for the power
deposited in a sample by a quasi-static magnetic field which are
applicable to our problem have been derived in the literature (38,
64). However, only a very general expression is necessary to
accomplish the present goals. The average power P expended in the
sample is defined as


P(B)= f I ( T') (A Ti)dV [5.71

where the integral is over the volume of the sample, and the sum is
over an arbitrary number of linear vector functions (of space
variables) 4,. BT is the actual magnetic field with rotating component

B. Linear functions have the following useful properties:


,(B, +B2=) (B)+j,,(Bj [5.8]


U,(cB,) = c,(B,).






With only these minimal requirements for the description of the
power in terms of the magnetic field, the relative values for the total
power deposited by a pair of coils which are simultaneously driven
can be obtained.

P(B, +B )= f2 ,(ti' +1 ).,"(,i +kTi2)dV [5.10a]


= j[,(B.i)+[(BT)] ,(I ) + (BTi2 )]dV [5. 1 0hb]


= f x|, (A', )f + 14 (ATi )12 + [4, (AI,). i (h2T) + !: (AIA) 4i (Ap. )]dV
[5.10c]

^ (B)2i2- 2 +,(BT) 2 12+[ ( ).(A2T )(ii2- + i,i.)]dV
[5.10d]
Since the currents are not functions of position they can be removed
from the integral and resistances can be defined from the functions
above, thus

P(B, +B,)= R. ,.'2+R2 i,12+R12(i,'I +i+,*i2) [5.11]

where R, =J J|i( ,) 2dV,R2 =J X ,(2T)12dV and
R, = fJ ,(Br)" (B2T)dV. It should be clear that the resistances R,
and R, are the isolated coil resistances in the presence of the sample
but with the other coil absent. Note that the condition of zero mutual
inductance (eq. [5.4] ) does not necessarily imply that this shared








power term is zero. This derivation also demonstrates that the field

functions which produce the shared power can be orthogonal in

space and/or in time. By electrically phasing them different by 90

orthogonality is obtained regardless of the spatial overlap of the
fields If two adjacent loops are combined so that their currents are

in phase, it is apparent that the combination looks much like a loop

twice as large with a corresponding increase in depth penetration

over a single coil. If the loops are driven with opposite phases then

the net field turns back on itself resulting in less depth penetration

than a single coil. The power expression above makes these effects

explicit and describes the net effect of the vector addition of the field

in terms of noise equivalent resistance. From the form of the power

expressions it is clear that by the triangle inequality, R1 + R2 > 2R12.

This ensures that the total noise resistance cannot be negative.

Although the resistance discussed here is from the inductive
loading from the sample it should be pointed out that all effective

losses will behave the same since all are representable as quadratic

forms of current. Ohmic losses in separate probes have no intrinsic

correlation however orthogonal modes which occupy the same

physical conductors in general will. Typical implementations of
these, such as the birdcage, phase orthogonal modes by 90 which

eliminates the effect from observation.

The calculation of the efficiency for two coils (eq. [5.3]) is
repeated, this time including the noise correlation. In the following it
is assumed that i, = ie'O and isolated resistances are equal (R1 =
R2=R). This implies that








B Ali, + ie 'e"I
1P R1, +Rji,e'j+R (i,,12e-"+ i, 2e')
B I + ee'"
^- Br~[5.12]
SR 2(l+ acose)


where O = R2- and will be called the correlation coefficient. In
R
general it is not true that optimum SNR is obtained by setting the
electrical angle equal to the negative of the field crossing angle, since
now a more complex function of electrical angle 0 is obtained. From
eq 5.12, it appears that in all two element cases the possible gain
approaches infinity as the correlation coefficient ac approaches 1,
unless 0 is zero. Of course, for a truly to reach 1 the fields would have
to be exactly the same and 4 is necessarily zero for all locations.
Therefore while large gains may be realized there is a real limit on
the gain.
For the general case of three coils the power calculation done for
two coils can be easily modified with the result that the total power
is
|i'2 [Rt + R2 + R3 + 2cos0 R12 + 2cos(4f-0)R23 + 2cosW R13]. [5.13]
If we again make the assumption that R1 = R2 = R3 = R and further
assume that the magnitudes of the shared resistances are equal (R] 2
= Ri3 = R23) as is required for the worst case analysis of total
correlation, we find again that ac = 1 for total noise correlation where
a is defined as R12/R. Note that the total power can mathematically
reach the value 9Ii(R when all electrical angles are chosen as 0 0 and








for total correlation (a( = 1). This case implies a maximum gain of one

for the system.
For the three element volume coil in figure 5-5, it is not
possible to obtain this mathematical maximum for the noise. For this
case, the defined orientation for correlation from coil 1 to coil 3 is
opposite that of 1 to 2 and 2 to 3, thus we obtain the total power


|if R[3 + 2cx(cosO + cos(t-O) cosW)]. [5.14]


For the same phase angles chosen in the case of no correlation (0 =-
TR/3, W =- 27T/3) we obtain a net gain of 3/2 for the worst case of

total correlation (c( = 1). It is interesting to note that even for total
correlation one obtains a GQ gain relative to a single coil element. For
arbitrary correlation, the total power is |ij3R[l + o:] for the above

choice of phase angles and is the largest attainable power for a given
current magnitude, but as we will show in a later section this choice
produces the highest SNR in the center as well.
There has been considerable debate about noise correlation in
relation to mutual inductance (37,38). Coupling and correlation have
been treated as distinct and not necessarily related. Some empirical
evidence suggests that it is impossible to have zero crosstalk and
still have noise correlation. It has been recently claimed that the
presence of noise correlation between coils necessarily implies that a
signal path exists between these coils (40). This was disproved in
chapter 4. Furthermore it was shown in chapter 4 that the effects of
coupling can be eliminated in other ways. Maximal isolations for
overlapped planar coils on the order of 10 to 20 dB have frequently








been observed for sample dominant loading conditions where both

coils are equally loaded by a single large sample. For sample

dominant loading in which two separate samples were used to load
the coils independently, the attainable isolation was much greater.

These results were obtained using two loops, each on a different

substrate. The isolation between coils is measured as the degree of

overlap is varied. Furthermore the author has found that the zero
mutual inductance overlap for the unloaded system does not give

maximum isolation under loaded conditions. It is clear, then, that a

significant amount of signal may be transferred from one coil to

another when we have the condition of substantial noise correlation.
In the next section we show real examples in which excellent gains

are obtained in the presence of substantial coupling caused by the
shared resistance of noise correlation supporting the theoretical
developments in chapters 4 and 5.


Applications and Examples


Three Phase Volume Coil


It is of interest to know how the new three phase design
compares in performance to standard designs. The techniques

derived above will be used to obtain approximate limits for
performance comparisons. Coupling will be ignored for this example.
It is assumed that the sample accounts for essentially all of the noise
but that the sample is not extremely close to the coil elements. Under
these conditions we conclude that a standard 120 arc saddle coil can








be approximated by two 600 arc saddle type coils which are slightly
overlapped for minimal coupling. This produces approximately a 60
physical angle (0) and an electrical phase difference (0) equal to 0

completes the construction. Since the currents are in phase the
elements which overlap will essentially cancel each other in the
region of interest and the result is approximately an equivalent
larger saddle coil. The three phase coil (Fig. 5-5) is shown in
comparison with the same electrical phase angles discussed above.
From eq. 5.5 we see that the gain is 3/2 compared to the 60 arc

saddle. The 3-phase coil (figure 5-5) with optimal phase angles
produces -3 gain (see eq. 5.6) relative to a single 60 arc saddle coil.
Multiplication of the 120 arc linear saddle case by 2 shows we
obtain exactly the same performance in the center for the standard
quadrature saddle coil and the 3 phase coil. These results are
summarized in Table 5-1. Although the analysis is not exact it is

clear that the greater gain of the GQ coil may not correspond to
improved SNR over other designs.


Four Phase Volume Coil


Consider the case where two saddle type coils with reduced arc

lengths are oriented for minimal coupling at a 45 physical angle for
0i. This coil is depicted in Fig. 5-4. Again assume that a loading

sample accounts for essentially all of the noise but that there is
substantial spacing between the coil elements and the sample.
Further, take R1 = R2. An estimate is obtained for the increase in








SNR of the optimally phased pair and a single saddle-type coil with

arcs which are approximately twice as long or about 90.


Table 5-1. Approximate GQ gain comparison of a standard linear
saddle coil, a quadrature saddle coil, and a three phase volume coil
all with a 60 arc saddle type coil as reference.


independent coil case correlated noise case



equivalent Vl+cos(60) = 3/2 1 +cos(60) 3 2

linear saddle (120 0)


equivalent

quadrature saddle (120 0) V3 -3
l+cx


3 phase (60 0) -3



Table 5-2 indicates the GQ gains over a single 45 arc saddle type
coil.
Consider the phased pair of 45 arc saddle type coils for a
typical correlation coefficient. For ax = 0.3, the GQ gain maximum
occurs at about 0 = 75 and is about 1.32. If it is assumed that the

sample prohibits coverage of more of the azimuthal angle than a 90
saddle coil, then a gain of 15% in SNR is obtained by using the
phased pair since the linear saddle with 0.3 correlation coefficient
produces a relative gain of 1.15.









Table 5-2. Approximate GQ gain comparison of a linear 90 arc
saddle type coil, a phased pair of 45 arc saddle coils, a quadrature
90 arc saddle type coil, and a four phase coil all referenced to a
single 45 arc saddle type coil.


independent coil case

equivalent

linear saddle (90) fl+cos(45) = 1.307


phased pair


Vr2 (0 = 45)


equivalent quadrature

(90) saddle V" 2 -l + cos(45)


4 phase coil


noise correlation case



l+cos(45) 1.307
1 +Cc Vl+a



Vl + cos(45 + 0)
Jl +acos(O)



V2 1l+cos(45)_ 1.85
Vl+a l~

2
Sl+a cos(45)


A perpendicular pair can be added to these coils to obtain another
[ 2 increase in SNR by phasing at 90. This requires selection of 0 = -
45 for the phased pair to provide uniformity. The GQ gains are

indicated in Table 5-2. From this analysis it is concluded that the 4
phase coil has improved SNR compared to a quadrature (90) saddle








coil in the range of 8% (a = 0) to 17% (a = 1). Furthermore the

homogeneity and SNR of the 4 phase coil is essentially that of a

quadrature 8 element birdcage (15) since the current distribution is
approximately the same.
These principles are tested by using a 4 element array each of

which was a curved surface coil with about a 50 arc. This is one
half of the cylinder of coils depicted in fig. 5-6.




















Figure 5-6. An eight coil array seen from the end, with each element
rotated 45 with respect to the previous element


The Q's of these coils were reduced to about 4 (from about 60) by

adding resistors in parallel with capacitors. This made the isolation
20 dB or better for each pair. The coils were placed around a
nonloading sample and an image was made using a 4-way combiner
which added their signals at 45 increments. Images were also

produced with each individual coil with the other coils in place and








still connected through the combiner. The two empty ports of the
combiner network were terminated with 50 2 loads. Because of the

resistors this is a case of essentially no correlation of noise. Fig. 5-7

shows each of the individual coils images, the combination of all coils

in rotating mode and in anti-rotating mode. Table 5-3 shows

isolations and table 5-4 shows SNR values.


Table 5-3. Isolation measurements (dB) of 4 element half-cylindrical
array under loading condition used in images of fig. 5-7.


Coil 1 Coil 2 Coil 3 Coil 4


Coil 1


Coil 2 20

Coil 3 20

Coil 4 20


Table 5-4. SNR values for 4
A70 for all images of fig. 5-7


20 20

23 20


21

phase coil measure at position L41,


SNR Gain of rotating mode over individual element


Coil 1

Coil 2

Coil 3

Coil 4


90.9

110.1

95.5

109.7


This same concept was applied to an 8-phase coil with
elements as in fig. 5-6. In this case, however, local samples (7 M
sodium chloride) were produced for each coil to preferentially load












































Figure 5-7. Top four images from individual coils-
bottom left GQ addition, bottom right anti-phase








only it and reduce correlation of noise between coils. Isolation was

not as high as for the 4 element case nor was reduction of correlation

as nearly complete. Fig. 5-8 shows the 8-way GQ addition of all the

loops in fig. 5-6, while figures 5-9a and 5-9b show the images made

by each individual coil. Tables 5-5 and 5-6 show relevant data.


Table 5-5. Isolation measurements for 8 element array loaded by


sample used in images of fig. 5-8

Coil 1 Coil 2 Coil 3 Co

Coil 1 26.0 18.3


'il 4

22.4


Coil 5

25.0


Coil 6

21.7


Coil 7

17.7


Coil 8

24.7


Coil 2

Coil 3

Coil 4

Coil 5


22.6

18.5

23.0


Coil 6 21.7 25.2

Coil 7 17.7 23.2

Coil 8 24.7 17.8


22.6 18.5 23.0 25.2 23.2 17.8

- 23.4 17.7 27.8 25.2 22.3


28.0 16.9

25.0


27.8 16.9 25.0


25.2 27.8

22.3 23.8


17.9 26.7

22.2 17.1


21.8 23.8

17.9 22.2

26.7 17.1

22.0


It has been shown with these examples that the assertion of multiple
coils producing greater than -2 gain in the case of no noise

correlation is valid and in the next section the claims for correlated
noise are addressed.




































Figure 5-8. GQcombination of eight coils







































Figure 5-9a. Individual images from four coils
used with coils of figure 5-9b to make
figure 5-8




























Figure 5-9b. Individual images from four coils
used with coils of figure 5-9a to make
figure 5-8


11 *1k









SNR measurements made at
Corrected values use extra
combination.


raw SNR

255

214

248

258

255

260

267

207


location R38. P38 in images
losses observed in network


Raw Gain Corrected SNR


1.65

1.96

1.70

1.63

1.65

1.62

1.57

2.03


Corrected Gain

1.65

1.96

1.84

1.98

1.71

1.82

1.87

2.27


Surface Coils

All of the examples of GQ coils given so far have been volume

coils, but the two element coil case is applicable and most useful for

surface coils. Consider Fig. 5-10, which shows the directions of the

B 1 fields (at a certain location) produced by two overlapping surface

coils. It is obvious that the physical angle 4 changes as the POI

moves along the symmetry axis. This is confirmed by fig. 5-11

which shows a sequence of images obtained by changing the

combination phase to various antiquadrature values. Note that the

location of cancellation moves along the central axis.


Table 5-6.
of fig. 5-8.
utilized for


Coil 2

Coil 3

Coil 4

Coil 5

Coil 6

Coil 7

Coil 8

8-phase




























Figure 5-10. Representative field lines from two overlapped surface
coils crossing at angle 4 on the symmetry axis




Equation 5-12 implies that for regions where the angle 4 approaches

180 the gain can be greater than V2 and where the fields cross at
angles that approach 0 the maximum gain will be less than -/2.
This further implies that one can choose the phase angle which

produces the optimum SNR for a given depth. This conclusion was
experimentally verified with a pair of identical surface coils with
strong noise correlation. By phasing the pair at 145 we were able to
obtain a raw gain in SNR of 59%. Fig. 5-12 shows these three
images, table 5-7 shows relevant parameters for these coils, and
Table 5-8 shows actual and predicted gains for various cases.































I % I


Figure 5-11. A sequence of images produced with different
combiner angles indicating the different crossing angles at different
locations


I----- mm-
















































Figure 5-12. Surface coil images -- top: right coil; middle: left coil;
bottom: 145 combination of both coils












Table 5-7. Bench measurements of individual coils and phased pair
under conditions of magnet test from images of fig. 5-11.




individual coils:


Resistance of untuned loop measured with vector impedance

meter
RI = R2 = 18 Q

Relative current measured in loop with network analyzer and

magnetic field probe
I1I = 12 = 28.4 dB




pair of coils:
isolation before combination = 12.5 dB => k = 0.237, (3 = 180

Resistance of series aiding (0 0) combination R= 49 Q

Resistance of series opposing (180 0) combination R= 21 Q


Currents measured in loops with 145 phase difference with

same power as for individual case


I1 = -28.7 dB 12 = 29.1 dB










Table 5-8a. Actual SNR measured at R41 P58 for coils of fig. 5-12.

SNR

Coil 1 21.5

Coil 2 21.3

Combined coil 145 34.1




Table 5-8b Actual versus predicted gains


Actual Gain 1.59

Maximum gain predicted from current measurements 1.89

Maximum gain predicted from theoretical considerations 1.71


In this case three different coils were constructed on a

standard spine coil form from MRI Devices Corporation of Waukesha,
WI. First only the left loop was tuned, in the next case another board

had only the right loop tuned and finally on another board both loops
were tuned and combined so that their currents differed by 1450.
The tests were performed with an open positioning table, so that the

placement of the coils and sample were extremely reproducible. The
loader was a Plexiglas box of saline and copper sulfate of dimensions
6.5 cm x 35.5 cm x 46 cm and was positioned so that the saline was
about 3 cm from and centered above the coils. The condition that RI
= R2 = R was therefore satisfied, so that eq 5.12 was used to compute
the expected gains from this system. The correlation coefficient x








was computed from the value of resistance measured, giving a =

0.39. The actual gain was lower than the predicted gain because of
small combiner losses and because the field crossing conditions for
maximum gain to be realized do not precisely occur in our coil
design. The agreement of the current measurement (which includes

combiner losses) gain and theoretical gain is fair.


Summary of Attainable Gains


Below is a description of various domains which differ in terms

of attainable SNR. Descriptions are based on a transmission model.


1. No power deposited in sample. Ohmic coil losses only.


The total power expended in driving the coils is the sum of the
individual ohmic power losses in each coil. Reciprocally there is no
correlation of noise. Therefore if n equivalent coils can be uniformly

positioned around a sample with correctable coupling one can obtain
\n- gain in SNR over the unit coil.


2. Some power deposited in sample.


The amount of gain depends on specific conditions. Noise
correlations will never be total because some power goes into ohmic
losses. Some gain is acheivable for any relative orientation.


3. All power deposited in sample.








a. non-uniform fields in sample


For non-uniform fields the gains again depend on specific conditions.
The degree of correlation is arbitrary and since the fields are non-
uniform it is possible to obtain large gains if the dominant fields in
the sample are much different in crossing angle than the angle at the
POI.


b. uniform fields in sample


This narrow domain is the common ideal case for volume coils and is
examined in detail.
Uniformity of the magnetic fields allows removal of the B3T

field terms from the spatial volume integral. This results in
modification of eq 5.10d to




P(B, + B2)= [| r2 + 121i2T 2 + BIT -B2T(1i' + i2 )]If X, (l)2dV .

[5.15]


For Bfl and BA1 equal except for a rotational change in position by
angle 4 we obtain a correlation coefficient of a = coso. Equation 5.12
becomes
B Al + ee'le [5
VP VR 2(1 + cos )cosO) [5.16]







t-2
This expression is maximized by 0 = -0 and the gain is +COIs--
1 + cos2 (|

For 0=0 the coils noises are totally correlated which results in no gain,
whereas for O=lt/2, the attainable gain is V2. Furthermore for the n-

phase volume coil in which we continue the assumption of all power
to the sample and uniform fields throughout the load, we find that
the maximum gain remains fixed at V2 for n>2 and cannot be
increased.
Conclusion


It was demonstrated that standard quadrature reception is a
special case of a more general principle which describes weighting
and phasing for several coils to obtain optimum SNR at a given
location. This principle was demonstrated for POI's which were all
at locations where the coils all provided approximately equal
magnitudes of signal. The method could easily be extended to other
regions. The concept of independence of acquisitions led to the
consideration of choosing a particular phase and weight for each coil
in an array and then combining the signals into a single channel data
stream. The result is optimum performance at some specific location
and less than optimum performance elsewhere. The same array
used in parallel acquisition provides the ability to perform the
combining in software on a pixel by pixel basis leading to optimal
performance everywhere. The number of channels is limited
however, and it may be that each element of a parallel acquisition
should be a GQ array.








The approximate theoretical performance of three and four
phase cylindrical volume coils was examined and it was found that
the four phase design appears to provide both excellent homogeneity
and SNR. Theoretically, the cylindrical geometry allows for canceling
of mutual inductance; however, it was previously shown that
excellent performance can be obtained even with significant coupling
if compensation is made for the phase and magnitude of the coupled
signal. This implies that arrays such as the abdominal array
described by Hayes et al. (63) can be quite effective even though the
elements appear to be coupled. Isolations on the order of 10 to 20
dB have proved to be sufficient for good performance on real GQ
coils. The correlation coefficient was derived by the use of long
wavelength approximations that may not be justified at frequencies
substantially higher than 64 MHz which is currently the highest
frequency commonly used in clinical MRI. The usage of mutual
inductance also must be assessed at higher frequencies. This method
is also helpful for comparisons of standard designs.
The two element coil analysis (eq. 5.12) demonstrated that a

pole in SNR occurs in the case of total noise correlation. This leads to
potentially high SNR gain even for coils which have very strong noise
correlation if the field lines at the POI do not differ by exactly the
phase of correlation (00 or 180). Furthermore, for a pair of surface
coils, a depth dependent variation of geometric phase angle can be
produced. This suggests that a variable phasing system could be
produced which would allow optimization at a particular depth. It
might also be useful, in some instances, to consider transmission with
one phase and reception with an unrelated phase to produce a given





93


excitation profile and possibly a different reception profile. This

could prove useful for some kinds of surface coil localization and

subcutaneous fat suppression.













CHAPTER 6

INDEPENDENT SIGNAL ACQUISITION
Introduction

The previous 3 chapters deal with optimal combination of

signals into a single channel. It was known that each coil received

signal from various regions of the sample and noise from similar

regions. The relative phase and amplitude of the signal acquired by

each coil was used with overall correlation parameters to define the

single combination which produced the highest SNR at a particular

location. This choice of a single point for optimization is restrictive;

however, there historically has always been a trade-off between SNR

and field of view. Note that the philosophy of knowledge of the
problem was highly relevant to the solution. Optimal combination

required an approximate knowledge of the coil field characteristics,

crosstalk and intrinsic noise correlation.
This chapter deals with increasing the knowledge available for

reconstruction of the image by tagging the signals from each probe in

a recognizable way, so that they can, in software reconstruction, be

optimally added for all locations in the resultant image. This is

theoretically equivalent to an optimum combiner designed for each
POI. Roemer et al. (39) were pioneers in this area and their work

resulted in the most general method for this process. Their system
employs a separate receiver for each coil. Obviously the signals from

each coil remain distinct, allowing a separate reconstruction for each