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SIGNALTONOISE 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 SignaltoN oise Ratio.............................................................................. 15 Radiofrequency 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 SIGNALTONOISE 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 signalto 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 multicoil 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 trombonelike 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 mulicoil 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 birdcage 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 zdirection with S, = m, being the 271 projection of the spin onto the zaxis. The quantum number ms may take on the values of {s,sl,s2 ..... 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 antiparallel with the field and the other of lower energy which is parallel. The energy difference is AE= hyB 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 105 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=MH, 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 timevarying 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 11 demonstrates the relationship of the magnetic moment precession, the perturbing rf magnetic induction BI1 and the main fixed magnetic induction Bo. M NMR Figure 11. 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 spinlattice 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 onedimensional 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 onedimensional 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 oftheart 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" electromagnet 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 digitalto 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 signaltonoise 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 12. 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 12. 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 signaltonoise 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 fluctuationdissipation 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 13. Sinland noise ... mean here Noise mean and 13 std. dev here Figure 13 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 multistage 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 stateofthe art preamplifier has approximately 0.5 dB noise figure and 30 dB gain. Figure 14 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 14. Quadrature phase detection receiver The signal is first downconverted 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 downconversion 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 38, 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 (810) 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 540cm. 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 38 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 SignaltoNoise Ratio Since the fundamental theme of this work is enhancement of the signaltonoise 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 skineffect conduction pattern of current at radio frequencies. The skin depth decreases as f1/2 (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 21 shows the voltages on the components which are responsible for the nonconservative Efield losses. C 2C L L/2 2,C L/2 V + V/2 + Figure 21. 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) Radiofrequency 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 nonelectrically 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 22. x Figure 22. 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 (ap)+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 regionspecific 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 signaltonoise 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 AldermanGrant resonator is a twoleg 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 counterrotation 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 nonpreferentially 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 pixelbypixel 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 nonoptimum 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 signaltonoise sense, this process is nonlossy. 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 aboveusing independent receivers all the way to the atod 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 (4650) 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 (5254). 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 multielement 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 analogtodigital 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. lowcost 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 31. Inner copper tubing Outer copper tubing Figure 31. 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 32 shows a circuit model of the coil and figure 33 shows a simplified version. LI C(x) L2z+ L(x) cTrUNET I ~ L1 C(x) L2+ L(x) Figure 32. Circuit model of the adjustable coil LT(x) 0 L / 'ICT(X) SCTUNET T Figure 33. 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 34, 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 34. 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 35, it can be observed that the rate of decrease in capacitance is slowed with this method. .c C Figure 35. 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. 36, 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 37 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 36. 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 38a and 38b 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 37. 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 38. 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, 378384 (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 nonvolumetric 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 intercoil 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 talkthis can also be referred to as couplingcross talk between two coils is nonzero 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 crosstalk and noise correlation are, in principle, separable. Consider the circuit model in figure 41. 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 41. 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 crosstalk, 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 crosstalk 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 crosstalk are entirely separable whereas crosstalk 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 tonoise 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 42. (~B) Figure 42. 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 43 depicts two identical reduced arc saddle coils which have been oriented for zero mutual inductance. Figure 43. 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 yaxis 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 41. Table 41. 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 44 shows an electrical block diagram of the coils in figure 43. 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 44. Electrical block diagram of the coils of figure 43 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 45 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 45. 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. 46a graphically demonstrates the field addition under these circumstances. O=7C/2 coupling, [=0 'N, S= Cos'r 2k2 Figure 46a. 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 46b 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 1k Figure 46b. 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 43, 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 ej or equivalently 1 + Ka R2 c Re" K R, a = R the optimum gain is once again obtained. 1c 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 +ii, '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 signaltonoise 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 socalled "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 nonvolume 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 timemultiplexed (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 zdirection, 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 51. 00 z____ Figure 51. 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 zaxis, 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 52. Z ZO0 Figure 52. 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 falloff 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 microimaging 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. 53 from an end view with a vector representing the magnetic field at the center of the coil. ( ) Figure 53. 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 antirotating 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. 54 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 54. 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] V25.5 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. 55. For complete independence the power will have ( )" Figure 55. 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+ ee' + 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 V3. 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 crosssectional 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 timeaverage 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 quasistatic 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 Ji( ,) 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(4f0)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 55, 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(tO) 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. 55) 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 3phase coil (figure 55) 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 51. 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. 54. 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 saddletype coil with arcs which are approximately twice as long or about 90. Table 51. 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 52 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 52. 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 52. 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. 56. Figure 56. 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 4way 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. 57 shows each of the individual coils images, the combination of all coils in rotating mode and in antirotating mode. Table 53 shows isolations and table 54 shows SNR values. Table 53. Isolation measurements (dB) of 4 element halfcylindrical array under loading condition used in images of fig. 57. Coil 1 Coil 2 Coil 3 Coil 4 Coil 1 Coil 2 20 Coil 3 20 Coil 4 20 Table 54. SNR values for 4 A70 for all images of fig. 57 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 8phase coil with elements as in fig. 56. In this case, however, local samples (7 M sodium chloride) were produced for each coil to preferentially load Figure 57. Top four images from individual coils bottom left GQ addition, bottom right antiphase 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. 58 shows the 8way GQ addition of all the loops in fig. 56, while figures 59a and 59b show the images made by each individual coil. Tables 55 and 56 show relevant data. Table 55. Isolation measurements for 8 element array loaded by sample used in images of fig. 58 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 58. GQcombination of eight coils Figure 59a. Individual images from four coils used with coils of figure 59b to make figure 58 Figure 59b. Individual images from four coils used with coils of figure 59a to make figure 58 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. 510, 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. 511 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 56. of fig. 58. utilized for Coil 2 Coil 3 Coil 4 Coil 5 Coil 6 Coil 7 Coil 8 8phase Figure 510. Representative field lines from two overlapped surface coils crossing at angle 4 on the symmetry axis Equation 512 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. 512 shows these three images, table 57 shows relevant parameters for these coils, and Table 58 shows actual and predicted gains for various cases. I % I Figure 511. A sequence of images produced with different combiner angles indicating the different crossing angles at different locations I mm Figure 512. Surface coil images  top: right coil; middle: left coil; bottom: 145 combination of both coils Table 57. Bench measurements of individual coils and phased pair under conditions of magnet test from images of fig. 511. 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 58a. Actual SNR measured at R41 P58 for coils of fig. 512. SNR Coil 1 21.5 Coil 2 21.3 Combined coil 145 34.1 Table 58b 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. nonuniform fields in sample For nonuniform 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] t2 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 tradeoff 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 