Radio frequency coils

0001The present invention relates to radio frequency coils for magnetic resonance imaging equipment. The invention further relates to the art of designing radio frequency coil structures or flow patterns to meet prescribed field properties such as amplitude, orientation, and spatial variation.

0002In early magnetic resonance imaging systems, saddle coils were used to create B₁ fields for both transmit and receive functions. Typically, two rectangular shaped loops of copper tubing or the like were bent around and mounted to a dielectric cylinder. For optimum uniformity, the rectangular loops commonly subtended an arc of about 120°. The length of the RF coil was about equal to its diameter. Lumped element capacitors were used to tune and match the coil to the selected magnetic resonance frequency.

0003With properly phased radio frequency current flow through this saddle pair, a symmetric about isocenter B_x field was created, where the effective axis of the loops is along a defined x-direction, orthogonal to y and z-directions. Typically, the axis of the cylinder along which the B₀ field lies is defined as the z-direction. A second pair of saddle coils were often tuned to resonate at a like frequency and mounted 90° offset from the first pair of saddle coils to create a B_y field. When this second coil pair was excited with radio frequency currents 90° out of phase with the first saddle coil pair, a circularly polarized radio frequency field was created. This arrangement, called a quadrature radio frequency coil, provided a signal-to-noise ratio advantage in magnetic resonance studies of about 1.4:1 over a linearly polarized coil.

0004Although such quadrature saddle RF coils provided adequate performance in early commercial magnetic imaging systems, the B₁ field suffered from non-uniformity problems, particularly in the transmit mode. Because the soft tissue contrast-to-noise ratio is more important than uniformity in some imaging applications, some non-uniformity of the B₁ field could be tolerated for the receive function. However, a principal role of the transmitter coil was to give uniform excitation of all dipoles in the field.

0005For greater RF uniformity, birdcage coils have become the RF coil of choice in most modern magnetic resonance imaging equipment, particularly for transmission. The birdcage coil improves upon uniformity relative to the saddle coil. Although the birdcage coil has intrinsically high B₁ uniformity through its transverse plane, it has relatively few degrees of design freedom. To adapt a birdcage coil to a particular application, its length can be adjusted as can its diameter. Its frequency is, of course, tuned to the resonance frequency and its impedance matched to the rest of the system.

0006Simple loop coils have been used in various forms to make surface or localized coils for the receive function. Surface coils sit in proximity to the target region of interest and give a high local B₁/sensitivity factor for signal-to-noise advantage, but have relatively poor uniformity. Circular and rectangular loop shapes have been used to detect B₁ fields along the axis of the loop. Crossed loop pairs, such as Figure-8's, have been used to detect B₁ fields orthogonal to the loop axis, particularly in a region of a mid-plane of a pair of crossed coils. By combining loop and crossed-loop shapes, quadrature surface coils, including quadrature planar coils, have been created. In addition to loop-type coils, saddle and birdcage coils have also been adapted as surface receiving coils. The exact configuration of the coils is adapted to provide a close fit to the head, torso, knee, or other imaged anatomical body part.

0007To improve the signal-to-noise ratio and depth of coverage on larger volume targets, arrays of individually resonant single loop structures have been mounted around a coil form or on multiple coil forms. The sensitivity fall-off of such surface coil arrays can be tolerated, within reason. For example, in spine imaging, a coil array is needed only on the backside of the subject because the spine is not too distant from that surface. To image the abdomen, for example, one needs coils which have penetrating B₁ at both the top and bottom sides of the circuit and, preferably, on lateral sides of the subject as well. In general, each element of the array is a single simple loop or a combination of the single loop and a crossed pair to create a locally quadrature field. Loop geometries, again, had only limited degrees of design freedom, typically length or diameter.

0008In addition to adjusting the size characteristics of the prior art saddle, birdcage, or loop coils, the resonant point of the coil is adjusted. The resonant point of the coil is determined with a network analyzer. A tuning capacitance is adjusted until the resonance frequency matches the selected magnetic resonance frequency. An electrical matching network, commonly a parallel capacitance at the feed point, is used to match the coil to the desired impedance of the magnetic resonance imaging system, e.g., 50 Ohms.

0009Typically, the saddle, birdcage, and loop coils have been designed by trial and error. The basic configuration of each coil is, of course, known. The relative size of the coils is selected in accordance with the size of the region to be imaged. Various adjustments are often made to the coil, particularly size adjustments, or folding, or bending the coil. Each time the physical properties of the coil are altered, the coil is analyzed to determine its actual characteristics. The adjustments are as reliable as the skill and experience of the designer and are not based on analytic or numerical analysis or guidance.

0010According to the invention there is provided a magnetic resonance imaging device in which a main field magnet generates a temporally constant magnetic field through an examination region, radio frequency and gradient magnetic field coils generate radio frequency pulses for exciting magnetic resonance of a portion of a subject within the temporally constant magnetic field and encoding such resonance, a receiver demodulates received encoded magnetic resonance signals, a reconstruction processor reconstructs the demodulated magnetic resonance signals into an image representation, and a radio frequency coil for at least receiving the encoded magnetic resonance signals, the radio frequency coil characterized by: <ul id="ul0001" list-style="none" compact="compact"><li>a first pair of loops disposed symmetrically to either side of a central plane of symmetry; and,</li><li>a second pair of loops disposed symmetrically to either side of the plane of symmetry.</li></ul>

0011Further according to the invention there is provided a method of magnetic resonance imaging in which a temporally constant magnetic field is generated through a cylindrical bore of a magnetic resonance imaging system, a localized coil which carries at least a radio frequency coil designed to receive magnetic resonance signals from a region contiguous thereto, a patient and the localized coil are inserted into the bore, magnetic resonance is excited within a portion of the patient contiguous to the localized coil and encoded with magnetic field gradients, the encoded magnetic resonance is received with the localized coil, demodulated, and reconstructed into an image representation, the method characterized by designing the radio frequency coil including: <ul id="ul0002" list-style="none" compact="compact"><li>a static problem formulation step in which static coil geometry and vector current density components are defined;</li><li>a static current solution step in which a set of current density expansion coefficients which define preselected B₁ field characteristics of the radio frequency coil are obtained, the set of current density expansion coefficients defining a continuous current density function;</li><li>a discretization step in which the continuous current density function is discretized;</li><li>a loop connection step in which current loops are defined which mimic the discretized current density function; and,</li><li>a high frequency solution step in which current carrying monopoles or V-dipoles are defined along the current loops, reactive elements are added to adjust resonance frequencies and matching characteristics, and a feed point from which received resonance signals are conveyed for demodulation and reconstruction is selected.</li></ul>

0012One advantage of the present invention is that it enables RF coils to be designed and built which have a prescribed spatial behavior of the B₁ field.

0013Another advantage of the present invention is that it facilitates the design of multi-turn structures which produce useful B₁ field characteristics at relatively high magnetic resonance frequencies, e.g., 64 MHz.

0014Another advantage of the present invention is that it allows for determination of a proper resonance point in magnetic resonance applications for a radio frequency coil structure with proposed feeding, tuning and matching schemes, and preferably derived from a static or quasi-static inverse solution.

0015Another advantage of the present invention is that it provides an analytical/numerical technique, as opposed to an experimental procedure, for optimizing B₁ coil characteristics.

0016The invention will now be described by way of example with reference to the accompanying drawings in which: <ul id="ul0003" list-style="none" compact="compact"><li>FIGURE 1 is a diagrammatic illustration of a magnetic resonance imaging system in accordance with the present invention;</li><li>FIGURE 2 is a flow chart outlining the radio frequency coil design technique;</li><li>FIGURE 3 is a plot of feed point input impedances versus frequency;</li><li>FIGURE 4 illustrates a coil pair from the local coil of FIGURE 1 laid out flat;</li><li>FIGURES 5A and 5B are two and four loop solutions generated by the present design technique;</li><li>FIGURE 6A illustrates the spatial characteristics of the coil of FIGURE 4 across the plane of symmetry;</li><li>FIGURE 6B is a three-dimensional representation of the spatial characteristics of the coil of FIGURE 4;</li><li>FIGURE 7 is another alternate embodiment of the radio frequency coil construction of FIGURE 4, but with a sharp drop-off in coil sensitivity at its longitudinal edges;</li><li>FIGURE 8 is another alternate embodiment of a radio frequency coil construction in accordance with the present invention, particularly adapted to planar coils; and,</li><li>FIGURE 9 illustrates shape changes in the coils loop as the region of interest is covered more uniformly.</li></ul>

0017With reference to FIGURE 1, a main magnetic field control <b>10</b> controls superconducting or resistive magnets <b>12</b> such that a substantially uniform, temporally constant magnetic field is created along a z-axis through an examination region <b>14</b>. A magnetic resonance echo means applies a series of radio frequency (RF) and magnetic field gradient pulses to invert or excite magnetic spins, induce magnetic resonance, refocus magnetic resonance, manipulate magnetic resonance, spatially and otherwise encode the magnetic resonance, to saturate spins, and the like to generate magnetic resonance imaging and spectroscopy sequences. More specifically, gradient pulse amplifiers <b>20</b> apply current pulses to selected ones or pairs of whole body gradient coils <b>22</b> to create magnetic field gradients along x, y, and z-axes of the examination region <b>14</b>. A digital radio frequency transmitter <b>24</b> transmits radio frequency pulses to a whole body RF coil <b>26</b> to transmit RF pulses into the examination region. Each typical radio frequency pulse is composed of a packet of immediately contiguous pulse segments of short duration which taken together with each other and any applied gradients achieve a selected magnetic resonance manipulation. The RF pulses are used to saturate, excite resonance, invert magnetization, refocus resonance, or manipulate resonance in selected portions of the examination region. For whole body applications, the resonance signals can be picked up by the whole body RF coil <b>26</b>.

0018For generating images of limited regions of the subject, a surface or localized coil is placed contiguous to the selected region. For example, an insertable head coil <b>30</b> is inserted with a selected brain region of interest at the isocenter of the insertable coil. A local, radio frequency coil <b>32</b> receives magnetic resonance signals emanating from the region of interest. More specifically, resonance is excited by the radio frequency coil <b>26</b> and the resultant magnetic resonance signals are received by the localized radio frequency coil <b>32</b>. Alternately, the localized coil assembly <b>30</b> can include its own gradient coils and an RF shield between the RF and gradient coils. The local radio frequency coil <b>32</b> can be connected with the transmitter <b>24</b> to operate in both transmit and receive modes.

0019A sequence control <b>34</b> is loaded with a selected magnetic resonance imaging sequence, such as a gradient echo, spin echo, gradient and spin echo, fast spin echo, echo planar imaging, or other magnetic resonance imaging sequence and controls the gradient amplifiers <b>20</b> and digital transmitter <b>24</b> in accordance therewith. More specifically, the sequence control causes the gradient amplifiers and digital transmitter to create gradient and radio frequency pulses at appropriate times and magnitudes for the selected imaging sequence. The sequence controller is indexed by a clock to step phase-encoding or other variables from repetition to repetition of the selected sequence.

0020After magnetic resonance is induced, the localized radio frequency coil <b>32</b> receives the magnetic resonance imaging signals which are conveyed to a receiver <b>36</b>. Typically, the magnetic resonance signals received during application of a read gradient are demodulated, digitized <b>38</b>, and converted into a digital data line or view. The views or digital data lines are typically stored in temporary storage or memory <b>40</b> until they are reconstructed by a reconstruction processor <b>42</b> into an image representation. The image representation is stored in an image memory <b>44</b>. Depending on the nature of the selected sequence, the image representation may be a slice image, a volume image, a series of images, or the like. A video processor <b>46</b> withdraws selected portions of the image representation and reformats the data in appropriate format to generate a selected display on a monitor <b>48</b>, such as a CRT, active matrix, CCD, or other conventional video display device.

0021With reference to FIGURE 2, the RF coil <b>30</b> is designed on a theoretical basis using a series of analytical steps to realize a physical distributed RF coil design. First, in a static problem formulation step <b>50</b>, the static coil geometry and vector current density components are defined. Using the example of a head coil, a circularly cylindrical geometry is selected with current density components J_φ and J_z. Preferably, the coil is constrained to be of a selected, finite length. The current density components are expressed as an expansion series. In the illustrated circularly cylindrical example, Fourier sine and cosine series with unknown expansion coefficients to be determined are advantageous.

0022Second, a static currents solution step <b>52</b> obtains a set of coefficients for the current density component expansion which yields a preselected set of B₁ field characteristics. In the preferred embodiment, stored magnetic energy is minimized subject to a set of field constraints. In this manner, techniques previously used for magnetic resonance imaging gradient coil design are adapted to the static B₁ field problem.

0023Third, a discretization step <b>54</b> discretizes the continuous current density function. In the preferred embodiment, the current for each loop of wire is held to preselected constraints, and a stream function technique is applied. For simplicity of design, equal, constant loop currents are preferred. However, when multiple feeds are provided to each of a multiplicity of loops or loop segments, a plurality of different loop current values can be assigned.

0024Fourth, a current loop connection step <b>56</b> condenses the current density function into a plurality of discretized coil loops. When all loops are constrained to a constant current, a static solution can be found in which the loops of wire would be connected in series to produce a B₁ field which mimics that produced by the continuous current density. However, the self-resonant point of an RF structure is lowered as more turns are added in series. For high field magnetic resonance imaging equipment in which higher resonance frequencies, e.g., 64 MHz are required, the resonance frequency of the RF structure is typically too low. Accordingly, in the preferred embodiment, to keep the inductance of the structure low, the loops are connected in parallel and impedance elements are distributed in the circuit to obtain a proper distribution of current to each parallel leg, element, and loop of the circuit. An appropriate RF feed point or points are selected. Preferably, the RF feed point or points are selected in accordance with the symmetry of the coil.

0025Fifth, a high frequency solution step <b>58</b> is used to select a magnetic resonance frequency solution. Given the prescribed wire connection scheme, initial lumped elements, and feed point, a method of moments or other full wave field solution technique is applied to study the resonance characteristics of the structure. A solution is obtained at various frequencies in the range of interest, e.g., 10-100 MHz for commercial whole body NMR systems. A feed point input impedance versus frequency (FIGURE 3) is plotted to identify the self resonances. At each self resonance point, the high frequency currents are analyzed to determine which resonance yields a preferred B₁ field characteristic, particularly a symmetric B₁ about the origin and having uniformity consistent with the constraints. In the preferred embodiment, one searches for the resonance that gives substantially equal current amplitude in each leg with minimum phase variation along the path of the leg. In this process, one suitably modifies the lumped elements in the circuit to yield the optimal B₁ characteristics. At least one lumped element is incorporated and adjusted to match the circuit to the connecting cable characteristic impedance. It should be noted that the other resonance modes may be useable in other applications, such as in examining magnetic resonance signals from other dipoles with different resonance frequencies.

0026With reference to FIGURE 4, a cylindrical winding pattern for the radio frequency coil <b>32</b> designed in accordance with the method of FIGURE 2 is illustrated The coil has two coil portions <b>60</b>, <b>62</b> of like construction which are disposed 180° apart on opposite sides of a dielectric cylinder. For simplicity of illustration, portion <b>60</b> is described in detail and it will be appreciated that a similar description applies to portion <b>62</b>. The first coil section includes a double inner loop <b>64</b>_<b>1</b>, <b>64</b>_<b>2</b> disposed symmetrically on either side of a central or z=0, plane through the isocenter of the localized coil <b>30</b>. These loops are completed at their outside edges with a capacitor or other load <b>66</b>_<b>1</b>, <b>66</b>_<b>2</b> which adjusts the resonance frequency to a preselected frequency.

0027The coil portion <b>60</b> further includes an outer pair of loops <b>68</b>_<b>1</b>, <b>68</b>_<b>2</b> also disposed symmetrically on either side of the central plane. The loops are again completed at their outer edge by the frequency adjusting impedances <b>66</b>_<b>1</b>, <b>66</b>_<b>2</b>. Connections <b>70</b> are provided between the inner and outer loops in order to create a parallel interconnection. The interconnections <b>70</b> are direct interconnections but impedances may be used, particularly when a larger number of loops are connected.

0028A cable <b>72</b> extending to the receiver is connected at one end with a matching impedance network <b>74</b>. The matching impedance network <b>74</b> is connected with a pair of symmetric feed points <b>76</b>. To assure that the current distribution in coil portions <b>60</b> and <b>62</b> match, the two portions are connected in series by a series interconnection <b>78</b>.

0029Looking now in greater detail to the coil design technique discussed above in conjunction with FIGURE 2, the mathematical formulation of the energy/inductance optimized radio frequency (RF) coil is viewed as a superposition of two parts. The first part <b>50</b>, <b>52</b>, <b>54</b> discusses the mathematical development of the energy/inductance minimized static field configuration which satisfies all the symmetry conditions and field qualities that a magnetic field generated by an RF coil is to have. Using the discrete current pattern which is generated in the first part, the second part <b>56</b>, <b>58</b> discusses the mathematical methodology of resonating such a current configuration using the Method of Moments (MOM) technique.

0030In an axial bore system, one is typically interested in producing RF fields transverse to <maths id="math0001" num=""><math display="inline"><mrow><msub><mrow><mtext>B</mtext></mrow><mrow><mtext>0</mtext></mrow></msub><msub><mrow><mtext> =B</mtext></mrow><mrow><mtext>0</mtext></mrow></msub><mtext>z</mtext></mrow></math><img file="EP0803737A2_D0001.tif" /></maths>, i.e., B_1x, B_1y fields. As an example, a circular cylindrical finite length coil structure is designed below.

0031The geometrical configuration of the finite length cylindrical RF coil is shown in FIGURE 1, where <b>L</b> is the total length of the coil, while <b>a</b> is the coil radius. In order to generate the B_x component of the RF field, the current density distribution is viewed as a vector superposition of two components; one along the axial z-direction and the other along the azimuthal direction. Thus, the general expression of the current density distribution is:<maths id="math0002" num="(1)."><math display="block"><mrow><msup><mrow><mover accent="true"><mrow><mtext>J</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>a</mtext></mrow></msup><mtext>(</mtext><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover><msub><mrow><mtext>)=[j</mtext></mrow><mrow><mtext>φ</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>a</mtext></mrow></msup><mtext>(φ,z)</mtext><msub><mrow><mover accent="true"><mrow><mtext>α</mtext></mrow><mo>^</mo></mover></mrow><mrow><mtext>φ</mtext></mrow></msub><msub><mrow><mtext>+j</mtext></mrow><mrow><mtext>z</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>a</mtext></mrow></msup><mtext>(φ,z)</mtext><msub><mrow><mover accent="true"><mrow><mtext>α</mtext></mrow><mo>^</mo></mover></mrow><mrow><mtext>z</mtext></mrow></msub><mtext>]δ(ρ-a)</mtext></mrow></math><img file="EP0803737A2_D0002.tif" /></maths>

0032The goal is to find the appropriate current density distribution which will generate a static field component along the x-direction with the specified uniformity. Specifically, both components of the current density are to possess certain symmetries, in order to generate a B_x component which is symmetric along the xz, yz, and xy planes. Since we are interested in a finite current distribution and with the help of the continuity equation:<maths id="math0003" num=""><math display="block"><mrow><mtext>(</mtext><mover accent="true"><mrow><mtext>∇</mtext></mrow><mo>→</mo></mover><mtext> · </mtext><msup><mrow><mover accent="true"><mrow><mtext>J</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>a</mtext></mrow></msup><mtext>=0),</mtext></mrow></math><img file="EP0803737A2_D0003.tif" /></maths> the expressions for both components of the current density can be written as:<maths id="math0004" num=""><math display="block"><mrow><mtable><mtr><mtd><mrow><mtable><mtr><mtd><mrow><mtable><mlabeledtr><mtext>(2a),</mtext><mtd><mrow><msub><mrow><mtext>j</mtext></mrow><mrow><mtext>φ</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>a</mtext></mrow></msup><mtext>(φ,z)=cosφ</mtext><apply><sum /><lowlimit><mtext>n=1</mtext></lowlimit><uplimit><mtext>∞</mtext></uplimit><mrow><msub><mrow><mtext>c</mtext></mrow><mrow><mtext>n</mtext></mrow></msub><msub><mrow><mtext>sink</mtext></mrow><mrow><mtext>n</mtext></mrow></msub><mtext>z for |z|≤</mtext><mfrac><mrow><mtext>L</mtext></mrow><mrow><mtext>2</mtext></mrow></mfrac></mrow></apply></mrow></mtd></mlabeledtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><mtable><mlabeledtr><mtext>(2b),</mtext><mtd><mrow><msub><mrow><mtext>j</mtext></mrow><mrow><mtext>z</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>a</mtext></mrow></msup><mtext>(φ,z)=sinφ</mtext><apply><sum /><lowlimit><mtext>n=1</mtext></lowlimit><uplimit><mtext>∞</mtext></uplimit><mrow><mtext></mtext><mfrac><mrow><msub><mrow><mtext>(-c</mtext></mrow><mrow><mtext>n</mtext></mrow></msub><mtext>)</mtext></mrow><mrow><msub><mrow><mtext>k</mtext></mrow><mrow><mtext>n</mtext></mrow></msub><mtext>a</mtext></mrow></mfrac><msub><mrow><mtext>cosk</mtext></mrow><mrow><mtext>n</mtext></mrow></msub><mtext>z for |z|≤</mtext><mfrac><mrow><mtext>L</mtext></mrow><mrow><mtext>2</mtext></mrow></mfrac></mrow></apply></mrow></mtd></mlabeledtr></mtable></mrow></mtd></mtr><mtr><mtd><mrow><mtable><mlabeledtr><mtext>(2c),</mtext><mtd><mrow><msub><mrow><mtext>j</mtext></mrow><mrow><mtext>z</mtext></mrow></msub><msup><mrow><mtext> </mtext></mrow><mrow><mtext>a</mtext></mrow></msup><mtext>(φ,z)= 0 for |z|≤</mtext><mfrac><mrow><mtext>L</mtext></mrow><mrow><mtext>2</mtext></mrow></mfrac><mtext></mtext></mrow></mtd></mlabeledtr></mtable></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></math><img file="EP0803737A2_D0004.tif" /></maths> where c_n are the Fourier coefficients, and<maths id="math0005" num=""><math display="block"><mrow><msub><mrow><mtext>k</mtext></mrow><mrow><mtext>n</mtext></mrow></msub><mtext> = </mtext><mfrac><mrow><mtext>(2n-1)π</mtext></mrow><mrow><mtext>L</mtext></mrow></mfrac><mtext> .</mtext></mrow></math><img file="EP0803737A2_D0005.tif" /></maths>

0033Because the coordinate system is cylindrical, the B_x component of the RF field can be expressed in terms of radial B_ρ and azimuthal B_φ components of the magnetic field as:<maths id="math0006" num="(3)."><math display="block"><mrow><msub><mrow><mtext>B</mtext></mrow><mrow><mtext>x</mtext></mrow></msub><msub><mrow><mtext> = B</mtext></mrow><mrow><mtext>ρ</mtext></mrow></msub><msub><mrow><mtext>cosφ - B</mtext></mrow><mrow><mtext>φ</mtext></mrow></msub><mtext>sinφ</mtext></mrow></math><img file="EP0803737A2_D0006.tif" /></maths>

0034Considering the Fourier transform of the two components of the current density distribution, and using the continuity equation to relate both components in the Fourier domain, the expression of the B_x component of the RF field is:<maths id="math0007" num=""><img file="EP0803737A2_D0007.tif" /></maths> with:<maths id="math0008" num=""><img file="EP0803737A2_D0008.tif" /></maths> and I_0,1, K₁ are the modified bessel functions of the first and second order, respectively.

0035Furthermore, the expression of the stored magnetic energy W is:<maths id="math0009" num="(6)."><math display="block"><mrow><mtext>W=</mtext><apply><int /><lowlimit><mtext>v</mtext></lowlimit><uplimit /><mrow><mover accent="true"><mrow><mtext>A</mtext></mrow><mo>→</mo></mover><mtext>·</mtext><mover accent="true"><mrow><mtext>J</mtext></mrow><mo>→</mo></mover><msup><mrow><mtext>d</mtext></mrow><mrow><mtext>3</mtext></mrow></msup><mtext>x</mtext></mrow></apply><mtext>=-</mtext><mfrac><mrow><msup><mrow><mtext>a</mtext></mrow><mrow><mtext>2</mtext></mrow></msup><msub><mrow><mtext>µ</mtext></mrow><mrow><mtext>o</mtext></mrow></msub><msup><mrow><mtext>L</mtext></mrow><mrow><mtext>2</mtext></mrow></msup></mrow><mrow><mtext>16</mtext></mrow></mfrac><mtext></mtext><apply><sum /><lowlimit><mtext>n=1</mtext></lowlimit><uplimit><mtext>∞</mtext></uplimit><mrow><mtext></mtext><apply><sum /><lowlimit><msup><mrow><mtext>n</mtext></mrow><mrow><mtext>/</mtext></mrow></msup><mtext>=1</mtext></lowlimit><uplimit><mtext>∞</mtext></uplimit><mrow><msub><mrow><mtext> c</mtext></mrow><mrow><mtext>n</mtext></mrow></msub><mtext>c</mtext><msub><mrow><mtext>​</mtext></mrow><mrow><msup><mrow><mtext>n</mtext></mrow><mrow><mtext>/</mtext></mrow></msup></mrow></msub><apply><int /><lowlimit><mtext>-∞</mtext></lowlimit><uplimit><mtext>∞</mtext></uplimit><mrow><msub><mrow><mtext>dkI</mtext></mrow><mrow><mtext>1</mtext></mrow></msub><msup><mrow><mtext>​</mtext></mrow><mrow><mtext>/</mtext></mrow></msup><msub><mrow><mtext>(ka)K</mtext></mrow><mrow><mtext>1</mtext></mrow></msub><msup><mrow><mtext>​</mtext></mrow><mrow><mtext>/</mtext></mrow></msup><msub><mrow><mtext>(ka) ψ</mtext></mrow><mrow><mtext>n</mtext></mrow></msub><mtext>(k)ψ</mtext><msub><mrow><mtext>​</mtext></mrow><mrow><msup><mrow><mtext>n</mtext></mrow><mrow><mtext>/</mtext></mrow></msup></mrow></msub><mtext>(k)</mtext></mrow></apply></mrow></apply></mrow></apply></mrow></math><img file="EP0803737A2_D0009.tif" /></maths>

0036The task is to find the current density distribution which generates a B_x component of the RF field according to the constraint equation:<maths id="math0010" num="(7),"><math display="block"><mrow><msub><mrow><mtext>B</mtext></mrow><mrow><mtext>x</mtext></mrow></msub><mtext>(</mtext><msub><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>j</mtext></mrow></msub><msub><mrow><mtext>) =B</mtext></mrow><mrow><mtext>xSC</mtext></mrow></msub><mtext>(</mtext><msub><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>j</mtext></mrow></msub><mtext>) for j=1, ..., N</mtext></mrow></math><img file="EP0803737A2_D0010.tif" /></maths> where B_xSC(<b>r</b>_j) is the constraint value of B_x at the point <b>r</b>_j and N is the number of constraint points.

0037In order to achieve this, the energy minimization mechanism is employed as applied for gradient coil design (U.S. Patent 5,296,810 to Morich) and construct the functional <img file="EP0803737A2_D0011.tif" /> as:<maths id="math0011" num=""><img file="EP0803737A2_D0012.tif" /></maths> Minimizing <img file="EP0803737A2_D0011.tif" /> with respect to c_n, a matrix equation for c_n' for every n becomes:<maths id="math0012" num="(8)."><math display="block"><mrow><apply><sum /><lowlimit><msup><mrow><mtext>n</mtext></mrow><mrow><mtext>/</mtext></mrow></msup><mtext>=1</mtext></lowlimit><uplimit><mtext>∞</mtext></uplimit><mrow><mtext> c</mtext><msub><mrow><mtext>​</mtext></mrow><mrow><msup><mrow><mtext>n</mtext></mrow><mrow><mtext>/</mtext></mrow></msup></mrow></msub><mtext></mtext><mfenced open="{" close="}"><mrow><mtext>-</mtext><mfrac><mrow><mtext>aLπ</mtext></mrow><mrow><mtext>2</mtext></mrow></mfrac><mtext></mtext><apply><int /><lowlimit><mtext>-∞</mtext></lowlimit><uplimit><mtext>∞</mtext></uplimit><mrow><msub><mrow><mtext> dkI</mtext></mrow><mrow><mtext>i</mtext></mrow></msub><msup><mrow><mtext>​</mtext></mrow><mrow><mtext>/</mtext></mrow></msup><msub><mrow><mtext>(ka)K</mtext></mrow><mrow><mtext>i</mtext></mrow></msub><msup><mrow><mtext>​</mtext></mrow><mrow><mtext>/</mtext></mrow></msup><mtext>(ka)ψ</mtext><msub><mrow><mtext>​</mtext></mrow><mrow><msup><mrow><mtext>n</mtext></mrow><mrow><mtext>/</mtext></mrow></msup></mrow></msub><msub><mrow><mtext>(k)ψ</mtext></mrow><mrow><mtext>n</mtext></mrow></msub><mtext>(k)</mtext></mrow></apply></mrow></mfenced></mrow></apply><mtext> = </mtext><apply><sum /><lowlimit><mtext>j=1</mtext></lowlimit><uplimit><mtext>N</mtext></uplimit><mrow><msub><mrow><mtext> λ</mtext></mrow><mrow><mtext>j</mtext></mrow></msub><mfrac><mrow><mtext>1</mtext></mrow><mrow><mtext>ρ</mtext></mrow></mfrac><mtext></mtext><apply><int /><lowlimit><mtext>-∞</mtext></lowlimit><uplimit><mtext>∞</mtext></uplimit><mrow><msub><mrow><mtext> dkcoskz</mtext></mrow><mrow><mtext>j</mtext></mrow></msub><msub><mrow><mtext>K</mtext></mrow><mrow><mtext>1</mtext></mrow></msub><msup><mrow><mtext>​</mtext></mrow><mrow><mtext>/</mtext></mrow></msup><msub><mrow><mtext>(ka)ψ</mtext></mrow><mrow><mtext>n</mtext></mrow></msub><msup><mrow><mtext>(k) [(kρ)cos</mtext></mrow><mrow><mtext>2</mtext></mrow></msup><msub><mrow><mtext>φI</mtext></mrow><mrow><mtext>0</mtext></mrow></msub><msub><mrow><mtext>(kρ) -cos2φI</mtext></mrow><mrow><mtext>1</mtext></mrow></msub><mtext>(kρ)]</mtext></mrow></apply></mrow></apply></mrow></math><img file="EP0803737A2_D0013.tif" /></maths>

0038Using a compact matrix notation and putting an upper threshold (M) to the infinite summations, Equation (8) becomes:<maths id="math0013" num="(9)."><math display="block"><mrow><apply><sum /><lowlimit><msup><mrow><mtext>n</mtext></mrow><mrow><mtext>/</mtext></mrow></msup><mtext>=1</mtext></lowlimit><uplimit><mtext>M</mtext></uplimit><mrow><mtext> c</mtext><msub><mrow><mtext>​</mtext></mrow><mrow><msup><mrow><mtext>n</mtext></mrow><mrow><mtext>/</mtext></mrow></msup></mrow></msub><mtext>C</mtext><msub><mrow><mtext>​</mtext></mrow><mrow><msup><mrow><mtext>n</mtext></mrow><mrow><mtext>/</mtext></mrow></msup><mtext>n</mtext></mrow></msub></mrow></apply><mtext> = </mtext><apply><sum /><lowlimit><mtext>j=1</mtext></lowlimit><uplimit><mtext>N</mtext></uplimit><mrow><msub><mrow><mtext> λ</mtext></mrow><mrow><mtext>j</mtext></mrow></msub><msub><mrow><mtext>D</mtext></mrow><mrow><mtext>jn</mtext></mrow></msub></mrow></apply><mtext>⇒</mtext><munder accentunder="true"><mrow><mtext>J</mtext></mrow><mo>̲</mo></munder><msup><mrow><mtext>​</mtext></mrow><mrow><mtext>a</mtext></mrow></msup><munder accentunder="true"><mrow><mtext>C</mtext></mrow><mo>̲</mo></munder><mtext>=</mtext><munder accentunder="true"><mrow><mtext>λD</mtext></mrow><mo>̲</mo></munder><mtext> or </mtext><munder accentunder="true"><mrow><mtext>J</mtext></mrow><mo>̲</mo></munder><msup><mrow><mtext>​</mtext></mrow><mrow><mtext>a</mtext></mrow></msup><mtext> = </mtext><munder accentunder="true"><mrow><mtext>λDC</mtext></mrow><mo>̲</mo></munder><msup><mrow><mtext>​</mtext></mrow><mrow><mtext>-1</mtext></mrow></msup></mrow></math><img file="EP0803737A2_D0014.tif" /></maths>

0039Employing the constraint equation, the final matrix solution for the continuous current distribution is given by:<maths id="math0014" num="(10),"><math display="block"><mrow><munder accentunder="true"><mrow><mtext>J</mtext></mrow><mo>̲</mo></munder><msup><mrow><mtext>​</mtext></mrow><mrow><mtext>a</mtext></mrow></msup><mtext>=</mtext><munder accentunder="true"><mrow><mtext>B</mtext></mrow><mo>̲</mo></munder><msub><mrow><mtext>​</mtext></mrow><mrow><mtext>x</mtext></mrow></msub><mtext>[</mtext><munder accentunder="true"><mrow><mtext>DC</mtext></mrow><mo>̲</mo></munder><msup><mrow><mtext>​</mtext></mrow><mrow><mtext>-1</mtext></mrow></msup><munder accentunder="true"><mrow><mtext>D</mtext></mrow><mo>̲</mo></munder><msup><mrow><mtext>​</mtext></mrow><mrow><mtext>T</mtext></mrow></msup><msup><mrow><mtext>]</mtext></mrow><mrow><mtext>-1</mtext></mrow></msup><munder accentunder="true"><mrow><mtext>DC</mtext></mrow><mo>̲</mo></munder><msup><mrow><mtext>​</mtext></mrow><mrow><mtext>-1</mtext></mrow></msup></mrow></math><img file="EP0803737A2_D0015.tif" /></maths> where <u>B</u>_x is the matrix representation of the field constraint points and ^T denotes the transpose of the matrix.

0040Thus, employing the energy minimization mechanism defines the continuous current distribution which generates the desired field behavior for the B_x component. Discretizing the continuous current density using the stream function technique (see U.S. Patent No. 5,296,810 to Morich), one obtains the discrete current pattern which is a close approximation of the continuous current density behavior. In order to validate the energy minimization mechanism independently, the B_x component of the RF field is re-evaluated by employing the Biot-Savart law to the discrete loop pattern.

0041We now proceed with the design of the first prototype RF coil. The radius of the coil is chosen by way of example to be a=15.25 cm, while the length of the coil is L=30 cm. Preferably, three constraint points are used in order to define the quality of the B_x component inside a 20 cm Diameter Spherical Imaging Volume (DSIV). The first constraint point sets the magnitude of the x-component to 23.5µT. The second constraint defines the on-axis variation of the magnetic field to within 10% from its ideal value and at a distance of 10 cm from the center of the RF coil. The final constraint limits the off-axis variation of the x component of the magnetic field to a highest of 30% at the borders of the 20 cm DSIV. This set of constraints are displayed in TABLE 1. <tables id="tabl0001" num="0001"><table frame="all"><title>TABLE 1</title><tgroup cols="4" colsep="1" rowsep="1"><colspec colnum="1" colname="col1" colwidth="39.37mm" /><colspec colnum="2" colname="col2" colwidth="39.37mm" /><colspec colnum="3" colname="col3" colwidth="39.37mm" /><colspec colnum="4" colname="col4" colwidth="39.37mm" /><thead valign="top"><row rowsep="1"><entry namest="col1" nameend="col1" align="center">n</entry><entry namest="col2" nameend="col2" align="right">ρ_i</entry><entry namest="col3" nameend="col3" align="right">z_i</entry><entry namest="col4" nameend="col4" align="center">B_xsc</entry></row></thead><tbody valign="top"><row><entry namest="col1" nameend="col1" align="right">1</entry><entry namest="col2" nameend="col2" align="char" char=".">0.001</entry><entry namest="col3" nameend="col3" align="char" char=".">0.000</entry><entry namest="col4" nameend="col4" align="char" char=".">0.000023500</entry></row><row><entry namest="col1" nameend="col1" align="right">2</entry><entry namest="col2" nameend="col2" align="char" char=".">0.100</entry><entry namest="col3" nameend="col3" align="char" char=".">0.000</entry><entry namest="col4" nameend="col4" align="char" char=".">0.0000211500</entry></row><row rowsep="1"><entry namest="col1" nameend="col1" align="right">3</entry><entry namest="col2" nameend="col2" align="char" char=".">0.001</entry><entry namest="col3" nameend="col3" align="char" char=".">0.100</entry><entry namest="col4" nameend="col4" align="char" char=".">0.000017000</entry></row></tbody></tgroup></table></tables>

0042Employing the energy minimization approach, the continuous current density is obtained for the linear B_x mode of the RF coil. FIGURES 5A and 5B show a two and four discrete loop pattern, respectively.

0043The theoretical basis for the electromagnetic analysis is the solution of Maxwell's equations for the electric field expressed in terms of the vector and scalar potentials. All wavelength and time dependence is taken into consideration. The Lorentz gauge condition is used to eliminate the scalar potential, and with harmonic time dependence, the electric field is written:<maths id="math0015" num="(11),"><math display="block"><mrow><msup><mrow><mover accent="true"><mrow><mtext>E</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>scat</mtext></mrow></msup><mtext>(</mtext><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover><mtext>)=</mtext><mfrac><mrow><mtext>-iµ</mtext></mrow><mrow><msub><mrow><mtext>4πωε</mtext></mrow><mrow><mtext>0</mtext></mrow></msub></mrow></mfrac><mtext></mtext><apply><int /><lowlimit><mtext>c</mtext></lowlimit><uplimit /><mrow><mtext>[∇</mtext><msub><mrow><mtext>​</mtext></mrow><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow></msub><mtext>(∇</mtext><msub><mrow><mtext>​</mtext></mrow><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow></msub><mtext>·</mtext><mover accent="true"><mrow><mtext>I</mtext></mrow><mo>→</mo></mover><mtext>(</mtext><msup><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>/</mtext></mrow></msup><mtext>)ψ(</mtext><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover><mtext>,</mtext><msup><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>/</mtext></mrow></msup><msup><mrow><mtext>))+β</mtext></mrow><mrow><mtext>2</mtext></mrow></msup><mover accent="true"><mrow><mtext>I</mtext></mrow><mo>→</mo></mover><mtext>(</mtext><msup><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>/</mtext></mrow></msup><mtext>)ψ(</mtext><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover><mtext>,</mtext><msup><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>/</mtext></mrow></msup><mtext>)]dℓ</mtext></mrow></apply></mrow></math><img file="EP0803737A2_D0016.tif" /></maths> where ψ(<b>r</b>,<b>r</b>') is the three-dimensional free-space Green function:<maths id="math0016" num="(12),"><math display="block"><mrow><mtext>ψ(</mtext><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover><mtext>,</mtext><msup><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>/</mtext></mrow></msup><mtext>)=</mtext><mfrac><mrow><msup><mrow><mtext>e</mtext></mrow><mrow><mtext>-iβR</mtext></mrow></msup></mrow><mrow><mtext>4πR</mtext></mrow></mfrac></mrow></math><img file="EP0803737A2_D0017.tif" /></maths> where R is the distance between the observation point <b>r</b> and the source point <b>r</b>', <b>I</b>(<b>r</b>') the current vector, and β a wave number. The integration is performed along the current path C.

0044For materials with high conductivity, the general condition:<maths id="math0017" num="(13),"><math display="block"><mrow><msup><mrow><mover accent="true"><mrow><mtext>E</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>scat</mtext></mrow></msup><mtext>(</mtext><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover><mtext>)+</mtext><msup><mrow><mover accent="true"><mrow><mtext>E</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>inc</mtext></mrow></msup><mtext>(</mtext><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover><mtext>)=0</mtext></mrow></math><img file="EP0803737A2_D0018.tif" /></maths> can be used to good approximation on the surfaces and inside of the conductors. In the present application, the condition implies that the sum of the scattered field and the incident field vanishes at the surface of the perfectly conducting wire and also interior to the wire. This leads to Pocklington's integral equation:<maths id="math0018" num="(14)."><math display="block"><mrow><mfrac><mrow><mtext>-iµ</mtext></mrow><mrow><msub><mrow><mtext>4πωε</mtext></mrow><mrow><mtext>0</mtext></mrow></msub></mrow></mfrac><mtext></mtext><apply><int /><lowlimit><mtext>c</mtext></lowlimit><uplimit /><mrow><mtext>[∇</mtext><msub><mrow><mtext>​</mtext></mrow><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow></msub><mtext>(∇</mtext><msub><mrow><mtext>​</mtext></mrow><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow></msub><mtext>·</mtext><mover accent="true"><mrow><mtext>I</mtext></mrow><mo>→</mo></mover><mtext>(</mtext><msup><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>/</mtext></mrow></msup><mtext>)ψ(</mtext><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover><mtext>,</mtext><msup><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>/</mtext></mrow></msup><msup><mrow><mtext>))+β</mtext></mrow><mrow><mtext>2</mtext></mrow></msup><mover accent="true"><mrow><mtext>I</mtext></mrow><mo>→</mo></mover><mtext>(</mtext><msup><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>/</mtext></mrow></msup><mtext>)ψ(</mtext><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover><mtext>,</mtext><msup><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>/</mtext></mrow></msup><mtext>)]dℓ</mtext></mrow></apply><mtext>=-</mtext><msup><mrow><mover accent="true"><mrow><mtext>E</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>inc</mtext></mrow></msup><mtext>(</mtext><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover><mtext>)</mtext></mrow></math><img file="EP0803737A2_D0019.tif" /></maths> The current <b>I</b>(<b>r</b>') in the above expression is the unknown function to be found.

0045The method of moments is utilized to reduce Pocklington's integral equation for wire segments to a system of simultaneous linear algebraic equations in terms of the unknown current. The conducting body is modeled by an approximation in which it is subdivided into wire segments (monopoles). The approximation is made as accurate as necessary by increasing the number of such monopoles. Any two consecutive monopoles are defined as a V-shaped dipole over which a testing function is defined, satisfying the boundary condition that the current vanishes at both ends of the dipole. The n-th testing function is non-zero only when the V-shaped dipole corresponds to the n-th V-shaped dipole, and zero otherwise. The testing function of choice is <b>W</b>_n(<b>r</b>), the n-th piecewise sinusoidal testing function are defined by:<maths id="math0019" num=""><img file="EP0803737A2_D0020.tif" /></maths> ℓ_n-1 is the vector connecting the two points <b>r</b>_n-1 and <b>r</b>_n, and ℓ_n the vector connecting the two points, <b>r</b>_n and <b>r</b>_n+1. ℓ_n-1 and ℓ_n are the unit vectors along the n-th V-shaped dipole. The current <b>I</b>(<b>r</b>') is expanded in terms of the testing function:<maths id="math0020" num="(16),"><math display="block"><mrow><mover accent="true"><mrow><mtext>I</mtext></mrow><mo>→</mo></mover><mtext>(</mtext><msup><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>/</mtext></mrow></msup><mtext>) = </mtext><apply><sum /><lowlimit><mtext>n=1</mtext></lowlimit><uplimit><mtext>N</mtext></uplimit><mrow><msub><mrow><mtext> I</mtext></mrow><mrow><mtext>n</mtext></mrow></msub><msub><mrow><mover accent="true"><mrow><mtext>W</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>n</mtext></mrow></msub><mtext>(</mtext><msup><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>/</mtext></mrow></msup><mtext>)</mtext></mrow></apply></mrow></math><img file="EP0803737A2_D0021.tif" /></maths> where the I_n is the unknown current coefficient.

0046The expansion of the electric field <b>E</b>^inc(<b>r</b>) is made in terms of the same set of testing functions procedure, corresponding to an implementation of Galerkin's method with Richmond-Schelkunoff's piecewise sinusoidal bases.

0047The substitution of the expansions into Equation (4) yields a linearized problem:<maths id="math0021" num="(17),"><math display="block"><mrow><apply><sum /><lowlimit><mtext>n=1</mtext></lowlimit><uplimit><mtext>N</mtext></uplimit><mrow><msub><mrow><mtext> Z</mtext></mrow><mrow><mtext>mn</mtext></mrow></msub><msub><mrow><mtext>I</mtext></mrow><mrow><mtext>n</mtext></mrow></msub></mrow></apply><msub><mrow><mtext> = V</mtext></mrow><mrow><mtext>m</mtext></mrow></msub></mrow></math><img file="EP0803737A2_D0022.tif" /></maths> where Z_mn is the generalized impedance matrix between segments m and n (m, n=1, ..., N),<maths id="math0022" num="(18),"><math display="block"><mrow><msub><mrow><mtext>Z</mtext></mrow><mrow><mtext>mn</mtext></mrow></msub><mtext> = </mtext><apply><int /><lowlimit><mtext>(</mtext><msub><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>m-1</mtext></mrow></msub><mtext>,</mtext><msub><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>m</mtext></mrow></msub><mtext>,</mtext><msub><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>m+1</mtext></mrow></msub><mtext>)</mtext></lowlimit><uplimit /><mrow><mtext></mtext><msub><mrow><mover accent="true"><mrow><mtext>W</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>m</mtext></mrow></msub><mtext>(</mtext><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover><mtext>)·</mtext><msub><mrow><mover accent="true"><mrow><mtext>E</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>n</mtext></mrow></msub><msup><mrow><mtext>​</mtext></mrow><mrow><mtext>scat</mtext></mrow></msup><mtext>(</mtext><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover><mtext>)dℓ</mtext></mrow></apply></mrow></math><img file="EP0803737A2_D0023.tif" /></maths> and the m-th generalized voltage matrix element is:<maths id="math0023" num="(19)."><math display="block"><mrow><msub><mrow><mtext>V</mtext></mrow><mrow><mtext>m</mtext></mrow></msub><mtext> = - </mtext><apply><int /><lowlimit><mtext>(</mtext><msub><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>m-1</mtext></mrow></msub><mtext>,</mtext><msub><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>m</mtext></mrow></msub><mtext>,</mtext><msub><mrow><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>m+1</mtext></mrow></msub><mtext>)</mtext></lowlimit><uplimit /><mrow><mtext></mtext><msub><mrow><mover accent="true"><mrow><mtext>W</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>m</mtext></mrow></msub><mtext>(</mtext><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover><mtext>)·</mtext><msup><mrow><mover accent="true"><mrow><mtext>E</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>inc</mtext></mrow></msup><mtext>(</mtext><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover><mtext>)dℓ</mtext></mrow></apply></mrow></math><img file="EP0803737A2_D0024.tif" /></maths> The expression for <b>E</b>_n^scat(<b>r</b>) is obtained from Equation (11) by the substitution:<maths id="math0024" num=""><math display="block"><mrow><mover accent="true"><mrow><mtext>I</mtext></mrow><mo>→</mo></mover><mtext>(</mtext><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover><mtext>)→</mtext><msub><mrow><mover accent="true"><mrow><mtext>W</mtext></mrow><mo>→</mo></mover></mrow><mrow><mtext>n</mtext></mrow></msub><mtext>(</mtext><mover accent="true"><mrow><mtext>r</mtext></mrow><mo>→</mo></mover><mtext>) .</mtext></mrow></math><img file="EP0803737A2_D0025.tif" /></maths> The integrations in Equations (18) and (19) are performed along the m-th V-shaped dipole. The voltage matrix elements V_m are determined by the location of the external power source. They are scaled to unit value if the corresponding dipoles are directly connected to the source; otherwise they are set to zero. This approximation assumes a delta function for the incident electric field <b>E</b>^inc(<b>r</b>), and the numerical results for it are accurate for the frequencies and the voltage gaps considered.

0048The unknown current coefficients are obtained by inverting the linear matrix problem of Equation (17):<maths id="math0025" num="(20)."><math display="block"><mrow><msub><mrow><mtext>I</mtext></mrow><mrow><mtext>n</mtext></mrow></msub><mtext>=</mtext><apply><sum /><lowlimit><mtext>m=1</mtext></lowlimit><uplimit><mtext>N</mtext></uplimit><mrow><msup><mrow><mtext> (Z</mtext></mrow><mrow><mtext>-1</mtext></mrow></msup><msub><mrow><mtext>)</mtext></mrow><mrow><mtext>nm</mtext></mrow></msub><msub><mrow><mtext>V</mtext></mrow><mrow><mtext>m</mtext></mrow></msub></mrow></apply></mrow></math><img file="EP0803737A2_D0026.tif" /></maths> The magnetic field behavior and the effective circuit impedance for circuit analysis is found following the current calculation.

0049Applying this solution to the dimensions of an exemplary RF coil system are as follows. The set of V-shaped dipoles are chosen for wire radii of 1.5 mm and subject to the restriction that the lengths of the V-shaped dipoles are less than or equal to 20% of the wavelength. The current pattern for the coil is specified by one quadrant out of the full plane of the current surface, in view of its symmetry. The number of coordinates of the optimized wire paths in the first quadrant may be reduced to 25 points, in order to handle the numerical demands for the Z_mn inverse-matrix computations, but maintaining accurate current patterns. The radius of the coil is 0.1525 m, and the length 0.3 m. The current paths, connections, and feed locations are shown in FIGURE 4. The number of testing functions for the total system is 206.

0050Lumped capacitors <b>66</b>_<b>1</b>, <b>66</b>_<b>2</b> are distributed over the current tracings to tune the system to the desired frequency. The coil frequency spectrum is determined by calculation of the coil input impedance:<maths id="math0026" num="(21)."><math display="block"><mrow><msub><mrow><mtext>Z</mtext></mrow><mrow><mtext>input</mtext></mrow></msub><mtext> = </mtext><mfrac><mrow><msub><mrow><mtext>V</mtext></mrow><mrow><mtext>input</mtext></mrow></msub></mrow><mrow><msub><mrow><mtext>I</mtext></mrow><mrow><mtext>input</mtext></mrow></msub></mrow></mfrac></mrow></math><img file="EP0803737A2_D0027.tif" /></maths> The insertion of a tuning capacitor (capacitance <b>C</b>) implies the diagonal matrix element Z_mm associated with the m-th V-shaped dipole is altered by:<maths id="math0027" num="(22)."><math display="block"><mrow><msubsup><mrow><mtext>Z</mtext></mrow><mrow><mtext>mm</mtext></mrow><mrow><mtext>/</mtext></mrow></msubsup><msub><mrow><mtext> = Z</mtext></mrow><mrow><mtext>mm</mtext></mrow></msub><mtext>-</mtext><mfrac><mrow><mtext>i</mtext></mrow><mrow><mtext>ωC</mtext></mrow></mfrac></mrow></math><img file="EP0803737A2_D0028.tif" /></maths>

0051In addition, the input impedance <b>74</b> is adjusted to the impedance of the external source. A matching capacitor (capacitance <b>C</b>_F) is used, and the input current I_input is modified by:<maths id="math0028" num="(23)."><math display="block"><mrow><msubsup><mrow><mtext>I</mtext></mrow><mrow><mtext>input</mtext></mrow><mrow><mtext>/</mtext></mrow></msubsup><msub><mrow><mtext> = I</mtext></mrow><mrow><mtext>input</mtext></mrow></msub><msub><mrow><mtext> + iωC</mtext></mrow><mrow><mtext>F</mtext></mrow></msub><msub><mrow><mtext>V</mtext></mrow><mrow><mtext>input</mtext></mrow></msub></mrow></math><img file="EP0803737A2_D0029.tif" /></maths>

0052The magnetic field is evaluated by a quasi-static Biot-Savart calculation, since the capacitive distribution has been designed to generate currents with relatively constant phases. The field is examined at each resonant frequency found in the aforesaid input-impedance frequency spectrum as shown in FIGURE 3. For the tuning scheme, the matching capacitance is chosen to be <b>C</b>_F=100 pF to begin with, and the tuning capacitances <b>C</b> are varied from 20 pF to 30 pF in the frequency-range of 10 MHz to 100 MHz. As seen from the spectrum, the system resonates at 64 MHz, which is the third resonant peak counting upward from 10 MHz, when <b>C</b> is 23 pF. When resonance at 64 MHz is achieved, the resulting linear mode B_x of the RF coil is re-evaluated to see that it has spatial characteristics consistent with the static solution (FIGURES 6A and 6B). The input impedance must be adjusted to match the external electronics. It was found that the change in the capacitance value of <b>C</b>_F is not sensitive to the resonant frequencies. Thus, <b>C</b>_F was adjusted to 142 pF with Z_input= 50.27 Ω while maintaining the resonant nature of the system.

0053Based on the results of the Method of Moments theory, Cu foil is etched with the pattern of the proposed RF coil and attached into a 29 cm diameter acrylic former. The coil resonates at 63.725 MHz using 28 pF tuning capacitors. The reflection coefficient is -13 dB when the coil is unloaded and -29 dB when the coil is loaded. As a next step, the coil is inserted into a 1.5 T whole body MRI unit. The coil is operated in a transmit as well as receive mode configuration in the same experiment. The MRI sequences used with the coil include a spin-echo (SE) sequence with TE= 30 msec. and TR= 50 msec. and FOV= 25 cm.

0054With reference to FIGURE 7, in some applications it is advantageous to limit the sensitivity of the coil along the z or axial direction. When constraints are placed on the design such that the coil sensitivity falls off rapidly at the end of the coil, counter-rotating coil sections <b>80</b>, <b>82</b> are added to the ends of the coil. The counter-rotating sections carry currents in an opposite direction of rotation from coils <b>64</b> and <b>68</b>.

0055With reference to FIGURE 8, the coil need not be constrained to lie on a circularly symmetrical cylinder. Rather, the coil can be constrained to other shapes such as oval cylinders, D-shaped cylinders, planes, shapes which conform to various segments of the body, and the like. When the currents are constrained to lie on a plane, the windings take on the distribution shown in FIGURE 8.

0056When the four loop coil of FIGURE 5B is constrained to become more uniform along the central plane, the coil becomes more squared off at the corners as illustrated in solid line in FIGURE 9.

0057Various alternate embodiments will immediately suggest themselves to those of ordinary skill in the art. Although described with reference to cylindrical bore magnets, the same design techniques can be utilized to design radio frequency coils for use with other types of magnets, such as C-magnets. Rather than connecting the coils in parallel, each of the coils can be tuned and matched with independent pickups to form an array coil structure. Currents in some legs of the coil can be permitted to be different from those flowing in other legs. A second coil of like construction can be mounted 90° rotated on the cylinder to make a cylindrical quadrature coil. Quadrature coils can also be designed for planar, biplanar, elliptical, and other geometries. The coils can also be used in transmit modes, using all or a portion of the loops. In the multiple loop embodiments, the multiple loops can be connected in parallel and the parallel arrangement separately fed from other individual or multiple loops of the structure. It will further be appreciated that in some embodiments, negative or counter-rotating current loops will be called for. Such loops are advantageously formed as a closed circuit so as to create the required counter-flowing currents according to Lenz's law.