GB2407388A - Method of designing asymmetric radio frequency coils for magnetic resonance - Google Patents

Abstract

Time harmonic methods for designing asymmetric radio frequency (RF) coils for magnetic resonance applications are described. Such methods can also be used to design symmetrically positioned RF coils in MRI machines. In addition, methods for converting complex current density functions into discrete capacitive and inductive elements are provided. An open ended rf coil system which provides an asymmetric rf field comprising inductive and capacitive elements is presented.

Description

METHOD OF DESIGNING ASYMMETRIC RADIO FREQUENCY COILS

FIELD OF THE INVENTION

This invention relates to radio frequency coils for magnetic resonance (MR) applications. In particular, the invention is directed to methods for designing asymmetric radio frequency coils for magnetic resonance imaging (MRT) machines.

Ilowevcr, the invention is not limited thereto as tile methods can be used to design radio frequency coils which are symmetrically positioned.

The radio frequency coils may be used for transmitting a radio frequency field, receiving a magnetic resonance signal, or both transmitting a radio frequency field and receiving a magnetic resonance signal. When the radio frequency coil serves a transmitting (or field generating) function, it will normally be combined with a shield to reduce magnetic interference with external components of the magnetic resonance imaging system.

BACKGROUND OF THE INVENTION

In magnetic resonance imaging (MRI) applications, a patient is placed in a strong and homogeneous static magnetic field, causing the otherwise randomly oriented magnetic moments of the protons, in water molecules within the body, to process around the direction of the applied field. The part of the body in the homogeneous region of the magnet is then irradiated with radio-frequency (RF) energy, causing some of the protons to change their spin orientation. The net magnetization of the spin ensemble is notated away from the direction of the applied static magnetic field by the applied RF energy. The component of this net magnetization orthogonal to the direction of the applied static magnetic field acts to induce measurable signal in a receiver coil tuned to the frequency of precession. This is the magnetic resonance (MR) signal.

The useful RF components are those generated in at plane at 90 degrees to the direction of the static magnetic field. The same coil structure that generates the RF field can be used to receive the MR signal or a separate receiver coil placed close to the patient may be used. In either case the coils are tuned to the Larmor precessional 3() frequency t'0 whereby = yBo Id if is the gyTomagnciic ratio for a specific nuclide and

Be is the applied static magnetic field.

A desirable property of radio frequency coils for use in MR is the generation of homogeneous RF fields over a prescribed region. Normally this region is central to the coil structure for transmission resonators. A well known example of transmission resonators is the birdcage resonator, details of which are given by Hayes et. al. in The Journal of Magnetic Resonance, 63, 622 (1985) and U.S. Pat. No. 4,694,255.

In some circumstances it is desirable to generate a target field over an asymmetric region of the coil structure, i e., a region that is asymmetric relative to the mid-lengtl1 point of the longitudinal axis of the coil structure. This is potentially advantageous for patient access, conformation of the coil structure to the local anatomy of the patient and for use in asymmetric magnet systems.

One method that is known in the art for generating homogeneous fields over a volume that is asymmetric to the coil structure is to enclose one end of the cylindrical structure, a so-called 'end-cap' or dome structure (details of which are given by Meyer and Ballon in The Journal of Magnetic Resonance, 107, 19 (1995) and by Hayes in SMRM 5th annual meeting, Montreal, Book of Abstracts, 39 (1986)). These designs were applied to structures that surrounded only the head of a patient and, by their nature, prevent access to the top of the head. The limited access also makes these structures problematic for whole-body imaging as they substantially reduce access from one end of the magnet.

It is an aim of the present invention to provide a general systematic method for producing a desired radio frequency field within a coil, using a full-wave, frequency specific technique to first define a current density on at least one cylindrical surface and subsequently to synthesize a coil pattern from the current density.

It is a particular aim of this invention to provide a method of designing coil structures that generate desired RF fields within certain specific, and asymmetric portions of the overall coil structure, preferably without substantially limiting access from one end of the structure. Asymmetric radio frequency coils can be used in conventional MR systems or in the newly developed asymmetric magnets of U.S. Patent No. 6,140,900.

It is a further particular aim of the present invention to use complex current densities in the full-wave, frequency specific method.

SUMMARY OF THE I.TV3TION

In one broad form, the invention provides a method for manufacturing a radio frequency coil structure for a MR device having a cylindrical space, preferably with open ends, comprising the steps of selecting a target region over which a transverse RF magnetic field of a predetermined frequency is to be applied by the coil structure, the target region being preferably asymmetrically located relative to the mid-length point of the longitudinal axis of the cylindrical space, calculating current density at the surface of the cylindrical space required to generate the target field at the predetermined fiequency, synthesizing a design for tile coil structure from the calculated current density in accordance with one of the methods discussed below, and forming a coil structure according to the synthesized design.

lO Preferably, the method for calculating the current density uses a time harmonic method that accounts for the frequency of operation of the RF coil structure and makes use of a complex current density.

The resultant RF coils can be used as transmitter coils, receiver coils, or both transmitter and receiver coils. As discussed above, when the coil serves a transmitting function, it will normally be combined with a shield to reduce magnetic interference with external components of the magnetic resonance imaging system. To avoid redundancy, the following summary of the method aspects of the invention is in terms of a RF coil system which includes a main coil (corresponding to the "first complex current density") and a shielding coil (corresponding to the "second complex current density"), it being understood that these methods can be practiced with just a main coil.

In accordance with a first method aspect of the invention, which can be used under "mild" coil length to wavelength conditions, i.e., conditions in which the coil length is less than about one-fifth of the operating wavelength, a method for designing apparatus for transmitting a radio frequency field (e.g., a field having a frequency of at least 20 Megahertz, preferably at least 80 Megahertz), receiving a magnetic resonance signal, or both transmitting a radio frequency field and receiving a magnetic resonance signal is provided which comprises: (a) defining a target region (e.g., a spherical or ellipsoidal region preferably having a volume of at least 30 x 103 cm3 and asymmetrically located relative to the midpoint of the RF coil, e.g., the ratio of the distance between the midpoint of the target region and one end of the coil to the length of the coil is less than or equal to 0.4) in which the radial magnetic component (e.g., Bx, By' or combinations of Bx and By) of the radio frequency field is to have desired values (e.g., substantially uniform magnitudes), said target region surrounding a longitudinal axis, i.e., the common longitudinal axis of the magnetic resonance imaging system and the RF coil; (b) specifying desired values for said radial magnetic component of the radio frequency field at a preselected set of points within the target region (e.g., a set of points distributed tlu-oughout the target region or a set of points distributed at the outer boundary of the target region) ; (c) defining a target surface (e.g., the shield coil surface) external to the apparatus on which the magnetic component of the radio frequency field is to have a desired value of zero at a preselected set of points on said target surface; (d) determining a first complex current density function, having real and imaginary parts, on a first specified cylindrical surface (i.e., the main coil surface) and a second complex current density, having real and imaginary parts, on a second specified cylindrical surface (i.e., the shield coil surface), the radius of the second specified cylindrical surface being greater than the radius of the first specified cylindrical surface (e.g., the radius of the second surface can be 10% greater than the radius of the first surface) by: (i) defining each of the complex current density functions as a sum of a series of basis functions (e.g., triangular and/or pulse or sinusoidal and/or cosinusoidal functions) multiplied by complex amplitude coefficients having real and imaginary parts; and (ii) determining values for the complex amplitude coefficients using an iterative minimization technique (e.g., a linear steepest descent or a conjugate gradient descent technique) applied to a first residue vector obtained by taking the difference between calculated field values obtained using the complex amplitude coefficients at the set of preselected points in the target region and the desired values at those points and a second residue vector equal to calculated field values obtained using the complex arnp]itude coefficients at the preselected set of points on the target surface; and (e) converting said first and second complex current density functions into sets of capacitive elements (as understood by persons skilled in the art, such capacitive elen ents will in general have some inductive and resistive properties) and sets of inductive elements (as understood by persons skilled in the art, such inductive elements will in general have some capacitive and resistive properties) located on the specified cylindrical surfaces by: (i) converting each of the first and second complex current density functions into a curl-free component Jcur-free and a divergence-free component JU'v-free using the relationships: Jour/ pree = V/ and Jdv_ flee = V X S where 1 and S are functions obtained from the respective first and second complex current density functions through the equations: Vet/ = V J -V2s = V X J and -V2(nS) =n VX J where n is a vector normal to the respective first and second specified cylindrical surfaces and J is the respective first and second complex current density functions; (ii) calculating locations on the respective first and second cylindrical surfaces for the respective sets of capacitive elements by contouring the respective or functions; and (iii) calculating locations on the respective first and second cylindrical surfaces for the respective sets of inductive elements by contouring the respective functions neS (sir and n.S are referred to herein and function as "streaming functions").

In accordance with a second method aspect of the invention, which can be used under "non-mild" coil length to wavelength conditions, i.e., conditions in which the coil length can be greater than about one-fifth of the operating wavelength, a method for designing apparatus for transmitting a radio frequency field or both transmitting a radio frequency field and receiving a magnetic resonance signal is provided which comprises: (a) defining a target region in which the radial magnetic component of the radio frequency field is to have desired values, said target region surrounding a longitudinal axis, i.e., the common longitudinal axis of the magnetic resonance imaging system and the RF coil; (b) specifying desired values for said radial magnetic component of the radio frequency field at a preselected set of points within the target region; (c) defining a target surface external to the apparatus on which the magnetic component of the radio frequency field is to have a desired value of zero; (d) determining a first complex current density function, having real and imaginary parts, on a first specified cylindrical surface and a second complex current density, having real and imaginary parts, on a second specified cylindrical surface, the radius of the second specified cylindrical surface being greater than the radius of the first specified cylindrical surface: (i) defining each of the complex current density functions as a sum of a series of basis functions (e.g., sinusoidal and/or cosinusoidal functions) multiplied by complex amplitude coefficients having real and imaginary parts; and (ii) determining values for the complex amplitude coefficients by simultaneously solving matrix equations of the form: LA] (aC)+ |AIS (as)= Be (Eq. I) LA2 1(a)+ BANS l(aS)= BS where AC,As,Ac,andA2s are transformation matrices between current density space and magnetic field space whose components are based on time hannonic Green's functions, ac and as are vectors of the unknown complex amplitude coefficients for the first and second complex current density functions, respectively, Bc is a vector of the desired values for the radial magnetic field specified in step (b), and Bs is a vector whose values are zero, said equations being solved by: (1) transforming the equations into functionals that can be solved using a preselected regularization technique, and (2) solving the functionals using said regularization technique to obtain values for the complex amplitude coefficients; and (e) converting said first and second complex current density functions into sets of capacitive elements and sets of inductive elements located on the specified cylindrical surfaces (preferably, this step is performed using the methods of step (e) of the first method aspect of the invention; also the methods of said step (e) can be used independent of either the first or second method aspects of the invention to convert a complex current density function to sets of capacitive and inductive elements, i. e., to a manufacturable coil structure).

In accordance with certain preferred embodiments of this second method aspect of the invention, the regularization functional is chosen so as to minimize the integral of the dot product of the first complex current density function with itself over the first specified cylindrical surface and to minimize the integral of the dot product of the second complex current density function with itself over the second specified cylindrical surface. Other regulanzation functonals that can he used include: (l) the second derivative of the complex current density functions, and (2) other functionals besides the dot product that are proportional to the square of the complex current density functions.

l 0 In accordance with other preferred embodiments, the complex amplitude coefficients are chosen so that the first and second complex current density functions each has zero divergence.

The parenthetical statements set forth in connection with the summary of the first method aspect of the invention also apply to the second method aspect of the invention except where indicated. In connection with all of its method aspects, the invention also preferably includes the additional step(s) of displaying the locations of the sets of inductive elements on the first and second specified cylindrical surfaces and/or producing physical embodiments of those sets of elements.

There is also disclosed herein, apparatus for use in a magnetic resonance system for transmitting a radio frequency field (e.g., a field having a frequency greater than, for example, 20 megahertz, and preferably greater than 80 megahertz), receiving a magnetic resonance signal, or transmitting a radio frequency field and receiving a magnetic resonance signal, the apparatus and the magnetic resonance imaging system having a common longitudinal axis. The apparatus comprises: (a) a support member (e.g., a tube composed of fiberglass, TEFLON, or other materials, which preferably can be used as a substrate for a photolithography procedure for printing conductive patterns and which does not substantially absorb RF energy) which defines a bore (preferably a cylindrical bore) having first and second open ends which are spaced from one another along the longitudinal axis by a distance T (the open ends are preferably both sized to r eceive a patient's body part which is to be imaged, e.g., the head, the upper torso, the lower torso, a limb, etc.); and (b) a plurality of inductive elements (e.g., copper or other metallic tracks or tubes) and a plurality of capacitive elements (e.g., distributed or lumped elements) associated with the support member (e.g., mounted on and/or mounted n and/or mounted to the support member); wherein if used as a transmitter, the apparatus has the following characteristics (the "if used as a transmitter" terminology is used in defining both transmitting and receiving RF coils since by reciprocity, coils that transmit unifonn radio frequency fields, receive radio frequency fields with a unifonn weighting function): (i) the apparatus produces a radio frequency field which has a radial magnetic component which has a spatially-varying peak magnitude whose average value is Ar avg; (ii) the apparatus has a homogenous volume within the bore over which the spatially-varying peak magnitude of the radio frequency field has a maximum deviation from Ar am which is less than or equal to 15% (preferably less than or equal to 10%); (iii) the homogeneous volume defines a midpoint M which is on the longitudinal axis, is closer to the first end than to the second end, and is spaced from the first end by a distance D such that the ratio D/L is less than or equal to 0.4 (preferably less than or equal to 0.25); and (iv) at least one of said inductive elements (e.g., l, 2, 3, ... or all) comprises a discrete conductor (e.g., a copper or other metallic track or tube) which follows a sinuous path such that during use of the apparatus current flows through a first part of the conductor in a first direction that has both longitudinal and azimuthal components and through a second part of the conductor in a second direction that has both longitudinal and azimuthal components, said first and second directions being different.

An example of such a sinuous path is shown in Figure 15 discussed below.

The homogenous volume is preferably at least 30 x 103 cm3 and/or L is at least 1.0 meter.

The first and second open ends preferably have transverse cross-sectional areas Al and A2, respectively, which satisfy the relationship: IAi -A2l/A < 0.1.

where Al is preferably at least 2 x 103 cm2.

The lengths of coils and of cylindrical spaces associated therewith are defined in terms of the inductive elements making up the coil. In particular, for a horizontally oriented coil, the length constitutes the distance along the longitudinal axis from the leftmost edge of the leftmost inductive element to the rightmost edge of the rightmost inductive element, corresponding definitions applying to other orientations of the coil.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be described by way of example with reference to the drawings in which: FIGURE I illustrates a general layout of a cylindrical radio frequency coil system of length L and radius a. Changing the value of D changes the degree of symmetry of the 'target region' relative to the coil structure; FIGURE 2 shows a current distribution on the surface of a cylinder calculated using a quasi-static formulation, where the vertical axis is the z-axis; FIGURE 3 shows plots of J', and Jz along the length of a cylinder in the =0 and =/2 planes respectively for a quasi- static formulation; FIGURES 4A and 4B show plots of J.,, and Jz along the length of a cylinder in the =0 and =/2 planes respectively for the real solutions of Table 4 (19OMHz). In FIGURE 4A, zOS=20mm, while in FIGURE 4B, zOs=50mm; FIGURE 5 shows the current distribution on the surface of a cylinder calculated from the real solution of Table 4 (zOs=20mm; 1 90MHz); FIGURE 6 shows the current distribution on the surface of a cylinder calculated from the real solution of Table 4 (zOs=50mm; 1 90MHz); FIGURE 7 shows a full conductor model of the contours of the symmetric current density distribution shown in FIG. 3.

FIGURE 8 shows a current density distribution for a 320mm length coil with radius 100mm having a homogeneous volume 50mm from the center of the coil towards the coil's lower end.

FIGURE 9 shows contour lines for n.S for the current density of FIG. 8.

FIGURE 10 shows contour lines for for the current density of FIG. 8.

FIGURE 11 shows the contour lines for the stream function n.S calculated using the second method aspect of the invention.

FIGURE 12 shows a normalized magnetic field for a representative zsyll., lletric coil in the transverse plane determined using the second method aspect of the invention.

The upper curve shows the distribution in the x direction and the lower curve shows the distribution in the y direction.

FIGURE 13 shows a transverse magnetic field along the z-axis for a representative asymmetric coil determined using the second method aspect of the invention.

FIGURE 14 is a circuit-type layout for a representative asymmetric coil.

FIGURE 15 is a 3D perspective of an inductive geometry for a representative asymmetric coil. A partial shield is included for illustrative purposes.

FIGURE 16 is a flow chart illustrating the first method aspect ofthe invention.

FIGURE 17 is a flow chart illustrating the second method aspect of the invention.

FIGURE 18 is a flow chart illustrating a preferred method for calculating a coil geometry from a current density.

The foregoing drawings, which are incorporated in and constitute part of the specification, illustrate the preferred embodiments of the invention, and together with the description, serve to explain the principles of the Invention. It is to be understood, of course, that both the drawings and the description are explanatory only and are not restrictive of the invention.

DETAILED DESCRIPTION OF EMBODIMENTS OF THE INVENTION

As discussed above, the present invention relates to RF coils having prescribed properties and to methods for designing these and other types of RF coils. Figures 16 18 illustrate the overall numerical procedures of the invention with reference to the various equations presented below.

The methods of the invention as described below are preferably practiced on a digital computer system configured by suitable programming to perform the various computational steps. The programming can be done in various programming languages known in the art. A preferred programming language is the C language which is particularly well-suited to performing scientific calculations. Other languages which can be used include FORTRAN, BASIC, PASCAL, C++, and the like. The program can be embodied as an article of manufacture comprising a computer usable medium, such as a magnetic disc, an optical disc, or the like, upon which the program is encoded.

The computer s,tsterr. can cor.prisc a general purpose scientirc computer and its associated peripherals, such as the computers and peripherals currently being manufactured by DIGITAL EQUIPMENT CORPORATION, IBM, HEWLETT PACKARD, SIAN MICROSYSTEMS, SGI or the like. For example, the numerical -1 1 procedures of the invention can be implemented in C-code and performed on a work station. The system should include means for inputting data and means for outputting the results of the RF coil design both in electronic and visua] form. The output can also be stored on a disk drive, tape drive, or the dike for further analysis and/or subsequent display.

Qiasi-,Static Arialy.ss In outline, the object ofthe design process is to produce structures that generate target RF transverse fields (that is, Bx or By or combinations thereof) over a defined volume inside a cylinder. The value of Bx (for example) is specified at a number of chosen points inside this region and in one embodiment, inverse methods are used to compute the corresponding current distribution.

The geometry of the coil is shown in Figure 1. Surface current densities on the cylinder are defined as: J(r) = Jo (, z) + Jz (, z)z for p = a and -L12 < z < L12 ( 1) where r = p p + zz; L and a are the length and radius of the cylinder.

As is known in the art, in one formulation for the case of a symmetric coil structure, J(r) can be written as a summation of M sinusoidal basis functions: J,(5,z) = CossiJcm sin(kmz) (2) m=1 M _ r Jz(',z) = sinews cm cos(kmz) (3) n=1 kma where Jcm are the current coefficients and km=(2ml)nlL.

However, this expression is only applicable to problems where the magnetic field is symmetrical about z=O. Magnetic fields are expressed in terms of magnetic vector potentials defined as the convolution of surface current density with the corresponding Green's function in the spatial domain: A(r)= | G(rlr')J(r') d'dz' for Ma (4) where the dashed elements indicate source points and the uncashed the field points.

The corresponding generalized Green's function in cartesian coordinates is: e-Jko|rr I G(r I r') = (5) where r and r' are the position vectors of field and source points respectively; ko is the wave-number given by k,7-2;c/?:o; and T is the free-space wavelength of the

propagating field.

We consider first an initial quasi-static simplification in which the wavelength is assumed to be very long, or the frequency near zero. Equation (5) then approaches: G(r l r') = (6) |r-r | In cylindrical coordinates one solution of this is: +00 G(rlr') =-Ad, elP( JdEcos[k(zz')]Ip (kp)Kp(kp') (7) 7T p=-X o where r (r') are again defined as the position vectors of field (source) respectively. IP and KP are the modified Bessel functions of pth order. This solution is for the case where observation points are internal to the cylinder, i.e., A' > p. The vector potentials may then be expressed as: L/2 27r 4'Z LJ J G(rlr)JG (r')sin(-i')d679lz' (8) L12 2rr At (r) = 4 J J G(r I r ') J./ (r I) cos(45 - 46 t)d ' dz ' (9) L12 2'r Az(r)= 4 1 JG(rlr)Jz(r)di dz (lO) Substituting equation (7) into the above equations and carrying out the integration over the cylinder's surface, we obtain the expression for (8)-(lO) in the form of Fourier integral (series) or in terms of spectral variable k (m). The relevant magnetic fields are calculated from the curl of the vector potentials to be: Bp(r) = J e7P) JkcosEz J(m,k) I'p(kp)K'p(ka)dk (11) P=-= 0 Bar) = - J pew [cosEz J,b(m,k)Ip(kp)K'p(ka) dk carp ',= () ( 1 2) Writing tile currents as summation of (2) and (3) and using the following relationship: Bx = Bp costs - B4, sing (13) we obtain Bx (r) =- [cos kz As, Jam Him (k)K ', (ka)[kp cos2 iIo (kin)-cos 2f I, (kp)]dk (14) where 10If, (k) sin(k-km)L/2 sin(k+km)L12 ( 15) (k-km)L/2 (k + km)L/2 The result described in equations (14) and (15) is known in the art as a quasi- static solution, being only valid in applications where the free space wavelength is much longer than the length of the cylinder generating the RF fields (see for example, H. Fujita, L. S. Petropoulos, M. A. Morich, S. M. Shvartsman, and R. W. Brown 'A hybrid inverse approach applied to the design of lumped-element RF coils,' IEEE Trans. B'omedicalEngineering, vol-46, pp. 353-361, Mar. 1999).

A target region is then defined as the volume within which a homogenous Bx is required. In particular, a target field value (Bx,) is defined as the fixed magnitude of the desired homogenous field Bx. The target region is, for example, specified as the volume inside a sphere of radius R centered at z=O. As an example, a target specification could be that R is set to be R ().7a and max()-min(B,,) is specified at less than 5%. If we set Bx(r)--Bx(r2)=Bx(r3)= = Bx(rN)= Bx' where rat, r2, r3. torn are the target field points chosen inside the target region, a set of N linear equations is obtained from equation (14).

In particular, the following matrix equation is obtained: BX' X[1]NX] [T] NXM [ Cm]Mx! (16) where [T] is a matrix containing the values of BX at the defined observation points due to each of the current componcuts; Jo,, , | is a column vector of the current coefficients JC/ .JC2 J.3 Jm alla fI] ;S a unity column vector. If the number of linear equations equals the number of unknowns (M--N),

then the numerical solution for [Jam] can be easily obtained by multiplying both sides of (16)by [T] As an example of the quasi-static method, a coil of dimension: a=100 mm, L=320 mm was designed. The specified target field in spherical coordinates was: I BX! = BX | rl r2 r3 | 1 3 01 03 1 1 23.5 AT 1 50 50 50 (mm) 90 90 45 1 90 045 The current density coefficients up to M=3 were then calculated from the quasi- static expressions in equations (14) to be: JC IJC2 IJC3 1 43.834274291992190 -48.744728088378910 36.222351074218750 The corresponding resultant current distribution on the surface of the cylinder is shown in Figure 2 and illustrates the current paths required for the coil. In the graph of Figure 3, the current components J.,, and J: are plotted along the z-direction for l=0 and =TE/2 respectively. Contour plots (not shown) demonstrate that BX is homogeneous within the target region.

Time Harmonic Analysis In accordance with the invention, instead of using the quasi-static approach, a time harmonic, full wave method is used to calculate the required current density. This is essential in cases where the operating wavelength and the dimension of the cylinder are of the same order, as the prior art quasi-static method described in the above section is no longer valid. However, the mathematical derivation of a time-harmonic expression for BX in the spectral domain or an equivalent of (14) is extremely difficult.

A more direct approach is to defies Bx directly in terms of the vector potentials of the gcnera]izcd Green's function of equation (5) rather than its spectral equivalent given in equation (7): B = Liz _ BAY (17) Fez where Az are Ay are the vector potentials calculated from equation (4) . (Note that numerically, the calculation of equation (17) is carried out using a Gaussian-Quadrature integration routine over the prescribed 2-D cylindrical surface integral.) In the time-harmonic case, the wavelength or frequency of operation is taken into account and the wavenumber (ko) is non-zero. This leads in many instances to the current density and its constituent coefficients being complex (that is, consisting of both real and imaginary components). In accordance with the invention, the calculated cun-ent densities for the generation of specific transverse RF fields are such complex quantities.

To illustrate the use of complex current densities, current densities were calculated on the same cylinder as the quasi-static example discussed above at three different frequencies: 190 MHz, 300 MHz and 500 MHz. The same target field was used, i.e., the homogeneous volume was symmetrically located. Equation (16) was modified to include complex current densities and a complex version of the matrix [T] was obtained using equation (17). The solution for [J] then follows the same procedures for the quasi-static example except that complex quantities are now included. The calculated current coefficients are shown in the following table: | Frequency(MHz) | Jcl | Jc2 | Jc3 l 1 38.6895+jl.9310 1 -42.722512.1026 1 35.0854+jl.6609 300 33.4992+j5.4433 -36.6628j S.7392 35.1309+j4.9677 1 500 23.8764+j 13.0168 -25.3327j 12.4308 39.07675+j 14.0778 l Table I. Computed current coefficients (M=3) at different frequencies.

From the above table, it is clear that as the frequency increases, the maginary part of the current densities becomes larger.

Basis Functonsfar Asymmetric Systems In a preferred embodiment of the invention, open-ended asymmetric structures are designed for a time hannonic system. For the design of asymmetric coil structures, surface current densities (Amp/meter) on a cylinder are defined In the same manner as for symmetric structures: J(r) = Jo (I), z)() + Jz (I), z)z Gloria and -L/2 < z < Ll2 (18) The relationship between Jz and J, under mild coil length/wavelength ratios, may be approximated as: J (lt},z)=- a| dZ did (19) Or alternately Jz and J0 may be decoupled and treated individually for the purposes of optimization. Currents on the coil are descretized along z and are represented by sub-domain basis functions.

For asymmetric structures a new set of basis functions must be specified, and in one embodiment Jz and J, are written as a summation of triangular Tz() and pulse Ilz () functions respectively as follows:

M

Jz (' z) = cos JcmTz (Z-Zm (20) m=l

M

J(5,z) = I singed Jcm {(Iz (Z Zm) flz(z Zm)} (2]) z n=l where the Jcm are the current coefficients and Tz (z-Zm) = Iz Zm 1/ lZ |Z-Zrn | < IZ (22) 0 elsewhere Hz (Z-Z., ) = 1 Zm 1 < 1z /2 (23) () elsewhere and Zm =mlz- 2 Zm =(m+2)1z-2 andlz =LI(M+I).

Magnetic fields are expressed in terms of magnetic vector potentials defined as the convolution of surface current density with the corresponding Green's function in the spatial domain: L/2 2rr A(r)= | | G(rlrl)J(r1) dfIdz I formica (24) -L12 0 As previously, the corresponding generalized Green's function in Cartesian coordinates is: e-Jko|rr I G(r I rl) = I I (25) where r and r' are once again the position vectors of field and source points respectively; ho is the wave-number given by ko=21\o; and X0 and is the free-space

wavelength of the propagating field.

The transverse field is written as:

B = Liz _ Ay (26) where Az are Al, are the vector potentials calculated by equation (24).

In accordance with the invention, the homogeneous field is preferably asymmetrical about the z=0 plane. The target region is defined as a spherical volume of radius R centered at z=zO,. Our first rr.cthod of obtaining a set of current coefficients that produces the required homogeneity is as follows.

Let (r/, r:, r....rN) be the N field positions chosen inside the target region and Bx(rn) (n=l..N) be the magnetic field at these points. The field at each target point is expressed in terms of M number of current coefficients as follows:

M

B r (rn) = I' Con 17 (27) 171'''1 where C., represents the field contributioil at rn due to the basis function at z -- zn The residue (Rn) is defined to be the difference between Bx(rn) and the specified

target field Bx, i.e.,

RN ( J., J2 JN) = Bx (rn) - Bx, n= I..N (28) A functionfis defined as:

N

f( l,J2 JN) Worn (Jl,J2 JO) (29) n=1 In order to suppress large current variations in the solution, it is sometimes useful to also include the change of current amplitude as a constrained condition. In this N-] case a term wham, is added to equation (28), where m' is the difference between the i=, adjacent current elements given by m,= J! - Jim; W is the weight used for bringing m' to the same order of magnitude as Rn.

Our task is to find a set of {Jl,J2...JM} such that f is minimum, or fnn=min{f(J,,J2...JM)}. There are various ways in which this function may be minimized. In one embodiment, a linear steepest descent method is used for each variable. An advantage of using this minimization technique is that it allows either a real or a complex solution to be readily obtained.

Examples of the first method aspect of the invention Examples of the invention, designed using the time harmonic, full wave method oithe first method aspect of the invention are now detailed.

A coil of dimension: a=100 mm, L=320 mm was designed at different frequencies: 0 MHz, 190 MHz and 300 MHz. A set of target points (m) was defined in spherical coordinates with respect to zOS, where in terms of the parameters of Figure 1, lzosl + |D| = L/2 The radial positions of these points were rn=53.3mm; their angular positions are given in the following table:

N

1 Go 0 - _. _ 2 45 0 3 90 0 _ _ _. . 4 45 30

_ _

GO 45 6 45 45 7 45 60 8 OO 90 9 90 90 45 120

_

11 45 135 12 45 150

_

13 _ OO 180 Table 2. Angular positions of target points in spherical coordinates centered at zOS.

The specified target field was BXI=23 S AT. The requirement for homogeneity was specified as: max(Bl)-min(Bx)< 5% mside a spherical region of radius R 0.7a for ZoS=20 mm and R 0.55a for z0s =50 mm. Eight current coefficients (Ad) were used for each computation. Tables 3 to 5 show the calculated current coefficients for the different zOS values at the three operating frequencies. For the non-static cases, the real (Re) and imaginary (Im) components of the complex currents are shown separately.

These coefficients are for use with the triangular and pulse functions discussed above.

Zos Jcl Jc2 Jc3 JO I 20mm 220.68 -188.62 -7.60 -42.38 1 50mm 1763.41 -959.25 -163.87 -52.15: IZOS IJC5 IJC6 IJC7 IJC8 1 20mm -29.62 1 -36. i 0 1 -25.10 -62.20 __ 1 1 50mm -18.58 1-35.02 1-9.71 -69.13 Table 3. Computed current coefficients (M--8) for quasi-static case.

Zos(mm) Re/IIn Jcl Jc2 Jc3 Jc4 Re 226.65 -195.19 17.93 -42.11 Im - 159. 73 +71.66 -5.96 -4.30 Re 815.32 -280.51 -68.98 0.22 Im -505.05 167.50 5. 94 -2.68 _ _ R e/Im Jcs Jc6 Jc 7 Jc8 Re -24.42 -33.19 -20.73 -55.24 Im +0.65 -1. 77 -0.79 -1.81 Re -33.27 -30.49 -9.94 -63.67 Im 0.21 0.98 0.47 0.44 Table 4. Computed complex current coefficients (M=8) at 190 MHz.

| Zos (mm) Re/Im Jcl Jc2 Jc3 Jc4 Re 194.73 -14.29 -22.91 Im 117.21 -21. 10 -1.15 Re 634.66 -48.97 23.80 Im 0.92 3.09 9.78 4.87 Zos (mm) Re/Im Jc5 Jc6 Jc7 Jc8 Re -23.01 -29.57 -10.18 -51.95 Im -1.32 -4.98 0. 61 -10.29 Re -36.71 -27.10 -8.53 -53.72 _ Im 4.19 3.67 2.50 2.32 Table 5. Computed complex current coefficients (M=8) at 300 MHz.

The transverse magnetic field (Bx) inside the cylinder was examined for each of the above cases. Computed results show that the resultant Bx values meet the required homogeneity for all the above cases and they show similar characteristics for a given zOS For the 190 MHz case, the real components of the current densities JO and Jz are plotted along the z-drection in the =0 and =/2 planes on the graphs of Figure 4A and 4B which correspond to zOS=20mm and zOs---50mm respectively. In both cases, J; t and Jz along z are of an oscillatory nature. It is seen that both components experience a surge in magnitude near the positive end of the coil (i.e., the end closest to the end where the target region is located) .

Figures 5 and 6 show current distributions on the surface of a cylinder from the read solutions of Table 4 for zu. equal to 2()mm and 50mm, respectively.

Plots of the Id field created by the currents of Figure 4A and Figure 4B over the volumep < SOmm, -160mm < z < 160mrn (not shown) revealed a homogeneous region in both cases that was asymmetrical about the center of the coil and satisfied the criteria for homogeneity. It was also found that there was a larger volume of homogeneity in the zOs=20mm case than in the zOs=50mm case.

Examples of the second method aspect of the invention As a representative illustration of the second method aspect of the invention, the following sets of basis functions are used for the current density components on the surface of a cylinder with radius pa (m) and length L (m): I N M q k z+ 2p m) a (30) amnpq cos (nit + 2) cos ( m 2) q=0 p=0 n=1 n=p+l J = cmnpqsin(n+ 2)sin(km 2) (31) q=0 p=0 n=1 m=p+l where k tam pair (32) and the complex coefficients Umnpq and Cmnpq are to be calculated.

This form of the basis functions was selected based on the following rationale.

If the specified field has only one vector component that does not vary with it, then it can be reasoned that the q term in the basis functions is unnecessary as the axis can be set in any direction in the xy plane. However for simplicity, = 0 is set to coincide with the x-axis and the q term wi]l be necessary for any specified field that has a y-component.

For specified fields in MRI applications, it is expected that the current density J is anti-symmetrical at 18() , so coefficients of even harmonics of will be zero and hence, n = 1,3,5.... Both components of J have a z dependence described with ken+ p m) where p--O,l and m is an integer series commencing alp. The index p=0 describes basis functions that have Jz = 0 at z = +L12 whereas the index p=l describes basis functions with a J. = 0 at z = +L /2.

If the specified field is symmetric with respect to the z = 0 plane, then all lO coefficients of even m terms will equal zero and hence m can be assigned m = 1,3,5,.... However, all terms of m are required if the specified region is asymmetric with respect to the z = 0 plane as in the example coil design presented herein.

The next step in the method is to calculate the coefficients of the basis functions in equations (30) and (31) such that a specified homogeneous field is generated.

lS Because the time harmonic Green's functions are of a complex variable, the coefficients of the current density J are complex as well. If the specified field is that of a circularly polarized field, then the real and imaginary components would be the same albeit spatially separated by 90D, So if J(, z) produces Kax in the DSV, J(l - / 2, z) would produce Kay where K is just a constant. The real part of the current that would

produce a circularly polarized field is then:

J linear = real {J (I), z) } - imag {J(l - 7r / 2, z) and this is the same current that would produce a linearly polarized field. Hence only Bxax is needed to be specified in the DSV to produce J(,z) and (33) can be used to obtain the final current density with only a real component.

Transr1liLLing coils must ha-ye a shield to prevent eddy currents in external conductors and to provide a suitable RF ground. A shield may be approximated by specifying B -- O on a cylinder with a radius slightly larger than the shield radius. A 3() current is approximated on the shield by the same set of basis functions as those for the inner coil except that the shield current now has troth div-fTee and curl-free components (see below).

If (at) and (cC) denote column vectors of the JO and Jz coefficients of the coil current (see equation (30)), (aS) and (es) denote column vectors of the JO and Jz coefficients ofthe shield current (see equation (31)), and [-] denotes a matrix, then the field in the DSV due to the cod] currents is found in matrix fond, as: [A](aC) + [B](cC) = (BXC! ) (34) [C] (aS) + [D] (CS) = (Bail) (3 5) where the summation of the vectors (BXC) and (BSx' ) results in the target field vector(Bx,)within the DSV. Similar matrix equations result when applying Bx=O, By = 0 and B' = 0 as the condition at specified points near the outside of the surface of the shield cylinder: [](ac)+ [L](cC)+ [M](a)+ [N](c) (O) (36) The matrix equations are usually not square because the number of points where the magnetic field is specified does not usually equal the number of unknown coefficients. The rank of the matrices are usually less than the number of unknown coefficients as well, and so a regularization method must be used for a solution.

It should be noted that equations (34), (35), and (36) can be written in more compact form as Eq. (I) above.

Regularzaton The regularization method chosen as a non-limiting example waste minimize some functional in terms of the current density and impose the extra conditions onto the matrix equations such as (34) to (36). Numerous functionals are available and it is convenient to choose a functional such that the resulting current density is easier to implement. The functional: min JJ Jds (37) results in a current density similar to one of minimum power and hence is a useful choice. When the current density equations (30) and (31) are substituted into the above equation (37) and differentiated to find the coefficients for a minimum: baud (3 8) |-(J J)ds = 0 (39) So ilk/ the resulting matrix equation is just some constant multiplied by the identity matrix.

This is because the basis functions are orthogonal. For a current density with no divergence, the wire implementation depends on the streamfunction neS, where S=-n d cos(n+q)sin(k z+(2p-m)7) q=0 p=0 n=1 m=p+l 2 where n is normal to the cylindrical surface dmnpq = Po (poEmamnpq + 'icmnpq) (4 l) Hence minimizing the variation in S over the surface would minimize the variation in its contours that ultimately determine the wire positions: min J(V V(n S))2ds (41A) sO where V V(n S) gives a quantitative measure of the stream-function's variation.

In this case, because the basis functions on the coil form a set with zero divergence, the dumpy coefficients of S reduce to: Pq km (42) Applying the condition (42) results in a matrix with diagonal terms: [R]=1TL:<k 2 +k/n) [I] (43) [R] being the regularization matrix, and [1] being the identity matrix.

Matrix solution An iterative method was used as a non-limiting example to calculate the coil coefficients and shield coefficients separately so that the error in each could be adjusted more exactly by adjusting two scalar penalty values X, and \2 Firstly, the initial coil currents were calculated using the expression: 10(a, )=([A] [A]+\,[R,]) [A] (Bx,) (44) where [A'] is the combination of [A] and [B] of (34) using the relationship of equation (42) and [R.] is the regularization matrix calculated by applying (37) to the coil currents.

The field very close to the shield due solely to the coil currents is: 15(bs/c)=[K](a' ) (45) where [K'] is the combination of [K] and [L] of (36). Shield currents are calculated to

negate this field:

(s)=-([M']T[M']+2[R2]) [M] (bloc) (46) where (s)T=((aS)T (CS)T), [M']=[[M] [A]] of(36)and [R2] is the regularization matrix calculated by applying (37) to the shield currents. The field in the DSV due to the shield currents is: (bDSY/.; ) = [C](s) (47) where [C'] = [[C] [D]] of (35). The field in the DSV due to the coil currents is: (hDSV/C)=[A](a' ) (48) The error in the DSV is then: (bE) = (Bx)-(bDsv/c)-(bDsv/s) (49) The new coil coefficient estimate is: (a,C) = (a,C)+([A']T[A']+[Rl]) [A']T(bE) (50) The calculation then loops back to (45) and is terminated when the difference in the error between two consecutive iterations falls below a certain prcdefined limit (e.g., 0.01). 'the error sin.', here is defined as: EDSV fib (51) and the shield error 5 iS defined as the error in the field produced outside the shield: E5 =|b5,C +[M](s)| (52) The penalty scalars \ and \2 affect the current density solution as well as determining the condition of the matrices to be inverted in equations (44), (46) and (50). Typically, the matrix problem becomes unstable (that is, where the matrix is tending towards being singular) below a certain value of \.

Examples illustrating the aspects of the invention zn which current densities are converted to discrete coil structures By whatever method the complex current density functions are obtained, the next step in the process of the overall design method is to synthesize the coil structure form the current density.

In a preferred embodiment, a stream function method is employed to calculate the positions and dimensions of conductors such that the stream function of the current distribution in the conductors closely approximates the stream function of the calculated current density.

Quasz-static case It can be seen from equations (2) and (3) that the divergence of the quasi-static current density is zero: V J=0 (53) In true electromagnetic systems, the divergence is not zero but: V J=-jc9p (54) where p is the charge density. However, for this analysis, the charge distribution is assumed to be negligible, which is true for static and low frequency systems.

Because of zero divergence, a function S can be calculated that represents the magnetic flux produced from the current density as: VxS-J (55) where from equations (2) and (3): S =-n cos km cos km z (56) where n is the radial unit vector and is normal to the cylindrical surface. Note that If J is a surface current density, S will only have a component perpendicular to the surface containing J. and hence only the magnitude of S is needed. The magnitude of S is a function in two dimensions.

The current distribution in the conductors should be such that it produces the same function S as that produced by the calculated current density J. This is done by finding the contours of neS and generating appropriate currents in each of the wires placed in the positions of the contours. If the contour 'heights' are chosen at equal increments between minimum and maximum, the currents in each of the wires following the contours should be equal. If the contour 'heights' are not equally spaced, the currents in the wires have to generated accordingly in proportion to the contour spacing.

By way of an example, consider equation (56) with six real coefficients (M=6): cm=47.3518, 212.791, 314.615, 687.005, -3946.81 and 5365.36.

The values of the stream function dictates the shape and position of the contours and hence, is decided iteratively by investigating the electromagnetic simulations each contour configuration produces. For example, the contours may be taken at equal fractions of the streamfunction's maximum value, resulting in equal current in all the wires. Alternatively, if the maximum is normalized to one, contours could be taken at +0.697, +0.394 and +0.091. Because the interval (0.091) between zero and neS=O.O91 does not equal the interval (0.303) between neS=O.O91 and neS=0. 394, the current in the lowest 0.091 contour will have to be 0.3 (i.e., 0. 091/0.303) of the current flowing in the 0.394 contour and 0.697 contour (these two contours should have equal current).

The next task is to derive a workable model from these contours.

Design decisions in this process include: 1. A magnetic symmetry plane at =0 is used for the stream function in the range -L/2<z<L/2 and -7<<7.

2. Each contour Is implemented as a conductor. Two cases were examined: one where the conductors are implemented as circular wires (tithes) of specified diameter and the other where the conductors are implemented as metal strips of specified width.

The diameter or width of a conductor determines its self-inductance and hence its complex impedance. It is possible to adjust the current levels in the conductors by adjusting the diameter or width of the conductors accordingly.

3. The contours are connected depending on the sign of the stream function. If two contours are in close proximity and the stream function value has the same sign for both contours, then the contours are connected in parallel. If the contours have a stream function value of differing sign, the contours are connected in series.

4. Capacitors are added at the connections of the contours connected in series and also at the mid-point of contours that are not connected to other conductors. However, there is a freedom to add capacitors wherever the designer considers suitable. The values of the capacitors are determined from an iterative approach using repeated r.1 1 1.

1rCtuCIlcy sWt'Cps 111 ttlt'UVICt:, slmulaLloll.

S. The source points are at the extreme ends of the contour, that is, either at both ends or at one end: either at z=L/2 and z=-L/2 or only at z=L/2.

A full implementation of this method results in the conductor configuration shown in Figure 7 for the current density defined by the six real coefficients set forth above, i.e., cm=47.3518, 212.791, 314.615, 687.005, -3946.81 and 5365.36.

In a preferred embodiment, asymmetric structures are synthesized.

When triangular basis functions are employed to approximate the current density, only the current-density values are provided at points within the domain of each triangular function. Thus the problem is to find the appropriate contours of a two dimensional grid-like data set that will best approximate the current density that the data set is rlleant to reprcscut. In the quasi-static problei-ll, the currerri density has zero divergence because of the nature of the basis functions and thus the electric field at the surface is parallel with the current density. In the non-quasi-static case however, the electric field can have a component perpendicular to the current density and so an approach has to include measures for the approximation of the charge density. In accordance with the preferred embodiments of the invention, the curl-free and divergence-free components of the current density are found separately and each Is approximated in turn.

Current-densify components The current density J can be split into its ctrl-free and divergence-free components by writing: J=VxS+V&7 (57) From Laplace's law, it follows that V2=V J (58) and the curl-free component of J is: JcurI-free V {7 (59) The divergence-free component can be found by applying curl to both sides of equation (57) to give: VxVxS =VxJ If the divergence of S is made zero: V S=0 then: -V2s = V X J (60) The divergence-free component is: J dv-free = v x s (61) Since J,v-free is only on the surface, and to be free of singularities in the differentiation normal to a surface, it is best if nxS = 0 Hence, (60) also becomes a Poisson equation over the surface of J: _V2(n S)=n Vx J (62) The task then is to solve the two Poisson equations (58) and (62) for and n.S and thereby using the contours of and the contours of n.S to approximate the entire current density J. Note that J'v-free is the current density that directly influences the magnetic field tangential to the surface of the conductor. Jcur-free is a measure of the quantity of rate of change of charge density and this quantity directly influences the electric field normal to the surface. In other words, J,v-free can be thought of as the inductive component of the circuit and Jcu'riee as the capacitive component of the cucuit.

Once S and are calculated, the contours and configuration of the coils can he designed. l he contours of n.S dictate the cun-ent configuration. Once n.S is found, its contours and hence the coil pattern can be calculated in a similar manner to the quasi- static case.

As examples of the above procedures, Figures 9 and 10 show the results of determining inductive and capacitive elements for complex current density functions obtained using the first method aspect of the invention and shown in Figure 8. The coil had a homogeneous region 50mm from the center towards the lower end; the length of the coil was 320mm and its radius was 100mm. The operating frequency was 190MHz.

A contour diagram of neS is plotted in Figure 9 and demonstrates the paths for the conducting strips and/or wires. The contours of sir will be demarcation curves of varying areas of capacitance.

One mechanism to increase capacitance is to make an RF shield a contoured surface rather than just a cylindrical sheet, such that at regions of higher capacitance, the shield is closer to the coil than at regions where the capacitance is lower. Another mechanism is to add dielectric material of high permittivity at regions of higher capacitance compared to other regions, varying according to hi.

Figure 10 shows a contour plot of for the same current density as shown in Figure 8. The contours in this case are not current paths but are markers to indicate the regions of increasing charge density and hence capacitance. The region of highest capacitive effect is in the center of the contours. Note that Jcur-free is actually perpendicular to these contour lines.

As further examples of the above procedures, Figures 11 through 15 show the results of determining inductive and capacitive elements for complex current density functions obtained using the second method aspect of the invention.

The design objective was to produce an RF coil 20cm in diameter, 25cm in length with a DSV diameter of I Ocm, offset by 2.5cm from the Z = 0 plane. (The Z = 0 plane passes half-way along the length of the coil). The coil was designed at the frequency of 1 90MHz such that it could be tested in an available MRI machine.

The second method of the invention was used with the constraint that Jz = 0 at z = +L/2. This constraint is a factor on how the resulting coil is excited because no current distribution flows out from the edge of the cylinder. After obtaining coefficients of the current density, the function IS was calculated. The contours of the stream function neS for half the coil are shown in Figure ll. These are the preliminary patternsfor conductor positions. The resulting current density was tested using a commercial method of moments package (e.g., the FEKO program distributed by EM Software & Systems, South Africa). The coil current density is approximated by Hertzian dipoles while the shield current density is ignored. Instead, a metallic shield approximated by triangles as per the method of moments is positioned where the shield should be. The normalized magnetic field of the coil in the transverse (x,y) plane is shown in Figure 12. The field varies within 10% over a distance of 13cm in the x-direction (upper curve) and 12cm in the y-direction (lower curve) . The field variation along the z-axis is shown in figure 13 which shows a 10% variation over a distance of l tom shifted along the z-axis by 2. 5cm. These simulated results generated by Hertzian dipoles approximating coil currents and a metal cylinder for the shield agree with the original field specification and target volume specification in the inverse program.

The coil patterns were then converted to conductor patterns. For this case, 2.2mm diameter wires were used for the conducting paths. Lumped elements were added to the model and adjusted such that the current distribution at the resonant frequency of l90MHz approximated the original calculated current distribution. The resulting coil is shown in Figure 15 with the circuit diagram shown in Figure 14. The values for the capacitors are: C1=3.9pF, C2=4.7pF, C3=3.3pF, C4=2.7pF, C5=2.2pF, C6=l OpF and C7 and C8 are variable capacitors.

When the coil was constructed, it realized an unloaded Q of 139, measured from the 3dB down power points either side of the resonant frequency In all of the examples presented throughout this specification, a homogenous target Reid has been specified, but non-homogeneous target fields may be required in some circumstances and these are readily handled by the methodology disclosed herein.

The invention has been described with particular reference to asymmetric radio frequency coils and a method for designing such coils, in which the section of interest (the DSV) can be placed at an arbitrary location within the coil. It will also be understood by those skilled in the art that various changes in form and detail may be made without departing from the scope of this invention.

Claims (35)

1. A method for designing apparatus for use in a magnetic resonance system for receiving a magnetic resonance signal having a predetermined radio frequency, said apparatus and said magnetic resonance system having a common 1onghdina] axis, said method comprising designing the apparatus by treating the apparatus as a transmitter of a radio frequency field having the predetermined radio frequency and then designing said transmitter by: (a) defining a target region in which the radial magnetic component of the radio frequency field is to have desired values, said target region surrounding said longitudinal axis; (b) specifying desired values for said radial magnetic component of the radio frequency field at a preselected set of points within the target region; (c) determining a complex current density function J. having real and imaginary parts, on a specified cylindrical surface by: (i) defining the complex current density function as a sum of a series of basis functions multiplied by complex amplitude coefficients having real and Imaginary parts; and (ii) determining values for the complex amplitude coefficients using an iterative minimization technique applied to a residue vector obtained by taking the difference between calculated field values obtained using the complex amplitude coefficients at the preselected points and the desired values at those points; and (d) converting said complex current density function J into a set of capacitive elements located on the specified cylindrical surface and a set of inductive elements located on the specified cylindrical surface by: (i) converting the complex current density function into a curl-free component Jcur-free and a divergence-free component J<v-free using the relationships: J curl- free = v {1/, and Jdv_free = V X S where 1 and S are functions obtained from the complex current density function through the equations: V2v=V J -3z -V2S = V x J and -V2(n S)=n VxJ, where n is a vector nonnal to the specified cylindrical surface; (ii) calculating locations on the specified cylindrical surface for the set of capacitive elements by contouring the function 1; and (iii) calculating locations on the specified cylindrical surface for the set of inductive elements by contouring the function UPS.

2 The method of Claim I wherein the basis functions are triangular and pulse functions.

3. The method of Claim I or 2 wherein the iterative minimization technique is selected from the group consisting of linear steepest descent and conjugate gradient descent.

4. The method of any preceding Claim wherein (i) the set of inductive elements on the specified cylindrical surface define first and second ends for the apparatus and (ii) the target region has a midpoint that is closer to the first end than to the second end.

5. The method of any preceding Claim comprising the additional step of displaying the locations on the specified cylindrical surface for the set of inductive elements.

6. The method of any preceding Claim comprising the additional step of producing the set of inductive elements and the set of capacitive elements on the specified cylindrical surface.

7. A method for designing apparatus for use in a magnetic resonance system for transmitting a radio frequency field or both transmitting a radio frequency field and receiving a magnetic resonance signal, said apparatus and said magnetic resonance system having a common longitudinal axis, said method comprising: (a) defining a target region in which the radial magnetic component of the radio frequency field is to have desired values, said target region surrounding said longitudinal axis; (b) specifying desired values for said radial magnetic component of the radio frequency field at a preselected set of points within the target region; (c) defining a target surface external to the apparatus on which the magnetic component of the radio frequency field is to have a desired value of zero at a preselected set of points on said target surface; (d) determining a first complex current density function, having real and imaginary parts, on a first specified cylindrical surface and a second complex current density, having read and imaginary parts, on a second specified cylindrical surface, the radius of the second specified cylindrical surface being greater than the radius of the first specified cylindrical surface by: (i) defining each of the complex current density functions as a sum of a series of basis functions multiplied by complex amplitude coefficients having real and imaginary parts; and (ii) determining values for the complex amplitude coefficients using an iterative minimization technique applied to a first residue vector obtained by taking the difference between calculated field values obtained using the complex amplitude coefficients at the set of preselected points in the target region and the desired values at those points and a second residue vector equal to calculated field values obtained using the complex amplitude coefficients at the preselected set of points on the target surface; and (e) converting said first and second complex current density functions into sets of capacitive elements and sets of inductive elements located on the specified cylindrical surfaces by: (i) converting each of the first and second complex current density functions into a curl-free component Jcur,-frec and a divergence-free component Jv-free using the relationships: Jcurl free = V&u 7 and JO fine =VX S where and S are functions obtained from the respective first and second complex current density functions through the equations: V2v = V J. -VZS = V x J. and -V2(n S)=n VxJ, where n is a vector normal to the respective first and second specified cylindrical surfaces and J is the respective first and second complex current density functions; (ii) calculating locations on the respective first and second cylindrical surfaces for the respective sets of capacitive elements by contouring the respective functions DIN; and (iii) calculating locations on the respective first and second cylindrical surfaces for the respective sets of inductive elements by contouring the respective functions US.

8. The method of Claim 7 wherein the basis functions are triangular and pulse functions.

9. The method of Claim 7 or 8 wherein the iterative minimization technique is selected from the group consisting of linear steepest descent and conjugate gradient descent.

10. The method of any one of claims 7 to 9 wherein (i) the set of inductive elements on the first specified cylindrical surface define first and second ends for the apparatus and (ii) the target region has a midpoint that is closer to the first end than to the second end.

11. The method of any one of claims 7 to 10 comprising the additional step of displaying the locations of the sets of inductive elements on the first and second specified cylindrical surfaces.

12. The method of any one of claims 7 to 11 comprising the additional step of producing the sets of inductive and capacitive elements on the first and second specified cylindrical surfaces.

13. A method for designing apparatus for use in a magnetic resonance system for receiving a magnetic resonance signal having a predetermined radio frequency, said apparatus and said magnetic resonance system having a common longitudinal axis, said method comprising designing the apparatus by treating the apparatus as a transmitter of a radio frequency field having the predetermined radio frequency and then designing said transmitter by: (a) defining a target region in which the radial magnetic component of the radio frequency field is to have desired values, said target region surrounding said longitudinal axis; (b) specifying desired values for said radial magnetc component of the radio frequency field at a preselected set of points within the target region; (c) determining a complex current density function J. having real and imaginary parts, on a specified cylindrical surface by: (i) defining the complex current density function as a sum of a series of basis functions multiplied by complex amplitude coefficients having real and imaginary parts; and (ii) determining values for the complex amplitude coefficients by solving a matrix equation of the form: [Al(ac) = B where A is a transformation matrix between current density space and magnetic field space whose components are based on time harmonic Green's functions, ac is a vector of the unknown complex amplitude coefficients, and B is a vector of the desired values for the magnetic field specified in step (b), said equation being solved by: (1) transforming the equation into a functional that can be solved using a preselected regularization technique, and (2) solving the functional using said regularization technique to obtain values for the complex amplitude coefficients; and (d) converting said complex current density function into a set of capacitive elements located on the specified cylindrical surface and a set of inductive elements located on the specified cylindrical surface.

14. The method of Claim 13 where the regularizaton functional is chosen so as to minimize the integral of the dot product of the complex current density function with itself over the specified cylindrical surface.

15. The method of Claim 13 or 14 where the complex amplitude coefficients are chosen so that the complex current density function has zero divergence.

16. The method of claim 13 wherein step (d) is performed by: (i) converting the complex current density function into a curl-free component Jcur-free and a divergence-free component Jdv-free using the relationships: Jcuri free = v 5v and Jdv Free = V X S. where and S are functions obtained from the complex current density function through the equations: V2v=V J. -V2S=VxJ and --V9(n S)=n VxJ, where n is a vector normal to the specified cylindrical surface; (ii) calculating locations on the cylindrical surface for the set of capacitive elements by contouring the function hi; and (iii) calculating locations on the cylindrical surface for the set of inductive elements by contouring the function IS.

17. The method of any one of claims 13 to 16 wherein the radio frequency field has a wavelength I, the inductive elements on the cylindrical surface define a longitudinal length L, and L>0.27.

18. The method of any one of claims 13 to 17 where the predetermined radio frequency is at least 80 megahertz.

19. The method of any one of claims 13 to 18 wherein (i) the set of inductive elements on the specified cylindrical surface define first and second ends for the apparatus and (ii) the target region has a midpoint that is closer to the first end than to the second end.

20. The method of any one of claims 13 to 19 comprising the additional step of displaying the locations for the set of inductive elements on the specified cylindrical surface.

21. The method of any one of claims 13 to 20 comprising the additional step of producing the set of inductive elements and the set of capacitive elements on the specified cylindrical surface.

22. A method for designing apparatus for use in a magnetic resonance system for transmitting a radio frequency field or both transmitting a radio frequency field find receiving, a ma',etic resonance signal, said apparatus and said lma,lletic resonance system having a common longitudinal axis, said method comprising: (a) defining a target region in which the radial magnetic component of the radio frequency field is to have desired values, said target region surrounding said longitudinal axis; (b) specifying desired values for said radial magnetic component of the radio frequency field at a preselected set of points within the target region; (c) defining a target surface external to the apparatus on which the magnetic component of the radio frequency f old is to have a desired value of zero; (d) determining a first complex current density function, having real and imaginary parts, on a first specified cylindrical surface and a second complex current density, having real and imaginary parts, on a second specified cylindrical surface, the radius of the second specified cylindrical surface being greater than the radius of the first specified cylindrical surface by: (i) defining each of the complex current density functions as a sum of a series of basis functions multiplied by complex amplitude coefficients having real and imaginary parts; and (ii) determining values for the complex amplitude coefficients by simultaneously solving matrix equations of the form: LA1C | (aC)+ LAIS (a s) = B c LA2 (a)+ LA2S I(a S) = B S where A'C,AS,A2c,andA2s are transformation matrices between current density space and magnetic field space whose components are based on time harmonic Green's functions, ac and as are vectors of the unknown complex amplitude coefficients for the first and second complex current density functions, respectively, Bc is a vector of the desired values for the radial magnetic field specified in step (b), and Bs is a vector whose values are zero, said equations being solved by: (1) transforming the equations into functionals that can be solved using a preselected regularization technique, and (2) solving the functionals using said regularization technique to obtain values for the complex amplitude coefficients; and -4() (e) converting said first and second complex current density functions into sets of capacitive elements and sets of inductive elements located on the specified cylindrical surfaces.

23. The method of Claim 22 where the regularization functional is chosen to as to minimize the integral of the dot product of the first complex current density function with itself over the first specified cylindrical surface and to minimize the integral of the dot product of the second complex current density function with itself over the second specified cylindrical surface.

24. The method of Claim 22 or 23 where the complex amplitude coefficients are chosen so that the first and second complex current density functions each has zero divergence.

25. The method of Claim 22 wherein step (e) is performed by: (i) converting each of the first and second complex current density functions into a curl-free component Jcur-free and a divergence-free component J<v-free using the relationships: J curl free=V&l/' and Jdiv free =VX S where sir and S are functions obtained from the respective first and second complex current density functions through the equations: V2v=V J -V2S = V x J. and -V2(n S)=n VxJ, where n is a vector normal to the respective first and second specified cylindrical surfaces and J is the respective first and second complex current density functions; (i) calculating locations on the respective first and second cylindrical surfaces for the respective sets of capacitive elements by contouring the respective functions if; and (iii) calculating locations on the respective first and second cylindrical surfaces for the respective sets of inductive elements by contouring the respective functions n.S.

26. The method of any one of claims 22 to 25 wherein the radio frequency field has a wavelength X, the inductive elements on the first cylindrical surface define a longitudinal length Lo, the inductive elements on the second cylindrical surface define a longitudinal length L:, and Lo >0.2X, and L2 > 0.2 \.

27. The method of any one of claims 22 to 26 where the frequency of the

radio frequency field is at least 80 megahertz.

28. The method of any one of claims 22 to 27 wherein (i) the set of inductive elements on the first specified cylindrical surface define first and second ends for the apparatus and (ii) the target region has a midpoint that is closer to the first end than to the second end.

29. The method of any one of claims 22 to 28 comprising the additional step of displaying the locations of the sets of inductive elements on the first and second specified cylindrical surfaces.

30. The method of any one of claims 22 to 29 comprising the additional step of producing the sets of inductive and capacitive elements on the first and second specified cylindrical surfaces.

31. A method of converting a complex current density function J into sets of capacitive and inductive elements located on a specified cylindrical surface comprising: (i) converting the complex current density function into a curl-free component Jcur-tree and a dvergence-free component J.'v free using the relationships: J cur/ free = V {I and J do free = V x S where and S are functions obtained from the complex current density function through the equations: V2y/=V J' -V2$ =VxJ and -V (;} S) = ". . \7 x I, where n is a vector normal to the specified cylindrical surface; (ii) calculating locations on the cylindrical surface for the set of capacitive elements by contouring the function ye; and (iii) calculating locations on the cylindrical surface for the set of inductive elements by contouring the function nets.

32. The method of (claim 31 comprising the additional step of displaying the locations of the set of inductive elements on the specified cylindrical surface.

33. The method of Claim 3l or 32 comprising the additional step of producing the set of inductive elements and the set of capacitive elements on the specified cylindrical surface.

34. Apparatus for use in a magnetic resonance system for transmitting a radio frequency field, said apparatus being designed according to the method of any one of the preceding claims.

35. A method for designing apparatus for use in a magnetic resonance system for transmitting a radio f equency field and/or for receiving a magnetic resonance signal, said method being substantially as hereinbefore described with reference to the accompanying drawings.