Abstract:
The present invention relates to a method of discretization of the continuous current solution of a gradient coil design that allows satisfaction of the target field quality characteristics as well as other characteristics such as minimization of the energy/inductance, minimization of the residual eddy current effect, minimization of the thrust forces on the coil and cold shields, coil resistance thus the power dissipated by the coil, etc. The method of optimized gradient coil design can be applied to the design of axial or transverse gradient coils. The method of this invention includes the steps of defining at least one, and more commonly numerous performance characteristics of the desired gradient coil, concurrently varying discretization parameters to develop numerous possible hypothetical gradient coil designs, evaluating the designs to determine whether the defined performance characteristics are met by each design and selecting one design.

Description:
TECHNICAL FIELD 
   The invention relates to a new method of gradient coil design for use in magnetic resonance imaging (MRI) for spatial encoding of the MRI signal. The invention more specifically relates to a new method of optimized gradient coil design for both transverse and axial gradient coil. 
   BACKGROUND OF THE INVENTION 
   Magnetic resonance imaging (MRI) is a medical diagnostic imaging technique used to diagnose many types of medical conditions. An MRI system includes a main magnet for generating a main magnetic field through an examination region. The main magnet is arranged such that its geometry defines the examination region. The main magnetic field causes the magnetic moments of a small majority of the various nuclei within the body to be aligned in a parallel or anti-parallel arrangement. The aligned magnetic moments rotate around the equilibrium axis with a frequency that is characteristic for the nuclei to be imaged. An external radiofrequency (RF) field applied by other hardware within the MRI system perturbs the magnetization from its equilibrium state. Upon termination of the application of the RF pulse, the magnetization relaxes to its initial state. During relaxation the time varying magnetic moment induces a detectable time varying voltage in the receive coil. The time varying voltage can be detected by the receive mode of the transmit coil itself, or by an independent receive only coil. 
   MRI systems are made of many hardware components that work in conjunction with specialized software to produce the final images  FIG. 1  illustrates an MRI system with the front cover removed so the main hardware components can be seen. Magnet  12  is the main hardware component of MRI system  10  and is responsible for producing the uniform main magnetic field, B 0 . Magnets used in MRI systems can produce either a horizontal or a vertical magnetic field. 
   Within the volume defined by main magnet  12  there is at least one gradient coil  14 . Gradient coils  14  produces substantially linear spatially varying magnetic fields within the imaging volume that are coincidental with the direction of the main magnetic field but vary along the three orthogonal directions (x, y, z) of the Cartesian coordinate system. Initially, gradient coils included only single primary gradient coil unit  18 , however, the presence of spurious and spatially dependent eddy currents on the magnet&#39;s Dewar structure necessitated the need to shield the gradient magnetic field at the magnet&#39;s vicinity. Secondary shielding gradient coil  20  is added to the primary gradient coil&#39;s structure with a required minimum gap between the primary and secondary gradient coil structures. 
   Self-shielded gradient coils of prior art are usually designed using the following general steps:
         1. The shield coil is assumed to be much longer than the primary coil;   2. The continuous current densities on both the primary and shield coil are derived to deliver the desired field quality characteristics of the gradient coil;   3. The continuous current density on the shield coil is truncated to match the desired length of the system;   4. The continuous current densities are discretized using the standard techniques, described below; and   5. The discretized solution is verified to ensure the solution satisfies the required characteristics.       

   One of the problems in past methods of gradient coil design is the discretization procedure that approximates the continuous current densities by a number of current carrying conductors (step 4 of the above method). The shape of the current carrying conductors determines the electromagnetic properties of the gradient coil. These include the gradient field strength, field quality characteristics inside the Field-of-View (FoV), linearity and uniformity along the principal axis and orthogonal axes respectively, leakage of the magnetic field into the region where eddy currents might be induced, coil resistance, slew rate, etc. The output of the discretization procedure determines the positions of the current carrying conductors and their minimum widths, which govern the coil resistance, thus the power dissipated by the coil and the possible slew rate. For a self-shielded gradient coil the level of the eddy currents is sensitive to discretization procedure When designing the gradient coil it is also important to be able to minimize the net thrust Lorentz force exerted on the coil due to the presence of the main magnet field. This force is sensitive to the position of the current carrying conductors, especially near the end of the coil. 
   The common practice of previous gradient coil designs is to derive the continuous current densities that provide the desired characteristics of the gradient coil. It is sometimes difficult to find a continuous solution that satisfies the field quality characteristics within the FoV and where the net thrust force is nullified by constraint. Depending on the magnet configuration the nullification of the thrust force can lead to an energy penalty or even to reversed current patterns on either the primary coil or the shield coil, which in fact leads to the energy/inductance penalty. 
   Another problem that arises in previous methods of gradient coil design is that eddy currents are induced in the worm bore or the cold shield of the magnet. Eddy currents in the cold shield result in distortions of the images. Because of the proximity of the cold shield to the magnet&#39;s coils the forces on the cold shield could be significant. These significant forces on the cold shield may lead to the vibration of the cold shield and even quenching of the magnet. 
   In previous gradient coil design methods the continuous current solution that satisfies the field quality characteristics within the FoV is obtained using either minimum inductance or target field method of Turner. In the design of an axial gradient coil, the continuous current densities and available power supply determine the minimum number of turns on both the primary and shield coil according to the method of Turner: 
                       ∫   0     L   P       ⁢         f   φ     (   P   )       ⁡     (   z   )       ⁢           ⁢     ⅆ   z         =       N   P     ⁢     I   P         ,         ∫   0     L   S       ⁢         f   φ     (   S   )       ⁡     (   z   )       ⁢           ⁢     ⅆ   z         =       -     N   S       ⁢     I   S                 (     Equation   ⁢           ⁢   1     )               
In this two-part equation, where the first portion of the Equation deals with the primary coil and the second portion deals with the shield coil, f φ   (P,S) (z) is the φ-components of the continuous current density on primary coil (P) and shield coil (S), L P,S  is the half-length of the primary/shield coil, N P,S , I P,S  are the number of turns and the current on half of the primary/shield coil and the current in the primary/shield, respectively. Ideally the currents on the primary and the shield coil should be equal to each other. In practice the latter is difficult to achieve (I P ≠I S ) because of the free parameters in the problem: the geometry of the coil is determined by the space available and usually is at premium.
 
   The positions of the current centroids (commonly called hoops) Z n,m   (P,S)  are usually determined by the integrated current density according to the following equation 
                         ∫   0     Z   n     (   P   )         ⁢         f   φ     (   P   )       ⁡     (   z   )       ⁢           ⁢     ⅆ   z         -       (     n   -   0.5     )     ⁢     I   P         =   0     ,     
     ⁢           ∫   0     Z   m     (   S   )         ⁢         f   φ     (   S   )       ⁡     (   z   )       ⁢           ⁢     ⅆ   z         +       (     m   -   0.5     )     ⁢     I   S         =   0             (     Equation   ⁢           ⁢   2     )               
Within Equation 2, n is an integer between 1 and the number of turns on the primary coil, and m is an integer between 1 and the number of turns on the secondary coil (1≦n≦N P , 1≦m≦N S ).
 
   In designing a transverse gradient using previous methods, the current paths on the primary/shield coil are defined by the following equation: 
                       φ   =       Φ   n     ⁡     (   z   )                       Φ   n     ⁡     (   z   )       =     arc   ⁢           ⁢     cos   ⁡     (       S   n     /       f   z     ⁡     (   z   )         )                       S   n     =         n   -   0.5     N     ⁢       f   z     ⁡     (     z   eye     )                       (     Equation   ⁢           ⁢   3     )               
Here f z (z) is the z-component of the continuous current density, Z eye  is the so called eye-position of the coil where the φ-component of the continuous current density is equal zero:
 
                 f   φ     ⁡     (     z   eye     )       =       0   :       f   z     ⁡     (     z   eye     )         =       max     z   ∈     [     0   ,   L     ]         ⁢     {       f   z     ⁡     (   z   )       }           ,         
N is the number of loops in one quadrant of the coil, and n is an integer: 1≦n≦N.
 
   At φ=0 Eq. (3) determines the positions of Z-intercepts: the positions where the current paths intercept the cardinal axis as it is shown in  FIG. 2 . In  FIG. 2  N-initial and N-final Z-intercepts are shown. Information about a transverse gradient can be expressed through the Z-intercepts and the current paths. 
   SUMMARY OF THE INVENTION 
   The present invention relates to an improved method of discretization of the continuous current solution of a gradient coil design that allows satisfaction of the target field quality characteristics as well as other characteristics such as minimization of the energy/inductance, minimization of the residual eddy current effect, minimization of the thrust forces on the coil and cold shields, coil resistance thus the power dissipated by the coil, etc. The method of optimized gradient coil design can be applied to the design of axial or transverse gradient coils. The method of this invention includes the steps of defining at least one, and more commonly numerous performance characteristics of the desired gradient coil, concurrently varying discretization parameters to develop numerous possible hypothetical gradient coil designs, evaluating the designs to determine whether the defined performance characteristics are met by each design and selecting one design. The design selected will often be the design that offers the best design solution by meeting numerous, if not all of the defined performance characteristics. 
   The method of gradient coil design of this invention allows designing a genuinely finite length gradient coil where the truncation of the continuous current densities is not involved. The new discretization technique allows for a gradient coil to be designed while taking into account predetermined desired characteristics of the gradient coil, without compromising the field quality characteristics, residual eddy currents, balancing the forces exerted on the gradient coil and the cold shield of the magnet. 
   It is an object of this invention to provide a method of gradient coil design wherein the method allows satisfaction of the target field quality characteristics as well as other characteristics such as minimization of the energy/inductance, minimization of the residual eddy current effect, minimization of the thrust forces on the coil and cold shields, coil resistance thus the power dissipated by the coil, etc. 
   It is yet a further object of this invention to provide an improved method of gradient coil design that is applicable to the design of both axial and transverse gradient coils. 
   It is a further object of this invention to provide for gradient coils designed by use of the method of this invention. 
   These and other objects of the present invention will become more readily apparent from a reading of the following detailed description taken in conjunction with the accompanying drawings wherein like reference numerals indicate similar parts, and with further reference to the appended claims. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The preferred embodiment of the invention, illustrative of the best mode in which applicant has contemplated applying the principals is set forth in the following description and is shown in the drawings and is particularly and distinctly pointed out and set forth in the appended claims. The invention may take physical form in certain parts and arrangements of parts, numerous embodiments of which will be described in detail in the specification and illustrated in the accompanying drawings which form a part hereof, and wherein: 
       FIG. 1  is a Magnetic Resonance Imaging (MRI) system of Prior Art; 
       FIG. 2  is a chart indicating the Z-intercepts in one of four quadrants of a transverse gradient of the design of a transverse gradient coil designed by the Prior Art method; 
       FIG. 3  is a series of graphical illustrations of the positions of the current centroids on the primary coil and shield coil as a function of discretization parameters σ P,S , wherein FIG.  3   a  represents positions of the first hoop,  FIG. 3   b  represents positions of the tenth hoop, and  FIG. 3   c  represents positions of the last hoop; 
       FIG. 4  is a series of graphical illustrations of the gradient strength non-linearity over a 50 cm×40 cm FoV as a function of the discretization parameters σ P,S  when the current is varied, wherein  FIG. 4   a  represents I=470 Amps (A),  FIG. 4   b  represents I=480 A, and  FIG. 4   c  represents I=490 A; 
       FIG. 5  is a series of graphical illustrations of the gradient strength non-uniformity over a 50 cm×40 cm FoV as a function of the discretization parameters σ P,S  when the current is varied, wherein  FIG. 5   a  represents I=470 A,  FIG. 5   b  represents I=480 A, and  FIG. 5   c  represents I=490 A; 
       FIG. 6  is series of graphical illustrations of the gradient strength at the isocenter as a function of the discretization parameters σ P,S  when the current is varied, wherein  FIG. 6   a  represents I=470 A,  FIG. 6   b  represents I=480 A, and  FIG. 6   c  represents I=490 A; 
       FIG. 7  is a series of graphical illustrations of the inductance of the coils with predetermined widths as the current is varied, wherein  FIG. 7   a  represents I=470 A,  FIG. 7   b  represents I=480 A and  FIG. 7   c  represents I=490 A; 
       FIG. 8  is a series of graphical illustrations of the maximum value of the residual eddy current effect as the current is varied, wherein  FIG. 8   a  represents I=470 A,  FIG. 8   b  represents I=480 A and  FIG. 8   c  represents I=490 A; 
       FIG. 9  is a series of graphical illustrations of the variation of the residual eddy current effect as the current is varied, wherein  FIG. 9   a  represents I=470 A,  FIG. 9   b  represents I=480 A, and  FIG. 9   c  represents I=490 A; 
       FIG. 10  is a series of graphical illustrations of the dependence of the z-component of the total thrust force exerted on the gradient coil by the magnetic field of an existing 1.5 T magnet while the current is varied, wherein  FIG. 10   a  represents I=470 A,  FIG. 10   b  represents I=480 A and  FIG. 10   c  represents I=490 A; 
       FIG. 11  is a series of graphical illustrations of the dependence of the z-component of the total thrust force exerted on the cold shield by the magnetic field of an existing 1.5 T magnet while the current is varied, wherein  FIG. 11   a  represents I=470 A,  FIG. 11   b  represents I=480 A, and  FIG. 11   c  represents I=490 A; 
       FIG. 12  is a graphical illustrations of the positions of the turns on half of the primary and shield coils when σ P =0.8, σ S =0.9, and I=480 A; 
       FIG. 13  is a graphical illustration of an example of z-components of the continuous current densities on the primary and shield coils that satisfy some field quality characteristics and the gradient strength of G x =30 mT/m; 
       FIG. 14  is a series of graphical illustrations of the position of the various z-intercepts on the primary coil as a function of the parameters 0≦σ≦1 and Δ, wherein  FIG. 14   a  shows the first initial z-intercept,  FIG. 14   b  shows the first final z-intercept,  FIG. 14   c  shows the last initial z-intercept; and  FIG. 14   d  shows the last final z-intercept; 
       FIG. 15  is a series of graphical illustrations of the position of the various z-intercepts on the shield coil as a function of the parameters 0≦σ≦1 and A, wherein  FIG. 15   a  shows the first initial z-intercept,  FIG. 15   b  shows the first final z-intercept,  FIG. 15   c  shows the last initial z-intercept; and  FIG. 15   d  shows the last final z-intercept; 
       FIG. 16  is a series of graphical illustrations of a comparison of the current paths on the primary coil when σ P =0.5, and when Δ P =0 as shown in  FIG. 16   a  and when Δ P =500 as shown in  FIG. 16   b;    
       FIG. 17  is a series of graphical illustrations of the dependence of the net thrust force, measured in lbs., at G=30 mT/m exerted on the primary gradient coil as shown in  FIG. 17   a  and on the shield gradient coil as shown in  FIG. 17   b  placed inside of an existing magnet; 
       FIG. 18  is a series of graphical illustrations of the current paths on half of a quadrant of the primary coil as shown in  FIG. 18   a  and on half of a quadrant of the shield coil as shown in  FIG. 18   b  of the gradient design using predetermined parameters; 
       FIG. 19  is a graphical illustration of the field quality characteristics within the 50 cm (x/y)×40 cm (z) FoV of the gradient coil design; and 
       FIG. 20  is a flowchart of the method of this invention as used to design a transverse gradient coil. 
   

   DETAILED DESCRIPTION OF THE INVENTION 
   Referring now to the drawings wherein the showings are for purposes of illustrating numerous embodiments of the invention only and not for purposes of limiting the same, the figures illustrate the novel idea of an improved method of discretization of the continuous current solution of a gradient coil design. 
   The present invention puts forth an improved method of gradient coil design using an improved discretization process of the continuous current solution. The present invention applies to the design of both axial and transverse gradient coil design. 
   Axial Gradient Coil 
   The method of designing an axial gradient coil of embodiments of the invention, wherein the axial gradient coil having a primary coil and a shield coil for use in magnetic resonance imaging, includes defining at least one performance characteristic of the axial gradient coil, concurrently varying at least two parameters within at least one equation, wherein each result of the varied at least two parameters within the at least on equation produces a different hypothetical axial gradient coil design, evaluating at least one performance characteristic of each hypothetical axial gradient coil design produced by the varied parameters; and selecting one of the hypothetical axial gradient coil designs. 
   In embodiments of the method of designing an axial gradient coil using the method of this invention, desired performance characteristics of the gradient coil design are often defined. The desired performance characteristics are not limited by the method of this invention, and are commonly determined with consideration being given to the specifications and design characteristics of the MRI system in which the gradient coil will be operating. The desired performance characteristics may be determined to be specific ranges, to be greater than or less than certain values, or may even be determined in general terms. 
   Once design characteristics have been determined, in a preferred embodiment of the current invention, the design of an axial gradient coil can be optimized in part by use of the following two-part equation, wherein the first portion of the equation applies to the primary coil design and the second portion applies to the shield coil design 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         ∫ 
                         0 
                         
                           Z 
                           n 
                           
                             ( 
                             P 
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             f 
                             φ 
                             
                               ( 
                               P 
                               ) 
                             
                           
                           ⁡ 
                           
                             ( 
                             z 
                             ) 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           z 
                         
                       
                     
                     - 
                     
                       
                         ( 
                         
                           n 
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                             σ 
                             P 
                           
                         
                         ) 
                       
                       ⁢ 
                       I 
                     
                   
                   = 
                   0 
                 
                 , 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     
                       
                         ∫ 
                         0 
                         
                           Z 
                           m 
                           
                             ( 
                             S 
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             f 
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                           ⁡ 
                           
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                   = 
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                 ( 
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   4 
                 
                 ) 
               
             
           
         
       
     
   
   Within Equation 4, n is an integer between 1 and the number of turns on the primary coil (1≦n≦N P ) and m is an integer between 1 and the number of turns on the shield coil (1≦m≦N S ). Within Equation 4, σ P,S  (discretization parameters) are greater than 0 and less than 1 (0≦σ P,S ≦1) and σ P,S  (discretization parameters) and I (current parameter) are the parameters of interest. 
   Using Eq. (4), the current parameter I and discretization parameters σ P,S  are concurrently varied numerically to investigate the tradeoffs in achieving the desired goals: field quality characteristics within the FoV, minimum energy/inductance, minimum coil resistance, minimum eddy current effect and its variation over the surface of the FoV, minimum net thrust force exerted on the coil and on the cold shield (due to the eddy currents). In certain embodiments, the current parameter I is varied reasonably close to the maximum available current (I MAX ). 
   Each set of concurrently varied parameters provides a different hypothetical gradient coil design, wherein each different design offers different performance characteristics. Using the method of the present invention, the position of the current centroids in the hypothetical designs are very sensitive to the variation of the discretization parameters σ P,S . The positions of the current centroids are less sensitive to the variation of the discretization current I. This results in the sensitivity of the coil characteristics to the variation of the discretization parameters σ P,S  and current I. It is foreseen by this invention that the position of the current centroids may be limited to incorporate various predetermined design and system parameters. For example, the position of the current centroids may be required to have a minimum separation to allow for a predetermined width off the conductors on the primary or shield coil. 
   The performance characteristics of the various hypothetical designs are then evaluated. The performance characteristics of each design is typically compared to the earlier defined desired performance characteristics for the final design found through the method of this invention. The hypothetical design that is evaluated as best providing the desired defined performance characteristics is often selected as the final design. 
   The following hypothetical example illustrates the method of the present invention, and in no way limits the invention. All chosen values are for example purposes only. For the current example, I MAX  will be assumed to be 500 Amps (I MAX =500 A) and will be varied reasonably close to that value. The discretization parameters σ P,S  will be varied between 0.1 and 0.9. In actual practice of the method of the present invention, no limits exist on the value of I MAX , or the discretization parameters σ P,S , and the parameters may be determined due to design and system parameters of the circumstances at hand. 
     FIGS. 3-11  illustrate various characteristics of the gradient coil design of the present example through theoretical calculations, simulations and models using the method of this invention.  FIGS. 3   a  through  3   c  show the sensitivity of the current centroids z-position on the primary coil and shield coil as a function of the discretization parameters σ P,S  (values varying between 0.1 and 0.9) when the current I=480 A.  FIG. 3   a  illustrates the z-positions of the first current centroid on the primary and shield coil as a function of σ P,S  when I=480 A.  FIG. 3   b  illustrates the z-positions of the tenth current centroid on the primary and shield coil as a function of σ P,S  when I=480 A.  FIG. 3   c  illustrates the z-positions of the last current centroid on the primary and shield coil as a function of σ P,S  when I=480 A. 
     FIGS. 4-6  illustrate the tradeoffs of the variations of the discretization parameters σ P,S  and current I and the effects on the sensitivity of the coil characteristics.  FIGS. 4   a  through  4   c  illustrate the gradient strength non-linearity over a 50 cm×40 cm FoV as a function of the discretization parameters when the current is varied ( FIG. 4   a , I=470 A;  FIG. 4   b , I=480 A and  FIG. 4   c , I=490 A).  FIGS. 5   a  through  5   c  illustrate the gradient strength non-uniformity over a 50 cm×40 cm FoV as a function of the discretization parameters when the current is varied ( FIG. 5   a , I=470 A;  FIG. 5   b , I=480 A and  FIG. 5   c , I=490 A).  FIGS. 6   a  through  6   c  illustrates the gradient strength at the isocenter as a function of the discretization parameters when the current is varied ( FIG. 6   a , I=470 A;  FIG. 6   b , I=480 A and  FIG. 6   c , I=490 A). 
   For the purposes of the current example, the minimum separation between the centroids is limited to be greater than 9.2 mm on the primary coil and greater than 17.3 mm on the shield coil. These minimum separations allow for the width of the conductors on the primary coil to be 8 mm and the width of the conductors on the shield coil to be 10 mm.  FIGS. 7   a  through  7   c  illustrate the inductance of the coils with these predetermined widths as the current is varied ( FIG. 7   a , I=470 A;  FIG. 7   b , I=480 A and  FIG. 7   c , I=490 A). 
   The residual eddy current effect (RECE) within the example above is defined as: 
                   RECE   ⁡     (   r   )       =           B   z   CS     ⁡     (   r   )           B   z   Coil     ⁡     (   r   )         ⁢     |   FoV     ⁢     ×   100   ⁢   %               (     Equation   ⁢           ⁢   5     )               
It is desirable that RECE and its variation over the FoV are as small as possible.  FIGS. 8   a  through  8   c  and  FIGS. 9   a  through  9   c  illustrate the dependence of the RECE on the surface of the FoV within the example.  FIGS. 8   a  through  8   c  show the maximum value of RECE, while  FIGS. 9   a  through  9   c  show the variation of the RECE considering the variations of current ( FIGS. 8   a  and  9   a , I=470 A;  FIGS. 8   b  and  9   b , I=480 A; and  FIGS. 8   c  and  9   c , I=490 A).
 
     FIGS. 10   a  through  10   c  and  FIGS. 11   a  through  11   c  illustrate the dependence of the z-component of the total thrust force as exerted on the gradient coil and the cold shield under the varying currents.  FIGS. 10   a  through  10   c  illustrate the dependence of the z-component of the total thrust force exerted on the gradient coil of this example by the magnetic field of an existing 1.5 T magnet, considering the varying currents ( FIG. 10   a , I=470 A;  FIG. 10   b , I=480 A and  FIG. 10   c , I=490 A).  FIGS. 11   a  through  11   c  illustrate the dependence of the z-component of the total thrust force exerted on the cold shield of this example by the magnetic field of an existing 1.5 T magnet, considering the varying currents ( FIG. 1   a , I=470 A;  FIG. 11   b , I=480 A and  FIG. 11   c , I=490 A). 
   In the current example, considering the characteristics and tradeoffs shown in  FIGS. 3-11 , σ P  is chosen to equal 0.8 and σ S  is chosen to equal 0.9. The current is chosen to equal 480 A. With these assigned values,  FIG. 12  illustrates the theoretical positions of the turns on half of the primary and shield coils using the method of this invention. Table 1, below, illustrates the properties of the chosen coil design. 
   
     
       
             
           
             
             
           
         
             
               TABLE 1 
             
           
           
             
                 
             
             
               Properties of the gradient coil of the example 
             
           
        
         
             
               Property 
               Value 
             
             
                 
             
             
               Electrical Length (P/S) [mm] 
               993.0/1050.00 
             
             
               Resistance [mΩ] 
               61.1/38.89  
             
             
               Energy @ Gz = 30 m T/m 
               50.00 
             
             
               Current [A] for Gz = 30 m T/m 
               478.60 
             
             
               Inductance [μH] 
               436.64 
             
             
               Slew rate [T/m/s] (losses not included) @ Gz = 
               166.0 
             
             
               30 m T/m and V = 1200 V 
             
             
               Rise time [sec] (losses not included) 
               181.0 
             
             
               DC Dissipated power [kW] @ Gz = 30 m T/m 
               22.9 
             
             
               Thrust force on the Coil [lbs] @ Gz = 30 m T/m 
               −8.0 
             
             
               Thrust force on the Cold Shield [lbs] @ Gz = 
               10.1 
             
             
               30 m T/m 
             
             
               RECE_max [%]/RECE_variation [%] 
               −0.05/0.33   
             
             
               Global non-linearity [%] at z = 0.2 m 
               −6.92 
             
             
               Local non-linearity [%] at z = 0.2 m 
               −25.3 
             
             
               Max/min Non-uniformity [%] at ρ = 0.25 m 
               1.1/−1.1  
             
             
                 
             
           
        
       
     
   
   Transverse Gradient Coil 
   Further embodiments of the invention involve a method of designing a transverse gradient coil for use in magnetic resonance imaging which includes defining at least one performance characteristic of the transverse gradient coil, concurrently varying at least two parameters within at least one equation wherein each set of varied parameters produces a different hypothetical transverse gradient coil design, evaluating at least one performance characteristic of each hypothetical transverse gradient coil design produced by the varied parameters; and selecting one of the hypothetical transverse gradient coil designs. 
   In further embodiments, similar to the above method for designing the axial gradient coil, in the method of designing a transverse gradient coil using the method of this invention, desired performance characteristics of the gradient coil design are often defined. The desired performance characteristics again are not limited by the method of this invention, and are commonly determined with consideration being given to the specifications and design characteristics of the MRI system in which the gradient coil will be operating. The desired performance characteristics may be determined to be specific ranges, to be greater than or less than certain values, or may even be determined in general terms. 
   Once design characteristics have been determined, in a preferred embodiment of the current invention, a transverse gradient coil can be designed using a further preferred embodiment of the method of this invention, we consider the following set of equations: 
   
     
       
         
           
             
               
                 
                   
                     
                       φ 
                       = 
                       
                         
                           Φ 
                           n 
                         
                         ⁡ 
                         
                           ( 
                           z 
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           Φ 
                           n 
                         
                         ⁡ 
                         
                           ( 
                           z 
                           ) 
                         
                       
                       = 
                       
                         arc 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             
                               
                                 S 
                                 n 
                               
                               / 
                               
                                 
                                   f 
                                   z 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   z 
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         S 
                         n 
                       
                       = 
                       
                         
                           
                             n 
                             - 
                             σ 
                           
                           N 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 f 
                                 z 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   z 
                                   eye 
                                 
                                 ) 
                               
                             
                             - 
                             Δ 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   6 
                 
                 ) 
               
             
           
         
       
     
   
   In the set of equations labeled Eq. (6), two discretization parameters are introduced that govern the positions of the z-intercepts and current paths, thus the gradient coil properties. These discretization parameters are defined as: 0≦σ≦1 and Δ. The parameter Δ can also be used to match the currents on the primary and shield coils of the gradient coil during discretization of the continuous current densities. 
   Each set of concurrently varied parameters provides a different hypothetical gradient coil design, wherein each different design offers different performance characteristics. The performance characteristics of the various hypothetical designs are then evaluated. The performance characteristics of each design are typically compared to the earlier defined desired performance characteristics for the final design found through the method of this invention The hypothetical design that is evaluated as best providing the desired defined performance characteristics is often selected as the final design. 
   Again, the following hypothetical example illustrates the method of the present invention, and in no way limits the invention. All chosen values are for example purposes only. For this example, the maximum current is again limited to 500 A (I max =500 A), and the gradient strength is predetermined to be 30 mT/m (G x =30 mT/m). In actual practice of the method of the present invention, no limits exist on the value of I MAX , or the desired gradient strength (G x ) and the values may be determined due to design and system parameters of the circumstances at hand. 
     FIGS. 13-17  illustrate various characteristics of the gradient coil design of the present example through theoretical calculations, simulations and models using the method of this invention.  FIG. 13  illustrates an example of z-components of the continuous current densities on the primary and shield coils that satisfy some field quality characteristics and the gradient strength of G x =30 mT/m. When limiting the maximum current to 500 A (I max =500 A), the current densities shown in  FIG. 13  can be discretized with the number of primary coils being 18 (N p =18) and the number of secondary coils being 10 (N s =10). 
     FIGS. 14   a  through  14   d  and  FIGS. 15   a  through  15   d  illustrate the sensitivity of the z-intercepts on the primary coil and shield coil, respectively to the variation of the parameters 0≦σ≦1 and Δ.  FIGS. 14   a  through  14   d  show the position of the various z-intercepts on the primary coil as a function of the parameters 0≦σ≦1 and Δ ( FIG. 14   a  shows first initial z-intercept;  FIG. 14   b  shows first final z-intercept,  FIG. 14   c  shows the last initial z-intercept; and  FIG. 14   d  shows the last final z-intercept).  FIGS. 15   a  through  15   d  show the position of the various z-intercepts on the shield gradient coil as a function of the parameters 0≦σ≦1 and A ( FIG. 15   a  shows first initial z-intercept;  FIG. 15   b  shows first final z-intercept,  FIG. 15   c  shows the last initial z-intercept; and  FIG. 15   d  shows the last final z-intercept). 
   It is apparent from  FIGS. 14   a  through  14   c  and  FIGS. 15   a  through  15   d  that while the positions of the first z-intercepts are sensitive to the variation of the parameters σ and Δ, the positions of the last z-intercepts are extremely sensitive to these variations. The positions of the last z-intercepts are especially sensitive to the parameters Δ P  and Δ S . Illustrating this sensitivity,  FIGS. 16   a  and  16   b  show a comparison of the current paths on the primary coil when σ P =0.5, and when Δ P =0 ( FIG. 16   a ) and when Δ P =500 ( FIG. 16   b ). 
     FIGS. 17   a  and  17   b  illustrate the dependence of the net thrust force, measured in lbs., at G=30 mT/m exerted on the primary ( FIG. 17   a ) and shield ( FIG. 17   b ) gradient coil placed inside of an existing magnet. 
   Investigation using an embodiment of the method of this invention as applied to the above example allows the identification of a set of discretization parameters that give an optimized gradient coil design, taking into consideration the desired parameters and trade-offs of the various characteristics of the gradient coil design as determined by the design specifications. The further limitations and characteristics shown below are desired for the gradient design of the above example and are determined by the existing and desired system characteristics and limitations: 
   −28%≦Gradient Strength Non-Uniformity≦−32% 
   |Gradient Strength Non-Linearity|≦6% 
   |Trust Force on the Gradient|≦120 lbs 
   Current for G=30 mT/m&lt;500 A 
   |RECE MAX |&lt;0.8%, RECE Variation &lt;0.4% 
   Tables 2 and 3, below, shows an example of the discretization parameters that allow for a gradient design meeting the desired characteristics, which were determined by using an embodiment of this invention: 
   
     
       
             
           
             
             
             
             
             
             
             
           
             
             
             
             
             
             
             
           
         
             
               TABLE 2 
             
           
           
             
                 
             
             
               Discretization parameters allowing the gradient coil to meet the 
             
             
               predetermined desired characteristics 
             
           
        
         
             
                 
                 
                 
                 
                 
                 
               Current 
             
             
               Δ P   
                 
               Δ S   
                 
               Non- 
               Non- 
               [A] for G = 
             
             
               [A/m] 
               σ P   
               [A/m] 
               σ S   
               linearity[%] 
               Uniformity[%] 
               30 m T/m 
             
             
                 
             
           
        
         
             
               0 
               0.6 
               300 
               0.4 
               2.93 
               −31.27 
               480.09 
             
             
               0 
               0.7 
               0.0 
               0.6 
               4.29 
               −31.78 
               472.68 
             
             
               300 
               0.6 
               600 
               0.3 
               3.14 
               −29.45 
               473.37 
             
             
               600 
               0.4 
               0.0 
               0.6 
               −0.73 
               −29.32 
               488.08 
             
             
               600 
               0.5 
               300 
               0.5 
               1.07 
               −30.31 
               478.99 
             
             
               600 
               0.6 
               0.0 
               0.7 
               2.09 
               −30.64 
               473.30 
             
             
               900 
               0.3 
               0.0 
               0.6 
               −2.07 
               −28.66 
               488.77 
             
             
               900 
               0.4 
               300 
               0.5 
               −0.35 
               −29.62 
               479.99 
             
             
               900 
               0.5 
               0.0 
               0.7 
               0.53 
               −29.90 
               474.78 
             
             
               900 
               0.5 
               600 
               0.4 
               1.36 
               −30.55 
               471.86 
             
             
               1200 
               0.3 
               300 
               0.5 
               −1.66 
               −28.98 
               480.51 
             
             
               1200 
               0.4 
               600 
               0.4 
               −0.03 
               −29.89 
               472.69 
             
             
                 
             
           
        
       
     
   
   
     
       
             
           
             
             
             
             
             
             
             
             
           
             
             
             
             
             
             
             
             
           
         
             
               TABLE 3 
             
           
           
             
                 
             
             
               Discretization parameters allowing the gradient coil to meet the 
             
             
               predetermined desired characteristics 
             
           
        
         
             
                 
                 
                 
                 
                 
                 
               Thrust Force [lbs] 
               Thrust Force [lbs] 
             
             
                 
                 
                 
                 
                 
                 
               on the Gradient 
               on the Cold Shield 
             
             
               Δ P  [A/m] 
               σ P   
               Δ S  [A/m] 
               σ S   
               RECE MAX  [%] 
               RECE VAR  [%] 
               at G = 30 mT/m 
               at G = 30 mT/m 
             
             
                 
             
           
        
         
             
               0 
               0.6 
               300 
               0.4 
               0.538 
               0.344 
               92.08 
               34.04 
             
             
               0 
               0.7 
               0.0 
               0.6 
               −0.211 
               0.392 
               70.32 
               68.76 
             
             
               300 
               0.6 
               600 
               0.3 
               0.278 
               0.199 
               46.28 
               10.34 
             
             
               600 
               0.4 
               0.0 
               0.6 
               0.037 
               0.228 
               99.71 
               48.61 
             
             
               600 
               0.5 
               300 
               0.5 
               0.509 
               0.282 
               98.61 
               31.10 
             
             
               600 
               0.6 
               0.0 
               0.7 
               −0.151 
               0.267 
               71.47 
               62.28 
             
             
               900 
               0.3 
               0.0 
               0.6 
               −0.307 
               0.249 
               106.75 
               45.04 
             
             
               900 
               0.4 
               300 
               0.5 
               0.061 
               0.141 
               105.57 
               27.62 
             
             
               900 
               0.5 
               0.0 
               0.7 
               −0.562 
               0.153 
               78.30 
               59.01 
             
             
               900 
               0.5 
               600 
               0.4 
               0.604 
               0.24 
               106.12 
               5.58 
             
             
               1200 
               0.3 
               300 
               0.5 
               −0.229 
               0.35 
               112.22 
               24.47 
             
             
               1200 
               0.4 
               600 
               0.4 
               0.148 
               0.063 
               112.70 
               2.48 
             
             
                 
             
           
        
       
     
   
   From Tables 2 and 3 it can be seen that the following discretization parameters allow for a gradient coil design meeting the desired characteristics: 
   σ P =0.6, Δ P =300 A/m 
   σ S =0.3, Δ S =600 A/m 
   The current paths on half of a quadrant of the primary coil ( FIG. 18   a ) and shield coil ( FIG. 18   b ) of the gradient design using the parameters given in the above example are illustrated in  FIGS. 18   a  and  18   b .  FIG. 19  illustrates the field quality characteristics within the 50 cm (x/y)×40 cm (z) FoV of the gradient coil design of the above example. Only half of the corresponding volume is illustrated in  FIG. 19 , as the other half is identical due to the symmetry of the coil design. 
   The characteristics of the gradient coil design of the above example are shown in Table 4, below. 
   
     
       
             
           
             
             
           
         
             
               TABLE 4 
             
           
           
             
                 
             
             
               Characteristics of Gradient Coil Design of Example 
             
           
        
         
             
               Property 
               Value 
             
             
                 
             
             
               Electrical Length (P/S) [mm] 
               1291/1455 
             
             
               Resistance (P/S) [mΩ] 
               74.16/36.57 
             
             
               Energy @ G = 30 m T/m 
               50.96 
             
             
               Current [A] for G = 30 m T/m 
               473.37 
             
             
               Inductance [μH] 
               417.70 
             
             
               Slew rate [T/m/s] (losses not included): V = 1200 V, 
               174.12 
             
             
               G = 30 m T/m 
             
             
               Rise time [sec] (losses not included) 
               172.30 
             
             
               DC Dissipated power [kW]: G = 30 m T/m 
               24.81 
             
             
               Thrust force on the Gradient Coil [lbs]: G = 30 m T/m 
               46.28 
             
             
               Thrust force on the Cold Shield [lbs]: G = 
               10.34 
             
             
               30 m T/m 
             
             
               RECE_max [%]/RECE_variation [%] 
               0.278/0.199 
             
             
               Global non-linearity max [%] at z = 0.25 m 
               +3.14 
             
             
               Local non-linearity [%] at z = 0.25 m 
               −12.2 
             
             
               Non-uniformity [%] at ρ = 0.2 m 
               −29.45 
             
             
                 
             
           
        
       
     
   
   A flowchart of an embodiment of the method of this invention as used to design a transverse gradient coil is shown in  FIG. 20 . As shown in the flowchart of  FIG. 20 , the discretization parameters are varied, and the field quality characteristics are searched over the range of the varied discretization parameters (see block  32 ) with consideration of the continuous current densities (see block  30 ). The thrust force on the gradient coil is searched over the range of the varied discretization parameters (see block  34 ) with consideration of the continuous current densities (see block  30 ). The RECE and the thrust force on the cold shield are searched over the range of the varied discretization parameters (see block  36 ) with consideration of the continuous current densities (see block  30 ). After these factors have been searched, all other characteristics may be searched over the range of varied discretization parameters (see block  40 ). The tradeoffs of the design characteristics can then be determined (see block  38 ). The various design options output by the varied discretization parameters are searched (see block  42 ) and a final solution is chosen (see block  44 ). 
   The present application generally provides a method of designing a gradient coil for use in magnetic resonance imaging (MRI). The method of the various embodiments of the invention can be used to design an axial or a transverse gradient coil. In embodiments of the invention, in designing an axial gradient coil, at least one performance characteristic of the axial gradient coil design is defined. In certain embodiments, the at least one defined performance characteristic may include one or more of the following: at least one field quality characteristic within a field of view, minimum energy, minimum inductance, minimum coil resistance, minimum eddy current effect, minimum eddy current variation over a field of view, minimum net thrust force exerted on the coil, and minimum net thrust force exerted on the cold shield due to eddy currents. Discretization parameters σ P,S  and current parameter I are concurrently varied within the following two equations: 
   
     
       
         
           
             
               
                 ∫ 
                 0 
                 
                   Z 
                   n 
                   
                     ( 
                     P 
                     ) 
                   
                 
               
               ⁢ 
               
                 
                   
                     f 
                     φ 
                     
                       ( 
                       P 
                       ) 
                     
                   
                   ⁡ 
                   
                     ( 
                     z 
                     ) 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ⅆ 
                   z 
                 
               
             
             - 
             
               
                 ( 
                 
                   n 
                   - 
                   
                     σ 
                     P 
                   
                 
                 ) 
               
               ⁢ 
               I 
             
           
           = 
           
             0 
             ⁢ 
             
                 
             
             ⁢ 
             and 
           
         
       
     
     
       
         
           
             
               
                 ∫ 
                 0 
                 
                   Z 
                   m 
                   
                     ( 
                     S 
                     ) 
                   
                 
               
               ⁢ 
               
                 
                   
                     f 
                     φ 
                     
                       ( 
                       S 
                       ) 
                     
                   
                   ⁡ 
                   
                     ( 
                     z 
                     ) 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ⅆ 
                   z 
                 
               
             
             + 
             
               
                 ( 
                 
                   m 
                   - 
                   
                     σ 
                     S 
                   
                 
                 ) 
               
               ⁢ 
               I 
             
           
           = 
           0. 
         
       
     
   
   Discretization parameters σ P,S  are varied between 0 and 1. Within the equations, f φ   (P,S) (z) is the φ-components of the continuous current density on said primary coil (P) and said shield coil (S), n is an integer between the value of 1 and the number of turns on said primary coil and m is an integer between the value of 1 and the number of turns on said shield coil. Each set of varied discretization parameters and current parameter produces a different hypothetical axial gradient coil design. At least one performance characteristic of each hypothetical axial gradient coil design produced by the varied discretization parameters and the varied current parameter are evaluated. One of the hypothetical axial gradient coil designs is selected. In certain embodiments, the selected design will commonly meet at least one of the defined performance characteristics. 
   Also described herein is a method of designing an axial gradient coil of further embodiments of the invention, wherein the axial gradient coil having a primary coil and a shield coil for use in magnetic resonance imaging which includes defining at least one performance characteristic of the axial gradient coil, concurrently varying at least two parameters within at least one equation, wherein each result of the varied at least two parameters within the at least on equation produces a different hypothetical axial gradient coil design, evaluating at least one performance characteristic of each hypothetical axial gradient coil design produced by the varied parameters; and selecting one of the hypothetical axial gradient coil designs. Again, in certain embodiments, the at least one defined performance characteristic is selected from the group consisting of at least one field quality characteristic within a field of view, minimum energy, minimum inductance, minimum coil resistance, minimum eddy current effect, minimum eddy current variation over a field of view, minimum net thrust force exerted on the coil, and minimum net thrust force exerted on the cold shield due to eddy currents. In certain embodiments, the selected axial gradient coil design meets the at least one defined performance characteristic. 
   In designing a transverse gradient coil of further embodiments of the invention, at least one performance characteristic of the transverse gradient coil design is defined. In certain embodiments, the at least one performance characteristic may include one or more of the following: at least one field quality characteristic within a field of view, minimum energy, minimum inductance, minimum coil resistance, minimum eddy current effect, minimum eddy current variation over a field of view, minimum net thrust force exerted on the coil, and minimum net thrust force exerted on the cold shield due to eddy currents. Discretization parameters σ and Δ are concurrently varied within the following equation: 
               S   n     =         n   -   σ     N     ⁢     (         f   z     ⁡     (     z   eye     )       -   Δ     )         ,         
wherein the previous equation is used to solve the equation: Φ n (z)=arccos(S n /f z (z)), which is in turn used to solve the equation: φ=Φ n (z). Discretization parameter σ is varied between the values of 0 and 1. Within the equations, f_(z) is the z-component of the continuous current density, Z eye  is the position where the φ-component of the continuous current density is equal zero, N is the number of loops in one quadrant of the coil, and n is an integer wherein 1≦n≦N. Each set of varied discretization parameters produces a different hypothetical transverse gradient coil design. At least one performance characteristic of each hypothetical transverse gradient coil design produced by the varied discretization parameters is evaluated. One of the hypothetical transverse gradient coil designs is selected. In certain embodiments, the selected design will commonly meet at least one of the defined performance characteristics.
 
   Also described herein is a method of designing a transverse gradient coil for use in magnetic resonance imaging which includes defining at least one performance characteristic of the transverse gradient coil, concurrently varying at least two parameters within at least one equation wherein each set of varied parameters produces a different hypothetical transverse gradient coil design, evaluating at least one performance characteristic of each hypothetical transverse gradient coil design produced by the varied parameters; and selecting one of the hypothetical transverse gradient coil designs. Again, in certain embodiments, the at least one defined performance characteristic is selected from the group consisting of at least one field quality characteristic within a field of view, minimum energy, minimum inductance, minimum coil resistance, minimum eddy current effect, minimum eddy current variation over a field of view, minimum net thrust force exerted on the coil, and minimum net thrust force exerted on the cold shield due to eddy currents. In further embodiments, the selected transverse gradient coil design meets the at least one defined performance characteristic. 
   In the foregoing description, certain terms have been used for brevity, clearness, illustration and understanding; but no unnecessary limitations are to be implied therefrom beyond the requirements of the prior art, because such terms are used for descriptive purposes and are intended to be broadly construed. Moreover, this invention has been described in detail with reference to specific embodiments thereof, including the respective best modes for carrying out each embodiment. It shall be understood that these illustrations are by way of example and not by way of limitation.