Patent Publication Number: US-2005127913-A1

Title: Lc coil

Description:
BACKGROUND  
      In magnetic resonance imaging, the rate of data acquisition is limited by how rapidly fields can be changed within the field of view. However, bounds are generally placed on how rapidly fields can be changed within an often larger region, such as a patient. With reduced fields outside the field of view, data acquisition can be acclerated.  
     SUMMARY  
      This invention provides a coil for a magnetic resonance imaging machine with two adjacent regions carrying different currents at the interface. Currents in the two regions at the interface can be in opposite directions.  
      The interface separating the two adjacent regions can be planar and the regions can be mirror images of each other across the interface. Current in one region and the opposite of current in the other region can be mirror images of each other across the planar interface. The current density, or volume current density integrated over the thickness of the coil, can be constant in each region.  
      The two adjacent regions of the coil can pass directly under and conform to a support surface, which can be flat. The cross-section of the coil can contain an arc of a circle. 
    
    
     BRIEF DESCRIPTION OF THE FIGURES  
       FIG. 1  shows the first embodiment of the coil, side view ( FIG. 1A ) and axial view ( FIG. 1B ).  
       FIG. 2  shows the second embodiment of the coil, side view ( FIG. 2A ) and axial view ( FIG. 2B ). 
    
    
     DETAILED DESCRIPTION  
       FIG. 1  shows a coil within a magnetic resonance imaging machine  10 . The coil has region  3  carrying current  5  and adjacent region  4  carrying current  6 . Currents  5  and  6  differ at the interface  7  between the regions  3  and  4 . The coil has a flat lower part  8  passing under a support surface  9  of the magnetic resonance imaging machine  10  and a partial cylindrical upper part  11  with axis  12 .  
      Throughout this specification, the term “LC coil” refers to this invention, the terms “axial”, “z direction”, and “along z ” refer to the direction of a specified axis of the LC coil, and “LC z  coil” refers to an LC coil designed to produce an axial gradient of an axial magnetic field. Terms “LC x  coil” and “LC y  coil” refer to LC coils designed to produce gradiy ents of an axial magnetic field orthogonal to the axis. The term “scanner” refers to a magnetic resonance imaging machine.  
     1 First Embodiment  
      The first embodiment ( FIG. 1 ) of an LC z  coil has region  3  carrying current  5  and adjacent region  4  carrying current  6 . Currents  5  and  6  differ at the interface  7  between the regions  3  and  4 . The first embodiment has a flat lower part S 1    8  passing a distance Δ y  under a flat support surface  9  of a scanner  10  and upper part S 2    11  a partial cylinder of radius R , angle (1+2ε φ )π, ε φ ∈[0, ½], and axis  12  coinciding with the scanner axis  13 . The axis  12  is also called the coil axis. The length of the first embodiment along its axis  12  is l z  and the dimension of S 1    8  are l x ×l z , with l x≦2 . The distance between S 1    8  and the axis  12  is y c  . A surface detection coil  14  is located between S 1    8  and the support surface  9 .  
      1.1 Definition of Coordinates  
      Define rectangular coordinates (x, y, z) with x parallel the support surface  9 , y perpendicular to the support surface  9 , and z along the coil axis  12 . The flat section S 1    8  is in the plane y=−y c  and is parallel to the support surface  9  in the plane y=−y c +Δ y . The plane x=0 perpendicular to the support surface  9  contains the coil axis  12  given by the line x=0 and y=0. Centers of fields of view are arranged to lie within the plane z=0.  
      Cylindrical coordinates (r, φ, z) are related to coordinates (x, y, z) by the transformation 
 
 x=r  cos φ  (1a) 
 
 y=r  sin φ  (1b) 
 
 The inverse transformation is  
             r   =         x   2     +     y   2                 (     2   ⁢   a     )               φ   =         cos     -   1       ⁢     x   r       =       sin     -   1       ⁢     y   r                 (     2   ⁢   b     )             
 
 The partial cylindrical section S 2    11  is at r=R and covers the angular range 
 
φ∈ I   φ =[−ε φ π, (1ε φ )π]  (3) 
 
      The coil dimensions 
 
 l   x =2 R  cos(ε φ π)  (4a) 
 
and 
 
 y   c   R  sin (ε φ π  (4b) 
 
 1.2 Current Density 
 
      The coil carries a current density  
                 J   →     ⁡     (     r   →     )       =     {               J   ⁡     (   z   )       ⁢     x   ^       ,             r   →     ∈     S   1                     J   ⁡     (   z   )       ⁢     φ   ^       ,             r   →     ∈     S   2                       (   5   )             
 
 where J(z) is the current density profile and 
 
 {right arrow over (r)} =( x, y, z )  (6) 
 
 1.3 Field 
 
      Using the Biot-Savart law, the total field  
                 B   ⁡     (     x   ,   y   ,   z     )       =       ∫       -     l   z       /   2         l   z     /   2       ⁢           ⁢       ⅆ     z   ′       ⁢     J   ⁡     (     z   ′     )       ⁢     g   ⁡     (     x   ,   y   ,     z   -     z   ′         )             ⁢     
     ⁢   with           (   7   )                       g   ⁡     (     x   ,   y   ,   z     )       =       ⁢             μ   0     ⁡     (     y   +     y   c       )         4   ⁢   π       ⁢       ∫       -     l   x       /   2         l   x     /   2       ⁢           ⁢       ⅆ     x   ′       ⁢     1       [         (     x   -     x   ′       )     2     +       (     y   +     y   c       )     2     +     z   2       ]       3   /   2               +                     ⁢           μ   0     ⁢   R       4   ⁢   π       ⁢       ∫     I   φ       ⁢           ⁢       ⅆ     φ   ′       ⁢       R   -     r   ⁢           ⁢     cos   ⁡     (     φ   -     φ   ′       )               [       R   2     +     r   2     -     2   ⁢   Rr   ⁢           ⁢     cos   ⁡     (     φ   -     φ   ′       )         +     z   2       ]       3   /   2                             (   8   )             
 
 Under conditions  
                  x        ,            y   +     y   c            ⁢       &lt;&lt;     l   x       /   2               (     9   ⁢   a     )                      y   +     y   c            ⁢       &lt;&lt;     l   z       /   2             (     9   ⁢   b     )                     x   2     +     y   2         ⁢     &lt;&lt;   R             (     9   ⁢   c     )             
 
 the function  
                     g   ⁢     (     x   ,   y   ,   z     )       ≈       ⁢           μ   0       2   ⁢   π       ⁢       y   +     y   c           2   ⁢       π   ⁡     (     y   +     y   c       )       2       +     z   2           +                     ⁢         μ   0       4   ⁢   π       ⁢         R   2         [       R   2     +     z   2       ]       3   /   2         [         (     1   +     2   ⁢     ɛ   φ         )     ⁢   π     +                         ⁢           l   x     ⁢   y   ⁢         x   2     +     y   2             R   3       ⁢     (         3   ⁢     R   2           R   2     +     z   2         -   1     )       ]                 (   10   )             
 
      If the projection of the field of view onto the x−y plane is a rectangle L x ×L y  centered about (x 0 , y 0 ) under conditions 
 
 L   x /2 , |x   0   |&lt;&lt;R   (11a) 
 
 L   y /2 , |y   0   |&lt;&lt;R, l   z /2  (11b) 
 
and 
 
 y   0   ≦−y   c +Δ y   +L   y /2  (11c) 
 
 and the coil parameter 
 
 y   c   ≦≦R, l   z /2  (12) 
 
 then conditions (9) are satisfied within the field of view. 
 
 1.4 Field with (14) 
 
      Define the function sgn by  
               sgn   ⁢           ⁢   z     =     {           1   ,     z   ≥   0                   -   1     ,     x   &lt;   0                       (   13   )             
 
 Under conditions (9) with 
 
 J ( z )= J   0   sgn z   (14) 
 
 the field  
                       B   ⁢     (     x   ,   y   ,   z     )       ≈       ⁢       b   ⁡     (     x   ,   y   ,   z     )       -       1   2     [       b   ⁢     (     x   ,   y   ,     z   -       l   z     /   2         )       +                         ⁢     b   ⁡     (     x   ,   y   ,     z   +       l   z     /   2         )       ]           ⁢     
     ⁢   with           (     15   ⁢   a     )                       b   ⁡     (     x   ,   y   ,   z     )       =       ⁢             μ   0     ⁢     J   0       π     ⁢     Tan     -   1       ⁢     z     y   +     y   c           +                     ⁢           μ   0     ⁢     J   0     ⁢   z       2   ⁢         R   2     +     z   2             [       (     1   +     2   ⁢     ɛ   φ         )     +                       ⁢           l   x     ⁢   y   ⁢         x   2     +     y   2             π   ⁢           ⁢   R       ⁢     (       1     R   2       +     1       R   2     +     z   2           )       ]                 (     15   ⁢   b     )             
 
 using (2a). 
 
      Under conditions (9) and 
 
| z|≦≦l   z /2 , R   (16) 
 
 with (14), the field  
               B   ⁡     (     x   ,   y   ,   z     )       ≈           μ   0     ⁢     J   0       π     ⁢     Tan     -   1       ⁢     z     y   +     y   c                   (   17   )             
 
 the gradient  
                   G   z     =       ∂   B       ∂   z                  (     x   ,   y   ,   z     )     =     (       x   0     ,     y   0     ,   0     )               (     18   ⁢   a     )                       ⁢     ≈         μ   0     ⁢     J   0         π   ⁡     (       y   0     +     y   c       )                   (     18   ⁢   b     )             
 
     2 Second Embodiment  
      The second embodiment ( FIG. 2 ) of an LC z  coil has region carrying current  17  and adjacent region  16  carrying current  18 . Currents  17  and  18  differ at the interface  19  between the regions  15  and  16 . The second embodiment has a flat upper part S 1    20  passing a distance Δ y  under a flat support surface  21  of a scanner  22  and lower part S 2    23  a partial cylinder of radius R , angle (1+2ε φ )π, ε φ ∈[0, ½], and axis  24  coinciding with the scanner axis  25 . The axis  25  is also called the coil axis. The length of the second embodiment along its axis  24  is l z  and the dimension of S 1    20  are l x ×l z , with l x ≦2R . The distance between S 1    20  and the axis  24  is y c . A surface detection coil  26  is located between S 1    20  and the support surface  21   
      2.1 Definition of Coordinates  
      Define rectangular coordinates (x, y, z) with x parallel the support surface  21 , y perpendicular to the support surface  21 , and z along the coil axis  24 . The flat section S 1    20  is in the plane y=−y c  and is parallel to the support surface  21  in the plane y=−y c +Δ y . The plane x=0 perpendicular to the support surface  21  contains the coil axis  24  given by the line x=0 and y=0. Centers of fields of view are arranged to lie within the plane z=0.  
      Cylindrical coordinates (r, φ, z) are related to coordinates (x, y, z) by the transformation (1). The inverse transformation is (2). The partial cylindrical section S 2    23  is at r=R and covers the angular range 
 
φ∈ I   φ =[−(1−ε φ )π, −ε φ π]  (19) 
 
      The coil dimensions l x  and y c  are given by (4a) and (4b).  
      2.2 Current Density  
      The coil carries a current density  
                 J   →     ⁡     (     r   ⇀     )       =     {               J   ⁡     (   z   )       ⁢     x   ^       ,             r   →     ∈     S   1                     -     J   ⁡     (   z   )         ⁢     φ   ^       ,             r   →     ∈     S   2                       (   20   )             
 
 where J(z) is the current density profile. 
 
 2.3 Field 
 
      The first and second embodiments have different partial cylindrical sections S 2    11  and S 2    23 : current densities (5) and (20) for {right arrow over (r)}∈S 2  are carried in complementary angular ranges (3) and (19). Expressions for the field produced by the second embodiment can be obtained by modifying the expressions of Sec. (1.3).  
      The field  
               B   ⁡     (     x   ,   y   ,   z     )       =       ∫       -     l   z       /   2         l   z     /   2       ⁢       ⅆ     z   ′       ⁢     J   ⁡     (     z   ′     )       ⁢     g   ⁡     (     x   ,   y   ,     z   -     z   ′         )                   (   21   )             with                                 g   ⁡     (     x   ,   y   ,   z     )       =       ⁢           μ   0     ⁡     (     y   +     y   c       )         4   ⁢           ⁢   π       ⁢       ∫       -     l   x       /   2         l   x     /   2       ⁢     ⅆ     x   ′                           ⁢       1       [         (     x   -     x   ′       )     ′2     +       (     y   +     y   c       )     2     +     z   2       ]       3   /   2         -                     ⁢           μ   0     ⁢   R       4   ⁢           ⁢   π       ⁢       ∫     I   φ       ⁢       ⅆ     φ   ′       ⁢       R   -     r   ⁢           ⁢     cos   ⁡     (     φ   -     φ   ′       )               [       R   2     +     r   2     -     2   ⁢           ⁢   Rr   ⁢           ⁢     cos   ⁡     (     φ   -     φ   ′       )         +     z   2       ]       3   /   2                             (   22   )             
 
 r can be expressed in terms of x and y using (2a). Under conditions (9),  
                     g   ⁡     (     x   ,   y   ,   z     )       ≈       ⁢           μ   0       2   ⁢           ⁢   π       ⁢       y   +     y   c             (     y   +     y   c       )     2     +     z   2           -                     ⁢         μ   0       4   ⁢           ⁢   π       ⁢         R   2         [       R   2     +     z   2       ]       3   /   2         [         (     1   -     2   ⁢     ɛ   φ         )     ⁢   π     +                         ⁢           l   x     ⁢   y   ⁢         x   2     +     y   2             R   3       ⁢     (         3   ⁢           ⁢     R   2           R   2     +     z   2         -   1     )       ]                 (   23   )             
 
      If the projection of the field of view onto the x−y plane is a rectangle L x ×L y  centered about (x 0 , y 0 ) under conditions (11) and the coil parameter y c  satisfies (12), then conditions (9) are satisfied within the field of view.  
      2.4 Field with (14)  
      Expressions for the field and gradient produced by the second embodiment can be obtained by modifying the expressions of Sec. (1.4).  
      Under conditions (9) with (14), the field  
                       B   ⁡     (     x   ,   y   ,   z     )       ≈       ⁢       b   ⁡     (     x   ,   y   ,   z     )       -       1   2     [       b   ⁡     (     x   ,   y   ,   z     )       -       l   z     /   2       )     +                     ⁢     b   ⁡     (     x   ,   y   ,     z   +       l   z     /   2         )       ]           ⁢     
     ⁢   with           (     24   ⁢   a     )                       b   ⁡     (     x   ,   y   ,   z     )       =       ⁢             μ   0     ⁢     J   0       π     ⁢     Tan     -   1       ⁢     z     y   +     y   c           -                     ⁢           μ   0     ⁢     J   0     ⁢   z       2   ⁢         R   2     +     z   2             [       (     1   -     2   ⁢     ɛ   φ         )     +                       ⁢           l   x     ⁢   y   ⁢         x   2     +     y   2             π   ⁢           ⁢   R       ⁢     (       1     R   2       +     1       R   2     +     z   2           )       ]                 (     24   ⁢   b     )             
 
      Under conditions (9) and (16) with (14), the field B is (17) and the gradient G z  is (18)  
      3 MRI Using an LC Coil  
      3.1 Field Profiles  
      The three types of LC coil are LC x , LC y , and LC z . An LC x  coil produces an LC X  field 
 
 b ( {right arrow over (r)} , ι)= G   x (ι) {tilde over (x)} ( {right arrow over (r)} )  (25a) 
 
 with gradient  
               G   x     =       ∂   B       ∂   x               (     25   ⁢   b     )             
 
 An LC y  coil produces an LC y  field 
 
 B ( {right arrow over (r)}, t )= G   y ( t ) {tilde over (y)} ( {right arrow over (r)} )  (26a) 
 
 with gradient  
               G   y     =       ∂   B       ∂   y               (     26   ⁢           ⁢   b     )             
 
      An LC z  coil produces an LC z  field 
 
 B ( {right arrow over (r)} , ι)= G   z (ι) {tilde over (z)} ( {right arrow over (r)} )  (27
 
 with gradient  
               G   z     =       ∂   B       ∂   z               (     27   ⁢   b     )             
 
 The gradients are evaluated at the center of the field of view. 
 
      Magenetic resonance imaging with an LC z  coil requires that {tilde over (z)}({right arrow over (r)}) satisfy 
 
{tilde over (z)}=0 at the center of the field of view  (28a) 
 
θ {tilde over (z)}/θz =1 at the center of the field of view  (28b) 
 
θ {tilde over (z)}/θz ≧0 within the field of view  (28c) 
 
and 
 
{tilde over (z)} attains values for fixed x and y within the field of view that are unique within the region of sensitivity of the detection coil  (28d) 
 
 Magnetic resonance imaging with an LC x  coil requires that {tilde over (x)}({right arrow over (r)}) satisfy (28) with x z and {tilde over (x)} {tilde over (z)}. Magnetic resonance imaging with an LC y  coil requires that {tilde over (y)}({right arrow over (r)}) satisfy (28) with y z and {tilde over (y)} {tilde over (z)}. 
 
      The center of a field of view refers to the center of an image in field coordinates. For example, if fields linear in x and y are used with an LC z  field linear in {tilde over (z)}, the field of view is a rectangular box in field coordinates (x, y, {tilde over (z)}) and the center of the field of view refers to the center of the box.  
      3.2 Imaging Parameters  
      This section assumes the use of an LC z  field together with fields linear in x and y . Similar equations hold for other combinations of LC and linear fields.  
      3.2.1 Non-Oblique Imaging Parameters  
      In field coordinates (x, y, {tilde over (z)}) , the field of view is a box L x ×L y ×{tilde over (L)} z  centered about 
 
( x, y, {tilde over (z)} )=( x   0   , y   0 , 0) ( x, y, z )=( x   0   , y   0   , z   0 )  (29) 
 
 With choices for pixel numbers N x , N y , N z , and pixel sizes Δx, Δy, Δz , the imaging parameters are  
               L   x     =       N   x     ⁢   Δ   ⁢           ⁢   x             (     30   ⁢   a     )                 L   y     =       N   y     ⁢   Δ   ⁢           ⁢   y             (     30   ⁢   b     )                   L   ~     z     =       N   z     ⁢   Δ   ⁢           ⁢   z             (     30   ⁢   c     )                 k   x     =       n   x     ⁢   Δ   ⁢           ⁢     k   x               (     31   ⁢   a     )                 k   y     =       n   y     ⁢   Δ   ⁢           ⁢     k   y               (     31   ⁢   b     )                 k   z     =       n   z     ⁢   Δ   ⁢           ⁢     k   z               (     31   ⁢   c     )                 n   x     ∈     {         -     N   x       /   2     ,   …   ⁢           ,         N   x     /   2     -   1       }             (     32   ⁢   a     )                 n   y     ∈     {         -     N   y       /   2     ,   …   ⁢           ,         N   y     /   2     -   1       }             (     32   ⁢   b     )                 n   z     ∈     {         -     N   z       /   2     ,   …   ⁢           ,         N   z     /   2     -   1       }             (     32   ⁢   c     )                 Δ   ⁢           ⁢     k   x       =       2   ⁢   π       L   x               (     33   ⁢   a     )                 Δ   ⁢           ⁢     k   y       =       2   ⁢           ⁢   π       L   y               (     33   ⁢   b     )                 Δ   ⁢           ⁢     k   z       =       2   ⁢   π         L   ~     z               (     33   ⁢   c     )             
 
      Fourier reconstruction yields an image on a grid in field coordinates: 
 
( x, y, {tilde over (z)} )=( n   x   Δx+x   0   , n   y   Δy+y   0   , n   z   Δz )  (34) 
 
 The image can be scaled to coordinates (x, y, z) using the 1-to-1 mapping 
 
( x, y, z ) ( x, y, {tilde over (z)} ) within  F∩D   (35) 
 
 The resolutions in coordinates (x, y, z) are accurately given by 
 
(Δ x, Δy, Δz ) J ′=(Δ x, Δy, Δz (θ {tilde over (z)}/θz)   −1 )  (36) 
 
 where Jacobian  
               J   ′     =         ∂     (     x   ,   y   ,   z     )         ∂     (     x   ,   y   ,     z   ~       )         =     [         1       0       0           0       1       0           0       0           (       ∂     z   ~       /     ∂   z       )       -   1             ]               (   37   )             
 
 using the relation 
 
θ z/θ{tilde over (z)} =(θ {tilde over (z)}/θz ) −1   (38) 
 
 which follows from the fact that both θz/θ{tilde over (z)}and θ{tilde over (z)}/θz are taken at constant x and y . At the center of the field of view (29), the Jacobian J′is the identity matrix and the resolutions are Δx, Δy, Δz . 
 
 3.2.2 Double-Oblique Imaging Parameters 
 
      Coordinates (x′, y′, z′) and field coordinates ({tilde over (x)}′, {tilde over (y)}′, {tilde over (z)}′) are obtained from coordinates (x, y, z) and field coordinates (x, y, {tilde over (z)}) by a rotation by θ 1  about z and a rotation by θ 2  about x : 
 
( x′, y′, z ′)=( x, y, z ) R   ( x, y, z )=( x′, y′, z ′) R   −1   (39) 
 
and 
 
( {tilde over (x)}′, {tilde over (y)}′, {tilde over (z)} ′)=( x, y, {tilde over (z)} ) R   ( x, y, {tilde over (z)} )=( {tilde over (x)}′, {tilde over (y)}′, {tilde over (z)}′ ) R   −1   (40) 
 
 where the rotation matrix  
             R   =     [           cos   ⁢           ⁢     θ   1               -   sin     ⁢           ⁢     θ   1           0             sin   ⁢           ⁢     θ   1     ⁢           ⁢   cos   ⁢           ⁢     θ   2             cos   ⁢           ⁢     θ   1     ⁢           ⁢   cos   ⁢           ⁢     θ   2               -   sin     ⁢           ⁢     θ   2                 sin   ⁢           ⁢     θ   1     ⁢           ⁢   sin   ⁢           ⁢     θ   2             cos   ⁢           ⁢     θ   1     ⁢           ⁢   sin   ⁢           ⁢     θ   2             cos   ⁢           ⁢     θ   2             ]             (   41   )             
 
 and inverse matrix  
               R     -   1       =     [           cos   ⁢           ⁢     θ   1             sin   ⁢           ⁢     θ   1     ⁢           ⁢   cos   ⁢           ⁢     θ   2             sin   ⁢           ⁢     θ   1     ⁢           ⁢   sin   ⁢           ⁢     θ   2                   -   sin     ⁢           ⁢     θ   1             cos   ⁢           ⁢     θ   1     ⁢           ⁢   cos   ⁢           ⁢     θ   2             cos   ⁢           ⁢     θ   1     ⁢           ⁢   sin   ⁢           ⁢     θ   2               0           -   sin     ⁢           ⁢     θ   2             cos   ⁢           ⁢     θ   2             ]             (   42   )             
 
      Consider combining fields linear in x , y , and {tilde over (z)} according to (40) to create fields linear in {tilde over (x)}′, {tilde over (y)}′, and {tilde over (z)}′. In field coordinates ({tilde over (x)}′, {tilde over (y)}′, {tilde over (z)}′) , the field of view is a box {tilde over (L)} x′ × y′ ×{tilde over (L)} z′  centered about  
                     (     x   ,   y   ,     z   ~       )     =     (       x   0     ,     y   0     ,   0     )           ⇔           (         x   ~     ′     ,       y   ~     ′     ,       z   ~     ′       )     =     (         x   ~     0   ′     ,       y   ~     0   ′     ,       z   ~     0   ′       )                           ⇔           (       x   ′     ,     y   ′     ,     z   ′       )     =     (       x   0   ′     ,     y   0   ′     ,     z   0   ′       )                   (   43   )             
 
 With choices for pixel numbers N x′ , N y′ , N z′ , and pixel numbers Δx′, Δy′, Δz′, the imaging parameters are 
 
 {tilde over (L)}   x′   =N   x′   Δx′   (44a) 
 
 {tilde over (L)}   y′   =N   y′   Δy′   (44b) 
 
 {tilde over (l)}   z′   =N   z′   Δz′   (44c) 
 
               k     x   ′       =       n     x   ′       ⁢   Δ   ⁢           ⁢     k     x   ′                 (     45   ⁢   a     )                 k     y   ′       =       n     y   ′       ⁢   Δ   ⁢           ⁢     k     y   ′                 (     45   ⁢   b     )                 k     z   ′       =       n     z   ′       ⁢   Δ   ⁢           ⁢     k     z   ′                 (     45   ⁢   c     )                 n     x   ′       ∈     {         -     N     x   ′         /   2     ,   …   ⁢           ,         N     x   ′       /   2     -   1       }             (     46   ⁢   a     )                 n     y   ′       ∈     {         -     N     y   ′         /   2     ,   …   ⁢           ,         N     y   ′       /   2     -   1       }             (     46   ⁢   b     )                 n     z   ′       ∈     {         -     N     z   ′         /   2     ,   …   ⁢           ,         N     z   ′       /   2     -   1       }             (     46   ⁢   c     )                 Δ   ⁢           ⁢     k     x   ′         =       2   ⁢   π         L   ~       x   ′                 (     47   ⁢   a     )                 Δ   ⁢           ⁢     k     y   ′         =       2   ⁢           ⁢   π         L   ~       y   ′                 (     47   ⁢   b     )                 Δ   ⁢           ⁢     k     z   ′         =       2   ⁢   π         L   ~       z   ′                 (     47   ⁢   c     )             
 
      Fourier reconstruction yields an image on a grid: 
 
( {tilde over (x)}′, {tilde over (y)}′, {tilde over (z)} )=( n   x′   Δx′+{tilde over (x)}′   0   , n   y′   Δy′+{tilde over (y)}   0   , n   z′   Δz′+{tilde over (z)}   0 )  (48) 
 
 The image can be scaled to coordinates (x′, y′, z′) using the 1-1 mapping (35) and the transformations (39) and (40): 
 
( {tilde over (x)}′, {tilde over (y)}′, {tilde over (z)}′ )→( x, y, {tilde over (z)} )→( x, y, z )→( x′, y′, z′ )  (49) 
 
 The resolutions in coordinates (x′, y′, z′) are accurately given by 
 
(Δ x′, Δy′, Δz′ ) J′   (50) 
 
 where Jacobian 
 
 {tilde over (J)}′=R   −1   J′R   (51) 
 
 At the center of the field of view (43), the Jacobians J′ and {tilde over (J)}′ are the identity matrices and the resolutions Δx′, Δy′, Δz′. 
 
 3.3 Additional Parameters 
 
      The parameter κ is defined by  
             κ   =              B          max   P              B          max     F   ⋂   D   ⋂   P           ≥   1             (   52   )             
 
 where B| max P  and B| max F∩D∩P  indicate the maximum values of B attained over a region P , such as a patient, and over the intersection F∩D∩P , where F is the field of view and D is the region of sensitivity of the detection coil. A second parameter κ D  is defined by  
               k   D     =              B          max     D   ⋂   P                B          max     F   ⋂   D   ⋂   P           ∈     [     1   ,   k     ]               (   53   )             
 
 4 MRI Using First and Second Embodiments 
 
      Field Coordinate with (14)  
      The current densities of the first and second embodiments are (5) and (20). Under conditions (9) and (16) with (14), the field is (17) and field coordinate  
               z   ~     =     B     G   z               (     54   ⁢   a     )               ≈       (       y   0     +     y   c       )     ⁢     Tan   1     ⁢     z     y   +     y   c                   (     54   ⁢   b     )             
 
 Within a field of view F that is a rectangular box L x ×L y ×{tilde over (L)} z  in coordinates (x, y, {tilde over (z)}) centered about x 0 , y 0 , 0) , (9) is satisfied given (11) and (12), and (16) is satisfied given  
                      z        maxF     =       (         L   y     2     +     y   0     +     y   c       )     ⁢   tan   ⁢       L   z       2   ⁢     (       y   0     +     y   c       )         ⁢       &lt;&lt;     l             ⁢   z         /   2         ,   R           (     55   ⁢   a     )             
 
 and 
 
 {tilde over (L)}   z ≦π( y   0   +y   c )  (55b) 
 
 The field coordinate {tilde over (z)} (54) then satisfies (28). 
 
 4.2 Improved Fields 
 
      The ideal LC z  field 
 
 B   id ( {right arrow over (r)}, t )= G   z   {tilde over (z)}   id ( z; {tilde over (L)}   z )  (56) 
 
 where the field coordinate  
                   z   ~     id     ⁡     (     z   ;       L   ~     z       )       =     {                 L   ~     z     /   2     ,           z   &gt;         L   ~     z     /   2                 z   ,           z   ∈     [         -       L   ~     z       /   2     ,         L   ~     z     /   2       ]                     -       L   ~     z       /   2     ,           z   &lt;       -       L   ~     z       /   2                       (   57   )             
 
      For a field of view with {tilde over (z)} id ∈[−{tilde over (L)} z /2{tilde over (L)} z /2], the field B id  has the values κ=κ D =1 and a uniform resolution Δz(θ{tilde over (z)}/θz) −1  (36) along z for z∈[−{tilde over (L)} z /2, {tilde over (L)} z /2].  
      Curent density profiles J(z) that produce fields B better approximating the ideal fields B id  (56) over D than (17) can be calculated. The field B(x 0 , y 0 , z) is given by (7), where g(x 0 , y 0 , z) is (8) for the first embodiment and (22) for the second embodiment. The integral (7) can be approximated by a sum:  
               B   i     ≈     δ   ⁢           ⁢   z   ⁢       ∑     j   =     -   n       n     ⁢           ⁢       g     i   -   j       ⁢     J   j                   (   58   )             
 
 where 
 
 B   j   =B ( x   0   , y   0   , jδz )  (59a) 
 
 J   j   =J ( jδz )  (59b) 
 
 g   j   =g ( x   0   , y   0   , jδz )  (59c) 
 
 and J(z) either vanishes or is neglected for |z|≧nδz 
 
      Defining the matrix 
 
 G   ij   =δzg   1−j   (60) 
 
 and treating B j  and J j  as column vectors, the approximation (58) becomes the matrix equation  
               B   i     ≈       ∑     j   =     -   n       n     ⁢           ⁢       G   ij     ⁢     J   j                 (   61   )             
 
      For the first embodiment  
               G   ij     ⁢     =     (   8   )       ⁢             μ   0     ⁢   Y   ⁢           ⁢   δ   ⁢           ⁢   z       4   ⁢   π       ⁢       ∫       -     l   x       /   2         l   x     /   2       ⁢           ⁢       ⅆ     x   ′       ⁢     1       [       X   2     +     Y   2     +         (     δ   ⁢           ⁢   z     )     2     ⁢       (     i   -   j     )     2         ]       3   /   2               +           μ   0     ⁢   R   ⁢           ⁢   δ   ⁢           ⁢   z       4   ⁢   π       ⁢       ∫     I   φ               ⁢           ⁢       ⅆ     φ   ′       ⁢       R   -       r   0     ⁢   cos   ⁢           ⁢   Φ           [       R   2     +     r   0   2     -     2   ⁢     Rr   0     ⁢   cos   ⁢           ⁢   Φ     +         (     δ   ⁢           ⁢   z     )     2     ⁢       (     i   -   j     )     2         ]       3   /   2                         (   62   )             
 
 where  
             X   =       x   0     =     x   ′               (     63   ⁢   a     )               Y   =       y   0     +     y   c               (     63   ⁢   b     )               Φ   =       φ   0     -     φ   ′               (     63   ⁢   c     )                 r   0     =         x   0   2     +     y   0   2                 (     63   ⁢   d     )                 φ   0     =         cos     -   1       ⁢       x   0       r   0         =       sin     -   1       ⁢       y   0       r   0                   (     63   ⁢   e     )             
 
 Both terms are decreasing functions of i−j . In addition, the first term is positive for y 0 ≦−y c  and the second term is positive for r 0 ≦R , using  
             0   &lt;     R   -     r   0       ≤     R   -       r   0     ⁢           ⁢   cos   ⁢           ⁢     (     φ   -     φ   ′       )                 (     64   ⁢   a     )             and                           0   &lt;       (     R   -     r   0       )     2       =       R   2     +     r   0   2     -     2   ⁢   R   ⁢           ⁢     r   0                 (     64   ⁢   b     )                       ⁢     ≤       R   2     +     r   0   2     -     2   ⁢   R   ⁢           ⁢     r   0     ⁢           ⁢   cos   ⁢           ⁢     (     φ   -     φ   ′       )                                 
 
 Therefore, G ij  is a positive, decreasing function of i−j for y 0 ≦−y c  and r 0 ≦R , and the determinant  
               det   ⁢           ⁢   G     =       ∑     σ   ∈     S       2   ⁢   n     +   1                   ⁢           ⁢       (     sgn   ⁢             ⁢             ⁢   σ     )     ⁢     G       -   n     ,     σ     -   n           ⁢   …   ⁢           ⁢     G     n   ,     σ   n                     (   65   )             
 
 can be written as an alternating sum of decreasing terms. Consequently, det G≠0 and the inverse matrix (G −1 ) ji  exists. 
 
      For the second embodiment  
               G   ij     ⁢     ≈     (   23   )       ⁢           μ   0     ⁢   Y   ⁢           ⁢   δ   ⁢           ⁢   z       2   ⁢     π   ⁡     [       Y   2     +         (     δ   ⁢           ⁢   z     )     2     ⁢       (     i   -   j     )     2         ]           -           μ   0     ⁢     R   2     ⁢   δ   ⁢           ⁢   z       4   ⁢       π   ⁡     [       R   2     +         (     δ   ⁢           ⁢   z     )     2     ⁢       (     i   -   j     )     2         ]         3   /   2           ⁢           [           ⁢         (     1   -     2   ⁢     ɛ   φ         )     ⁢   π     +       (         3   ⁢     R   2           R   2     +         (     δ   ⁢           ⁢   z     )     2     ⁢       (     i   -   j     )     2           -   1     )     ⁢         l   x     ⁢     r   0     ⁢     y   0         R   3           ]               (   66   )             
 
 under conditions (9) on (x, y)=(x 0 , y 0 ) and using (63). Defining  
             W   =         (     1   -     2   ⁢           ⁢     ɛ   φ         )     ⁢   π     +         2   ⁢     l   x     ⁢     r   0         R   3       ⁢   min   ⁢     {       y   0     ,   0     }                 (   67   )             
 
 G ij  is a positive, decreasing function of i−j for 
 
0≦ Y≦≦R   (68) 
 
 and  
               n   ⁢           ⁢   δ   ⁢           ⁢   z     &lt;         RW   yo     2               (   69   )             
 
 and the determinant det G (65) can be written as an alternating sum of decreasing terms. Consequently, det G≠0 and the inverse matrix (G −1 ) ji  exists. 
 
      Using the inverse matrix (G −1 ) ji , the equation  
               J   j     ≈       ∑     i   =     -   n       n     ⁢           ⁢         (     G     -   1       )     ji     ⁢     B   i                 (   70   )             
 
 can be used to find J j  required to produce specified field values B i  . 
 
      Let D be such that 
 
 D ∈{( x, y, z ): | z|≦nδz}   (71) 
 
 and let {tilde over (b)}(z) be a smooth function better approximating the ideal field B id  over D∩λthan (17), where λis the line 
 
λ={( x, y, z ):  x=x   0   , y=y   0 }  (72) 
 
 With 
 
 B   i   ={tilde over (b)} ( iδz )  (73) 
 
 values of {tilde over (b)}(z) J j  can be calculated from (70) and a current density profile J(z) constructed by connect-the-dots. The smooth field B(x, y, z) (7) produced by J(z) better approximates B id  over D than (17). 
 
 4.3 Adjustable Field of View 
 
      The first and second embodiments can be designed with several fields of view 
 
η 1   {tilde over (L)}   z ≦η 2   {tilde over (L)}   z ≦ . . . ≦η N   {tilde over (L)}   z   (74) 
 
 in mind. Let fields G z {tilde over (z)} a , a=1, . . . , N , approximate ideal fields (57) over D : 
 
 {tilde over (z)}   a   |D≈{tilde over (z)}   id ( z; η   a   {tilde over (L)}   z )| D   (75) 
 
 The notation D means restricted to D . Define fields B a  by 
 
 B   1   =G   z   {tilde over (z)}   a=1   (76a) and, for a=2, . . . , N , 
 
 B   a   G   z ({tilde over (z)} a   {tilde over (z)}   a−1 )  (76b) 
 
 Current density profiles J a (z) producing fields approximating B a  can be found by the method of Sec. (4.2). With components a=1, . . . , N carrying J a (z) , the first k≦N components generate a field 
 
 B   (k)   =B   1   + . . . +B   k   =G   z   {tilde over (z)}   k   (77a) 
 
 with 
 
 B   (k)   |D≈{tilde over (z)}   id ( z; η   k   {tilde over (L)}   z )| D   (77b) 
 
 5 Advantage of LC Coil for MRI 
 
      The gradient of an LC coil with smaller fields outside the field of view can be switched more rapidly without violating a bound on the field rate of change. Consequently, larger regions of k -space can be covered within a given time.  
      It is to be understood that while the invention has been described in conjunction with the detailed description thereof, the foregoing description is intended to illustrate and not limit the scope of the invention, which is defined the scope of the appended claims. Other aspects, advantages, and modifications are within the scope of the following claims.