MRI 1.5 Tesla

Superconducting Magnets

©2004 Magneticasoft

Magnetic resonance imaging (MRI) superconducting magnets feature many specific and demanding characteristics: high magnetic field, very high homogeneity in the central region, large dimensions, high magnetic energy, and large stray field. Using niobium-titanium superconducting material at cryogenic temperatures (liquid helium), the electrical power needed to drive the magnet is zero once the magnetic field has been set.

This demonstration shows two types of magnets generating 1.5 teslas in a spherical volume of approximately 0.5 meters (20 inches) in diameter (the region of interest where the body image can be produced).

The first magnet is made of four pairs of coils all acting together and the second magnet uses five pairs of coils of which two pairs are set to reduce the stray field (a so-called magnet with active shielding).

All the coils are solenoid with specific dimensions and numbers of turns. They are assumed to be series connected; the current intensity is therefore the same in all the coils.

The proposed symbolic names of these magnetic systems are mri1 and mri2.

Magnetic System Definition

mri1 = 
        AXSMagneticSystem[
{AXSSolenoid[SuperNbTi8,  
              1240,  1280,  120,  -139.97284827108058,  486,  
              734.9996671332043], 
 
            AXSSolenoid[SuperNbTi8,  1240,  1280,  120,  
              139.97284827108058,  486,  
              734.9996671332043], 
 
            AXSSolenoid[SuperNbTi8,  1240,  1280,  
              200,  -330.13406451703946,  153,  
              734.9996671332043], 
 
            AXSSolenoid[SuperNbTi8,  1240,  1280,  200,  
              330.13406451703946,  153,  
              734.9996671332043], 
 
            AXSSolenoid[SuperNbTi8,  1240,  1280,  
              220,  -580.1892588159749,  847,  
              734.9996671332043], 
 
            AXSSolenoid[SuperNbTi8,  1240,  1280,  220,  
              580.1892588159749,  847,  
              734.9996671332043], 
 
            AXSSolenoid[SuperNbTi8,  1240,  1320,  
              250,  -875.5273994563552,  611,  
              734.9996671332043], 
 
            AXSSolenoid[SuperNbTi8,  1240,  1320,  250,  
              875.5273994563552,  611,  
              734.9996671332043]}, 
 
          SystemLabel  ->  "MRI 1.5 T superconducting magnet, 6th order \
homogeneity"];

mri2 = 
        AXSMagneticSystem[
{AXSSolenoid[SuperNbTi4,  
              1120,  1280,  60,  -108.29012662326866,  679. ,  
              766.7908523598545], 
 
            AXSSolenoid[SuperNbTi4,  1120,  1280,  60,  
              108.29012662326866,  679. ,  
              766.7908523598545], 
 
            AXSSolenoid[SuperNbTi4,  1120,  1280,  
              80,  -335.1989704952877,  659. ,  
              766.7908523598545], 
 
            AXSSolenoid[SuperNbTi4,  1120,  1280,  80,  
              335.1989704952877,  659. ,  
              766.7908523598545], 
 
            AXSSolenoid[SuperNbTi4,  1120,  1240,  
              180,  -697.642715356854,  1545,  
              766.7908523598545], 
 
            AXSSolenoid[SuperNbTi4,  1120,  1240,  180,  
              697.642715356854,  1545,  
              766.7908523598545], 
 
            AXSSolenoid[SuperNbTi4,  1700,  1860,  
              80,  -134.41807753795598,  -655,  
              766.7908523598545], 
 
            AXSSolenoid[SuperNbTi4,  1700,  1860,  80,  
              134.41807753795598,  -655,  
              766.7908523598545], 
 
            AXSSolenoid[SuperNbTi4,  1700,  1860,  
              80,  -887.6632666898012,  -633,  
              766.7908523598545], 
 
            AXSSolenoid[SuperNbTi4,  1700,  1860,  80,  
              887.6632666898012,  -633,  
              766.7908523598545]}, 
 
          SystemLabel  ->  "MRI 1.5 T superconducting magnet with active \
shield"];

The Superconducting Material

The superconducting materials have a zero resistivity when cooled in liquid helium (4.2°kelvin). They present a strong diamagnetic behavior, but this effect has a negligible influence on the production of the magnetic field.

Properties[SuperNbTi4]

Drawing the Magnetic Systems

DrawSystem[mri1];

DrawSystem[mri2];

DrawSystem2D[mri2];

Spherical Harmonics Field Calculation

Calculating the magnetic field using spherical harmonics is an important feature of Magnetica. Although the spherical harmonics method is an efficient way to calculate the magnetic field, it is restricted to space without a magnetic source. Consequently, three regions can be distinguished:

1. a central spherical region

2. a forbidden region (due to a fundamental cause: no mathematical convergence)

3. a remote region extending from a limiting outer sphere to infinity

The next graph shows the three regions. The forbidden region (yellow) is bordered by a remote region (brown) where the precision of calculations becomes insufficient.

DrawSHLimits[mri2];

The spherical harmonics method is based on a set of coefficients (the Legendre source coefficients) for Legendre polynomial development. The Legendre source coefficients measure both the intensity of the field, indicated by the zero-order coefficient, and the homogeneity of the field, shown by the coefficients of increasing order. The coefficients of the central and remote regions are two different sets of numbers (expressed by default in the current magnetic field unit).

VerboseLegendreCoefficients[mri1, 
      LegendreRadius -> {200, 2000}]

The high homogeneity characteristic of MRI magnets is evidenced by the small values of the low-order coefficients in the central region.

VerboseLegendreCoefficients[mri2, 
      LegendreRadius -> {200, 2000}]

Due to its active shield, this magnet not only shows quasi null coefficients in the central region (up to the 8th order) but also has very small values for the remote region coefficients (up to the 4th order).

Punctual Magnetic Field Calculation

FieldPoint[mri1, {0, 0}, FieldUnit -> Tesla]

This is the field intensity at the isocenter of the magnet (which is also the origin of the coordinates). This particular target point can be obtained by the function FieldPointOrigin.

FieldPointOrigin[mri2, ComponentsList -> All]

The option ComponentsList->All displays the contribution of each coil.

Field Map Graphics

2D Graphics: Field Functions along a Line

FieldMap[mri1, {Bz}, Line[{0, -500}, {0, 500}]];

FieldMap[mri2, {Bz}, Line[{0, -500}, {0, 500}]];

3D Graphics: Field Map, Contour Field Map, and Stray Field on a Plane

FieldMap[mri1, {Bz}, 
        Plane[{-200, -200}, {200, 200}]];

ContourFieldMap[mri2, {Bz}, 
        Plane[{-200, -200}, {200, 200}]];

The function DrawIsoFieldLines displays the isofield (magnitude) around the magnet for a set of magnetic field intensities.

DrawIsoFieldLines[mri1, {10, 5, 1}, AngularRange -> {0, 360}, 
        NumberOfPoints -> 45];

DrawIsoFieldLines[mri2, {10, 5, 1}, AngularRange -> {0, 360}, 
        NumberOfPoints -> 45];

This graph, compared to the previous one, shows a drastic reduction of the distance of the isofield line. (Safety regulations require a limit of 3 or 5 gauss outside the MRI medical examination room.)

Inductance and Magnetic Energy Calculation

The intensity of a magnetic field as well as the size of an MRI magnet make this device a large well of stored magnetic energy. In case of a technical mishap, this may become a problem. It is, therefore, important to calculate thoroughly the magnetic energy and the corresponding inductance.

The verbose function is used to obtain a textual, explanatory, and formatted output.

VerboseSystemInductance[mri1, InductanceUnit -> Milli Henry, 
      FluxUnit -> Weber, 
      MagneticEnergyUnit -> Kilo Joule]

For the mri1 magnet, the stored magnetic energy is 3.0 megajoules.

VerboseSystemInductance[mri2, InductanceUnit -> Milli Henry, 
      FluxUnit -> Weber, 
      MagneticEnergyUnit -> Kilo Joule]

For the mri2 magnet, the stored magnetic energy is 4.8 megajoules.

The 6×6 matrix inductance of mri1 follows.

VerboseInductanceMatrix[mri1, 
      InductanceUnit -> Milli Henry]