Читать книгу Essentials of MRI Safety - Donald W. McRobbie - Страница 65

An object’s shape adds a further layer of complexity with great significance for magnetic forces and torques, and hence for MRI safety. Figure 2.13 shows how magnetization of the object in an external B₀ generates “virtual poles”, resulting in a “de‐magnetization field” which reduces the apparent value of susceptibility χ_app:

(2.8)

Figure 2.13 Demagnetization field within objects magnetized by an external magnetic field: the flatter object (left) creates more ‘virtual poles’ and develops a greater degree of demagnetization.

where d_i is the demagnetizing factor. For simple shapes (ellipsoids of rotation, cuboids and cylinders) there are three demagnetizing factors: d₁ along the principal axis, d₂ and d₃ along the minor axes. Representative values are shown in Table A1.1 in Appendix 1. Figure 2.14 shows the values of 1/d₁ for the axial and 1/d₂ (=1/d₃) for the radial axes of cylindrical objects. It is the demagnetizing factor that determines the level of magnetization of a ferromagnetic object, rather than the value of χ. For an unsaturated sphere the internal B field is three times the external B₀. For elongated objects the internal B can be significantly greater. As a simple rule of thumb, for a cylinder whose length is aligned with B₀

(2.9a)

Figure 2.14 Reciprocal demagnetization factors for a cylinder: 1/d₁ (axial, red line), 1/d₂ (radial, blue line). For a cylinder d₃ = d₂.

If the object is rotated 90° to B₀

(2.9b)

Figure 2.15 shows the theoretical internal B for ferromagnetic cylindrical objects of various length‐diameter (l/d) ratios and a sphere as they approach 1.5 and 3 T shielded magnets. In each case the saturation value B_sat (=1.6 T) is reached at greater distances from the magnet for the more elongated objects.

Figure 2.15 Predicted internal B for a ferromagnetic sphere and cylinders of differing length/diameter ratios in the approach to 1.5 and 3 T MRI magnets. The material saturates at 1.6 T. The bore entrance is at 0.8 m. The dotted gray line indicates the B₀ field strength.

Demagnetizing factors and B_sat are crucial for determining the force and torque on different shaped objects within the scanner’s fringe field.