Magnetic Resonance Imaging with Inhomogeneous Background Fields

Haewon Nam

In order to generate an MR signal, the imaging object is first put in a relatively strong background field, ${\mathbf B}({\mathbf x})$. Then a radiofrequency (RF) pulse is used to excite precession in some nuclei within the object. Nuclei spin at the rate $\gamma \vert{\mathbf B}\vert$, where $\gamma = 4.26e7$ Hz/Tesla is the gyromagnetic ratio. The RF pulse is designed to have a particular bandwidth, so only nucleii that spin with rates within that bandwidth are excited by the RF. Traditionally/ideally, the background field is homogeneous and a linear gradient field is applied so that the RF excites spins only within a flat ``slice" of the object. In reality, however, we apply an inhomogeneous background field with a nonzero gradient of its own. By applying an RF pulse we can excite spins in an ``orange peel" along surfaces of equal field strength.

The nucleii at the bottom of an orange peel are exposed to the same field strength, $\vert{\mathbf B}_{o} - \Delta{\mathbf B}\vert$, and the spins at the top of the peel are exposed to a different field strength, $\vert{\mathbf B}_{o} + \Delta{\mathbf B}\vert$.

Therefore, the nucleii at the bottom of the peel spin with frequency

$\omega = \gamma \vert{\mathbf B}_{o} + \Delta{\mathbf B}\vert$, whereas the nucleii at the top spin with frequency, $\omega = \gamma \vert{\mathbf B}_{o} - \Delta{\mathbf B}\vert$. In order to detect a signal, the top and bottom

nucleii must not dephase too much, otherwise they will cancel each other out. For now, we estimate

that for reasonable image quality, spins may dephase by no more than $\pi/4$ during each data

acquisition.

The question of interest to us is (vaguely stated):

``How curved can an imaging surface be and still yield a decent signal?"

In this thesis, we study the effects of field inhomogeneity on several types of curvature of isochromat and also prove that inhomogeneous fields achieve min field strength in the interior of the imaging region.