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Analytical Sphere

The analytical magnetic field (including Lorentz Correction) produced by a spherical object of radius $ R_0$ and susceptibility $ \chi_{i}$ inside a medium of susceptibility $ \chi_{e}$ is given by

$\displaystyle B_z = \left\{ \begin{array}{lccl}
B_0 + \frac{\chi_{i} - \chi_{e}...
...uad & \mathrm{for} & r>R_0 \\
B_0 & & \mathrm{for} & r<R_0
\end{array} \right.$     (28)

where cylindrical coordinates are used with $ \theta$ being the angle to the positive $ z$ axis.

Figure 1: Sphere object model (left), analytical field (middle) and perturbation field calculation (right) for two sizes of sphere: $ R_0=64$ (top row) and $ R_0=8$ (bottom row).
\begin{figure}\begin{center}
\begin{tabular}{ccccc}
\psfig{figure=sph64.ps, widt...
...sfig{figure=sph8b0.ps, width=\figwidth}\\
\end{tabular}\end{center}\end{figure}

Figure 2: Plots of the $ B_z$ distribution along the z axis for spherical objects of radius $ R_0 = 2, 4, 8, 16, 32, 64$ voxels. Results are shown for the analytical calculation (theory), the perturbation calculation and the difference between the two. Note that the values are only calculated at the centres of the voxels -- hence the noticeably non-vertical sides in the $ R_0=2$ case.
\begin{figure}\begin{center}
\begin{tabular}{ccc}
\psfig{figure=sphere2zplot.ps,...
...g{figure=sphere64zplot.ps, width=\figwidth}\end{tabular}\end{center}\end{figure}

Figure 1 show the field distribution (less $ \mathrm{B}_0$) along the $ z$ axis of a sphere, comparing the analytical solution given in equation 28 with the solution calculated with the perturbation method. Figure 2 shows plots through the centres of these spheres for a range of different radii, $ R_0 = 2, 4, 8, 16, 32, 64$ mm, where in each case the voxel size is $ 1 \times 1 \times 1$ mm. This demonstrates the relationship between spatial extent and size of the error, where it can be seen that the maximum error is approximately equal to 1 ppm in all cases, with the spatial extent of significant errors covering 5 voxels or so. These results are very similar to those shown in [4,1,2,5] despite the range of different object models (e.g. boundary element methods vs voxels) and approximations used.


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Next: In-vivo Human Head Up: Validation and Results Previous: Validation and Results