D. Purgill, GE Medical Systems, 3200 N Grandview Blvd, Waukesha, WI 53 188. Magnetic field ... magnetic field at point 7 and the center of the imaging volume.
VRMS Homogeneity Definition: A Proposal T.J. Havens, T. Duby, J. Huang, GE Medical Systems, 3001 W Radio Dr, Florence, SC 29501 D. Purgill, GE Medical Systems, 3200 N Grandview Blvd, Waukesha, WI 53 188 Magnetic field homogeneity is a critical parameter in MRI system performance. The industry standard for reporting homogeneity has been shifting from listing peak to peak homogeneity on the surface of the imaging volume to specifying the volume root mean square homogeneity (henceforth VRMS) over several volumes. Unfortunately, no definition or standard for VRMS has been proposed, making it difficult to compare magnet homogeneities. This paper derives an equation for easily computing VRMS homogeneity values over spherical volumes of arbitrary radius inside the imaging volume in terms of the spherical harmonic expansion coefficients of the magnetic field. Quoting VRMS andor surface root mean squared (SRMS) over different volumes within the magnet allows the MRI system purchaser to estimate the volumes over which he or she may perform various scan sequences. Thus, the industry has been including homogeneity performance on more volumes within the imaging volume in product brochures. The ability to compute RMS and VRMS values over spherical volumes in a well-defmed and standardized manner will be of considerable benefit. Mathematically rigorous derivations of equations for computing these quantities follow below.
on which the field is measured, typically 45 cm DSV or larger on cylindrical magnets. Inserting equation 2 into equation 1 and taking the a results in equation 3 below:
A similar equation holds for SRMS. Note that the summation in equation 3 starts with n = 1, rather than n = 0. For spherical
volumes the above integrals can be evaluated analytically taking advantage of the orthogonality of spherical harmonic functions, producing equation 4 below:
The VRMS and (for completeness) SRMS homogeneities are defined by where the Neuman factor is defined by em =
where B , (7) and B , (0) are the z component of the total magnetic field at point 7 and the center of the imaging volume V, respectively. The integration is over the imaging volume V, or surface S enclosing V, respectively. These definitions conform to the standard engineering definition of root mean square. Inside the imaging volume the following spherical harmonics expansion holds:
where P,,, are the associated Legendre polynomials and R is the reference radius of the volume on which the expansion coefficients Cn,,, and Sn,m are computed. The best accuracy is achieved by making the reference radius, R, the largest volume
© Proc. Intl. Soc. Mag. Reson. Med. 10 (2002)
2 form=O
1 otherwise The dominant term in the spherical harmonics expansion for typical cylindrical magnets is (n, m) = (12, 0) which becomes important on larger homogeneity volumes, thus the M = 0 terms should at least include n = 12 if magnet homogeneity is to be quoted on larger volumes. In practice, due to magnetic field mapping on a finite number of points, expansion into spherical harmonics is truncated at certain Nand Mneglecting all higher order terms. Note also, that an n-th order term contributes more to SRMS than to VRMS: this is due to different weight of the (I / R)” dependence in both expressions. Finally, one may choose other normalization schemes for the expansion, altering the formula slightly, but producing the same numerical results.
Thus, equations 4 may be used to compute the VRMS and SRMS over any spherical volume of radius Y I R from the coefficients in the spherical harmonic expansion of the magnet homogeneity,,and summing the terms at the appropriate radius. These expansion coefficients may be computed from field maps acquired by mechanical mapping fixtures or by imaging system based means, allowing VRMS and SRMS values to be computed uniformly independent of the measurement system. It should be noted that equations (4) should be used for I > R only with extreme caution, as high accuracy is required in the field measurement if one is to extract coefficients for the required number of harmonics.
