BEE zyxwvutsrqpon TRANSACTIONSON MAGNEITCS, VOL. 28. N0.2, MARCH 1992 1517 zyx Application of Eddy Current Formulations to Magnetic Resonance Imaging W. Renhart, C.A. Magele, K.R. Richter Graz University of Technology Kopernikusgasse 24, A-8010 Graz, Austria P. Wach zyxwvutsrqp Abstruct - zyxwv In thls paper the calculatlon of 3D- Radlo Frequency ( RF 1 flelds wlth the ald of an eddy current fwmulatlon based zyxwvuts on flnlte elements zyxwvut Is de- mcrlbed. It Is shown that the dlsplacement current denslty In the flrst Maxwell equatlon can be taken Into account by making the conductlvlty e a complex qumtlty. The valldlty of thls fwmulatlon has been v~lfled by calculatlng a conductlng sphere Influ- zyxwvut meed by a homogenuous tlmeharmonlc RF-fleld. The numerlcal results are compared with an analytlcal molutlon. Flnally, the formulation descrlbed has been applied to calculate a 3D phantom wlth reglons of dlfferent conductlng materlals. For thls phantom measured results obtalned by MR-lmaglng exlst and are compared, too. I. INTRODUCTION MR-Imaging interpretation becomes more and more important in medical diagnostics. It depends on various parameters like RF-field intensity and pulse width [l]. Due to the conductivity of the human body, the RF-field induces eddy currents and leads to power absorptions. To avoid thermal destruction of biological material it must be ensured that the power absorbed remains limited. In general, the pulse width and the RF-field intensity control this effect. To opti- mize the parameters mentioned, the specific absorp- tion rate ( SAR and the total power absorbed should be known. In the past, several investigations con- cerning the RF-power deposition in homogeneous objects and circular layered cylinders were per- formed [2 -61. Such approximations were combined with the assumption of homogenous RF-fields. So the results obtained could not correspond to real MR-imaging conditions. They were of principle inter- est only. Especially, it was not possible to investi- gate 'hot spots' due to material inhomogenities. The finite element method formulated for the quasi- stationary case represents a very convenient possi- bility to attain this goal. As shown below, only a few modifications in existing eddy current formulations have to be carried out. Manuscript received July, 7. 1991. R. Stollberger University of Graz Auenbruggerplatz 9, A-8036 Graz, Austria I I. FORMULATION As a result of the presence of conducting mate- rial, eddy currents will be induced. Taking the .dis- placement current density into account, Ampere's Law becomes : 3D curl H = J + - 3t Substituting the constitutive relationships (1) J-aE (2) D=EE (3) wherein the conductivity a and the permittivity E are assumed to be scalar constants, eq. (1) becomes for the timeharmonic case : (4) curl H = CJ E +jus E. The right hand side can be rewritten to curl H = (6+JW& 1 E. zyxw (5 1 Now, the expression in brackets can be interpreted as a complex conductivity bc also depending on the frequency o : ac = a+joa. (6) So we end up with equation which are formally the same as in the eddy current case. Ill. FINITE ELEMENT FORMULATION The finite element formulation, modified by the complex conductivity ac is based on the well known A, V- CD formulation, presented in [7 1. In the con- ducting region the field is described by B = curl A (8) E = -jwA - gradV. (9) With the magnetic vector potential A and the electric scalar potential V, the following differential equations must be solved : : 0018-9464/92$03.00 Q 1992 IEEE