IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 4, JULY 1998 2177 Magnetic Field of Current Monopoles in Prolate and Oblate Spheroid Volume Conductors Jian-Cheng Lin and Dominique M. Durand Abstract— We derive an equation for the magnetic ﬁeld contributed from the volume current generated by a current monopole in two special ﬁnite volume conductor shapes, i.e., prolate spheroid and oblate spheroid. In deriving the equation, ﬁxed referential current monopoles are added to the ﬁnite system in order to balance the volume current generated by the current monopole under study. The location of these monopoles is chosen such that they produce an equipotential surface which coincides with the volume conductor and therefore do not contribute to the magnetic ﬁeld. The magnetic ﬁeld generated by arbitrarily distributed, but balanced, current monopoles within a prolate or oblate spheroid volume conductor can then be derived. The magnetic ﬁeld generated by a current dipole in these volume conductors can then be obtained from the monopole solution and is identitical to the previously derived dipole solution [1]. Index Terms—Current monopoles, magnetic ﬁeld, oblate, pro- late, spheroid. I. INTRODUCTION E LECTRICAL activity produced by the excitable heart tissue and the brain can generate a magnetic ﬁeld [2]–[6]. Information about sources generating this electrical activity can be obtained by measuring these magnetic ﬁelds. This is a typical inverse problem of identifying the electrical source from the measurable magnetic ﬁeld generated by the unknown electrical source. To solve this inverse problem, one must ﬁrst solve the forward problem, which is the calculation of the magnetic ﬁeld generated by known sources. Theoretical analytical studies have modeled the volume conductors in which these sources are located into simpliﬁed volumes, such as semi-inﬁnite volume, sphere, prolate spheroid, and oblate spheroid, and source currents as current dipole [1], [7]–[10]. Magnetic ﬁelds generated inside these models have been calcu- lated with dipole current sources using a quasi-static approach [1], [11]–[12]. Although the current dipole as a current source unit has its advantage in the study of the magnetic ﬁeld, for example the Dirac delta function expression for the current, current monopoles present several additional advantages. First, a current dipole can easily break down symmetries of a volume conductor due to its dipole orientation, making theoretical calculations of the magnetic ﬁeld much more difﬁcult [1], [7]. This is particularly true in the case of semi-inﬁnite volume and sphere volume conductors where symmetry can be utilized Manuscript received February 17, 1996; revised January 17, 1998. This work was supported by NIH Grant NS 32575. The authors are with the Department of Biomedical Engineering, Applied Neural Control Laboratory, Case Western Reserve University, Cleveland, OH 44106 USA (e-mail: dxd6@po.cwru.edu). Publisher Item Identiﬁer S 0018-9464(98)03377-9. by introducing the concept of current monopole to obtain simple results without complicated analytic calculations [13]. Second, a current dipole is, by deﬁnition, the contribution of two monopoles with equal amplitude and opposite polarity located at an inﬁnitesimal distance. A dipole can be a good approximation when the observation point is far from the source. Monopoles, however, can be located anywhere and can be combined to represent more accurately the source. Third, models with current dipoles as a current source cannot be used to study systems under external electrical stimulation, while those with current monopoles can. When there is external electrical stimulation, the currents through the anode and cathode leads of the stimulator contribute to the external magnetic ﬁeld. Those contributions must be substracted from the total magnetic ﬁeld in order to obtain the magnetic ﬁeld generated by the internal current source. Both currents passing through an anode and a cathode can be well approximated by current monopoles, but not a current dipole. There have been several previous attempts on ﬁnding mag- netic ﬁeld solutions for monopoles within semi-inﬁnite and sphere volume conductor, but those attempts were mainly based on the existence of a solution for a dipole current [9]–[10]. In a previous paper [13], symmetry analysis has been employed to solve the same problem with signiﬁcant simplicity for several volume conductors. However, this tech- nique cannot be extended to the conductors with prolate and oblate spheroid shapes since they do not share the same degree of symmetry as a semi-inﬁnite and sphere volume conductor (i.e., the semi-inﬁnite and sphere volume conductors always have axial symmetry regardless of the location of the current monopole within the conductors). Moreover, in the literature, there is no solution for the magnetic ﬁeld generated by current monopoles within prolate and oblate spheroid volume conductors. In this paper, we provide an expression for the magnetic ﬁeld generated by the volume current of a current monopole anywhere within the prolate and oblate spherical isotropic volume conductor. II. CALCULATION OF THE MAGNETIC FIELD The magnetic ﬁeld outside a volume conductor can be calculated using quasi-static approximation. Under a quasi- static condition [11], [14], the magnetic ﬁeld satisﬁes the Maxwell equation [11], [15] (1) where is the current density. In a biological system, the current density is expressed as the sum of the source current 0018–9464/98$10.00  1998 IEEE