IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 44, NO. 2, FEBRUARY 1997 177 Theoretical Study of Magnetic Field of Current Monopoles in Special Volume Conductor Using Symmetry Analysis Dominique M. Durand,* Member, IEEE, and Jian-Cheng Lin Abstract—We derive a formula for the magnetic field outside volume conductors having axial symmetry with radial and axial symmetrically distributed source currents. The magnetic field is shown to have components only along the cylindrical polar angle direction and its magnitude to depend only on the topological structure of the volume conductor and the location of the source current. With this formula, the magnetic field generated by the volume current of a current monopole within and on the symmetrical axis of several volume conductors (such as semi- infinite volume, infinite slab, sphere, infinite cylinder, semi-infinite cylinder, finite cylinder, prolate spheroid, and oblate spheroid) is shown to be equivalent to the magnetic field generated by a line current calculated using the Biot–Savart’s law. In the first three volume conductors, the monopole solution of the magnetic field allows the calculation of magnetic fields generated by arbitrarily distributed (and balanced for finite volume conductors) current monopoles. Index Terms— Biomagnetics, biomedical computing, current monopoles, magnetic fields, volume conductors. I. INTRODUCTION M EASUREMENTS of the magnetic fields produced by the human heart or brain can provide information about what electrical activities are taking place within these organs [2]–[6]. Calculations of the magnetic field outside these organs are necessary to provide information about the electrical sources located inside. Modeling analytical studies have simplified these human organs as some specific vol- ume conductors, such as semi-infinite volume, sphere, prolate spheroid, and oblate spheroid. Moreover, the source currents have been limited to current dipoles [1], [7]–[10]. 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, there are several disadvantages. First, a current dipole can easily breakdown symmetries of a volume conductor due to its orientation. Therefore, theoretical calculation of the magnetic field be- comes tedious and complicated and sometimes unnecessary without a symmetry consideration [1], [7]. This is particularly Manuscript received February 23, 1996; revised August 19, 1996. This work was supported by the National Institutes of Health under Grant NS 32572. Asterisk indicates corresponding author. *D. M. Durand is with the Department of Biomedical Engineering, R. 3510 CB Bolton Bldg., Case Western Reserve University, Cleveland, OH 44106 USA (e-mail: dxd6@po.cwru.edu). J.-C. Lin is with the Department of Biomedical Engineering, Case Western Reserve University, Cleveland, OH 44106 USA. Publisher Item Identifier S 0018-9294(97)00285-1. true in the case of semi-infinite volume and sphere volume conductor where simple results were reached through compli- cated analytic calculations [1], [7]. Second, a current dipole can not be a good approximation when the distance between the two ends of a line current source is not negligible. Third, models with current dipoles as current sources can not be used to study systems under external electrical stimulation. Since the currents through the anode and cathode of a stimulator contribute to the external magnetic field, these contributions should be subtracted from the total magnetic field in order to obtain the magnetic field generated by the internal current sources. Obviously, the volume currents passing through the anode and cathode cannot be modeled by a current dipole when the observation point is far away from the stimulation site. An alternate current source is the current monopole. Current monopoles offer several advantages. 1) A current monopole located anywhere within semi-infinite, infinite slab, and sphere volume conductors does not affect the axial symmetry of the volume conductors; 2) using superposition, one can use the magnetic field solution for monopoles to calculate the magnetic field solutions of source current with other type of configurations; 3) current passing through either anodic or cathodic electrodes can be well approximated by monopoles and their superposition. Previous attempts to find magnetic field solutions for monopoles within semi-infinite and sphere volume conductors have been published, but these derivations were mainly based on the existence of a solution for a dipole current [9], [10]. In this paper we apply rigorous symmetry analysis to find magnetic field solutions outside several types of volume conductors containing monopole current sources. II. THEORY AND SYMMETRY ANALYSIS In this section, we show that when a volume conductor has axial symmetry and source currents are radially distributed with axial symmetry, then the magnetic field outside the volume conductor has components only along the polar angle direction and its magnitude depends only on the topological structure of the volume conductor and the source current. We also clarify the concept of the current monopole and show how it should be applied to calculate the magnetic field of a real system. We refer to any biological or physical volume conductor system which contains current sources satisfying quasistatic condition as a real system. 0018–9294/97$10.00  1997 IEEE