On the magnetic field generated outside an inhomogeneous volume conductor by internal current sources

Abstract

The magnetocardiogram results from the detection of magnetic fields generated outside the body by electric current sources in the heart. Let the source current dipole moment per unit volume be J <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sup> , the conductivity <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</tex> , the electric potential <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V</tex> , and the electric field intensity <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</tex> . Then it may be shown that the magnetic field <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</tex> is given by either <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H = (1/4\pi) \int J^{i} \times \nabla(1/r) dv + \sum_{i} \int (g' - g")(E \times dS_{i}/r)</tex> , or <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H = (1/4\pi) \int J^{i} \times \nabla(1/r) dv + \sum_{i} \int (g' - g")V\nabla(1/r) \times dS_{i}</tex> , where the surface integral is over all surfaces S <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> separating regions of different conductivity, i.e., <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</tex> ' and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</tex> ", and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</tex> is the distance from the point of measurement to the element of volume or surface. The magnetic dipole moment <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> is given by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m = \frac{1}{2} \int r_{1} \times J^{i}dv - \frac{1}{2} \sum_{i} \int (g'-g")Vr_{1} \times dS_{i}</tex> , where r <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> is a radius vector from an arbitrary origin.