Identification of coeﬃcients in time-harmonic Maxwell’s equations
and its application to biomagnetic inverse problems
Takaaki Nara
The University of Tokyo
Imaging of electrical properties inside human bodies provides us with valuable pathological information
since the electrical conductivity and permittivity of cancerous tissues are much diﬀerent from those of normal
tissues [1]. Among several techniques, a method based on magnetic resonance (MR) has attracted attention
because the magnetic field can be measured using a usual MRI scanner not only on the surface but also
inside the human bodies, which enhances stability of the inverse problem for reconstruction of the electricalproperties. The governing equations for this biomagnetic inverse problem is the time harmonic Maxwellequations∇×H= γE + J ,

(1)

∇×E

= −iωµ0 H,(2)

lim (H × r − |r|E) = 0,(3)

with the Silver-Muller radiation condition|r |→∞

where σ and ϵ are the electrical conductivity and permittivity, respectively, which should be reconstructed
inside the human body D, γ is admittivity defined by γ = σ + iωϵ, ω is the Larmor frequency, µ0 ispermeability assumed to be constant, and J represents currents applied to the coils in the MRI scanner.
A measurable quantity using the MRI scanners is a limited component of the magnetic field, that is, thepositively rotating magnetic field H + = 21 (Hx + iHy ), where the z-axis is parallel to the body axis. Then,
our inverse problem is formulated as follows: given H + in D, determine γ in D.
In our previous paper [2], we propose a complex analysis-based method that explicitly reconstruct γ fromthe measured H + . Assuming that Hz = 0 when using a so-called birdcage coil in the MRI scanners and thatthe spatial derivative of γ with respect to the z-axis is small, we can derive a Dbar equation¯ z∂E=

ωµ0 H + ,(4)from the Faraday law (2), where ∂¯ = 21 (∂x + i∂y ). Hence, letting Ω be a region of interest (ROI) in a 2D
slice of the human body, for any w ∈ Ω, the z-component of the electric field can be expressed in terms of
the Cauchy integrals as∫∫ ∫Ez (ζ)ωµ0H + (ζ)
1dζ −dξdη, ζ = ξ + iη.
(5)Ez (w) =2πi ∂Ω ζ − wπΩ ζ −w
Therefore, substituting Equation (5) into the equation
4∂H + = iγEz(6)derived from the z-componen of the Ampere law (1) with ∇·H = 0, we have an explicit formula to reconstruct
γ in Ω from the measured H + in Ω and the boundary value of γ given as a priori information.

However, this formula breaks down when Ez = 0. Even when Ez ̸= 0, large errors are generated when |Ez |is small. To avoid this, we propose to use a physical constraint as a new regularization term for reconstructionof γ. To this end, first, using the components of the Faraday law and Ampere law that have not been used
so far for reconstruction, we derive an explicit formula for computing H − = 12 (Hx − iHy ) from the measured
H + , with which Hx and Hy are separately obtained. Then we derive a partial diﬀerential equation (PDE)for γ shere Hx and Hy are included as its coeﬃcients. Since this PDE is a constraint for γ, it can be used
in a regularization term for reconstruction of γ. Numerical simulations will be shown in which γ is wellreconstructed with the regularization term even where |Ez | is small.

References[1] Shin, J., Kim, M. J., Lee, J., Nam, Y., Kim, M-O., Choi, N., Kim, S., and Kim, D-H., Initial study onin vivo conductivity mapping of breast cancer using MRI, Journal of Magnetic Resonance Imaging, 42,371-378, 2015.
[2] Nara, T. and Fushimi, M., An explicit reconstruction method for magnetic resonance electrical property
tomography based on the generalized Cauchy formula, Inverse Problems, 33, 105005, 2017.