Understanding the Coefficient of a PDE in the context of Electrical Tissue Property Imaging

In this blog, I discuss how the coefficient of the elliptic partial differential equation \(\sum_{i=1}^3 \partial_i \big( a_{ij}\,\partial_j u \big) = 0\) in a body arises in the context of electrical tissue property imaging, where \(u\) denotes the electrical potential. In brief, bioimpedance is directly linked to this coefficient, and several companies , such as InBody and Sciospec, are actively developing bioimpedance-based devices. This blog is based on the book *Electromagnetic Tissue Properties MRI* (Imperial College Press) written by Jin Keun Seo, Eung Je Woo, Ulrich Katscher, and Yi Wang. 

The mathematical model for electrical tissue property imaging is derived from an appropriate reduction of Maxwell’s equations. In the time-harmonic regime, the electric field \( \mathbf{E} \), current density \( \mathbf{J} \), magnetic field \( \mathbf{H} \), and magnetic flux density \( \mathbf{B} \) satisfy the following relations:

• Gauss's law: \( \nabla \cdot \mathbf{B} = 0. \)
• Faraday's law: \( \nabla \times \mathbf{E} = -\mathrm{i}\,\omega \mathbf{B}. \) Here, \( \omega \) is the angular frequency.
• Ampère's law:  \(\nabla \times \mathbf{H}=\mathbf{J}=(\sigma + \mathrm{i}\,\omega \varepsilon)\,\mathbf{E}.\) Here, \( \sigma \) is the electrical conductivity, \( \varepsilon \) is the permittivity, and \( \mathbf{J} \) denotes the total current density, consisting of both the conduction current and the displacement current.

Nutritional, metabolic, or pathological changes modify the electrical properties of biological tissue, especially the conductivity \( \sigma \) and permittivity \( \varepsilon \). The combined quantity \( \gamma = \sigma + \mathrm{i}\,\omega \varepsilon \) is called the effective admittivity. In homogenized tissue modeling, an averaged form of Ohm’s law yields \[ \int_{\text{voxel}} \mathbf{J}(\mathbf{x})\, d\mathbf{x} ~\approx~  \underbrace{(\sigma + \mathrm{i}\,\omega \varepsilon)}_{\gamma}  \int_{\text{voxel}} \mathbf{E}(\mathbf{x})\, d\mathbf{x}, \] for all admissible field pairs \( (\mathbf{E}, \mathbf{J}) \).

At low frequencies (0 Hz \( \le \omega/2\pi \le \) 1 MHz), the relation \( \nabla \times \mathbf{E} = 0\) implies  the existence of an electric potential \( u \) satisfying \( \mathbf{E} = -\nabla u \).  Since \(\nabla \cdot \mathbf{J} = 0, \) the governing equation is essentially elliptic: \[ \nabla \cdot \big(  \underbrace{(\sigma + \mathrm{i}\,\omega \varepsilon)}_{\gamma} \,\nabla u \big)= 0, \]

At higher frequencies (10 MHz \( \le \omega/2\pi \le \) 1 GHz), the model transitions toward a Helmholtz-type equation for the magnetic field \( \mathbf{H} \): \[ -\nabla^2 \mathbf{H} = \left( \frac{\nabla(\sigma + \mathrm{i}\,\omega\varepsilon)} {\sigma + \mathrm{i}\,\omega\varepsilon} \right) \times (\nabla \times \mathbf{H}) - \mathrm{i}\,\mu_0 \omega (\sigma + \mathrm{i}\,\omega\varepsilon)\,\mathbf{H}. \]

Biological tissues and organs exhibit distinct electrical properties depending on their physiological functions and pathological states. In a similar way, the effective admittivity \( \sigma + i\omega \varepsilon \) also differs for the same type of orange—such as a fresh orange, a rotten orange, or orange juice—as shown in the figure below.

In this blog entry, we focus only on the low-frequency case, which leads to the classical admittivity equation used in electrical tissue property imaging. The central PDE relating the electric potential \( u \) to the admittivity distribution \( \gamma \) within a biological domain \( \Omega \) is \(\nabla \cdot (\gamma \nabla u) = 0\)  in a region occupying \(\Omega \).

How do we define \( \gamma \)? We usually use the following terminology:

• Pointwise admittivity: The admittivity at each point is assumed to be well-defined and independent of microscopic variations. It is often further assumed to be isotropic.
• Effective admittivity: The admittivity of a sample at a macroscopic scale (for example, the level of a voxel in an image), determined solely by its intrinsic electrical properties.
• Apparent admittivity: The admittivity that depends not only on the tissue’s electrical properties but also on the configuration of the measurement system.
• Equivalent admittivity: A representative admittivity that substitutes heterogeneous or complex structures with an equivalent homogeneous medium having comparable electromagnetic  behavior.

How can we measure the effective admittivity \( \gamma = \sigma + i\omega \varepsilon \)?

Using homogenization theory, the paper *Spectroscopic Imaging of a DiluteCell Suspension* by Habib Ammari, Josselin Garnier, Laure Giovangigli, Wenjia Jing, and Jin Keun Seo analytically demonstrates, for the first time, the fundamental mechanisms showing that the effective electrical properties of biological tissues—and their frequency dependence—reflect the underlying tissue composition and physiology. 

The MREIT technique has achieved the highest spatial resolution among conductivity imaging methods. The article “Magnetic Resonance Electrical Impedance Tomography(MREIT)” published in SIAM Review (2011) provides a comprehensive overview of MREIT technology, including its underlying principles, reconstruction algorithms, and experimental methods.

Conductivity Image

Since undestanding homogeneization-based  effective $\gamma$ is somewhat difficult for general reader, I will provide an intuitive defintion of effective admittivity $\gamma^{ef}$ for the unit cube. Assume that the effective admittivity in the cube is a
constant tensor, \[\gamma^{\mathrm{ef}}(\omega)= \begin{pmatrix} \gamma^{\mathrm{ef}}_{xx}(\omega) & \gamma^{\mathrm{ef}}_{xy}(\omega) & \gamma^{\mathrm{ef}}_{xz}(\omega) \\\gamma^{\mathrm{ef}}_{yx}(\omega) & \gamma^{\mathrm{ef}}_{yy}(\omega) & \gamma^{\mathrm{ef}}_{yz}(\omega) \\\gamma^{\mathrm{ef}}_{zx}(\omega) & \gamma^{\mathrm{ef}}_{zy}(\omega) & \gamma^{\mathrm{ef}}_{zz}(\omega)\end{pmatrix},\] defined implicitly by the relation \[\gamma^{\mathrm{ef}}(\omega)\int_{\Omega} \nabla u(\mathbf{r},\omega)\, d\mathbf{r}~\approx~\int_{\Omega} \gamma(\mathbf{r},\omega)\, \nabla u(\mathbf{r},\omega)\, d\mathbf{r},\] for all potentials \(u\) satisfying

\[\nabla\cdot \big( (\sigma(\mathbf{r}) + i\omega \varepsilon(\mathbf{r})) \nabla u \big) = 0\qquad \text{in a domain containing } \Omega.\]

Here, \( \gamma(\mathbf{r},\omega) = \sigma(\mathbf{r}) + i\omega \varepsilon(\mathbf{r}) \) denotes the pointwise admittivity, and \(\mathbf{r}=(x,y,z)\).

We define three electrical potentials \(u^x, u^y,\) and \(u^z\), where \(u^y\) is the potential induced as shown in the figure below, and \(u^x\) and \(u^z\) are defined in a similar manner. 

 

With the three electrical potentials \(u^x, u^y,\) and \(u^z\), we can provide an intuitive definition of the effective admittivity \( \gamma^{\mathrm{ef}}(\omega) \) without relying on homogenization theory.

Finally, I provide a visual explanation of equivalent conductivity.

At my imaging scale using MREIT system, it is impossible to resolve a cell membrane with a thickness on the order of \(0.00001\,\mathrm{mm}\). In my MREIT setup, the voxel size—representing the achievable spatial resolution—is approximately \(1\,\mathrm{mm}\). Therefore, my expression is 

The equivalent conductivity obtained from MREIT varies with the size of the hole, as shown in the figure below.