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Anisotropic electrical properties can be found in biological tissues such as muscles and nerves. Conductivity tensor is a simplified model to express the effective electrical anisotropic information and depends on the imaging resolution. The determination of the conductivity tensor should be based on Ohm's law. In other words, the measurement of partial information of current density and the electric fields should be made. Since the direct measurements of the electric field and the current density are difficult, we use MRI to measure their partial information such as B1 map; it measures circulating current density and circulating electric field. In this work, the ratio of the two circulating fields, termed circulating admittivity, is proposed as measures of the conductivity anisotropy at Larmor frequency. Given eigenvectors of the conductivity tensor, quantitative measurement of the eigenvalues can be achieved from circulating admittivity for special tissue models. Without eigenvectors, qualitative information of anisotropy still can be acquired from circulating admittivity. The limitation of the circulating admittivity is that at least two components of the magnetic fields should be measured to capture anisotropic information.

Noninvasive measurement of electrical properties for biological tissues can be useful in EEG/MEG and electromagnetic source imaging [

Microscopically, the conductivity of the biological tissues could be isotropic. However, depending on the imaging resolution, the conductivity of an imaging voxel can be anisotropic. Macroscopically, in other words, if several tissues with different electrical properties are combined in the imaging voxel, the conductivity of the imaging voxel differs when measured in different directions so that it becomes anisotropic. Especially in biological tissues, anisotropic electrical conductivity can be found in muscles and nerves [

The eigenvectors of the conductivity tensor at low frequencies (<1 kHz) can be inferred from a prior knowledge of the object or the diffusion tensor imaging [

Admittivity tensor denoted by

The admittivity tensor can be represented with six parameters:

Expressing its eigenvectors

We should note that three equations of Ohm’s law

Assume that the three eigenvectors are known. The effective admittivity,

However in MRI,

From time-harmonic Maxwell Equations, the relationship among the current density, electric fields, and the magnetic fields can be expressed as

Based on the work by Katscher et al. [

In this work, we investigated the relationship between the circulating admittivity and the admittivity tensor for simple cases. First, if the admittivity tensor is isotropic, the circulating admittivity is also isotropic and equal to the isotropic admittivity. Second, if two eigenvalues of the admittivity tensor are the same,

In addition, as it is considered in [

As a combination of last two cases, if the largest eigenvalue is much bigger than the two smaller eigenvalues of the admittivity tensor and the two smaller eigenvalues are the same, by measuring the circulating admittivity for several directions of the normal vector, all three eigenvalues could be estimated.

In the previous section, we determined the admittivity tensor under the assumption that the eigenvectors of the admittivity tensor were known

As shown in Figure

For numerical evaluation, a numerical phantom with anisotropic effective admittivity can be generated using periodic binary medium. According to homogenization theory, anisotropy can be derived from pointwise admittivity,

In this numerical experiment the imaging subject is the box

(a) Numerical phantom with anisotropic effective admittivity by stacking periodic binary medium, (b) placement of the imaging object inside the RF coil.

Driven by a birdcage coil at 3T (

Two phantoms with anisotropic admittivity were generated using straws as shown in Figure

(a) Two straw phantoms before filling saline water, (b) three phantoms filled with saline water.

Using a single-channel transreceive head coil, MR images were measured. The phantoms were located at the isocenter of the coil with the straw orientation of left-right. Only the phase of

Based on (

(a) and (d) depict the true conductivity and the relative permittivity values in the microscopic scale. (b)–(f) illustrate the true effective

True

Effective

Effective

True

Effective

Effective

In MRI, only partial information of the magnetic fields can be measured. Using conventional single-transmit channel MR scanner, the circularly polarized component of the magnetic fields,

Figure

Circulating conductivity and relative permittivity derived from

However, as shown in Figures

Circulating conductivity and relative permittivity derived from only

Circulating conductivity derived with only phase of

Experiment results: circulating conductivity of phantoms with measured

Saline water:

Saline water:

Straw phantom 1:

Straw phantom 1:

Straw phantom 2:

Straw phantom 2:

The circulating conductivities were determined from phantom experiments in which only the phase of

Using simulated magnetic fields,

Effective conductivity maps and circulating conductivity maps of one anisotropic voxel and one isotropic voxel: (a) effective conductivity map of anisotropic voxel,

For the isotropic voxel, the values of the effective conductivity map and circulating conductivity map were equal to the conductivity of the tissue. For the anisotropic voxel, the circulating conductivity map is also uniform along the direction

For a qualitative analysis, at each voxel, the maximum value, the minimum value, and the ratio of the maximum to the minimum of the circulating conductivity maps were computed with

The distribution of the circulating conductivity: (a) the maximum values for the circulating conductivity maps, (b) the minimum values for the circulating conductivity maps, (c) the ratio of the maximum to the minimum, (d) the ratio of two eigenvalues in the conductivity tensor.

Maximum

Minimum

Max/Min

Conductivity tensor is a simplified anisotropy model. Given three eigenvectors, the tensor can be estimated if the electric current densities and electric fields can be measured. In MRI, however, electric fields are hard to be measure without knowing or estimating the conductivity and permittivity of tissues. In this work, using MREPT formulae, the circulating admittivity is proposed as a measure to analyze the anisotropy of the tissues. Circulating admittivity was defined as the ratio of circulating current densities to the circulating electric fields, which can be determined from the magnetic fields. We did not fully investigate, but we derived the relationship between the admittivity tensor for special cases. Using numerical phantom simulations, we verified the relationship for the first two cases:

In this work, to deal with unknown eigenvectors, the circulating admittivity map (CAM) was proposed as a qualitative measure. The ratio of the maximum to the minimum conductivity was reduced but still anisotropic tissues can be separated from isotropic tissues.

In the conventional single-transmit channel MR scanner, the circularly polarized magnetic field,

Simulation results: circulating conductivity of simulated phantoms with measured

Experiment results: circulating conductivity of phantoms with measured

Noninvasive measurement of conductivity tensor at Larmor frequency could be achieved using MRI. Using measured

This work was supported by the National Research Foundation of Korea (NRF) Grant funded by Korea Government (MEST) (no. 2012-009903 to J. Lee, N. Choi, and D. H. Kim and no. 2011-8-1782 to Y. Song and J. K. Seo). The authors acknowledge Ulrich Katscher for beneficial discussions.