Magnetic Resonance Microscopy. Группа авторов

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      The above equation can be simplified to:

       (2.10)

      2.3.4 Mode Frequency

      To be used as a coil for MR imaging, the exploited mode of the resonator must be adjusted to the working frequency. In this framework estimating the mode frequency with a simple method is very useful for predesigning the ceramic probe. Methods with low computational costs provide an approximated value of the TE_01δ mode frequency for disk resonators. For example, a commonly used expression, derived from simulation data, is the following:

(2.11)

The expression in Equation 2.11 is valid with an accuracy around 2% or below in the following domain: [1]. A diversity of alternative simplified methods can be found in the literature, some extrapolated from experimental data [25,26], some using rough approximations about the boundary conditions at the dielectric interfaces [27] or implying the numerical solving of a transcendental equation [17], and others combining the above-mentioned approaches [21].

These estimation methods are valid for disk resonators, while the dielectric probe is described as a high-permittivity (ϵ_d = ϵ_rϵ₀) ring filled with a lower permittivity (ϵ_d = ϵ_r,sampϵ₀) sample. In the specific case of the TE_01δ mode, equating the field distribution of the probe and the disk resonators is a reasonable approximation. Indeed, between the disk and the probe, the TE_01δ mode features are affected in proportion to the electric and magnetic energies stored in the sample volume V_samp [28]. The mode frequency varies from the value f_disk for the disk to f_probe for the probe according to:

       (2.12)

As the region where the sample is located corresponds to the lowest values of the E-field distribution, the frequency shift due to the sample is limited. For the TE_01δ mode, the ceramic probe is expected to be robust with respect to the sample loading while the permittivity contrast and the sample volume (or sample over disk radii ratio) have reasonable values. This is demonstrated in Figure 2.4.

      Figure 2.4 Quantification of the TE_01δ mode frequency shift between the disk resonator and the dielectric probe (for both, the numerical simulations results were obtained with the CST Eigenmode Solver). The probe ring has its relative permittivity equal to 200, 500, and 800. The frequency variation is plotted as a function of the sample permittivity and for several discrete values of the radii ratio. Curves in dashed lines correspond to systematic frequency shift inferior to 5%. From [21].

      2.3.4 Application: Design Guidelines

One way of designing a ceramic probe as depicted in Figure 2.5 consists of studying how the achievable SNR varies with the probe dimensions and/or material properties for a given sample. In the following, the required field of view constrains the ring’s inner diameter and height, the ceramic material properties optimize the SNR value, and the outer diameter and the permittivity are used for tuning the resonance at the Larmor frequency. Therefore, designing the ceramic probe consists of the following steps:2.5

      Figure 2.5 Schematics of the sample, typically contained in a water tube.

      1 Define the required field of view: diameter Dsamp and length Lsamp. These two parameters define the inner radius of the dielectric ring rh, and its height L, respectively. The outer radius rd is kept as degree of freedom for tuning the mode at the Larmor frequency, and the material properties (permittivity ϵr and loss tangent tan δ) are used to optimize the achievable SNR.

      2 For a given list of permittivity values estimate the required outer radii list for tuning at the Larmor frequency (Ne elements in each list).

Figure 2.6 is an example of such a tuning curve, with a target frequency of 730 MHz (Larmor frequency at 17 T), for a resonator with height 10 mm.

      1 For a given list of loss tangent values (Nt values), estimate for each element of (and the associated outer radius for tuning) the corresponding SNR value (Ne × Nt values).

      Figure 2.6 Example of tuning the first TE_01δ mode frequency of a disk resonator with a given height through its outer radius for varying values of the permittivity.

This array of SNR values can be used as it is for optimizing the ceramic properties, or divided by the corresponding SNR of another probe for comparison. In Figure 2.7, we display the SNR gain (Equation 2.13) over a solenoid coil with the same inner dimensions loaded with the same sample, defined as follows:

       On the left: as a function of the ceramic relative permittivity (vertical axis) and its loss tangent (horizontal axis) for given sample properties (ϵr,samp=50, σsamp = 1 S/m).

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