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of magnetic and electric fields within the sample.

      We start this chapter by briefly discussing the state of the art of ceramic probes in MR applications. Next, we describe several theoretical tools developed to help design MR probes with optimal performance. Finally, MRM experiments with ceramic probes are detailed and discussed.

      2.2 State of the Art

      Take-home message: Dielectric resonators have been used in microwave engineering for decades. More recently, probes for electron paramagnetic resonance (EPR) and magnetic resonance imaging (MRI) applications at several scales have been designed as well. In EPR, several theoretical tools to quantify the transmit efficiency have been developed. The engineering of high-permittivity, low-loss materials has significantly contributed to the development of these probes.

Dielectric resonators have been used in microwave engineering for decades, since they serve as miniaturized circuit components like antennas and filters, which is of great interest for telecommunication applications [1–6]. More recently, probes for MR applications at several scales have been designed as well.

In EPR, dielectric resonators are used instead of metallic cavities since they have reduced dimensions for the same resonance frequency and much higher quality factors [7,8]. They are built with ferroelectric materials and used at microwave frequencies. Several theoretical tools have been developed to quantify the transmit efficiency of ceramic probes [7,9,10].

Prototypes of dielectric resonators probes for MRI have been proposed at several B₀ field strengths and for various applications, as listed in Table 2.1. Dielectric resonators can be used in transmit/receive mode or as transmitters only. In most cases, the constitutive dielectric material is a high- to very high-permittivity (relative permittivity ϵ_r≥100) ceramic based on calcium and barium/strontium titanates. One exception in this review is the probe for wrist clinical imaging made of water (ϵ_r ~ 80).

      The engineering of high-permittivity, low-loss dielectric materials has contributed significantly to the development of these probes. The high permittivity helps to confine the electromagnetic field within the resonator, while the low losses inside the material limit noise during the signal acquisition. The development of ferroelectric ceramic materials with perovskite crystalline structures has enabled the reachable permittivity to be increased while keeping the losses low [11,12].

Table 2.1 Literature review of dielectric probes for MRI.

	Application	B0(T)	Frequency (MHz)	∈r	Imaging zone (height x diameter)	Type of probe	Dielectric material	Exploited distribution	Tuning technique
[13]	Human breast	3	128	1000	15 cm × 10 cm	Volume	(BaSr)TiO3 + Mg	TE01δ (5 coupled disks)	Interdisk gap adjustment
[14]	Wrist	7	298	80	10 cm × 10 cm	Volume	Water	HEM11δ	Capacitors (coupling loops electronic network)
[15]	Cardiac (torso)	7	298	165		Surface	(BaSr)TiO3	TE01δ (array of 8 resonators)
[16]	Microscopy	14.1	600	156	24 mm × 4.8 mm	Volume	CaTiO3	TE01δ	Overlap with copper foils
[17]	Microscopy	14.1	600	323	4 mm × 3.8 mm	Volume	(BaSr)TiO3	2 coupled TE01δ modes	Overlap with copper strips
[18]	Microscopy	17.2	730	536	10 mm × 5.6 mm	Volume	(BaSr)TiO3 + Mg	TE01δ	Temperature
[19]	Microscopy	17.2	730	536.	10 mm × 5.6 mm twice	Volume	(BaSr)TiO3 + Mg	2 coupled TE01δ modes	Temperature

      2.3 Modeling and Design Guidelines

      Take-home message: It is possible to model the first TE mode of cylindrical dielectric resonators, at the cost of manually writing the mode’s field expression and numerically solving a set of equations.

Conventional MR coils, like the solenoid, have been extensively studied from a theoretical point of view and several design tools are now readily available to optimize their performance. So far, theoretical approaches have been proposed to describe the resonant modes of dielectric resonators, but rarely in the context of their use in MRI. This implies that the noise contributions of the ceramic coil need to be identified and quantified in order to estimate the achievable SNR. Provided that transmit power is enough to reach a desired flip angle (as in the case of MRM), the SNR can be evaluated as for other transceiver probes. In this section, we describe an approximate method to calculate the magnetic and electric fields of the first TE mode (TE_01δ) of a cylindrical resonator sketched in Figure 2.1; however, the methodology used to study this

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