MATCHING NETWORK
12. DESIGN OF EPR IN VIVO IMAGING EXPERIMENTS
The frequency dependence ofSIN at constant B1 ranges (Table 1) from co-114 to co11/4 if resonator resistance dominates the noise, and depends on
3. FREQUENCY DEPENDENCEOFEPRSENSITIVITY 147
either00or00112if sample loss dominates the noise. Typically, one might seek to match the resonator size to the portion of the animal to be studied, and then lower the frequency to get the desired depth of penetration. To make this quantitative, assume X-band (9.1 GHz) as the reference frequency, and assume the sample size allows one to keep the resonator size constant with frequency (case I). Note that the filling factor is kept constant in this example. Because of interest in in vivo EPR at 1 GHz and 200 MHz, the results are presented for two steps: 9.1 to 1 GHz, and then 1 GHz to 200 MHz to achieve penetration depth. Further, we consider three possibilities for the dominant noise source, resonator resistance (Table 1), sample loss (Table 2) and RF source phase noise (section 5).
Source phase noise is proportional to Q.JP, just as is the EPR signal (Eqs. (I) and (2». To calculate the SIN when phase noise dominates, divide Eq. (2), by Q...JP. The resulting frequency dependence of the SIN is"00. For cases 1 and 2, " is constant, so the SIN is proportional to 00. Note that for case 3 the filling factor depends on 003, which is a dominant effect, making SIN very poor at low frequency relative to higher frequency when the sample size is kept constant and RF source phase noise dominates. The effect of phase noise is calculated with the assumption that the power used at the higher frequency was that where the phase noise just began to dominate, and that the phase noise of the sources at both frequencies are the same. The predictions are summarized in Table 3.
The frequency dependence of the phase noise is strikingly different for case 3 relative to cases I and 2, because of the filling factor effect.
The numerical examples in Table 3 reveal orders of magnitude difference in SIN depending on the relative scaling of sizes of sample and resonator and on the dominant noise source. Not surprisingly, the best results are obtained if the sample size and the resonator size are scaled with inverse frequency.
When the sample is limited in size, the best SIN could be achieved at the highest frequency compatible with the size and loss of the sample, because this will give the highest filling factor . If the resistance of the materials of which the resonator is constructed dominate the system noise, the EPR SIN can be larger at low frequency than at high frequency. If sample loss dominates the noise, the low-frequency EPR SIN decreases only by the square root of the frequency ratio, resulting in a relatively small SIN penalty for low-frequency in vivo EPR. If RF source phase noise dominates the spectrometer system noise, then the SIN scales linearly with frequency for cases 1 and 2, but SIN scales with 004for case 3, because of the filling factor effect for a fixed sample size. Elimination of the source phase noise problem would substantially improve SIN. For example, compare case 2 for scaling of sample and resonator size with inverse of frequency from 9.1 GHz to 0.2 GHz. If source phase noise were eliminated the SIN would improve by
0.15/0.022
=
6.8 if the sample loss then became dominant, and by 2.6/0.022=
118 if the resonator resistance then became dominant. An improvement of 118 in voltage SIN corresponds to about 41 dB, an isolation of RF source phase noise from the EPR signal, which is readily achievab le with a crossed-loop or other bimodal resonator.The real situation for animal samples will be somewhere between the extremes in the Tables. For example, there will be a limited range of application of the functional dependences in Table 2. At sufficiently high frequency , the depth of penetration will be limited by combinations of conductivity losses, dielectric concentration effects (lens effects), and for special geometries resonant effects (Roschmann, 1987; Gadian and Robinson, 1979; Foster, 1992; Zypman, 1996; Tofts , 1994; Petropoulos and Haacke, 1991; Mansfield and Morris, 1982). Heterogeneous tissue reduces the power loss in a given volume relative to the same mass in a homogeneous volume. All of these factors make the predictions of Table 2 very conservative over relatively small changes in frequency and inapplicable for large changes in frequency.
One might scale the resonator size with frequency (case 2) from X-band to L-band, in order to be able to study a mouse. But then there would be no further benefit in making the resonator large relative to the mouse, so then one would switch to keeping both the sample and resonator size constant (case 1) as the frequency is lowered by another factor of 5 to 200 MHz. For this, one would pay a signal amplitude price of about 17, ameliorated somewhat by the greater penetration depth of the RF at the lower frequency.
If the signal came from the center of the mouse, one might not lose much signal by lowering frequency , since the skin depth is proportional to00-112and starts to become significant for an animal the size of a mouse at frequencies above a few hundred MHz. Noise due to losses in the sample also increases with frequency for animal samples in this frequency range. The net result is that in practical applications, sensitivity (SIN) appears to increase approximately linearly with frequency for animal imaging in the RF range.
As measurements proceed to larger animals, it wilI be necessary to scale the resonator size to fit them, and the best results wilI be obtained if the wavelength is large compared with the size of the animal. If the organ of interest increases in size proportionately with the overall size of the animal, the situation is more like case 2, and SIN should not be much worse at lower frequency.
12.1 Dielectric and Conductivity Properties
Interaction of the dielectric and conductivity properties of the sample with the RF also affects SIN. Aqueous and conducting samples have
3.FREQUENCY DEPENDENCE OFEPRSENSITIVITY 149 frequency-dependent skin depths (Halpern and Bowman, 1989; Sueki et al., 1996). The conductivity of biological media such as muscle and bone have been tabulated (Johnson and Guy, 1972). The change with frequency is small enough that over the frequency range 200 - 2000 MHz the changes in conductivity will not substantially alter the general trends as a function ofro
displayed in Tables I and 2. The conductivity of biological media increases a factor of 1.8 from 40 to 200 MHz, which is a little slower than roO.4Ifwe use this frequency dependence of conductivity in row 4 of Table 2, we predict for the three cases roO .8
, roO .3
, and roO.8
• The very mild roO.3 dependence for case 2, where sample and resonator size scale inversely with frequency offers encouragement for low-frequency EPR imaging. On the other hand, for non-lossy and non-conductive samples, the ro 11/4dependence of Table 1, row 8, case 3, is a strong encouragement to high-frequency EPR for small samples.
The dielectric properties can produce phase changes that can null signals from certain locations in samples (Sueki et al., 1996), and can cause the magnitude of the RF magnetic field to be larger at the center of a homogeneous cylinder of high dielectric material than at the surface (Zypman, 1996; Tofts, 1994). The much larger penetration depth at low frequencies can result in many more spins in a large sample being visible at low frequency than at high frequency. This effect can reverse the general trend toward greater sensitivity at higher frequency. Heterogeneous samples appear to have greater penetration depth than homogeneous samples of the same type of material. The physical interactions that result in the losses mentioned here also result in reduction of the resonator Q, and the detected ESR signal is proportional to resonator Q.
12.2 Skin Depth
One should be cautious about using the standard skin depth formula as given after Eq . (3) as anything more than a qualitative indicator when dealing with animals. The skin depth as defined there assumes electromagnetic radiation incident on a plane surface of a homogeneous conducting body. Animals are neither homogeneous nor planar. The formula for inductive (magnetic) loss assumes a homogeneous body.
Vlaardingerbroek and den Boer (1996, page 248) point out that if a conductive spherical volume were divided into ten equal spheres with the same total volume, the power dissipated in the ten small spheres would be about 1/5 of the power dissipated in the larger sphere . Roschmann (1987) obtained good fit to experimental RF interaction data on humans with a tissue model in which small regions of high-conductivity (e.g. muscle) are
separated by "quasi-insulating" layers of lower conductivity (e.g., adipose tissue) .
Calculations on homogeneous bodies reveal that the value of the conductivity required to produce a certain current density (hence, power deposition) in the body increases from the planar body to the cylinder to the sphere (Petropoulos and Haacke, 199 I). In terms of apparent skin depth, one could restate these results as implying that for the same conductivity the skin depth increases in the order planar to cylindrical to spherical.