4.2.1. Introduction

Measurement of tissue dielectric properties is important because it provides information necessary for calculating RF power absorption by biological models and for constructing tissue-equivalent models. Experiments with tissue-equivalent models are useful in evaluating biological hazards as well as the EM heating patterns of devices used to produce hyperthermia. Also, many biophysical interaction mechanisms of EM fields with biological systems can be inferred from the characteristic behavior of tissue permittivity as a function of frequency.
For complete characterization of the dielectric properties of biological substances and to identify and characterize the various relaxation processes, the complex permittivity should be measured over a broad frequency band. The two principal broad-band measurement systems are frequency domain and time domain. In frequency-domain measurements, sweeping the frequency over the band of interest provides broad-band information; in time-domain measurements, broad-band information can be obtained from a single measurement of the pulse response of the material under test. Both methods require biological sample holders specially designed for evaluating the effect of parameters such as temperature and physiological factors on the measurements. Several measurement techniques are commonly used, each valid only in a specific frequency band. For example, at frequencies below 1 MHz--where all sample lengths, electrical paths, and connecting leads are short compared to a wavelength--a lumped-circuit approach is usually used. Typically a sample of the material under test is contained in a parallel plate or a coaxial capacitor.
Cormnercially available impedance-measuring bridges and vector voltmeters are used to determine the input impedance of the sample holder. This method can, in theory, be extended down to zero frequency; however, practical measurements on conductive biological solutions are difficult below 1 kHz because of electrode polarization effects. At frequencies above 10 MHz, on the other hand, measurements are less straightforward and the results are subject to greater error. In this frequency range, a distributed-circuit approach rather than a lumped-circuit approach is required because the sample size is usually a considerable fraction of a wavelength. The sample is often placed in or at the end of a section of coaxial transmission line or waveguide or in a microwave cavity. Coaxial-cable methods are usually used in the frequency range from 50 MHz to 10 GHz. Between 10 GHz and 100 GHz, waveguides are often used; above 40 GHz, free-space quasi-optical techniques are usually used.
Instrumentation problems initially limited time-domain techniques to the lower frequency range. With the advent of sampling oscilloscopes and stepfunction generators with very short rise times, however, time-domain methods now provide valuable measurements in the frequency range from 10 MHz to 10 GHz. The 10-GHz limit is due to the rise time of the step-voltage excitation of typical time-domain reflectometers (TDRs).
The various time-domain and frequency-domain techniques are reviewed below and some typical examples of broad-band methods are given. Also, in vitro and in vivo results are compared.

4.2.2. Low-Frequency Techniques

Impedance bridges and series or parallel resonant circuits are usually used to measure dielectric properties below 100 MHz. The sample holder is usually either a parallel-plate or coaxial capacitor, with the test material forming the dielectric between its plates. Since biological materials are conductive (lossy), the input impedance of the capacitor is complex and is usually represented by an equivalent circuit consisting of a parallel connection of a resistance and a capacitance. A typical bridge circuit with the equivalent circuit of the sample holder is shown in Figure 4.6 (Von Hippel, 1954).

Figure 4.6.
Bridge circuit for measuring dielectric properties of materials at frequencies below 100 MHz.

The bridge is like a Wheatstone bridge, but impedances are measured instead of resistance. Balancing the bridge requires adjusting one or more of the impedances (Z1, Z2, and Z3). If the complex permittivity of the material under test is given by

* = o (' - j") (Equation 4.6)

where o is the permittivity of free space, the admittance Y of the capacitor is given by Y = jC. For a lossy capacitor filled with the dielectric material under test,

Y = jKo (' - j ") (Equation 4.7)

where K is a constant dependent on the geometry of the sample holder. For example, K = A/d for an ideal parallel-plate capacitor, where A is the area of the plates and d is the separation between the plates. The imaginary and real parts of the admittance are hence given by

B = K o' (Equation 4.8)

G = Ko" (Equation 4.9)

The real part of the permittivity can thus be found from the imaginary part of the measured admittance, and the imaginary part of the permittivity can be found from the real part of the measured admittance. Although the capacitor sample holder seems easy to use, accuracy of measurements may be limited by a number of factors such as effects of the lead impedance (particularly at higher frequencies), effects of fringing fields at the edges of the electrodes, and electrode polarization effects.
Accuracy of bridge methods at frequencies above a few MHz is often limited by the self-inductance of the cell and its associated leads. Careful cell design and calibration to account for fringing capacitance and selfinductance are required to overcome this problem (Grant et al., 1978).
At lower frequencies, measurements are limited by electrode polarization, which is caused by the piling up of ions at the electrode-sample interfaces when direct or low-frequency current is passed through the measuring system. Measurement of dielectric properties of conductive materials is particularly restricted by electrode polarization, as is any measurement at frequencies where the signal period is long enough to permit ions to migrate over appreciable distances and accumulate at the electrode-dielectric interface. Large electrode separation would minimize this polarization but is undesirable because it increases the error due to stray fields.
Electrochemists have overcome electrode polarization problems by using four electrodes, two for applying the RF signal and two for picking up the potential difference within the material under test (Collett, 1959). Electrode-solution combinations that are nonpolarizing or only slightly polarizing are also used to minimize electrode polarization effects (Chang and Kaffe, 1952). These electrode-solution combinations are known as reversible electrode systems. An example of such a system is electrodes containing a layer of platinum black (Schwan, 1963b). The reversible electrodes reduce the polarization errors by providing a large effective area of electrode surface. This large area allows the migrating ions to spread out very thinly over the electrode surface so that the capacitance of the double layer, which is in series with the sample capacitance, is very large. This reduces measurement errors. Sandblasted platinum-black electrodes also minimize electrode polarization problems in biological applications. There is no known way to completely eliminate polarization problems, however, and some analytical procedures to calibrate for electrode polarization effects should always be incorporated into low-frequency measurement techniques (Grant et al., 1978).

4.2.3. High-Frequency Techniques

A distributed-circuit instead of a lumped-circuit approach must be used at frequencies above 100 MHz because the sample size nears a considerable fraction of a wavelength for these frequencies. In distributed-circuit techniques, the sample is typically placed in or at the end of a section of transmission line or waveguide or in a microwave cavity. Since these transmission line methods are broad-band, they are often preferred over the narrow-band cavity techniques.
In the transmission-line methods the complex reflection and/or transmission coefficients are measured instead of the sample impedance. In reflection methods, the sample holder is treated as a one-port network terminating a 50 ohm coaxial line. When transmission coefficients are measured, the sample typically fills the space between inner and outer conductors of a coaxial line with two low-dielectric beads confining the sample to the desired length. In the latest techniques, the scattering parameters (S-parameters) of the sample are measured with an automatic computer-based network analyzer such as the one shown in Figure 4.7 (Burdette et al., 1980; Iskander and DuBow, 1983). The key elements of the network analyzer system are a stable synthesizer, broadband and high-ratio directional couplers, and a computer-controlled processor capable of making corrections in real-time measurements and calculating changes in permittivity from measured changes in reflection and/or transmission coefficients. The coaxial sample holder is connected to the S-parameter device, which has two outputs. One output is proportional to the incident signal and is connected to the reference channel of the network analyzer. The other output provides a signal either reflected by the sample or transmitted through the sample. This output is connected to the test channel of the network analyzer. The analyzer, using a calibrated superhetrodyne receiver, provides a measurement of the reflection and transmission coefficients by comparing the amplitudes and phases of the reflected and transmitted waves, respectively, with those of the incident wave.

Figure 4.7.
Experimental setup for measuring S-parameters, using an automatic network analyzer.

With the reflection (S11) and transmission (S12) parameters measured, the real and imaginary parts of the complex permittivity * = o (' - j") can be determined from

(Equation 4.10)

(Equation 4.11)

where Im means imaginary part; Re, real part; , the complex reflection coefficient assuming the sample to be of infinite length; and P, the propagation factor. and P are given in terms of the S-parameters by

(Equation 4.12)


(Equation 4.13)


(Equation 4.14)

where is the propagation constant and L is the length of the sample under test . This measurement procedure provides enough information to obtain the complex permeability of the sample as well as the complex permittivity. To avoid resonance effects in these measurements, the sample length should be limited to less than a quarter of a wavelength at the highest frequency of operation. Typical sample holders suitable for these measurements at microwave frequencies are shown in Figure 4.8. For the lumped-capacitor holder in Figure 4.8b, only measurement of the reflection coefficient is required; and the calculations are made as described in the following section.

Figure 4.8.
Typical sample holders for measuring the dielectric properties of biological substances at microwave frequencies.
(a) Coaxial sample holder.
(b) Lumped capacitor terminating a section of a coaxial transmission line.

4.2.4. Time-Domain Measurements

Measurements over the broad frequency range necessary to characterize dielectric properties can be very time consuming and tedious unless automated frequency-domain techniques are available, but such techniques are generally not practical because a single RF oscillator will not work over a sufficiently wide frequency range. Several RF oscillators are usually required, one for each range of frequencies. A single technique capable of covering the frequency band from 100 kHz to the higher microwave frequencies with acceptable accuracy is therefore desirable. Time-domain techniques can provide such capabilities. Since their introduction in the late sixties, they have been widely used to measure dielectric properties of materials over broad frequency ranges. These techniques are also conceptually simple and experimentally straightforward, particularly when used in conjunction with modern data-acquisition systems. They are also less expensive than conventional frequency domain microwave dielectric spectroscopic systems. Below 10 GHz, time-domain measurements can be made with about the same accuracy as swept-frequency measurements, which is generally less than that obtainable with single frequency measurements. The strong decrease above 10 GHz in the spectral intensity of the exciting step-voltage generator in commercially available time-domain reflectometers limits their use to frequencies below 10 GHz.
In time-domain methods, the Fourier transform of the measured response of the dielectric sample to short-rise-time pulses is calculated. The dielectric properties over a wide frequency range can be obtained from this Fourier transform because the frequency spectrum of the short-rise-time pulses is very wide. The four essential parts of a time-domain system are a sub-nanosecond step-function generator, a broad-band sampling oscilloscope, a temperature-controlled sample holder, and a microcomputer for data processing. A typical time-domain reflectometer (TDR) system is shown in Figure 4.9 (Iskander and DuBow, 1983).
With this brief discussion of the relative merits of the frequency domain and time-domain techniques as background, a specific example of a time domain method used in our laboratory, called the lumped-element time-domain method (Iskander and Stuchly, 1977), is described next. Information about a system analogous to the automated microwave network-analyzer technique described in Section 4.2.3 is available in the literature (Nicholson and Ross, 1970). In the lumped-element time-domain method used in our laboratory, the sample holder is a small shunt capacitor terminating a section of coaxial transmission line. This sample bridges the gap between the low-frequency measurements, where lumped capacitors are often used, and the high-frequency measurements, where distributed elements such as a section of transmission line are used. The capacitor sample-holder consists of a cap screwed on the outer conductor of the coaxial line. The center conductor is made slightly shorter than the outer conductor to form a gap between the center conductor and the cap, which is the capacitor at the end of the transmission line. A schematic diagram of the sample holder is shown in Figure 4.8b.

Figure 4.9.
Typical experimental setup for time-domain measurement of complex permittivities.

The measurement procedure is to first replace the sample holder by a short circuit and obtain a reference signal, then to replace the sample holder and record the reflected signal at the sample interface. Both signals are digitized and their Fourier transforms calculated. The frequency dependence of the reflection coefficient is given by

(Equation 4.15)

where represents the Fourier transform; Vin and Vr, the incident and reflected voltages respectively; Vsc, the reflected voltage when the sample holder is replaced by a short circuit; Vo, the total voltage signal recorded on the TDR screen; and to, the propagation time between the sampling probe and the sample holder. The real and imaginary parts of the relative permittivity are calculated from the complex reflection coefficient in Equation 4.15 using the following relations:

(Equation 4.16)

(Equation 4.17)

where and are, respectively, the magnitude and phase of the frequency-domain reflection coefficient, and Co is the capacitance of the airfilled capacitor terminating the transmission line of characteristic impedance Zo.

4.2.5. Measurement of in Vivo Dielectric Properties

Most measurements of the dielectric properties of tissue have been made on excised samples. Making measurements in vivo, though, would be better for two main reasons. First, preparing samples that fit the sample holder properly is difficult; and second, the condition of the tissue deteriorates rapidly after it is removed from the body. How dielectric properties of excised tissue compare with those of tissue in a living body is difficult to determine.
Two procedures for measuring the dielectric properties of tissue in vivo are described in this section. Both use an open-ended coaxial transmission line placed in or on the tissue. The first technique is simpler but works only for higher permittivity tissues. The second technique is more complicated but can be used to measure the dielectric properties of the lower permittivity tissue like fat and bone.

Measurement of High-Permittivity Tissues--Two probes are available for measuring the dielectric properties of tissue in vivo. Both consist of a section of coaxial transmission line (see Figure 4.10): one with the center conductor extended (Burdette et al., 1980), and one without (Athey et al. 1982). During the measurement, the center conductor is pressed against the material being tested.
The primary theoretical basis for the concept of the in vivo probe measurement is found in an antenna modeling theorem (Burdette et al., 1980) that applies to a short monopole antenna (antenna length much less than 0.1 wavelength). This theorem relates the impedance of a short antenna operating at frequency and radiating in the material under test, to the impedance at frequency n and radiating into free space. For nonmagnetic materials, the theorem states
(Equation 4.18)


Z = antenna impedance

* = complex permittivity of the material being measured

= intrinsic impedance of the material being measured

= intrinsic impedance of free space

= index of refraction of the material being measured relative to free space

Figure 4.10.
In vivo dielectric probes for measuring dielectric properties of biological substances.
(a) Open-ended section of coaxial transmission line.
(b) A short electric monopole immersed in the material under test.
(c) The low-frequency (neglecting radiation resistance) equivalent circuits.

When a short monopole antenna is used as the probe, the probe impedance is given by

(Equation 4.19)

where A and C are constants determined by the probe's dimensions. This expression is valid when the probe length is less than 10% of the wavelength in the material being measured. Combining this expression with Equation 4.18 gives the following expressions for the resistance and reactance of the complex impedance Z(, *) = R + jX:

(Equation 4.20)

(Equation 4.21)

where tan is the loss tangent. In the above pair of equations all parameters except ' and are known or can be determined from experimental measurements. Because simultaneous solution of these equations is difficult, an iterative method of solution is usually used. The second terms in Equations 4.20 and 4.21 are small at low frequencies. When these terms are neglected, the following equations result:

(Equation 4.22)

(Equation 4.23)

Solutions to these equations are obtained by dividing Equation 4.22 by 4.23 to get tan = R/X; therefore, by measuring the input impedance of a short monopole antenna inserted into a material, we can calculate both the relative dielectric constant, ', and the conductivity, .

The other probe used for in vivo measurements of dielectric properties is a special type of the monopole antenna just described. An open coaxial line, placed in contact with a test sample, serves as a sensor. The equivalent circuit of the sensor consists of two elements (Figure 4.10): a lossy capacitor, C ( * ), and a capacitor, Cf, that accounts for the fringing field in the Teflon.
C ( * ) = Co *, where Co is the capacitance when the line is in air. This equivalent circuit is valid only at frequencies for which the dimensions of the line are small compared to a wavelength, so the open end of the line does not radiate. At higher frequencies, increased evanescent TM modes excited at the junction discontinuity cause Co to increase with frequency. When the evanescent modes are taken into account, C o should be replaced by Co + Af2 , where A is a constant dependent on the line dimensions.

Measurement of Low Permittivity Tissues--The probes just described work well for measuring the permittivity when it is high but not when it is low, such as in fatty tissue. For low-permittivity tissues better accuracy is obtained by extending the length of the center conductor of the coaxial transmission-line probe further into the tissue (Olson and Iskander, 1986). The analysis described for high-permittivity measurement is not valid here. A procedure for this case was developed with the following new features:
  1. A rigorous expression developed by Wu (1963) is used for the input impedance of the in vivo probe immersed in the material under test. The method of analysis, therefore, accounts for the radiation resistance of the probe for larger values of h/, where h is the length of the center-conductor extension, and is the wavelength.
  2. Because the mathematical expressions for this case are very complex, the dielectric parameters of the sample under test are determined by comparing the measured and calculated values of the input impedance, using an iterative two-dimensional (error surface) complex zerofinding routine. This procedure is illustrated graphically in Figure 4.11 (Olson and Iskander, 1986).

Figure 4.11.
Graphical illustration of the iterative procedure for calculating complex permittivity parameters by minimizing the difference between measured and calculated values of the input impedance of the in vivo dielectric probe. The minimum on the error surface | Z measured - Z calculated | indicates the most appropriate values ' and " that satisfy the measured value of the input impedance.

Except for these new features, the measurement procedure is like that described in Section 4.2.3. As with all other in vivo probes, special effort should be made to maintain good contact between the dielectric probe and the material under test. For low permittivities, a ground plane of approximately 12-cm radius is needed to fine tune the measured values of the input impedance. We evaluated the accuracy of the in vivo probe measurements for low permittivity materials by measuring the complex permittivity of known lossy (octyl alcohol) and lossless (heptane) materials. The measured results were all within less than ±5% of the measured values given in the literature.

4.2.6. Summary

At frequencies below 100 MHz, methods based on impedance bridges are satisfactory for measuring the permittivity of tissue samples. Above 250 MHz distributed-circuit methods must be used; with these, the sample holder is typically a section of transmission line or waveguide. The use of impedance-bridge techniques is bounded on the low-frequency end by electrode polarization; on the high-frequency end by the self-inductance of the cell and its leads.
Modern transmission-line techniques based on automatic network analyzers are remarkably accurate and relatively easy to use, for both in vitro and in vivo measurements. In all cases, however, the sensitivity of the measured permittivity to the experimental errors in the measured parameters should be analyzed to determine the advantages and limitations of a given method in a specified frequency band. Examples of such analyses are those developed for the time-domain measurements using the lumped capacitance method (Iskander and Stuchly, 1972). These calculations were later used with frequency-domain measurements using in vitro (Stuchly et al., 1974) and in vivo probes (Athey et al., 1982; Stuchly et al., 1982). Such uncertainty analyses lead not only to bounding the measurement errors but also to optimizing the parameters of the measuring cell, such as the value of the capacitance in the lumped-capacitance method (Iskander and DuBow, 1983). Uncertainty analyses should, therefore, be included in all measurement techniques, even when complicated expressions relating the measured parameters to the dielectric properties of the material under test are involved (Olson and Iskander, 1986).

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Last modified: June 24, 1997
October 1986, USAF School of Aerospace Medicine, Aerospace Medical Division (AFSC), Brooks Air Force Base, TX 78235-5301