# Determination of Bolus Dielectric Constant for  Optimum Coupling of Microwaves

Authors: Carey Rappaport
Journal: International Journal of Antennas and Propagation
 (2008)

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## Abstract

We find the complex dielectric constant of the bolus liquid with the best microwave impedance match to skin-covered fatty breast tissue.  The real dielectric constant and conductivity of the ideal medium are determined by minimizing the magnitude of the reflection coefficient from the two-interface frequency-dependent system, using published measured skin and breast fat dielectric characteristics in the 400 MHz to 10 GHz frequency range.  Error bounds are provided to indicate the coupling degradation when using real, nonideal media for the bolus liquid.  Two frequency regimes are identified.  Below 6.45 GHz, conventional liquid mixtures can approach the optimal coupling permittivity, but the reflection magnitudes for these best cases are high.  Above 6.45 GHz, perfect coupling is possible, but only for impractically high values of bolus dielectric constant and loss.


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## Body

## 1. Introduction

In cases where breast tumors occur in fatty healthy
breast tissue, microwaves can be used to take advantage of the   large dielectric
contrast difference between the tumor scatterer and its surrounding background to
detect the cancer. Several breast cancer
detection systems have been proposed [1–9], each   indicating great potential for
discriminating the high-water-content tumor volume
from the low-water-content fatty surrounding tissue.One open question is the choice of coupling fluid,
often referred to as bolus liquid, which allows the greatest transfer of waves
from the antennas through the skin and into the breast tissue.Published work investigating empirically
enhancing the coupling with suitable mixtures of glycerin and water [10, 11],
indicates that proper choice makes a significant difference in sensing
performance.Although the skin layer is thin (on average, 1.6 mm
[12]), it is a high-water-content biological tissue and thus has a significant effect
on the coupling into breast tissue.The
problem is additionally complicated since all biological tissue is frequency
dependent.This paper uses a simple one
dimensional impedance model to determine the best bolus complex dielectric
constant necessary to minimize reflection from skin-covered fatty breast tissue
and hence transfer the maximum percentage of microwave sensing signal into the
breast.In practice, it may not be possible to obtain a real
bolus liquid with the desired dielectric properties across the intended
frequency range.However, having the
information about the desired complex permittivity characteristics across large
frequency ranges provides potential guidance in the selection of bolus
liquid.


## 2. One Dimensional Impedance Model

Since
the skin layer covering the internal breast tissue is very thin compared to all
other length scales of the problem geometry, it is reasonable to model the wave
interaction as a three-layer one-dimensional model.A further simplification considers only normally
incident plane waves.While in reality
sensing waves are almost never uniform and normally incident on tissue
boundaries, this assumption selects the most important of the great number of
possible illumination cases.Following
the standard method [13], assume that the bolus with complex permittivity ε=εbε0 fills space to the left of the
origin, z<0, that fatty breast
tissue with ε=εfε0 fills
the space to the right of plane at z=Δz, and that skin with ε=εsε0 occupies the space of
thickness Δz between the two.The incident and reflected, right-going and
left-going, and transmitted waves in the three layers can be represented as (1)Eoe−jkbz+ΓEoe+jkbz,AEoe−jksz+BEoe+jksz,TEoe−jkfz, where the reflection and
transmission coefficients are given by Γand T, A and B are unknown skin layer coefficients, and kb=ωμ0εbε0=k0εb is the complex wave number for the bolus
layer, given in terms of the radian frequency and permeability and permittivity
of free space and complex dielectric constant.
Similar expressions hold for the wave numbers in the skin and fat
layers.Continuity of transverse
impedance at the two interfaces results in (2)1+Γ1−Γωμ0kb=A+BA−Bωμ0ks,Ae−jksΔz+Be+jksΔzAe−jksΔz−Be+jksΔzωμ0ks=ωμ0kf.Solving first forB in the latter equation,
substituting it into the former, cancelling the common factor of A, and then solving for the three-layer
reflection coefficient yield (3)Γ=ejksΔz(kb−ks)(kf+ks)−e−jksΔz(kb+ks)(kf−ks)ejksΔz(kb+ks)(kf+ks)−e−jksΔz(kb−ks)(kf−ks). The matched case, Γ=0,
occurs when (4)kbopt=ks(ejksΔz−e−jksΔz)ks+(ejksΔz+e−jksΔz)kf(ejksΔz+e−jksΔz)ks+(ejksΔz−e−jksΔz)kf.Equation
(4) identifies the best bolus dielectric constant, as long as its imaginary
part is negative.If it becomes
positive, the coupling medium must have gain rather than conductive loss and
hence is impractical.The
next step in the modeling process is to specify the wave numbers for the skin
and breast fat as functions of frequency.
Accurate measurements of these types of tissues have been conducted [14, 15].The Cole-Cole dispersive medium
relation, given by (5)εf,s(ω)ε0=ε∞+Δε11+(jωτ1)1−α1+σsjωε0, accurately
and compactly matches both the real dielectric constant and the conductivity of
a frequency-dependent medium.For breast
tissue with 85–100% fat in the 400 MHz to 20 GHz, frequency range [15] has determined coefficients for (5),
while for wet skin, one possible model is plotted in Figure 1.Table 1 gives the coefficients for these two
models.Table 1
Coefficients
for Cole-Cole models of tissues for 400 MHz–20 GHz.
Tissueε∞Δε1τ1 (ps)α1σs (S/m)Breast fat3.1451.70114.680.058260.03605Wet skin2.0489.50.190.710Measured (points) [14] and Cole-Cole model fitted (solid curve) dielectric values for wet skin in the microwave frequency range.


## 3. Optimal Dielectric Constant Results

Inserting
the Cole-Cole dielectric models for skin and breast fat into the formulas for
the wave numbers and then into (4) and choosing Δz=1.6 mm
[12] yield optimal values for kbopt,
which can be squared and divided by ω2μ0ε0 to give the complex dielectric
constant:(6)εbopt(ω)=εb′(ω)−jσ(ω)ωε0. The values for real relative
dielectric constant εb′ and conductivity σ, as a
function of frequency, are plotted in Figure 2.
For 10, 9.5, 9, 8.5, 8, 7.5, 7, and 6.5 GHz, the points corresponding to
the ordered pair (εb′, σ) for zero
reflection, as well as the elliptical contours corresponding to |Γ|=0.03 (arbitrarily chosen for graphical
clarity) are indicated.The contours
give an error bound showing the tolerance about the ideal complex values for
liquids with close—but not exact—dielectric matching characteristics.Figure 2 also provides a sense of impedance
variation of this three-layer geometry for wideband microwave systems.A bolus liquid which is perfect for one
frequency may be considerably mismatched at another.Figure 2
Exact complex dielectric values for bolus liquid with perfect match,|Γ|=0, and contours of constant reflection magnitude |Γ|=0.03 for 6.5 to 10 GHz, in steps of 0.5 GHz.While
the optimal complex permittivity values of Figure2 provide perfect coupling,
they are impractically high, both in real part and in conductivity.Even if such bolus fluids could be found, the
propagation loss from antenna to skin would eliminate any coupling advantages.Extending the reflection magnitude contours
to larger values and into the more physically realizable permittivity parameter
space requires separate plots for each frequency.This set is shown in Figure 3, for reflection
magnitudes of 0.1, 0.15, and 0.2, for 6.5 to 10 GHz.Also indicated in Figure 3 are the asymptotes
that define the limits of reflection magnitude.
For different frequencies, there are bolus fluids with εb′ and σ combinations that result in greater reflection,
but for εb′ and σ combinations to the left and below a given
asymptote, no frequency choice will couple with lower reflection than that
asymptote value.Observing the plots of
Figure 3 leads to simple rules for improving coupling to breast tissue.(1)
For frequencies above 9.5 GHz,
increaseσ and
reduce εb′.(2)
Between about 8 and 9.5 GHz, just increaseσ.(3)
Between 7 and 8 GHz, increaseεb′ and σ.(4)
Between 6.5 and 7 GHz, increaseεb′ and reduce σ.Contours of constant reflection magnitude|Γ|=0.1 (dashed curves), 0.15 (dotted curves), and 0.2 (solid curves) as indicated, for frequencies: 10.0, 9.5, 9.0, 8.5, 8.0, 7.5, 7.0, and 6.5 GHz, as well as the common asymptotes for |Γ|=0.1, 0.15, and 0.2, for practical values of bolus complex permittivity. The curve contours are extended from the centers shown in Figure 2.Just below 6.5 GHz, the imaginary part
of the optimal wave number (4) becomes positive, which makes the conductivity
given by (6) negative.For these cases,
the best bolus liquid is a pure dielectric with zero conductivity.Contours of constant reflection magnitude,
given by (3) with kb kept
purely real, are plotted in Figure 4. The best dielectric constant for any particular
frequency is found by picking the darkest contour (the lowest reflection
magnitude level) crossed by the vertical line through that frequency and
associating the dielectric constant at the crossing.The white regions of Figure 4 correspond
to|Γ|>0.4. Both
the minimum reflection magnitudes and their variation with dielectric constant
are much greater than those for frequencies above the critical value, found
numerically to be 6.45 GHz.Figure 4
Contours of constant reflection magnitude for lossless bolus matching to skin-covered fatty breast tissue. A perfect match occurs at 6.45 GHz for a pure dielectric bolus with dielectric constant of 135. The worst coupling occurs at 2.67 GHz, with reflection coefficient of 0.34 for the best choice dielectric constanε′=30.4.For
frequencies below 1 GHz or above 5 GHz, the optimal reflection magnitude is
below 0.2 for bolus dielectric constants of about 10 and 100,
respectively.However, in the
diagnostically useful frequency range of 2–4.5 GHz, the reflection magnitude
cannot be reduced below 0.3 forany choice of matching fluid.For real liquids with nonzero conductivity,
Figure5 indicates the error bounds about the ideal dielectric choice.Each contour indicates the region of possible
combinations of real dielectric constant and conductivity which would result in
reflection magnitudes not more than 30% above the ideal value for that
frequency.The horizontal maximum widths
of the contours correspond to the vertical elongation of the contours of Figure
4.Note that while there is considerable
tolerance for dielectric characteristics for 5.0 GHz, this is due to the reflection
magnitude contour of Figure 4 being quite long in the vertical direction.Figure 5
Contours of constant reflection magnitude at levels 30% above the best level determined for pure dielectric with no loss, for frequencies: 0.667, 1.33, 2.0, 3.0, 4.0, 5.0, and 6.0 GHz, as indicated on each curve on the figure. The 30% increased reflection contours have values: 0.17, 0.34, 0.42, 0.435, 0.37, 0.246, and 0.08, respectively. The optimal match dielectric constant for each frequency is indicated by the vertical line from the contour peak to the horizontal axis.The lowest frequency range is
examined in detail in Figure6, which shows several common reflection magnitude
contours for 400, 900, and 1300 MHz. The
|Γ|=0.55 contours include higher bolus
conductivity for 1.3 GHz than for the lower frequencies, but the lowest
displayed contour, |Γ|=0.25, is
attainable only for the two lower frequencies displayed.Also shown on Figure 6 are dielectric values
for water, saline, and several glycerin/water mixtures used in breast cancer
imaging   experiments, as reported by Meaney et al. [11].Figure 6
Low-frequency detail of Figure5, for 400 MHz (solid curves), 900 MHz (dotted curves), and 1.3 GHz (dashed curves) reflection magnitude contours of 0.25, 0.4, and 0.55. Labeled points indicate particular bolus fluids:  W1: water at 300 MHz, W2: water at 3 GHz, S1: 0.1 molal saline at 300 MHz, S2:  0.1 molal saline at 3 GHz, S3:  0.3 molal saline at 300 MHz [16], S4: 0.15 molal (0.9%) saline at 900 MHz, G1: 10:30 glycerin/water at 900 MHz, G2: 80:20 glycerin/water at 900 MHz, G3: 87:13 glycerin/water at 900 MHz, G4: 79:21 glycerin/water at 1.3 GHz, and G5: 87:13 glycerin/water at 1.3 GHz [11].


## 4. Conclusion

This
analysis has determined the best dielectric constant and conductivity
characteristics for a bolus liquid to match to skin-covered fatty breast tissue
for microwave in the 400 MHz to 10 GHz frequency range.The results are based on a simple 1D plane-wave model, using accurate frequency dependent measurements of dielectric
characteristic of skin and breast fat.Above
6.45 GHz, specific bolus dielectric values can match exactly to the skin/breast
layers, although their dielectric constants and conductivities must be very
high, in the 5–30 S/m range.About 10%
variation in real dielectric constant and 15% in conductivity result in
reflection magnitudes of only 0.03.Although
the optimal values are impractical for typical liquids, they indicate the
directions in complex permittivity space that bolus fluids should approach to
increase coupling efficiency.The
contours for larger reflection magnitudes of 0.1, 0.15, and 0.2 extend into
permittivity values typical of water-mixture bolus fluids.Below
6.45 GHz, the best bolus dielectric constants are purely real, and the lowest
possible reflections magnitudes are as high as 0.34 for 2.67 GHz.For practical liquids with moderate
conductivity, the reflection magnitude increases slowly, with only 30% reflection
increase for liquids with 6.5 S/m at 5 GHz, or 0.17 S/m at 1.33 GHz.Thus, there is much greater tolerance for
variation of real dielectric constant and conductivity from the optimal values
for bolus fluids below 6.45 GHz, but the reflection magnitudes are considerably
higher than for frequencies above 6.45 GHz.



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*Source: 359582-2008-03-27.xml*