276
10 Characterization of Atherosclerotic Lesions by Inversion. . .
the saline experiment. We cannot yet explain the origin of the two resonances, but
we are satisfied that they are not due to cavity resonances, because there are no finite
boundaries in the model.
Now, when we consider frequencies beyond the second resonance (which we
assume to be the last), which is the end of our experimental data, we return to
the condition of (10.11). We can only model this condition with VIC-3D ® , since
we have no experimental data for it, and we do that spanning the frequency range
500 MHz to 1 GHZ, with the results shown in Fig. 10.22. We used the same values
of σ = 1.016 S/m, , = 219.79 0 for the computation. Note that the response is as
predicted in (10.11); the resistance continues to increase, but the reactance reaches
its peak at about 640 MHz, and then descends through zero at roughly 875 MHz.
From (10.11), therefore, we estimate ω 0 ≈ 875 MHz.
Eventually, of course, the resistance will go to zero, but the frequency at which
this occurs may be so high that the coil may actually begin to radiate, and the
equivalent circuit of Fig. 10.21 will no longer be valid. In fact, a careful look at
the reactance curve of Fig. 10.22 will show a slight ‘glitch’ at 620 MHz and a more
pronounced one at about 1 GHz. There is a similar glitch in the resistance curve at
about 1 GHZ. These indicate that the circuit model may be breaking down in this
frequency range.
Finally, we show corresponding results when the saline is assumed to lie within
a slab of 38 mm (1.5 in) in Fig. 10.23. The results are virtually indistinguishable
from those of a halfspace over the frequency range 50–500 MHz, for which we
have experimental data. Beyond 500 MHz, however, there are little wiggles in the
computed response, which we believe are actuall cavity resonances that are strongly
damped due to the large conductivity of the saline. In any case, the qualitative shape
is the same as for the halfspace, so we will continue to rely on the halfspace model,
and the equivalent circuit of Fig. 10.21 to anlalyze the results of this experiment.
10.10.1 Summary
The circuit diagram of Fig. 10.21 has four independent parameters: the mutual
inductance, M, the virtual secondary inductance, L 1 , and the load elements, G L
and C L . From a field-theoretic viewpoint, which is what VIC-3D ® looks at, there
are also four independent parameters: the lift-off of the coil over the workpiece, the
depth of the workpiece, and the electromagnetic parameters of the saline, σ and .
By matching the VIC-3D ® -model with the measured data, we have found that the
lift-off is 0.55 mm, the workpiece extends to infinity in depth, i.e., that it is indeed a
half-space, and that σ = 1.016 S/m, , = 219.79 0 . The reason that we can easily
move between circuit models and field models is that the measurable is impedance,
which is equally at home in each model.
10 Characterization of Atherosclerotic Lesions by Inversion. . .
the saline experiment. We cannot yet explain the origin of the two resonances, but
we are satisfied that they are not due to cavity resonances, because there are no finite
boundaries in the model.
Now, when we consider frequencies beyond the second resonance (which we
assume to be the last), which is the end of our experimental data, we return to
the condition of (10.11). We can only model this condition with VIC-3D ® , since
we have no experimental data for it, and we do that spanning the frequency range
500 MHz to 1 GHZ, with the results shown in Fig. 10.22. We used the same values
of σ = 1.016 S/m, , = 219.79 0 for the computation. Note that the response is as
predicted in (10.11); the resistance continues to increase, but the reactance reaches
its peak at about 640 MHz, and then descends through zero at roughly 875 MHz.
From (10.11), therefore, we estimate ω 0 ≈ 875 MHz.
Eventually, of course, the resistance will go to zero, but the frequency at which
this occurs may be so high that the coil may actually begin to radiate, and the
equivalent circuit of Fig. 10.21 will no longer be valid. In fact, a careful look at
the reactance curve of Fig. 10.22 will show a slight ‘glitch’ at 620 MHz and a more
pronounced one at about 1 GHz. There is a similar glitch in the resistance curve at
about 1 GHZ. These indicate that the circuit model may be breaking down in this
frequency range.
Finally, we show corresponding results when the saline is assumed to lie within
a slab of 38 mm (1.5 in) in Fig. 10.23. The results are virtually indistinguishable
from those of a halfspace over the frequency range 50–500 MHz, for which we
have experimental data. Beyond 500 MHz, however, there are little wiggles in the
computed response, which we believe are actuall cavity resonances that are strongly
damped due to the large conductivity of the saline. In any case, the qualitative shape
is the same as for the halfspace, so we will continue to rely on the halfspace model,
and the equivalent circuit of Fig. 10.21 to anlalyze the results of this experiment.
10.10.1 Summary
The circuit diagram of Fig. 10.21 has four independent parameters: the mutual
inductance, M, the virtual secondary inductance, L 1 , and the load elements, G L
and C L . From a field-theoretic viewpoint, which is what VIC-3D ® looks at, there
are also four independent parameters: the lift-off of the coil over the workpiece, the
depth of the workpiece, and the electromagnetic parameters of the saline, σ and .
By matching the VIC-3D ® -model with the measured data, we have found that the
lift-off is 0.55 mm, the workpiece extends to infinity in depth, i.e., that it is indeed a
half-space, and that σ = 1.016 S/m, , = 219.79 0 . The reason that we can easily
move between circuit models and field models is that the measurable is impedance,
which is equally at home in each model.